B. G. TEUBNERS SAMMLUNG YOIS LEHRB&CHERN AUF DEM 'OEBIETE DER MATHEMATISCHEN WISSENSCHAFTEN MIT EmSCHLUSZ IHRER ANWENDUNGEN. BAND XI. THE DYNAMICS OF PARTICLES AND OF RIGID, ELASTIC, AND FLUID BODIES I BEING LECTURES ON MATHEMATICAL PHYSICS BY ARTHUR GORDON WEBSTER, A.B.CHARV.), PH.D.CBEROL.) PROFESSOR OF PHYSICS, DIRECTOR OF THE PHYSICAL LABORATORY, CLARK UNIVRRSITY, WORCESTER, MASSACHUSETTS. B R A / or THE UNlV'iTPSfT LEIPZIG: B. Gt. TEUBNEE. 1904. ... PREFACE. JLhe science of Dynamics may be variously classified. In by far the greater number of universities in both Europe and America it is dealt with by professed mathematicians, and is properly considered an essential part of mathematical discipline. Nevertheless it is but an application of mathematics to the most fundamental laws of Nature, and as such is of the highest importance to the physicist. The whole of modern Physics experiences the attempt to "explain" or describe phenomena in terms of motion, with conspicuous success in the departments of Light, Electricity, and the Kinetic Theory of Gases. It is therefore evident that no one can expect to materially advance our knowledge of Physics who is ignorant of the principles of Dynamics. It is nevertheless to be feared that this subject is often slighted by the physical student, partly on account of its difficulty, and partly because of the fact that the many excellent treatises on Dynamics existing in English address themselves chiefly to the mathe- matician, and often seem to lay more stress on examples in analysis or trigonometry than on the elucidation of physical laws. The aim of this book is to give in compact form a treatment of so much of this fundamental science of Dynamics as should be familiar to every serious student of physics (and in my opinion no less should suffice for the student of mathematics). The classical English treatises usually fill one or even two large volumes with one of the subdivisions of the subject, such as Dynamics of a Particle, Rigid Dynamics, Hydrodynamics or Elasticity. The student confronted with the five volumes of Routh, the three of Love, and the large work of Lamb is likely to be appalled at the size of the task before him. It is practically impossible for the physical student, while spending the necessary amount of time in the laboratory, to read through all these or similar works , and thus his knowledge of the whole subject generally remains fragmentary. The great work of Lord Kelvin and Tait, while treating the whole subject, is far too difficult for most students, though it must ever remain a mine of information for those sufficiently advanced. This book has grown out of the lectures which I have given at Clark Universitiy during the last fourteen years primarily to my own students of Physics. It is obvious that it leads to no particular examinations, from which we in America are to a large extent fortunately free. The text is not interrupted by examples for the student to work, which are found in great numbers in the usual treatises, and to which I could hardly add. The attempt has been made to treat what is essential to the under- standing of physical phenomena, leaving out what is chiefly of mathematical interest. Thus the subject of Kinematics is not treated as a subject by itself, but is introduced in connection with each subdivision of Dynamics IV PREFACE. as it comes up. The student is supposed to have a fair knowledge of the Calculus, but not of Differential Equations or the Higher Analysis. Many explanations are therefore necessary, some of which are given in the form of notes. Two opposing tendencies have at various times made themselves manifest in the treatment of Dynamics, both of which have been very fruitful. Lagrange, in the advertisement to his great work, the "Mecanique Analytique", proudly says, "On ne trouvera point de Figures dans cet Ouvrage. Les methodes que j'y expose ne demandent ni constructions, ni raisonnements geometriques ou meeaniques, mais seulement des operations algebriques, assujetties a une marche reguliere et uniforme. Ceux qui aiment 1' Analyse verront avec plaisir la Mecanique en devenir une nouvelle branche, et me sauront gre d'en avoir etendu ainsi la domaine." Lagrange's boast of having made Mechanics a branch of Analysis has been amply justified by the results obtained by means of his general method for solving mechanical problems, and his pleasure would have been greatly enhanced could he have foreseen the results of extending it to wider fields in the hands of Maxwell, of Helmholtz, and of J. J. Thomson. Nevertheless in attempting to do without figures or mental images we may rob ourselves of a precious aid. Thus Maxwell, speaking of the motion of the top, says that "Poinsot has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligible propositions supersede equations". There is certainly no doubt of the advantge, parti- cularly to the physicist, of having ideas take the place of symbols. The introduction by Hamilton of the notion of vector quantities was a great step in this direction, which has assumed very great value to the physicist, and it was to a particular case of this that Maxwell alluded, namely to the idea of the moment of momentum, or impulsive couple, as it was termed by Poinsot. The importance of this physical or geometrical conception may be seen from the use made of it, under the name of the Impulse, by Klein and Sommerfeld in their very interesting work on the Top. On the other hand this notion of impulse, while in this particular case a vector, is but one case of the general notion of the momentum in Lagrange's generalized coordinates. Will it not then be an additional advantage if, keeping both the analytical and the geometrical modes of expression, we attempt to introduce into Lagrange's analytical method geometrical analogies and terminology? This it is perfectly possible to do, for it turns out, as was shown by Beltrami, and beautifully worked out in detail by Hertz, that the properties of Lagrange's equations have to do with a quadratic form, of exactly the sort that represents the arc of a curve in geometry. Analytically it is of no importance whether the number of variables is more or less than three — how natural it is accordingly to employ the terminology of geometry, which must result in giving a more definite image of the quantities involved. For this reason I hope that no physicist will accuse me of having dragged in the subject of hyperspace into a physical treatise.- I have insisted that what is involved is merely a mode of speaking, and has the advantage of PREFACE. V logical consistency with the results of geometry, which is to most of us a physical subject. At all events this matter has been so introduced that it may be completely passed over by those to whom such analogies are repugnant. The advantage of a good terminology, as well as of clear physical conceptions, must be plain to all, and every physicist will acknowledge the indebtedness which our science owes in this respect to Kelvin and Tait. The work divides itself naturally into three parts, the first of which considers the Laws of Motion in general and those methods which are applicable to systems of all sorts. Although not addressed to students who are beginning Mechanics, it seemed necessary to begin at the beginning, and to explain the exhibition of Newton's Laws of Motion in mathematical form. For this purpose the Principle of Hamilton is of so universal application that it has been introduced near the beginning, and considerable attention devoted to it. I consider this principle, together with the equations of Lagrange, a very practical subject, of the highest importance for the physical student. The same may be said of the subject of Energy, upon which it has even been attempted to found the laws of Physics. Although such attempts seem doomed to fail, for the reason that the principle of Energy, though affording an integral, is insufficient to deduce the differential equations, the notion of Energy must remain one of the most important in Dynamics, and is therefore considered in every problem. The subject of Oscillations, of very great physical interest, with its accompanying phenomena of Resonance, is next taken up. After this follows a treatment of the so-called Cyclic Systems, from which, since the labors of Helmholtz and Hertz, it seems that Physics has so much to expect. In fact the first steps have been taken to explain the nature of Potential Energy by means of Motion, perhaps the chief desideratum of Physics. In this connection we way again point to the epochmaking work of Lord Kelvin, both in Mechanics and in the Theory of Light. The second part is devoted to the Motion of Rigid Bodies, particularly to their rotation, a matter of the greatest importance practically, especially to the engineer, but one which is often avoided by the physical student. To this subject Maxwell again called the attention of physicists, and created a charming instrumental demonstration in his celebrated Dynamical Top. To this the writer has ventured to add a small detail, which permits of a number of interesting additional verifications. A number of practical illustrations, of interest to the physicist and engineer, are also included. The third part divides itself from the other two from the fact that in it the differential equations are partial, while in the others they are ordinary. As a preparation for this subject is introduced the theory of the Potential Function, which introduces the most important mathematical theorems, and prepares for the subsequent chapters. Most of this chapter has already appeared in the author's treatise on the Theory of Electricity and Magnetism, but several/ matters have been added, especially on applications to Geodesy. Next follows the subject of Stress and Strain, with applications to the simpler problems of Elasticity, including the problem of de St. Venant *bn the flexion and torsion of prisms. Finally YI PREFACE. in Hydrodynamics the main questions of wave and vortex motion are taken up, with a brief account of the phenomena of the tides and of viscous fluids. Thus the student is prepared for the study of Sound, Light, and Electricity. The only work in English of which I know having the same purpose is Professor Tait's admirable treatise on Dynamics. While this book has been in preparation, there has appeared the first volume of Professor Gray's Treatise on Physics, the scope of which is much broader, but the aim of which is not greatly different from that of this book. I have however attempted to provide a treatise which would in not over a year's time offer to the student an amount of knowledge of Dynamics sufficient to prepare him for the study of Mathematical Physics in general. My obligations to previous authors are obvious, and where possible explicit mention is made. A list of works which have been of service to me is appended, but I wish particularly to acknowledge my indebtedness to Thomson and Tait, to Kirchhoff and to Appell. I am under great obligations to Dr. Margaret E. Maltby for valuable assistance in the preparation of the manuscript, and for frequent suggestions, and to Messrs. J. G. Coffin and J. C. Hubbard, Fellows of Clark University, for efficient aid in the preparation of the drawings. I take this opportunity of expressing my thanks to my colleague Professor William E. Story for his continual willingness during fourteen years to aid me by putting at my disposition his unusual knowledge in matters connected with Algebra. My thanks are due to the publisher for the fine mechanical execution of the work in the style for which the house of Teubner is noted. As the proof has been read only by myself, it is hoped that errors will be dealt with lightly. In conclusion I venture to hope that my attempt to make Dynamics more of an experimental science by subjecting some of its conclusions to quantitative experimental verification may deserve notice. Worcester, Mass., July 22, 1904. A. G. WEBSTEE. CONTENTS, PAET I, GENERAL PRINCIPLES. CHAPTER 1 Kinematics of a Point. Laws of Motion. Art. 1. 2. 3. 4. 5. 6. 7. 8. 9. Acceleration Dynamics . Kinematics . Scalars and Vectors Addition of Vectors Moments . Velocity . . Polar Coordinates Sector Velocity Page 3 3 Art. 10. Acceleration Components . . 1 1 . Moment of Acceleration Page 14 17 sors ... 4 ors . . . 5 7 12. Kepler's Laws 13. Physical Axiom's. Laws of Motion 18 20 9 14. Units . . . 26 s . . . . 11 . ... . 12 13 15. Derived Units and Dimensions 16. Universal Gravitation . . . 17. Absolute Systems .... 27 29 31 CHAPTER II. Important Particular Motions of a Material Point. 18. Constant Accelerations 19. Harmonic Motions . . 20. Central Forces 33 21. Constrained Motion 35 22. Plane Pendulum . . 38 23. Spherical Pendulum 41 45 48 CHAPTER III. General Principles. Work and Energy. 24. Work 56 25. Statics. Virtual Work . . 57 26. D'Alembert's Principle . . 63 27. Energy. Conservative Systems. Impulse 65 28. Particular Case of Force- function . 73 29. Calculus of Variations. Bra- chistochrone 77 30. Dependence of Line Integral on Path. Stokes's Theorem. Curl 84 31. Lamellar Vectors .... 87 32. Motion of the Center of Mass 89 33. Moment of Momentum . . 94 VIII CONTENTS. CHAPTER IV. Principle of Least Action. Generalized Equations of Motion. Art. Page 34. Hamilton's Principle ... 97 35. Principle of Least Action . 99 36. Generalized Coordinates. La- grange's Equations .... 108 37. Lagrange's Equations by di- rect Transformation. Various Reactions 115 Art. 38. Equation of Activity. Integral of Energy 39. Hamilton's Canonical Equat. 39a.Varying Constraint 40. Hamilton's Principle the most general dynamical principle 41. Principle of Varying Action CHAPTER V. Oscillations and Cyclic Motions. 42. Tautochrone for Gravity . . 144 43. Damped Oscillations . . . 148 44. Forced Vibrations. Resonance 152 45. General Theory of small Oscillations ...... 157 Vibration of a String of Beads. Continuous String . . . . 164 Forced Vibrations of General System ....... ".173 Cyclic Motions. Ignoration of Coordinates 175 of 46 47 48 49. Example. Three Degrees Freedom. General Case . . 50. Effect of Linear Terms in Kinetic Potential. Gyroscopic Forces 51. Cyclic Systems 52. Properties of Cyclic Systems. Reciprocal Relations . . . 53. Work done by the Cyclic and Positional Forces . . . . 54. Examples of Cyclic Systems Page 125 126 129 130 131 181 184 188 190 192 193 PART II. DYNAMICS OF RIGID BODIES. Systems of Vectors. CHAPTER VI. Distribution of Mass. Instantaneous Motion. 199 55. Translations and Rotations 56. Rotations about two Parallel Axes 201 57. Rotations about Intersecting Axes. Infinitesimal Rotations 202 58. Vector -couples 204 59. Statics of a Rigid Body . . 205 59 a. Parallel Forces. Force- couples 205 60. Reduction of Groups of Forces. Dualism 209 61. Variation of the Elements of the Reduction. Central Axis. Null- System 209 62. Vector- cross . . ... . 212 63. Complex of Double -lines. . 214 64. Composition of Screws . . 216 65. Work of Wrench in Produ- cing a Twist 220 66. Analytical Representation. Line Coordinates 221 67 Momentum Screw. Dyna- mics 224 68. Momentum of Rigid Body . 225 69. Centrifugal Forces .... 228 Moments of Inertia. Parallel Axes 229 Moments of Inertia at a Point. Ellipsoid of Inertia .... 229 72. Ellipsoid of Gyration ... 232 73. Ellipsoidal Coordinates . . 234 Axes of Inertia at Various Points 237 Calculation of Moments of Inertia . . 240 Analytical Treatment of Kine- matics of a Rigid System. Moving Axes 243 Relative Motion 247 78. Angular Acceleration . . . 248 79. Kinetic Energy and Momen- tum due to Rotation . 249 70 71 74 75 76 77 CONTENTS. IX CHAPTER VII. Dynamics of Rotating Bodies. Page 250 Art. 80. Dynamics of Body moving about a Fixed Axis . . . 81. Motion of a Rigid Body about a Fixed Point. Kine- matics 252 82. Dynamics. Motion under no Forces 256 84. Euler's Dynamical Equa- tions .... 260 85. Poinsot's Discussion of the Motion 261 86. Stability of Axes .... 264 87. Projections of the Polhode . 264 88. Invariable Line 265 89. Symmetrical Top. Constrained Motion 271 90. Heavy Symmetrical Top . . 274 91. Top Equations deduced by Lagrange's Method .... 277 92. Nature of the Motion ... 279 93. Precession and Nutation . 283 Art. page 94. Small Oscillations about the Vertical 288 95. Top Equations deduced by Jacobi's Method . . . . 297 96. Rotation of the Earth. Pre- cession and Nutation. . . 298 97. Top on smooth Table . . 302 98. Effect of Friction. Rising of Top 303 99. Motion of a Billiard Ball . 304 100. Pure Rolling 307 101. Lagrange's Equations ap- plied to Rolling. Noninte- grable Constraints . . . 313 102. Moving Axes . . . . . 316 103. Rotating Axes. Theorem of Coriolis 317 104. Motion relatively of the Earth 320- 105. Motion of a Spherical Pen- dulum 323 106. Foucault's Gyroscope . . 324 PART III. THEORY OP THE POTENTIAL, DYNAMICS OF DEFORMABLE BODIES. CHAPTER VIII. Newtonian Potential Function. 107. Point -Function 329 108. Level Surface of Scalar Point -Function . . . .. . 329 109. Coordinates 330 110. Differential Parameter . . 330 111. Polar Coordinates .... 334 112. Cylindrical, or Semi-polar Coordinates 335 113. Ellipsoidal Coordinates . . 335 114. Infinitesimal Arc, Area and Volume ., 338 115. Connectivity of Space. Green's Theorem .... 339 116. Second Differential Para- meter 344 117. Divergence. Solenoidal Vec- tors 347 118. Reciprocal Distance. Gauss's Theorem . . . . . . . 350 WEBSTER, Dynamics. 119. Definition and fundamental Properties of Potential . . 352 120. Potential of Continuous Dis- tribution 353 121. Derivatives ...... 355 122. Points in the Attracting Mass. 357 123. Poisson's Equation . . . 359 124. Characteristics of Potential Function 362 125. Examples. Potential of a homogeneous Sphere . . . 363 126. Disc, Cylinder, Cone . . 366 127. Surface Distributions . . 367 128. Green's Formulae .... 370 129. Equipotential Layers . . . 372 130. Gauss's Mean Theorem . . 374 131. Potential completely deter- mined by its characteristic Properties ...... 375 a* CONTENTS. Art. Page 132. Kelvin and Dirichlet's Prin- ciple 376 133. Green's Theorem in Ortho- gonal Curvilinear Coor- dinates 379 134. Stokes's Theorem in Ortho- gonal Curvilinear Coor- dinates 381 135. Laplace's Equation in Spheri- cal and Cylindrical Coord. 383 136. Logarithmic Potential . . 385 137. Green's Theorem for a Plane 386 138. Application to Logarithmic Potential 387 139. Green's Formula for Loga- rithmic Potential .... 388 140. Dirichlet's Problem for a Circle. Trigonometric Series 388 140 a. Development in Circular Harmonics 390 141. Spherial Harmonics . . . 393 142. Dirichlet's Problem for Sphere 395 143. Forms of Spherical Harmo- nics 395 144. Zonal Harmonics .... 397 145. Harmonics in Spherical Coordinates . 398 Art. Page 146. Development of Reciprocal Distance 398 147. Development in Spherical Harmonics 400 148. Development of the Potential in Spherical Harmonics . . 149. Applications to Geodesy. Clairaut's Theorem . . . 150. Potential of Tide - generating Forces 151. Ellipsoidal Homoeoids. New- ton's Theorem 409 152 Condition for Infinite Family of Equipotentials ... 153. Application to Elliptic Coord 154. Chasles's Theorem . . . 155. Maclaurin's Theorem . . 156. Potential of Ellipsoid . 157.. Internal Point .... 158. Verification by Differen- tiation 419 159. Ivory's Theorem .... 420 160. Ellipsoids of Revolution . 421 161. Development of Potential of Ellipsoid of Revolution . . 424 162. Energy of Distributions. Gauss's theorem .... 425 163. Energy in terms of Field . 426 402 404 408 410 411 413 414 415 418 CHAPTER IX. Dynamics of Deformdble Bodies. 164. Kinematics. Homogeneous Strain 427 165. Self- conjugate Functions. Pure Strain 430 166. Rotation 435 167. General Small Strain . . 436 168. Simple Strains. Stretches and Shears 439 168 a. Elongation and Compression Quadric ....... 441 169. Heterogeneous Strain . . 444 170. Stress . . - . 446 171. Geometrical Representation of Stress 450 171 a. Simple Stresses .... 451 172. Work of Stress in producing Strain 454 173. Relations between Stress and Strain 455 174. Energy Function for Isotro- pic Bodies 457 175. Stresses in Isotropic Bodies 460 176. Physical Meaning of the Constants . . 461 CHAPTER X. Statics of Deformable Bodies. 111. Hydrostatics 463 178. Height of the Atmosphere 465 179. Rotating Mass of Fluid . . 467 180. Gravitating, rotating Fluid 468 181. Equilibrium of Floating Body 471 182. Solid hollow Sphere and Cylinder under Pressure 183. Problem of de Saint -Yenant 184. Determination of Function for particular Cases . . . CONTENTS. XI CHAPTER XI. Hydrodynamics. Art. Page 186. Equations of Motion . . . 496 187. Hamilton's Principle ... 499 188. Equation of Activity . . . 501 189. Steady Motion 503 190. Circulation ...... 506 191. Vortex Motion 509 192. Vector Potential. Helm- holtz's Theorem .... 511 193. Velocity due to Vortex . . 514 194. Kinetic Energy of Vortex . 515 195. Straight parallel Vortices . 515 196. Irrotational Motion 620 Art. Page 197. Uniplanar Motion 521 198. Wave Motion 529 199. Equilibrium Theory of the Tides 535 200. Tidal Waves in Canals . . 538 201. Sound Waves 542 202. Plane Waves 543 203. Echo. Organ pipes . . . 545 204. Spherical Waves .... 547 205. Waves in a Solid .... 548 206. Viscous Fluids . 549 NOTES. Note I. Differential Equations 559 „ II. Algebra of Indeterminate Multipliers 562 „ III. Quadratic Differential Forms. Generalized Vectors 563 „ IV. Axes of Central Quadric 567 „ V. Transformation of Quadratic Forms 572 INDEX 580 LIST OF WORKS CONSULTED BY THE AUTHOB. Appell, Traite de Mecanique Rationelle. Boltzmann, Vorlesungen u'ber die Prin- zipe der Mechanik. Budde, Allgemeine Mechanik. Christiansen, Elemente der Theo- retischen Physik. Clifford, Elements of Dynamic. Despeyrous, Cours de Mecanique. Foppl, Vorlesungen iiber technische Mechanik. Gray, Treatise on Physics. Helmholtz, Dynamik diskreter Massen- punkte. Hertz, Die Prinzipien der Mechanik. Jacobi, Vorlesungen iiber Mechanik. Kirchhoff, Yorlesungen iiber mathe- matische Physik -Mechanik. vfLang, Theoretische Physik. • Klein und Sommerfeld , Theorie des Kreisels. Love, Theoretical Mechanics. Loudon, Rigid Dynamics. •Mach, Die Mechanik in ihrer Ent- wickelung. Mathieu, Cours de Physique Mathe- matique. •Maxwell, Treatise on Electricity and Magnetism. Poisson, Traite de Mecanique. Rausenberger, Lehrbuch der ana- lytischen Mechanik. Resal, Traite de Physique Mathe'- matique. Ritter, Lehrbuch d. Ingenieurmechanik. •Routh, Dynamics of a Particle. •Routh, Dynamics of a System of Rigid Bodies. Schell, Theorie der Bewegung und der Krafte. Somoff, Theoretische Mechanik. Sturm, Cours de Mecanique. Thomson and Tait, Treatise on Natural Philosophy. Yoigt, Elementare Mechanik. Yoigt, Kompendium der theoretischen Physik. Williamson and Tarleton, Elementary Treatise on Dynamics. Ziwet, Elementary Treatise on Theore- tical Mechanics. Boussinesq, Applications des Poten- tiels. Betti, Teorica delle Forze Newtoniane. Basset, Treatise on Hydrodynamics. Bjerknes, Yorlesungen iiber hydro- dynamische Fernkrafte. Clebsch, St. Yenant, Theorie de TElasti- cite des Corps Solides. Greenhill, Treatise on Hydrostatics. Helmholtz , Dynamik kontinuierlich verbreiteter Massen. Ibbetson, Mathematical Theory of Elasticity. • Lame , Le9ons sur la Theorie mathe- matique de 1'Elasticit^ des Corps solides. Lamb , Hydrodynamics. Levy, Theorie des Mare'es. Love, Treatise on the Mathematical Theory of Elasticity. Mathieu, Theorie de 1'^lasticite des Corps solides. Minchin, Treatise on Statics. Neumann, Das Logarithmische und Newtonsche Potential. • Peirce, Elements of the Theory of the Newtonian Potential Function. • Poincare , Le9ons sur la Theorie de TElasticite. Poincare, Theorie des Tourbillons. Poincare, The'orie du Potentiel New- tonien. Poincare , Figures d'Equilibre d'une Masse Fluide. • Rayleigh , Theory of Sound. Riemann, Schwere, Elektrizitat und Magnetismus. Routh, Treatise on Analytical Statics. Tarleton, Introduction to the Mathe- matical Theory of Attraction. Todhunter and Pearson, History of the Theory of Elasticity. Wien, Lehrbuch der Hydrodynamik. Williamson, Mathematical Theory of Stress and Strain. PART I GENERAL PRINCIPLES , Dynamics. CHAPTEK I. KINEMATICS OF A POINT. LAWS OF MOTION. 1. Dynamics. Dynamics or Mechanics is the science of motion. It is the fundamental subject of Physics, since it is the aim of scientists to reduce the characterization of all physical phenomena to description of states of motion. The problem of dynamics, according to Kirchhoff1), is to describe all motions occurring in nature in an unambiguous and the simplest manner. In addition it is our object to classify them and to arrange them on the basis of the simplest possible laws. The success which has attended the efforts of physicists, mathematicians, and astronomers in achieving this object, from the time of Galileo and Newton through that of Lagrange and Laplace to that of Helmholtz and Kelvin, constitutes one of the greatest triumphs of the human intellect. . 2. Kinematics. That which moves is matter. The properties of matter may be left for later consideration. We may, however, describe motions without considering the nature of that which is moved, — this forms a special branch of our subject known as Kinematics. Kinematics is merely an extension of geometry and may be called geometry of motion, for while in geometry we consider the properties of space, in Kinematics we consider also the idea of time, giving us another variable. Since the position of a point in space is known when its three rectangular Cartesian coordinates with respect to a definite system of axes are given, its motion is completely described if its coordinates are given for all instants of time, or are known functions of the time. Analytically The functions f1} f%, fz must be continuous, since in no actual motion does a point considered disappear in one position to reappear after 1) Kirchhoff, Vorlesungen fiber mathematische Physik. Mechanik, p. 1. 1* 4 - I. KINEMATICS OF A POINT. LAWS OF MOTION. a very small interval of time in a new position at a finite distance from the old. The functions are also supposed to have definite derivatives for every value of t. Since the motion of a point involves four variables, Kinematics was called by Lagrange Geometry of four dimensions. We shall not here discuss the nature of time, nor the mode of measuring it, reserving the latter until we have considered motions that actually occur in nature, upon which all methods for measuring time are based. We may accept the fact that the idea of time, like that of space, is the intuitive possession of us all. Its exact definition must depend on the science of dynamics. 3. Scalars and Vectors. In mathematics we have to consider two sorts of quantities, those which do not involve the idea of direction, called by Hamilton scalars (because they may be specified by numbers marked off on a scale), and those which do, called steps or vectors. The distance between two points xly y1} #1; #2, y.2, z.2 is a scalar, whereas the geometrical difference in position of the two points is known only when we specify not merely the length, but also the direction of the line joining them. This is usually done by giving its length s and the cosines of the angles made by the line with the three rectangular axes, cosA, cos/i, costs,* which in virtue of the relation 3) cos2;, + cos2/i -f cos2^ = 1, leaves three independent data. We may otherwise make the speci- fication by giving the three projections of the line upon the co- ordinate axes, = s cos yl = # — x Squaring and adding we have in virtue of relation 3) By the vector AB we mean the line in the direction from A to By and its projections have the sign of the coordinates of B minus those of A, the vector being defined as that which carries us from A to B. We may write symbolically pt- A -f AB=pt-B AB=pt'B—pt-A. AB is to be understood, vector AB. Similarly when we wish to specify that s is to be regarded as a vector (i. e. its direction is to be considered as well as length), we shall write "s. 2, 3, 4] SCALARS AND VECTORS. RESULTANT. We have from 4) and 5) cos I = — = x — j cos 11= — = COS Also multiplying the equations 4) respectively by cos %, cos /i, cos v, and adding, 7) sx cos k -\- sy cos /i + sz cos v = s. Whatever quantities are needed to completely specify a quantity are called its coordinates. A point has three, and we have seen that a vector also has three, which may be taken as sx, sy, sz. In this sense all vectors are to be considered as equal whose lengths are equal and directions parallel irrespective of the absolute positions of their ends. It is, however, sometimes necessary to distinguish vectors equal in this sense, but whose ends do not respectively coincide. To determine such a vector we must know not only its length and direction, but also the position of one end. It will therefore be specified by six coordinates, which may be the three coordinates of one end, xly y^, #±, with the projections, sx, sy, sz, or the co- ordinates of both ends, xify190lf %2> 2/2; #2- ^n anJ case there will be six coordinates. Such a vector may be called a fixed vector to distinguish it from the ordinary or free vector. 4. Addition of Vectors. To add two vectors means to take successively the steps denoted by them, their sum being a single step equivalent thereto. For example, (Fig. 1) The vectors AB and BC are called the components of AC, which is called their resultant, or geometrical sum. We may state the rule: Place the initial point of the second vector at the terminal point of the first, the resultant or geometrical sum is the vector from the initial point of the first component to the terminal point of the second. This con- struction gives us the so-called triangle of vectors. By continuing the process any number of vectors may be added, giving us the polygon of vectors. Fig. 1. 6 I. KINEMATICS OF A POINT. LAWS OF MOTION. The nature of the construction shows that the resultant is inde- pendent of the order of taking the components. Since a negative quantity is denned as that which added to a given positive quantity produces zero, the negative of A 13 must be BA, for hy the above rule, A+ AB = B, therefore A + AB The coordinates of BA are also the negatives of AB. The scalar length of a vector is called by Hamilton its tensor, so that the tensor of the negative of a vector is the same as that of the vector itself. It is evident from the definition of a vector that the projection of the sum of two vectors on any direction is the algebraic sum of the projections of the components. Projecting on the three directions of the coordinate -axes, and distinguishing the projections of the components by suffixes, we have for the projections of the resultant, Sx = Six + Szx, S = Sl + S2, S2 = (Six -f S2*)2+ (Sly + $2y)2 + (Siz + S2*)2, and for the sum of any number of vectors, 8) S2=(Is,)2+(ISj,)2+(IS2)2. We may easily find an expression for the projection of any vector ~s upon any direction, which is given by its direction cosines, cos A, cos p, cos v. We have for the angle # between two lines whose direction cosines are cos I, cos p, cos v, cos Zr, cos ^ , cos i/, cos & = cos I cos ^ + cos 11 cos ^ + cos v cos v\ but by 6), we have for "s, s s s, cosA' = — i cos//=— > cosi/=— i s s s BO that 9) s cos # = sx cos A + sy cos 11 -f sz cos Vj which is the expression for the projection. Taking for the direction of projection the direction of the vector itself, this becomes equation 7). 4, 5] PROJECTION. GEOMETRIC PRODUCT. MOMENTS. 7 If cos A, cos \iy cos v are the direction cosines of a second vector s~9 cos I = — 9 2W "2z = — 7 COS V = — 7 multiplying by s2 we have the expression symmetrical with respect to both vectors 10) St This expression, which may be denned either as the product of the tensors of the vectors and the cosine of their included angle , or as the tensor of either multiplied by the value of the projection on its direction of the other, is so important that it has received a special name, and will be called the geometric product of the two vectors. It is not a vector, but is essentially a scalar quantity, and its negative was called by Hamilton the scalar product of the vectors. The condition of perpendicularity of two vectors is that their geometric product vanishes. 2= 0. 5. Moments. Consider a fixed vector AB, Fig. 2. — The product of the length AB and the perpendicular distance of 0 from AB is called the moment of AB about 0. It is arithmetically equal to twice the area of the triangle OAB. The sign of the moment will change with the direction of AB. If we draw a line through 0 whose length is equal to the magni- tude of the moment and whose direction is perpendicular to the plane 0 A B, this line is called the axis of the moment, and in a certain way represents the latter. We shall draw it in such a direction that a person standing on 0 with his back against the axis would see motion from A to B as from right to left. 8 I. KINEMATICS OF A POINT. LAWS OF MOTION. The coordinates of the the vector AB and of 0. axis may be found from those of If we choose 0 for origin, OAB for the plane of (Fig. 3) and let the co- ordinates of A be x, y, the projections of AB, sx, sy9 we have for the area of the triangle GAB -xy-sxs,} Accordingly we have for m, the moment about 0 of a vector whose pro- jections are sX9 stj and whose initial point has the coordinates x, y, m = xsy— ysx. To find the moment of the resultant of two vectors drawn from the same initial point, whose plane contains 0, their projections being sj, Sy', sj, sx", Sy", sz", we have m = x(sv'+sv")-y(sx'+sx") Fig. 3. — ysx= m thus the moment of the resultant is equal to the sum of the mo- ments. If the plane of OAB is not one of the coordinate -planes, we may project the triangle OAB upon the three coordinate -planes, and obtain three moments mx, myj mz. If the direction cosines of the axis of m are cos a, cos /3, cos y, we have by the rule for the projection of areas, mx = m cos a, my = m cos /?, mz = m cos 7, •m' mz Therefore the moment m has three coordinates, mx, my, mz, and may itself be considered a vector m. Since the coordinates of the pro- jections of A and AB on the YZ plane are y, 2, sy, S2, we have by the preceding formula 12) = ys, - es. xsz, mz = xSy — In the language of Hamilton m is the vector product of the vector OA into the vector AB. We have evidently 5, 6] VECTOR PRODUCT. VELOCITY. 9 xmx + ymy + zmz = 0 sxmx + symy + szmz = 0, that is, the vector product of the two vectors is perpendicular to their plane. From the definition of moment, or by reference to Fig. 3, its magnitude or tensor is equal to the product of their tensors times the sine of the angle included between them. (It is to be noted that the projections of the first factor in the vector product follow each other in cyclic order in equations 12), those of the second factor in reverse order.) It is at once evident that the moment of the resul- tant of two vectors with the same initial point is the resultant of their individual moments. Thus moments are to be considered in all respects like vectors. It is evident that the moment of a vector, Sx, Sy, sz, with initial point x, y, s, about a point |, ??, J, has the projections: mx = (y- vj) s, - (e - Q sy 13) my = (z — g) sx — (x— !) sz mz = (x- fj) sy — (y - 7?) sx. 6. Velocity. As a second means of description of the motion of a point we may give the geometrical locus of the positions that it occupies at different instants. This is called the path of the point, and if it is straight, the motion is said to be rectilinear. This alone does not suffice to describe the motion, for the same path may be described with different speeds. We must therefore give something which shall determine what positions are reached at various instants. If we call s the distance the point has traversed in its path, counting from a fixed point, and give the value of s for every value of t s = = f; = Fs(t) together with the initial conditions x = x0^ y = yoj 2=2o} when t = t0. An integration of these three simultaneous equations would give us a description equivalent to 1). In equations 1), if t is any parameter, not necessarily the time, we have what is called the parametric representation of a curve. By the elimination of t, we may obtain two coordinates as functions of the third. If, on the contrary, we have only the path given, whereas the geometry of the motion is known, kinematically the description is incomplete, as the specification of the time is lacking. To remedy this defect of the geometrical representation, Hamilton introduced the Hodograph, which is a curve, the locus of a point related to the moving point on the path by having its position 1) It is to be noticed that in stating that velocity is a vector we assume the mode of composition of velocities as a matter of definition. 6, 7] HODOGRAPH. RADIAL AND TRANSVERSE VELOCITY. H vector with respect to a point taken as origin equal to the vector velocity of the moving point. Thus the radius vector of any point on the hodograph is parallel to the tangent at the corresponding point of the path. If X, Y, Z are the coordinates of a point on the hodograph, we have for the relation between the two curves, 1Qx ^ dx v dy 7 dz z."--ar Y===dt' z=di' so that having established the correspondence of point to point, we obtain the time from ds 20) <-/- y We shall call any vector which is related to another vector as the vector X, Y, Z is to the vector x, y, z, the velocity of the vector, and by a natural extension, shall call the locus of the end of the second vector drawn from a fixed origin the hodograph of the first vector. Thus we call Hamilton's hodograph the hodograph of the position vector of the first point. 7. Polar Coordinates. If a point moves in a plane it may be convenient to specify its position by means of polar coordinates. Let r be the distance of the point from the origin 0, cp the angle that the radius vector makes with a fixed line through the origin. If now the point moves from A to B (Fig. 4) in the time A£, describ- ing the space As, so that r turns through the angle Aqp, at the same O time increasing by Ar, we may re- Fig. 4. solve the velocity into two com- ponents, one proportional to AC, where AC is perpendicular to OB, the other proportional to CB. We have then the following vector equation ,. (AC ~ ^ . or, passing to the limit, and the transverse velocity, v(p = r -~r • The rate of increase of the angle qp is called the angular velocity ~- The vector equation 21) gives rise to the scalar equation 12 I. KINEMATICS OF A POINT. LAWS OF MOTION. that is, 22) dt} which might have been obtained from the expression for the lengths of the arc in polar coordinates. 8. Sector Velocity. Let the polar coordinates of a point at -the time t be r, qp, and let the area of the sector enclosed between the path, the fixed line of reference, and the radius vector be denoted by S. If £ denote the angle made by the tangent to the path, in the direction of motion with the direction of the radius vector from the origin, we have (Fig. 5) dr = ds cos e, rd

9 cos (as) = 1O. Acceleration Components. We may now find the com- ponent of the acceleration in the direction of the tangent to the path. The direction cosines of the tangent being, by § 6, »* vy v* ; J } V V V we have for the tangential component by 9) rtl rtl A* 28) a* = ax ^ 4. a JL 4. a _^ v y v v _ I \dx d*x dy d*y ~ ~v(dt ~dW ' ~dt ~dir But differentiating equation 17) dz dt dv ds dzs dx d*x dy ™ ^7^ ^7* ^7*2 Jj. J*2 I Jj. Jj.2 i J± dz d2 4t * (~l velocities at P and Q, the acceleration, lim — — , is in the plane of AS and AC, that is the vector acceleration is in the osculating plane. As we have already found the component parallel to the tangent, there remains only the component parallel to the principal normal. Since BC is proportional to the acceleration, DC is pro- portional to the tangential acceleration at, BD to the normal acceleration avy and since the angle at A is dr and the side AB is v9 BD = vdr vdr Also since ds = gdr OA\ v ds v'2 30 ) a= ——-==—. gdt 2 "h 2 "^ \dsz Now ,e have Differentiating this by s gives _ _ 9 idxd*x dy d*y dz ~ds \ds \ds I ~ (^ds^ds^ '^ ~ds~ds~ ~^~ds ( dz\ \ds) Therefore equation 32) reduces to or OQN 33) If j- = t> = 1, ds = dt and the right hand member of 33) becomes -the square of the acceleration. We thus have a kinematical definition of curvature , viz., the acceleration of a point traversing the curve with unit velocity. This agrees with the original expression 30), av = — ; for if v = 1, af = 0, the acceleration is entirely normal and av == — =x. 9 We may in like manner resolve the acceleration into components along the radius vector and at right angles to it. Let us consider the case of motion in a plane, that of XY. We will call the radial component of the acceleration, or the radial acceleration, ary and we d^T shall find that it is not equal to -^-> which is the scalar acceleration of the radial velocity. We will denote the component perpendicular to the radius or the transverse acceleration by a^ which is not equal to the angular acceleration -=--> nor to the acceleration of the trans- verse velocity, ~~- Differentiating the formulae for the change of coordinates gves dx dr . dcp dy dr . dm 10, 11] CURVATURE. RADIAL AND TRANSVERSE ACCEL. 17 Differentiating again d*r dcp dr lit lit dtp dr dt lit dt* ~ dt* The direction cosines of the radius vector are: cos(rx) = cos (pj cos(ry) = sinqp, so that we obtain by resolution, d*r 34) ar = - cos (r x) being less than the scalar acceleration of the radial velocity by the product of the radius vector and the square of the angular velocity. (If ,,- = 0, the motion is circular, and ar is the normal acceleration.) The direction cosines of a line perpendicular to r and in the direction of increasing

L dt* Let 18 I KINEMATICS OF A POINT. LAWS OF MOTION. If the motion is not in one plane, we have, differentiating the sector velocity components 24) The resultant of these is the moment of the acceleration. The fact that the moment of the acceleration is the exact time derivative of the moment of velocity leads to an important general principle of mechanics, the so-called Law of Areas. 12. Kepler's Laws. We may now obtain Newton's conclusions from Kepler's three laws of planetary motion, which were purely kinematical and hased on a great amount of observational material collected by Tycho Brahe. The first law states that the areas swept over by the radius vector drawn from the sun to a planet in equal times are equal. (The motion is in one plane.) That is dS ~dt = C0nst> dt* therefore from 37) the moment of the acceleration with respect to the sun is zero. Consequently the line of direction of the accelera- tion passes through the sun, or the acceleration is central. The second law states that the planets describe ellipses about the sun as a focus. The ellipse being always concave toward the focus, the acceleration is directed toward tbe sun. In order to deduce the quantitative meaning of the second law, we will use the polar equation of a conic section referred to the focus, 1 -j- e cos cp 1) If d is the distance from focus to directrix, e the eccentricity, by the definition of a conic section, r ed = p — = e, r = — * • d — r cos cp 1 -f- e cos cp When cos cp = 1 ~l + e COS qp = — 1, 11, 12] KEPLER'S LAWS. 19 and will find the value of the central acceleration. We have 34) but from Kepler's first law, ndS »dy 2 -7T- = T -3T = const. = h. say, dt dt J ' dcp _ h Now changing the variable from t to (p, dr dr dcp _ h dr _ , d / 1 \ dt dcp dt ~ r2 dcp ~ dcp\r / Differentiating by t, dt* dcp^ \r / dt r* dq From the equation of the path we obtain lie \r/~ -7Sn(P> d2 /I \ e 1 1 j — ^ I — I = COS Op = d(p2\r/ p p r Inserting this value above gives d*r _ h* W and finally, Thus the fact that the path is a conic section shows that the central acceleration varies inversely as the square of the length of the radius vector. The negative sign shows that the acceleration is toward the sun. The third law states that for different planets the squares of the times of describing the orbits are proportional to the cubes of the major axes. Since if T is the time of a complete period JiT is twice the area of the orbit. JiT= 20 I- KINEMATICS OF A POINT. LAWS OF MOTION. From which h* = ^(l-e*), 1 Now since |^ is by the third law constant for all the planets, the factor by which the inverse square of the radius vector is multiplied in order to obtain the central acceleration is the same for all the planets and depends only on the sun. We have thus obtained a complete kinematical statement of the law of gravitation for the planets. Newton tested the law of the inverse square by applying it to the motion of the moon about the earth, and comparing its accelera- tion with that of a body at the surface of the earth as directly observed. Supposing the moon's orbit to be circular, of radius a} with period J, since the tangential acceleration is zero, its velocity is constant, and equal to — ^r" Its acceleration, which is entirely normal, will accordingly be by 30) If the acceleration varies inversely as the square . of the distance, the acceleration experienced by a body at the earth's surface as will be given by where R is the earth's radius. Therefore Now we have T=27d. 7h. 43 m. = 39, 343 m., 2jrjR = 4-107 meters, a = 60 R, from which 2* • 60s • 4 • 107 meters _ Q „. meters 0/8 = (39,343 • 60 sec.)2 sec.2 Now terrestrial observations give for the mean acceleration of bodies at the earth's surface 9.82 - 2-> which by a more exact calculation sec. is in agreement with the predicted result. 13. Physical Axioms. Laws of Motion. It is necessary in order to pass from the kinematical specification of motion to the dynamical one to make use of knowledge drawn from a consideration of terrestrial phenomena. This knowledge is summed up by Newton in his three Axiomata sive Leges Motus. An axiom is defined by 12, 13] NEWTON'S LAWS OF MOTION. 21 Thomson and Tait1) as a proposition, the truth of which must he admitted as soon as the terms in which it is expressed are clearly understood. These physical axioms rest not on intuitive perception, but on convictions drawn from observation and experiment. The manner of summing up the results of our experience is to a great extent unimportant, provided that it is sufficiently all- embracing. We are not concerned with the metaphysical question of the causes of motions, but merely with the physical question of stating what is actually found to take place in nature. The statement may be made by means of a single analytical formula, as was done in different ways by Lagrange, Hamilton and Hertz, or we may consider the various assumptions upon which such formulae are founded, making detailed statements, employing conceptions with which we are familiar. This is what was done by Newton, and although his laws have received considerable criticism, they have, when properly understood, been generally admitted to be better than anything that has been proposed in their place. Lex I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum suam mutare. Every body persists in its state of rest or of uniform motion in a straight line, except in so far as it may be compelled by force to change that state. The property of persistence thus defined is called Inertia. This gives a criterion for finding whether a force is acting on a body or not, or in other words a negative definition of force. Force is acting on a body when its motion is not uniform. By uniform we mean such motion that the vector velocity is constant. If the body be a material point, that is a body so small that the distances between its different parts may be neglected, the motion is uniform if dx _ dy dz dt=~~Ci> d* = = C2> dt=C^ 38) that is Accordingly we see that the force and acceleration vanish together. Integrating the equations 38), x = c±t-\-di, y = c2t+d2, 0 = c3tf-fd3, 39) x — dl y — dz z — ds 1) Thomson and Tait, Natural Philosophy, § 243. 22 I. KINEMATICS OF A POINT. LAWS OF MOTION, the path is a straight line, and since it is traversed with constant velocity. We may on the other hand interpret the statement as giving us a means of measuring time. Intervals of time are proportional to the corresponding distances traversed ty a, material point not acted on by forces. Obviously this statement gives us an absolute definition neither of time nor of force, but only a relation between them. It is difficult or impossible for us to realize experimental conditions in which a body shall be withdrawn from the influence of all force. However we may approximate toward this condition, which must at any rate give us the ideal measurement of time. However we find in nature angular motions which, by an application of the first law, give us a practical means for the measurement of time. The second law gives us in a move positive manner than the first a measure of a force. Lex II. Mutationem motus proportionalem esse vi motrici im- pressae, et fieri secundum lineam rectam qua vis ilia imprimitur. Change of motion is proportional to force applied, and takes place in the direction of the straight line in which the force acts. By change of motion is meant acceleration. If all our experiments were made with a single body, there would be no advantage in the introduction of the term force over that of acceleration, the mul- tiplication of names being useless when no new ideas are thereby introduced. The convenience of the term force arises from the consideration of the third law. In the case of more than one body , the factor of proportionality mentioned above requires separate defini- tion for the diiferent bodies. Lex III. Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in paries contrarias dirigi. To every action there is always an equal and contrary reaction: or, the mutual actions of any two bodies are always equal and oppositely directed. If we have a certain action between two bodies 1 and 2, and if the actio were proportional only to the accelerations, we should have dt* ~dt^ ~dt*~ ~dt*~ ~dt^ ~ dt* which is not found to be the case. We must accordingly introduce a factor of proportionality, or (for symmetry) two factors, so that we write 13] ACTION AND REACTION. MASS. 23 Afr dt Experiment shows that the factors ml and m2 are constant for a given body that undergoes no changes other than those of position. These factors are called the masses of the hodies. The nature of the actions between the two bodies may be of any sort, and may be transmitted by the help of any number of intervening bodies. For instance , the actions of two heavenly bodies on each other ; trans- mitted we know not how, or the actions of two bodies kept at a fixed distance by means of a rod or string or connected by an elastic spring, or attracting or repelling each other by magnetic or electric agencies, are all illustrations of the third law. It is obvious that if we could observe the motions so as to obtain the coordinates of both bodies as functions of the time, equations 40) would enable us to determine the ratio of the masses. For example, consider the toy consisting of two horse-chestnuts or bullets connected by a string, and suppose this to be whirled about and projected into the air so that the two bodies describe complicated paths, the whole apparatus describing in general a parabolic path. If we take a series of photographs of it in rapid succession, by means of a kinetoscope or similar device, we may by measurement obtain the coordinates of the two bodies as functions of the time. This illustrates perfectly the dynamical measurement of mass and the means of obtaining the relative masses of the heavenly bodies. We have no means of defining the absolute mass of a body. As a further example of the third law, let us suppose the action is transmitted from one body to the other by means of a flexible string passing over frictionless pulleys, as in the case of Atwood's machine. The assumption here made is that the tension of the string is unchanged by passing over the pulleys. A more practical means of realizing the dynamical comparison of masses would be by experimentally establishing the equality of both sides of equations 40) with the same quantity. For example let the body be made to describe a horizontal circular path, say by means of a whirling machine. It will be found that it must be retained in this path by external means such as the tension of a string. Let this be passed over a pulley at the center of the path and exactly balance its pull against that of a weight suspended from it. The resultant of the components ^ ^' m~di?' *s ^v 30) equal to — ^-> 24 I. KINEMATICS OF A POINT. LAWS OF MOTION. where r is the radius of the circular path. The resultant is directed toward the center, and measures the effect of the tension of the string on the motion. If we repeat the experiment with another body for which the corresponding quantities are denoted by accents, making use of the same counterbalancing weight, the tensions of the string in the two cases are obviously equal and consequently we have, mv* m' r r' Measuring the velocities and radii therefore enables us to compare the masses. The vector denned by the product of the scalar quantity mass by the vector quantity acceleration, whose components are ,n\ -cr d*x -r-r d*y r? d^z 41) x=mf> Y=m-> Z=m is called the force acting upon the body, and is the vis impressa of the second law. The second and third laws taken together accord- ingly give us a complete definition and mode of measurement of force. The introduction of the new term is justified by the third law. For we find that force is capable of representing the dual nature of the interaction between two bodies, while the acceleration is not, there being two different accelerations for the two different bodies. The two sided nature of the action between two bodies is often expressed by calling it a stress. The equations 41) are called the differential equations of motion of the body. This statement needs some explanation. The introduc- tion of the term force has given us no explanation of the cause of motion, for whereas the second law tells us that the change of motion is proportional to the force applied, and we are accustomed to say that the force is the cause of the change, no additional knowledge of the motion is given us by this statement. When we say that a body moves because we push it, all we mean is that the motion and the push exist simultaneously. Were we accustomed to a different point of view, we might be as much struck with the fact that the body pushes back when it moves as that we push it. This is what the third law calls to our attention. It is undoubtedly true that our fundamental notions of dynamics are derived through what may be called the muscular sense, which is affected when we make ourselves one of the bodies of a system. We then perceive the reactions, and we have learned to correlate our perceptions to the motions of the other bodies of the system. Nevertheless , had we not possessed this extremely important sense, we might have elaborated the same system of dynamics that we now have merely by the sense of sight, as illustrated by the example of 13] FORCE. STATIC AND. KINETIC REACTIONS. 25 the two particles , fixing our attention on the facts embodied in equations 40). Had we been merely astronomers this is what we should have been obliged to do. We may perhaps doubt whether we should have in this way arrived at the conceptions of force which we possess with the aid of both senses. At any rate no one can doubt that an individual newly arrived in this world learns its properties as much through the muscular sense as through the more generally appreciated sense of sight. Let us now reverse the mode of looking at equations 41). Suppose that we find that under given conditions a certain agency will produce a certain force, as shown by the motion of some body, and suppose that as the circumstances are changed we can always measure the force. If then it is possible to submit a second body to the action of the same agent under similarly varying circumstances, we shall be able to find the motion of the second body. The equations 41) under these circumstances furnish merely another means of describing motions. We might go on obtaining still further descriptions by means of higher derivatives of the coordinates, but experience shows us that nothing is gained thereby, for, in the great majority of cases with which we have to deal, it is found that the components, X, Y, Z, are expressible as functions of only the coordinates of the bodies involved, or at most of the coordinates and their first time derivatives. There is a further advantage in the introduction of the notion of force, in that if a body be submitted to the action of two agencies at different times, so as to move under the influence of definite forces, and then be submitted to the action of both simultaneously, the force now found to be acting will be the resultant of the two original forces. This statement, that forces are compounded as vectors, being the equivalent of the so-called statement of the parallelogram of forces, is implicitly contained in Newton's second law of motion. Under certain circumstances, an agent which would under other conditions cause motion, may cause no motion. We then say that its effect is counteracted by that of some other agent, or otherwise, that the two forces are in equilibrium. According to the third law, the two forces are equal and opposite, either being the reaction with respect to the other. Such reactions are called static reactions, as opposed to the kinetic reactions exerted by bodies undergoing acceleration. As has been stated above, most of the forces which occur in nature depend only on the positions of the bodies upon which they act, or at most upon their positions and velocities, but not upon the higher derivatives of the coordinates. Forces of the former sort 26 I KINEMATICS OF A POINT. LAWS OF MOTION. are called positional forces, those of the latter motional forces. As an example of the latter, we know that a body moving through the air experiences a negative acceleration which is greater the greater the velocity of the body, and we say that the motion is retarded hy a force, which we call the resistance of the air. Supposing now X, Y, Z to be given functions of the coordinates and velocities, the integration of the differential equations 41) con- stitutes the problem of the mechanics of a single particle. It is in this sense that the problems of mechanics in general are to be considered. (See Note I.) Returning to the "change of motion" mentioned in the second law, it is customary to characterize the product of the mass by the vector velocity as the momentum of the body, a vector whose components are A^\ -n/r dx -~,r dy -,.- dz 42) JCrrr*»it< M» = mdl' M> = mw This is the motus whose rate of change measures the force, so that equations 41) may be written dM dM 14. Units. The specification of any quantity, scalar or vector, involves two factors, first a numerical quantity or numeric, and secondly a concrete quantity in terms of which all quantities of that kind are numerically expressed, called a unit. The simplest unit is that of the geometrical quantity, length. We shall adopt as the unit of length the centimeter, defined as the one -hundredth part of the distance at temperature zero degrees Centigrade, and pressure 760 millimeters of mercury, between two parallel lines engraved on a certain bar of platinum -iridium alloy, deposited in a vault in the laboratory of the "Comite International des Poids et Mesures", at Sevres, near Paris. This bar is known as the ({ Metre Prototype", and serves as the basis of length measurements for the civilized world J) (except the British Empire and Russia2). It was proposed by Maxwell to use a natural unit of length, namely the length of a wave of light corresponding to some well defined line in the spectrum of some element, at a definite temperature and pressure, as it is highly probable that such a wave-length is extremely constant. Measurements were carried out at Sevres by Michelson, with this end in view, which established the ratio between 1) See Gnillamne, La Convention chi Metre. 2) The United States yard is defined as 3600/3937 meters. 13, 14, 15J UNITS AND DIMENSIONS. 27 the above meter and the wave-length in air of a red cadmium ray as I^S^IGSA1) The unit of mass will be assumed to be the gram, denned as the one -thousandth part of a piece of platinum -iridium, deposited at the place above mentioned and known as the "Kilogramme Prototype". As the unit of time we shall take the mean solar second, obtained from astronomical observations on the rotation of the earth. The unit of time cannot be preserved and compared as in the case. of the units of length and mass, but is fortunately preserved for us by nature, in the nearly constant rotation of the earth. As the earth is gradually rotating more slowly, however, this unit is not absolutely constant, and it has been proposed to take for the unit of time the period of vibration of a molecule of the substance giving off light of the standard wave-length. To obtain such a unit would involve a measurement of the velocity of light, which cannot at present be made with the accuracy with which the mean solar second is known, 15. Derived Units and Dimensions. It can be shown that the measurements of all physical quantities with which we are acquainted may be made in terms of three independent units. These are known as fundamental units, and are most conveniently taken as those of length, mass, and time. Other units, which depend on these, are known as derived units. If the same quantity is expressed in terms of two different units of the same kind, the numerics are inversely proportional to the size of the units. Thus six feet is otherwise expressed as two yards, the numerics 6 and 2 being in the ratio 3, that of a yard to a foot. If we change the magnitude of one of the fundamental units in any ratio r, the numeric of a quantity expressed in derived units will vary proportionately to a certain power of r, r~n, the derived unit is then said to be of dimensions*) n in the fundamental unit in question. For instance, if we change the fundamental unit of length from the foot to the yard, r = 3, an area of 27 sq. ft. becomes 3 sq. yds., the numeric has changed in the ratio 3:27 = l:32 = r~"2, and the unit of area is of dimensions 2 in the unit of length. We may express this by writing [Area] = [£*]. The derived unit increases in the same ratio that the numeric of the quantity decreases. In our system the unit of area is the square 1) Travaux et Memoires du Bureau International des Poids et Mesures. Tome 11, p. 85. 2) The idea of dimensions of units originated with Fourier: Theorie ana- lytique de la Chaleur, Section IX. 28 I- KINEMATICS OF A POINT. LAWS OF MOTION. centimeter, written 1 cm2. In like manner the unit of volume is of the dimensions [Z3] and the unit is 1 cm3. The dimensions of velocity are pp » or as we write for convenience, Velocity = Length/ Time. Two quantities of different sorts do not have a ratio in the ordinary arithmetical sense, but such equations as the above are of great use in physics, and give rise to an extended meaning of the terms ratio and product. The above equation is to be interpreted as follows. If any velocity be specified in terms of units of length and time the numerical factor is greater in proportion directly as the unit of length is smaller, and as the unit of time is greater. For instance we may write the equation expressing the fact that a velocity of 30 feet per second is the same as a velocity of 10 yards per second or 1800 feet per minute. 30— = 10-^ = 1800-5^. sec. sec. mm. We may operate on such equations precisely as if the units were ordinary arithmetical quantities, for the ratio of two quantities of the same kind is always a number. For instance 30 yd. sec. 10 ft. sec." The ratio T is the number 3, while - — - = 1. Also it. sec. 1800 yd. min. Q CA 10 " " ~ftT ITecT ~ Such an expression as — '- is read feet per second. The unit of velocity is one centimeter -per -second, written, cm. - = cm. sec."1, sec. Since acceleration is defined as a ratio of increment of velocity to increment of time, we have , . -, [Velocity] [Length] r L [Acceleration] = ^..^ = L *e2j = [^ or the numeric of a certain acceleration varies inversely as the magnitude of the unit of length, and directly as the square of the unit of time. For instance, an acceleration in which a velocity of 10 feet per second is gained in 2 seconds is equal to one in which a velocity of 9000 feet per minute is gained in a minute, 10 ft. _ 10 ft. _ QQQQ ft. (2 sec.)2 4 sec.2"" min.2 15, 16] CHANGE OF UNITS. GRAVITATION. 29 The unit of acceleration is one centimeter -per -second per second, cm. written -- '2 = cm. sec.~2. (It is to be noted that in a derivative sec. such as -y^j-; the numerator being a differential of no matter what order is of the same dimensions as s, while the denominator being the square of a differential is of dimensions [T2]). Since momentum = mass • velocity, we have mr -i [Mass] • [Length] rML~\ [Momentum] = * rrimel = ~Y~ ' Since force = mass • acceleration, [~F -, _ [Mass] • [Length] __ [MlTi [Time2] ~" L~r*J The unit of force is one gram -centimeter -per -second per second. It is called a dyne. Moment of a force being force • length is of dimensions r2 The dimensions of an angular magnitude, being those of the ratio of two quantities of the same kind, arc and radius, are zero. Angular velocity being defined as -±gp- is of dimensions J • All physical equations must be homogeneous in the various units, that is, the dimensions of every term must be the same. This gives us a valuable check on the correctness of our equations. For an excellent account of the theory of dimensions the reader may consult Everett, The C. G. S. System of Units. 16. Universal Gravitation. We may now convert the kinematical statement of § 12 regarding the planetary motion into the dynamical one, that the sun attracts the different planets with forces proportional directly to the product of their masses and in- versely to the square of their distances from itself. From this we may pass to Newton's great generalization: Every particle of matter in the universe attracts every other particle, with a force whose direction is that of the line joininy the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other1), the factor of proportionality y being the same for all bodies. This is the law of Universal Gravitation, 1) Thomson and Tait, Treatise on Natural Philosophy, Part II, p. 9. 30 I KINEMATICS OF A POINT. LAWS OF MOTION. The numerical value of 7, the Newtonian constant of gravitation, depends upon the system of units used. Its dimensions are those of [Force] • [Length2] _ r Ls "1 [Mass2] ~\_WT*]' It is possible, and in astronomy is convenient to choose the units in such a manner as to make y equal to unity. If this were done, we should get a relation between the dimensions of mass, length and time, for by supposing that y has no dimensions, we should have Thus we should need only two fundamental units instead of three. This is an example of the somewhat arbitrary nature of the dimen- sions of physical quantities. What is not arbitrary however is the statement that every physical equation must be dimensionally homogeneous. For the purposes of physics it is customary to retain the three fundamental units, giving y the dimensions specified above. Determinations undertaken to ascertain the numerical value of y by terrestrial observations have been made in great numbers from the time of Cavendish to the present. One of the most accurate, that of Boys1), gives in the units which we have adopted, ' cm- gm. sec.2 that is, two spherical masses each of mass one gram with centers one centimeter apart attract each other with the force of y dynes.2) If two particles have coordinates xlfylf0lf x2, «/2> #2 an^ distance apart r12, the direction cosines of the line drawn from 1 to 2 are and, since the force exerted by 2 on 1 has the direction of this line, the equations of motion for 1 are A at" T! m* -V77T = V — - 1) Boys, Phil. Trans. 1895, I. 2) It will be shown later that homogeneous spheres attract each other as if their masses were all concentrated at their centers. 16, 17] GRAVITATION CONSTANT. THREE BODIES. 31 and for 2 are W2 dtz ~~f r122 r12 m^ = ?Y 2 r 12 '12 The integration of these six equations is easily carried out (see § 102), and gives us for the case of the sun and a planet a slight modification of Kepler's laws, for the sun does not remain absolutely at rest. If there are three bodies their equations of motion are similarly, v« — __. /• • I /vi/i *^ ^ «^ •*• I n/i/i K» dt* ~? -£ = 7 (^2 —r^ir + mz~-ir \ ri9. rt* ~d^ = r The problem of integrating these equations is known as ''the problem of three bodies" and has not been completely solved. The problem of the solar system is still more complicated, but by means of approximations, the perturbations of the different planets upon each other, causing slight variations from Kepler's laws, have been calculated. It is in this manner that the observations of astronomers from the time of Newton until the present have furnished the most brilliant verification of Newton's great discovery. 17. Absolute Systems. The above system of units, which has for its fundamental units the centimeter, gram, and second, is called the C. Gr. S. system, and was recommended by a committee of the British Association for the Advancement of Science in 1861. It is sometimes incorrectly spoken of as the absolute system of units. 32 I. KINEMATICS OF A POINT. LAWS OF MOTION. An absolute system is any system, irrespective of the magnitudes of the units, by which physical quantities can be specified in terms of the least number of fundamental units, which shall be independent of time or place, and reproducible by copying from standards. A system based on the foot, pound, and minute is just as much an absolute system as the C. G. S. system. The idea of an absolute system is due to Gauss.1) The ordinary method of measuring force, used by non- scientific persons and engineers, though very convenient, does not belong to the absolute system of measurements. The unit of force is taken as the weight of, or downward force exerted by the earth upon, the mass of a standard piece of metal, such as the standard pound or kilogram. To measure the force in absolute units, we must know what acceleration the earth's pull would cause this mass to receive, if allowed to fall. As stated above, the attraction according to the Newtonian law exercised by the earth is the same as it would be if the whole mass were concentrated in a very small region at its center. Consequently the more remote a body is from the center the less will be the earth's pull upon it, or its weight. If however we consider a region so small that its dimensions may be neglected in comparison with those of the earth, the force exerted2) upon a given body at any point of the region may be considered as constant, and exerted in a constant direction, called the vertical of the place. Dividing the weight, which is proportional to the mass of the body, by the mass, we find that the acceleration experienced by all bodies at a given place is the same. This was proved exper- imentally by Galileo, to the great astonishment and scandal of the philosophers of the time. (On account of the disturbing action of the air, this statement is exactly true only for bodies falling in vacua.} The value of this acceleration is denoted by g, and its value at the sea -level in latitude 45° is = 980.606 . Accordingly the force exerted by the earth on a mass of m grams is mg dynes, or the weight of a Mogram in latitude 45° = 980,606 dynes. Now the value of the acceleration g is not constant, but varies as we go from place to place on the earth's surface, ascend mountains or descend into mines. Accordingly, the weight of a kilogram is 1) Gauss, Intensitas vis magneticae terrestris ad mensuram dbsolutam revo- cata. Gottingen, 1832. Ges. Werke, V. p. 80. 2) For the effect of the earth's rotation, see § 104. 17, 18] II. IMPORTANT PARTICULAR MOTIONS OF A MATERIAL POINT. 33 not an invariable, or absolute standard of force. At the center of the earth, a kilogram would weigh nothing. Its mass is, however, invariable. The ordinary method of comparing masses by means of the balance is in reality a comparison of two forces, the weights of the bodies. As these are proportional to the masses, the method becomes one for the comparison of masses, being a statical one, as distinguished from the kinetic method of § 13. If, however, we should make use of a balance with arms so long that the two masses compared were situated in regions for which the values of g were different, equality of weights would not connote equality of masses. An instrument which shows the variable weight of a body as it changes locality is found in the spring -balance, another in the pendulum. The value of g at points on the earth in latitude A and h centi- meters above the sea -level, is given by the formula, originally given by Clairaut1), g = 980.62 - 2.6 cos 2 1 - 0.000003 h. For further information with regard to units, the reader may consult Everett's The C. G. S. System of Units. CHAPTER H. IMPORTANT PARTICULAR MOTIONS OF A MATERIAL POINT. 18. Constant Accelerations. Let us examine the motion of a particle experiencing a constant vertical downward acceleration g. If the axis of Z be taken vertically upward, we have for the equations of motion, Integrating with respect to t we have where VX9 Vy, Vz are constants representing the component velocities at the time t = 0. Integrating again, 3) x-x0 = Vxt, y-y, = Vyt, 0 - *0 = - gt* + V,t, 1) Everett, The C. G. S. System of Units, Chap. VI. The above constants are adopted by Helmert. WEBSTER, Dynamics. 3 34 II. PARTICULAR MOTIONS OF A POINT. where #0, yQ, #0 are the coordinates of the point at the time t = 0. Eliminating t between the first two of equations 3), we obtain which shows that the motion is in a vertical plane. (The twisted curves sometimes described by a base-ball, golf or tennis-ball or rifled shot are the results of actions due to the air and the rotation of the ball and not here contemplated.) If we choose this vertical plane for the plane of XZ, we shall have y = 0, Vy = 0, and the equation of the path is found by eliminating t between the first and third of equations 3) giving the equation of a parabola with axis vertical. If Vz is positive, the projectile will rise until ^- = 0, or The height reached at this point is projectile will rise until - = 0, or - = 0, that is x — #0 = 6) ,^^-V.i 2 16) becomes »> S+U-T' a parabola (Fig. 11). Since we may always express sinw# rationally in terms of sin#, cos 'x, when m is an integer, the elimination may always be performed and the curves will be algebraic. . 11. 2O. Central Forces. Having now dealt with two cases in which the acceleration passes through a fixed point, - - that of the motion of the planets and harmonic motions, it will be convenient to treat the general case. In § 12 we found the nature and magnitude of the acceleration by the differentiation of the equations expressing the motions. We will now consider the inverse problem, that of obtaining the equations describing the motion by integration of the differential equations of motion when the force is given. We have by § 10, 34) and 35) for the radial acceleration in the direction away from the center, 18) ar = ~ - dv and for the transverse acceleration, If the acceleration is central ay = 0 and we have by integration 20) ,**?_», Kepler's law of areas. 19, 20] CENTRAL FORCES. 39 It will now be convenient to change the independent variable from t to (p and at the same time to introduce the reciprocal of r, ^ -, dr dr dtp and introducing the value of -^ from 20) gives dr li dr , du dt rz dcp dcp Differentiating again and proceeding in like manner, so that finally, . . S+»= or as we may write it, Thus w.~ ! TT is given in terms of qp by an equation like equation 8), whose integral is u — i = a cos (cp — a) fi ... ah* or putting — = e, This is the equation of a conic section with which we started the investigation of § 12. In order to find the eccentricity e let us consider the initial circumstances, or the magnitude and direction of the velocity for a given position of the body. Let the body be projected from a point

e will be less than, equal to, or greater than 1, and the orbit will be respectively an ellipse, parabola, or hyperbola. The critical velocity, F, has a simple physical significance. Suppose we consider a particle falling from infinity straight toward the center of attraction. Its equation of motion is d*r _ j_ dt* ~ ~ T2" Multiply by -j-> both sides become exact derivatives and we may integrate, obtaining 20, 21] CONSTRAINED MOTION. 41 If it starts with no velocity the constant is zero, consequently F the velocity at a distance R is given by Therefore we may state the result by saying that the path will be an ellipse, parabola, or hyperbola according as the body is projected in any direction with a velocity less than, equal to, or greater than the velocity that it would acquire in falling from an infinite distance to the point of projection. 21. Constrained Motion. We have so far considered the moving particle as free to move in any direction. This is however by no means usually the case, since in the majority of cases with which we have to deal the particle forms part of a body which is possibly itself a part of a machine, and is guided by contact with other bodies to travel in certain definite paths, although the velocities with which it travels may be left undetermined. Such limitations to the freedom of movement of a body are known as constraints, and they are specified by certain equations having a geometrical significance. In the case of a single particle, the simplest constraint is that in which the particle is constrained to move upon a certain surface. For instance, if the surface is a material one, the particle may, during the whole motion, press against its inner, or concave side, the material preventing the particle from passing across the geometrical surface. The surface may itself be in motion, in this case the constraint is said to be varying, and the equation of the surface will contain the time. Let the equation expressing the constraint be 33)

^~.- If the surface is smooth, it is evident that it cannot affect a motion of the particle which would naturally take place on the surface. Consequently the reaction has no component tangential to the surface, but is in the direction of the normal. This is otherwise a definition of a smooth or frictionless surface. The components of the reaction Xlf Y^ Z± are accordingly proportional to the direction cosines of the normal to the surface cp = 0, so that we may write o7\ v idv v i^y 7 i^y f? ?-*?;? ri=^' zi = Aar When A has been determined as above we have for the magnitude of the reaction, As an example let us consider the motion of a particle acted upon by gravity and constrained to move on the surface of a fixed sphere of radius I. If the constraint is caused by attaching the particle to a fixed point by means of an inextensible string whose mass is negligible, we have the so-called ideal pendulum. The equation of constraint is and does not contain t, so that ~ = 0. If the ^-axis be taken vertically downward the equations of motion are 21] MOTION ON SPHERE. 43 ^-l^-lx dt* dx~ ^> d'g . 8 ^? adding and making use of 43) we may integrate at once and obtain where In is an arbitrary constant of integration. This integral gives us the square of the velocity and shows that it depends only upon the initial velocity and the height through which the particle has fallen, for if it has a velocity VQ when z = #0, we have to determine h. Making use of 44) in 41) we have 45) Jt- and from 38) 46) J? = 44 II. PARTICULAR MOTIONS OF A POINT. Multiplying the second of equations 42) by x and subtracting from it the first multiplied by y we obtain Fig. 12. M which expresses the fact that the horizontal compon- ent of the acceleration has no moment about the origin, as in § 12. We may therefore integrate, obtaining, 47) /c^_?/^==c where c is another constant of integration representing the moment of the horizontal component of the velocity about the origin and cor- responding to the h of § 20. It will be convenient to introduce polar coordinates such that (Fig. 12) x = Zsin-frcosqp, y = I sin # sin 9, z = 48) Differentiating we have dx = I (cos # cos + yF Csin^dtlj + ^W Csin 0 Now since 71 1-3-5 . . . 2w — 1 sm 0 If a be the maximum value of #, for which ^ = — > A; = sin — ? and the period is given by r»t\ or o 65) 2'=2 This is the formula which is used to correct our result 56) for finite oscillations. If cc is 1° the correction is less than one part in fifty thousand, and if a = 5° it is less than one in two thousand. 23. Spherical Pendulum. Let us now return to equations 53) and 54), which we will write 66^ t = + I -^L, cldz > where <&(z) = 2(l2 — £2) (gz -f li) — c2. As the integrals are real (P(V) must be positive for all values of z that occur in the motion. Substituting successively for £, — oo, — ?, #0, + I we find Accordingly the polynomial 3>(z) has three real roots. If we call these a, /3, 7 in the order of magnitude, they lie so that Fig. 13 is the graph of <&(2) as ordinate, with 2 as abscissa. SPHERICAL PENDULUM. 22, 23] Since we have, equating the coefficients of #, 49 from which ft 0. Since between ft and y ®(&) is negative, z cannot in the motion lie in this region (for ~\/(z) must be real). Now since, 66) Fig. 13. when 0 the mean position of the particle is below the center of the sphere. Since by 50) ^ _ cp always varies in the same sense and when z equals a or ft the path has a horizontal tangent, for — - — - = 0, while -~ is not equal to zero. If #0 is a root, that is if the particle was originally on one of the limiting circles, we must take the positive sign for the radical in the integrals if #0 = ft (so that ^| may be positive and e increase), the negative sign if £0 = a. The time of passing from the highest to the lowest point is T= r ids m J V*®' The meridian planes passing through the points of tangency with the parallels a and ft are planes of symmetry for the path. WEBSTER, Dynamics. 4 50 II. PARTICULAR MOTIONS OF A POINT. For if we consider two points P, Pf of azimuths qp, y>' lying on the same parallel and on opposite sides of a vertex A (Fig. 14), cldz and since the radical changes sign on passing through a vertex, cldz Therefore the points P, P' are symmetrical about A and the times cc of traveling the arcs PA and APr are equal to / -— — • In like J v*o*) $ manner it can be shown that the path is symmetrical about an upper vertex B. The path is accordingly composed of equal parts continu- ally repeated. It of course is not generally true that the path will be reentrant after Fig. 14. g°ing once around the sphere. We will now consider the horizontal projection of the path. Fig. 15 a. Fig. 15 b. 1°. Suppose both limiting parallels are below the equator, the projection of the circle e = cc is within that of z = /3, and the path 23] COMPARISON OF THEORY AND EXPERIMENT. 51 is similar to Fig. 15, the angle sub- tended at the center by two successive points of tangency being greater than a right angle, as we shall see. Figs. 14, 15 a, 15b, 15c, 16a, are reproductions of photographs of actual swings of a pendulum. A brass ball was swung by a string attached to a screw - eye, and carried a small in- candescent lamp. On the floor below, and at one side were placed cameras with open shutters, in a dark room. When the ball was swung, the light was turned on for a sufficient number of swings, and the path registered on the photographic plate. On the photograph Fig. 15c, the maximum and minimum radii were mea- sured, from which could be calcu- lated the roots a, /J, and thence y. Then from equa- Fig. 15 c. Fig. 15 d. 52 n. PARTICULAR MOTIONS OF A POINT. tion 67), by an arithmetical approximate quadrature, cp was calculated for a number of values of z, from which with the polar coordinates tp)r=~\/l/* — z*, the horizontal projection Fig. 15 d was drawn. It will be observed that it almost exactly coincides with the observed curve Fig. 15 c. From the projection and the values of z the per- spective Fig. 16 b was constructed, which in like manner nearly coincides with the observed Fig. 16 a. The eye is below the shaded square in the figure. Figures 15 d, 16 b were constructed by Mr. Joseph G. Coffin. Fig. 16 a. 2°. If /3 is negative, the projection of the circle z = a is still within that of z = ft, for since a + /3 > 0, the lower circle is farther Fig. 16 b. from the center than the upper. The projection of the path is also tangent to the equator. The angle AOB in Fig. 15 a has the value 69) *-f— a cldz Inserting the value y = - ~T~ * 23] we have 'ANGLE BETWEEN SUMMITS. 53 Now putting 0 = 1, writing we have 70) A = Y(l — a) (I — ft) , AS C J (l*- In order to find limits between which this integral lies for all possible values of a and ft, we notice that the coefficient of z in the last factor is positive, and that the value of the factor, varying always in the same sense as z, necessarily lies between the two values it would have when 0 had its extreme values, I and — I. But these are B2 and A? so that Substituting in the radical a value that is too great or too small will make the integral have an error in the opposite sense, therefore C- J (i*- p The polynomial under the radical being now of only the second degree, the integral can be easily calculated, as follows. idz _ j.^ r fa i r ^ 54 H. PARTICULAR MOTIONS OF A POINT. Accordingly we have therefore *P > —> as above stated.1) If in the integral 70) we substitute for the factor the greatest and least values that it takes during the motion, namely 0 = a and z = ft, we shall get closer limits between which ty lies. If we then make a and /3 approach I, *P will approach a right angle, so that the horizontal projection tends to be a closed curve. This case may also be treated directly. Our equations 40) were Now we have 0 =~J/Z2 — (x* + £/2) = l( 1 - — j^-J and developing by the binomial theorem, .- If now x and y are small with respect to I and we neglect small d* z quantities of the second order, z is constant. Then -^ = 0, and from the third equation above, i _ 9 T Inserting this value of "k in the first two gives _ dt* ' I > dt* the integrals of which are where a, 6, a, /3 are arbitrary constants, giving elliptic harmonic motion of the same period as that of the small plane harmonic motion. Another important case is that in which the two roots a and /3 are equal. We then have 2 and & constant, and 1) This treatment is taken from Appell, Mecanique Eationelle. The proof that *F > — -is due to Puiseux. 23] CONICAL PENDULUM. 55 The condition for equal roots is that (&(#) and '(#) have a common root. Now c2, If then (!>' (%) = 0, we have together with from which We accordingly have for the value of dcp _ c ~dt — i*-s* ~ We thus obtain for the time of revolution *(*.) = 2 fo*0 + A) (J»-V) -«•- c2 = 2 to, + A) (Z» - V) = g { The time of revolution of a conical pendulum is the same as that of a complete oscillation of a plane pendulum of length #0 performing small vibrations. As & approaches a right angle, #0 and therefore T approaches zero, that is the velocity increases without limit. We have in this case Now the centripetal acceleration in the circular motion is (§ 10), ^="9 An acceleration g directed downward together rig. 17. with the reaction R directed toward the center of the sphere will compound into an acceleration g tan & in a horizontal direction (Fig. 17). Accordingly if the particle is projected horizontally with the velocity v, it will describe a circle. 56 HI. GENERAL PRINCIPLES. WORK AND ENERGY. CHAPTER HI, GENERAL PRINCIPLES. WORK AND ENERGY. 24. Work. If a point be displaced in a straight line, under the action of a force which is constant in magnitude and direction, the product of the length of the displacement and the component of the force in the direction of the displacement, that is, the geometric product of the force and the displacement § 4, 10), is called the work done by the force in producing the displacement. If the components of the force F are X, Y, Z, and those of the displacement s are sx> s^ sz, the work W is 1) W=sFcos (Fs*) = Xsx + Ysy + Zsz. It is at once evident that if a force is resolved into components, the sum of the works of the components is equal to the work of the resultant, for if W2 = X2sx + Y2sy + Ztsg, Since work is denned as force x distance, we have for its dimensions, [Work] = [L] [^j = [ML2T~*]. The C. G. S. unit of work is the work done when a force of one dyne produces a displacement of one centimeter in its own direction. This unit is called the erg = gm • cm2 • sec~2. If the displacement be not in a straight line, and the force be not constant, the work done in an infinitesimal displacement ds is and the work done in a displacement along any path AS is the line integral 3) ^' The components of the force are supposed to be given as func- tions of s and the derivatives -f-> -A -. *- are known as functions of s as as as from the equations of the path. 24, 25] GEOMETRICAL CONSTRAINTS. 57 Understanding this, we may write B 4) WAB=fxdx + Ydy -f Zdz. A 25. Statics. Virtual Work. Suppose that we have a system of n material points. If they are entirely free to move, they require 3n coordinates for their specification. If however they are subjected to geometrical constraint, as explained in § 21 for a single particle, there must he certain relations satisfied by their coordinates. Let these equations of condition or constraint be 9>1 fa> Vl> gl> X*> 2/2, %• •-- Xn> yn, *«) = 0, 92 0*i, ft, *!, x*, 2/2, *a> - • • xn, y«, *„) = 0, 9k i, ft, *i, x*9 2/2, *a> • • • x*> y*> = °- Such constraints may be caused in a great variety of ways. Particles may be caused to lie on certain fixed or moving surfaces, may be connected by inextensible strings which may pass over pulleys, or by rigid links variously jointed. For instance, if two particles 1 and 2 are connected by a rigid rod of length ?, either particle must move on a sphere of radius I of which the other is the center, and we have the equation of condition 9 ~ («, - xtf + (y, - ytf + (Zi - stf -l* = 0. (We might have constraints defined by inequalities, e. g., if a particle were obliged to stay on or within a spherical surface of radius I the constraint would be only from without, and we should have (x - a)* + (y- &)2 + (g - c)2 - I2 ^ 0. We shall assume that the constraint is toward both sides, and is defined by an equation.) If any particle at xr, y,~, zr is displaced by a small amount so that it has the coordinates xr + $xr, yr + dyr, zr + $zr, in order that the constraints may hold we must have for each 9, yr, sr, . . .) = 0, (f (Xr + dxr, yr + Syr, 2r + §Zry . . .) = 0, and if g? be a continuous function, developing by Taylor's Theorem, r\ V (Xr + dXr, yr + 8yr, «i + 8*r, ' • •) = 9 (»r, yr, Zr, . . •) + 8 XT * + 58 HI. GENERAL PRINCIPLES. WORK AND ENERGY. Accordingly, taking account only of the terms of the first order in the small quantities 8xr, 8yr, dzr, and using equations 6) we have If a number of particles are displaced, we must take the sum of expressions like the above for all the particles, or as the conditions which must be satisfied by all the displacements dxr, dyr, dzr. There must be one such equation for each function cp. Such displacements, which are purely arbitrary, except that they satisfy the equations of condition, are called virtual, being possible, as opposed to the displacements that actually take place in a motion of the system. If the equations of constraint contain the time, t is supposed to be kept constant during the virtual displacement. The number of independent coordinates possessed by a system is called the number of degrees of freedom of the system, which may be otherwise defined as the number of data necessary to fully specify its position. Between the 3n changes dx, dy, dz, occurring in an equation, there are It linear equations, hence only 3^ — k of them may be taken arbitrarily, and this is the number of degrees of freedom of the system. It has long been customary to make a subdivision of the subject of Dynamics entitled Statics which deals with only those problems in which forces produce equilibrium. A system is in equilibrium when the impressed forces upon its various particles together with the constraints balance each other in such a way that there is no tendency toward motion of any part of the system. The Principle of Virtual Work is the most general analytical statement of the conditions of equilibrium of a system. It was used in a very simple form by Galileo, but its generality and its utility for the solution of problems in statics was first recognized by Jean Bernoulli, and it was made by Lagrange the foundation of statics.1) If the system consists of a single free particle, in order for it to be in equilibrium the resultant of all the forces applied to it, whose components are X = XXr, Y= I.Yr, Z = I.Zr) must vanish, 9) x=r=z=o. 1) For the history of the principle see Lagrange, Mecanique Analytique, Partie, Section I, §§16 and 17. 25] PRINCIPLE OF VIRTUAL WORK. 59 If we multiply these equations respectively by the arbitrary small quantities dx, dy, dz, and add, we get 10) Xdx + Ydy + Zdz = 0, which states that the work done in an infinitesimal displacement of a point from its position of equilibrium vanishes. The equation 10) is equivalent to the equations 9), for since the quantities dx, dy, dz are arbitrary, if X, Y, Z are different from zero, we may take dx, dy, dz respectively of the same sign as X, Y, Z, -- each product will then be positive, and the sum will not vanish. If the sum is to vanish for all possible choices of dx, dy, dz, X, Y, Z must vanish. If the particle is not free, but constrained to lie on a surface cp = 0, dx, dy, dz are not entirely arbitrary, but must satisfy Let us multiply this by a quantity A and add it to 10), obtaining We may no longer conclude that the coefficients of dx, dy, dz must vanish, for dx, dy, dz are not arbitrary, being connected by the equation 7). Two of them are however arbitrary, say dy and dz, I has not yet been fixed - - suppose it determined so that Then we have in which dy and dz are perfectly arbitrary, it therefore follows of necessity that the coefficients vanish. By the introduction of the multiplier A we are accordingly enabled to draw the same conclusion as if dx, dy, dz were arbitrary. If X, Y, Z refer to the resultant of the impressed forces only, not including the reaction, equations 9) do not hold, but if we suppose 10) to hold, we shall obtain the conditions for equilibrium. Elimin- ating A from the above three equations we get X Y Z dtp drp dx dy dz Now the direction cosines of the normal to the surface qp = 0 O r\ r\ are proportional to **r-> 7p? -^-j consequently, the components X, Y, Z being proportional to these direction cosines, the resultant is in the 60 HI. GENERAL PRINCIPLES. WORK AND ENERGY. direction of the normal to the surface. But owing to the constraint the motion can be only tangential, consequently the particle cannot move, and the applied forces together with the reaction produce equilibrium. The Principle of Virtual Work is as follows. If any system of as many bodies or particles as we please, each acted upon by any forces whatsoever, is in equilibrium, and a small arbitrary virtual displacement is given to each point of the system, the work done by all the forces will vanish (at least to the first order of small quantities). For instance a particle placed on a smooth surface under the action of gravity experiences a force mg vertically downward. If we displace it a distance ds the work done by the force will be mgdz, if the z coordinate is taken positively downward. We may write this and if this vanishes whatever the value of ds for all directions of displacement on the surface v -=- must be zero, that is the tangent plane to the surface is horizontal. But the particle is in equilibrium at such a point. Conversely, if the surface is not horizontal, dW will not vanish for all possible displacements, neither will the particle be in equili- brium. (It is to be noticed that in the neighbourhood of a point where the tangent plane is horizontal dz is proportional to ds2, so that the work, although vanishing to the first order, does not vanish to the second, z is in this case a maximum or minimum.) Simple illustrations of the principle of virtual work are furnished by the so-called mechanical powers. Consider in particular the pulley. The mechanical advantage or multiplying power as regards force, that is the ratio of the force sustained by the movable block to the tension on the cord, is equal to n, the number of cords coming from the movable block, for the fundamental assumption is that the tension of the cord is everywhere the same. If the end of the cord is displaced a small distance in its own direction, the block is displaced l/nih of that distance, consequently the work of the two equilibrating forces is equal in absolute magnitude, but one being positive and the other negative, their sum is zero. By means of this principle Lagrange gave a simple general proof of the principle of virtual work. He supposed each force applied to a point of the system to be replaced by a pull of a block of pulleys, the number of pulleys in each block being so chosen that the proper force could be produced by the tension of a single cord passing over all the pulleys and fastened to a weight at one 25] CONDITION OF EQUILIBRIUM. 61 end, the other being fixed. If now the forces are in equilibrium, an arbitrary small displacement of all the blocks will neither raise nor lower the weight at the end of the string. Thus by the applica- tion of the property of the pulley the principle was proved.1) We shall not here undertake to give a more formal proof of the principle, which may be given by an analysis of the various kinds of constraint, such a proof is found in Appell, Traite de Mecanique Bationelle. Tom. I, Chap. 7. If the forces X19 T19 Z± act upon the particle 1, X2, Y2, Z2 upon the particle 2, etc., the condition of equilibrium is 12) X, dx, + YJyi + Z^gt or as we may write it, is) 2j (Xdx + Ydy + Zd^ = °- This is the analytical expression of the Principle of Virtual Work. If the particles satisfy the equations of constraint 5) the dis- placements must satisfy the equations Multiplying the equations 14) respectively by J^9 Ag, . . . A*, and adding to 12) we have 2n = 0. Of the 3n quantities dxlf . . . d#n, only 3n — Jc are arbitrary, we may however determine the Jc multipliers A so that the coefficients 1) Ibid. § 18. 62 HI. GENERAL PRINCIPLES. WORK AND ENERGY. of the & other tf's vanish, then the coefficients of the 3n — ~k arbitrary <5X, ^^, ^2/2^ ^^2- We may therefore take any two of them arbitrarily. Suppose we assume dyl = dy2 = 0. We then have four linear equations in dx±, d£17 dx2, d£2, and in order that they may be satisfied for values 1) Lagrange, Mecanique Analytique, torn. I, p. 79. 2) See Note II. 25, 26] D'ALEMBERT'S PRINCIPLE. 63 of the d's other than 0, the determinant of the coefficients must vanish 0 , mt , 0 , m2 x, , ii • :' , o , o 0 0 r z \J • \J • Ju& j && = 0 or reducing, (%#! -f m2a;2) (iu2^3 — ojjtfg) = 0. The solution that applies is given by the vanishing of the first factor, that is, m x + m x =0 In like manner if we had assumed dx± = dx2 = 0, we should have obtained This equation with the preceding gives by the elimination of mlt w2, Hence the points lie in a vertical plane containing the center of the sphere. The two equations express the fact that a point dividing the line connecting the particles in the inverse ratio of their masses is vertically below the center of the sphere. The azimuth of the plane containing the particles is indeterminate on account of the symmetry about the vertical. 26. D'Alembert's Principle. The equations of motion of a particle may be written dt< 17) Zr — mr o, 0. Multiplying these equations respectively by the arbitrary quantities dxr, dyr, dzr, adding, and taking the sum for all values of the suffix ry belonging to the different particles of a system, This equation may be called the fundamental equation of dynamics, and is the analytical statement of what is known as d'Alembert's 64 HI. GENERAL PRINCIPLES. WORK AND ENERGY. Principle. Lagrange made it the basis of the entire subject of dynamics.1) We may interpret 18) in terms of the principle of virtual work by means of the introduction of the conception of effective forces due to d'Alembert. If a system of particles is not free, when acted on by certain impressed forces it will not take on the same motion as if there were no constraint, the reactions causing it to deviate from this natural motion. Having found the actual motion, we know the system of forces that would produce it, if there were no constraints. These are termed the effective forces and if we represent them by Xr', Yr', Z/, they are given by the equations d* x dz y d^ z Xr' = mr^, Yr' = mr^f, Zr' = mr-^- The equation 18) accordingly states that the reversed effective forces, - X', — Y', — Z' together with the impressed forces, X, Y, Z, will form a system in equilibrium. We may regard the principle from another point of view. When a body is set in motion with an acceleration, it reacts on the agent which produces the motion, and this kinetic reaction has the properties of any force whatsoever. For instance if the accelerating agency is due to contact with a second moving body, the second body is retarded by a force, and this force is the reaction of the first. This kinetic reaction is measured by the components and is thus in the opposite direction to the acceleration experienced by the body. The reaction is often termed the Force of Inertia, a very expressive term, representing in tangible form the fundamental property of inertia, possessed by all matter, this property being that matter reacts against, or in ordinary language resists, being put in motion. (By the use of the term resists we in no wise mean prevention of motion - - the use of the term has been objected to, and Maxwell2) has jokingly remarked that we might as well say that a cup of tea resists being sweetened, because it does not become sweet until we add sugar. The meaning here is precisely similar - we mean that matter does not move until it is moved by some agent external to itself. It is hardly likely that confusion can be caused by the use of such common phrases, which indeed seem to attribute volition to matter - - we shall accordingly make no attempt to avoid them.) We may thus define matter as that which can exert forces of 1) Lagrange, Mecanique Analytique, 1. 1, p. 267. The equation 18) although first explicitly given by Lagrange, will be referred as "d'Alembert's equation", as briefer than uLagrange's equation of d'Alembert's Principle". 2) Maxwell, Scientific Papers, Vol. II, p. 779. 26, 27] KINETIC REACTION. 65 inertia. This is the only universal definition of matter now possible. .-> "") (It is to be noticed that this definition includes the luminiferous , [ ether.) We may then state d'Alembert's Principle in these words: The impressed forces, together with the forces of inertia, form a system in equilibrium. Thus the principle is not new, but merely expresses Newton's third law of motion, embodying at the same time the other two, in the expression of the forces. The great service done by d'Alembert was in reducing the statement of a problem in motion to that of a statical problem. A practical advantage frequently of great use in applications is similar to that possessed by the principle of virtual work, namely, that the reactions of the constraints do no work, and may therefore be omitted from the equation 18), for it is evident that the reactions due to all constraints between bodies act equally in opposite directions on both, so that the work done in the motion of their common point of application vanishes. As a simple example of the meaning of force of inertia consider two locomotives pulling in opposite directions at the ends of a train, the pulls being transmitted by spring dynamometers. If the train remains at rest, the pull recorded on both dynamometers will be the same. If now one locomotive be given more steam, so that the train begins to move, the indications of the dynamometers will be found to be unequal, the greater pull being that of the locomotive on the side toward which the train is moving, the difference being found to be exactly equal (disregarding friction) to fche product of the mass of the train by the acceleration which it gains. Thus the difference of pull is balanced by the force of inertia, or kinetic reaction. Again, consider a person standing in a street -car, when the car starts. An acceleration is impressed on his body in the direction of the motion of the car. The kinetic reaction is thus directed horizont- ally to the rear. The force of weight of the person being vertically downwards, the remaining force, namely, the static reaction of the floor of the car, must be such as to equilibrate these two, and is found by the triangle of vectors to be directed upwards and inclined forwards. Thus the person must lean forward in order to preserve equilibrium. Similarly when the car stops, the acceleration being directed the other way, he must lean backward. This application of d'Alembert's Principle is a matter of common knowledge, where electric railroads are common. 27. Energy. Conservative Systems. Impulse. If in the equation of d'Alembert's principle, 18), we put for dx, dy, 82 the WEBSTER, Dynamics. 5 66 HI. GENERAL PRINCIPLES. WORK AND ENERGY. displacements which take place in the actual motion of the system in the time dt, dxr dyr dzr we obtain ™r \dt* ~dt~ ~*~ ~W ~~dt~ + ~dtr ~di. dxr dyr dz\ Now since mr~d£~dT = ^dt\mr\dT } I' the sum of the first three terms is the derivative of the sum and the equation may be written, omitting the factor dt, The expression dXr\* (dyr\* /d!*r\2} 1 ) + ( -W) + U) ) = the half -sum of the products of the mass of each particle by the square of its velocity, is called the Kinetic Energy of the system. It is one of the most important dynamical quantities. If we denote -j fj-j it by T, equation 20) has on the left -^- Since Xrdxr H- Yrdyr + Zrdzr is the work done upon the rth particle, the terms under the summa- tion sign on the right denote the total work done by the impressed forces in unit time, or the Activity ^) of the forces. The equation 20) is called the equation of activity, and states that the rate of increase of kinetic energy of the system is equal to the activity of the im- pressed forces. 1) The word "adio" is used by Newton, in a scholium on the third law, where he says, "If the activity of an agent (force) be measured by its amount and its velocity conjointly; . . . activity and counteractivity, in all combinations of machines, will be equal and opposite." The activity will sometimes be denoted dA 27] EQUATION OF ENERGY. 67 Integrating equation 20) with respect to t between the limits and ^, 21) The square brackets with the affixes tQ, ^ denote that the value of the expression in brackets for t = tQ is to be subtracted from the value for t = t1. The integral on the right of 21), which may be written Xrdxr -f Yrdyr -f Zrdzr, denotes the work done by the forces of the system on the particle mr during the motion from t0 to tlf and the sum of such integrals denotes the total work done by the forces acting on the system during the motion. The equation 21) thus becomes 22) Ttl - Tto to This is called the equation of energy, and states that the gain of kinetic energy is equal to the work done by the forces during the motion. The equation of energy assumes an important form in the particular case that the forces acting on the particles depend only on the positions of the particles, and that the components may be represented by the partial derivatives of a single function of the coordinates 7 r r In this case the expression 2r{XrdXr + Yrdyr + Zrdgr] =^ is the exact differential of the function U, and the integral that is the work done in the motion, does not depend upon the paths described by the various particles, but only on the initial and final configurations of the system, since 68 HI. GENERAL PRINCIPLES. WORK AND ENERGY. Ut = Ux8 . . . and where the affix 0 denotes the value of the coordinate at the time t0. The equation of energy then is 24) Tti-T,. = Utl-Uia. The function U is called the force -function, and its negative W= — U is called the Potential Energy of the system. Inserting W in 24) we have 25) Ttl + Wtl = T,, + Wta. The sum of the kinetic and potential energies of a system possessing a force -function depending only on the coordinates is the same for all instants of time. This is the Principle of Conservation of Energy. Systems for which the conditions 23) are satisfied are accord- ingly called conservative systems. The potential energy, being defined by its derivatives, contains an arbitrary constant. The functions T and W have one essential d xr dyr dzr difference, namely, T contains only the velocities, —^r\ ~^r> ~j^' ' ' •> while W does not contain the velocities, but only the coordinates. One important consequence of the equation of Conservation of Energy is that if at any time in the course of a motion, all the points of the system pass simultaneously through positions that they have occupied at a previous instant, the kinetic energy will be the same as at that instant, irrespective of the directions in which the particles may be moving, for T -f W is constant during the whole motion, and W depends only on the coordinates, consequently when all the coordinates resume their former values, the kinetic energy does the same. In other words, the work done on the system has been stored up or conserved, to the amount TF, and may be got out again by bringing the system back to its former configuration. For instance, a particle thrown vertically upward, or a pendulum swinging, have the same velocity when passing a given point whether rising or falling. As an example, consider a particle acted upon by gravity. We have 26) X = 0, F=0, Z=-mg, so that U= — mgz -f const. The equation of energy is 27) m (v* - t>0«) = or the velocity depends only on the vertical height fallen. Accord- ingly a particle, descending from a point A to another I?, constrained 27] CONSERVATION OF ENERGY. 69 to follow any curve, reaches B always with the same velocity, although the time occupied in the descent may be very different from one curve to another. The equation 27) might have been applied to immediately give us the integral equation 44) § 21. (In that equation , the Z-axis is drawn positively downward) The principle of virtual work, § 25, may evidently be expressed by saying that for equilibrium the potential energy of the system is a maximum or minimum, and a little consideration shows that for stable equilibrium it is a minimum.1) For instance in the above example the potential energy W= mgz + const, z being measured positively upward. If the particle is in equili- brium on a surface concave upwards, z and with it W is a minimum, the equilibrium being stable. If the concavity is down- wards, the equilibrium is unstable and W is a maximum.2) The question of stability of equilibrium will be discussed in § 45. It is possible to have a force -function denned by equations 23), which contains the time as well as the coordinates. The system is not then conservative, and it is not customary to speak of its potential energy. We have now dU dU —,(dUdxr dU dyr dU dz so that our equation of activity 20) is in this case dT _ dU dU ~3f**~dt^W In certain cases we may be able to assign the term -^-~ to a potential dW energy, as -- -rf-~.. If the forces depend on the velocities or on anything beside the coordinates, the system is not conservative. Such a case is that of motion with friction, where the friction, being a force that always tends to retard the motion, not only changes sign with the velocities but also depends upon the magnitudes of the velocities in such resisting media as the air and liquids. The dynamical theory of heat accounts for the energy that apparently disappears in non- conservative systems. 1) Dirichlet, Uber die Stabilitdt des Gleichgewichts. Crelle's Journal, Bd. 32, p. 85 (1846). 2) See Kirchhoff, Mechcmik, p. 34. 70 HI. GENERAL PRINCIPLES. WORK AND ENERGY. We shall see later, that whereas positional forces are usually conservative, and motional forces not, there are certain conservative motional forces. Kinetic energy being defined as 2 — mv2 is of the dimensions -7?r~ ' the same as those of work. Potential Energy is defined as work. The C. Gr. S. unit of energy is, therefore, the erg. We have in this chapter been concerned with the line integral of the force exerted on a moving point resolved in the direction of the motion of the point of application. This has been called the work of the force, and is physically a quantity of fundamental importance. We have occasionally to consider the time -integral of a force, that is, if F be a force always in the same direction, the quantity which has received the name of the impulse of the force during the time from £0 to ^. The effect of a force may be measured either by the work or by the impulse, but it is to be observed that the information obtained when one or the other of these two quantities is given is of a quite different nature. Supposing the force is constant in magnitude and direction, the work done is equal to the force times the distance moved, and a knowledge of the work tells us how far the point of application will be moved by the given force, while the impulse is equal to the force times the interval of time, and tells us how long the point will move under the application of the given force. If the force is variable, considering the significa- tion of a definite integral as a mean1), we may say that the work is the mean with respect to distance of the force multiplied by the length of the path, while the impulse is the mean with respect to the time multiplied by the duration of the motion. Thus the work answers the question "how far", while the impulse answers the question ahow long". The work is a scalar quantity, its element being the geometric product jf the force and the displacement. For the element of impulse, however, we have, using equation 7), § 3, Fdt = Xdt cos (Fx) + Ydt cos (Fy) -f Zdt cos (F0) thus the element is the component in the direction of the force of the vector whose components are ______ d!x=Xdt, dIy 1) See footnote, § 34, p. 98. 27] IMPULSE. IMPULSIVE FORCES. 71 For the whole impulse we may then take as definitions so that the impulse is a vector quantity. We thus lose the relation to the direction of the path, or of the force, in the case of a variable force, but on comparing with equations 43), § 13, dMx dMy dMz we have by integration so that the impulse of a force acting on a single particle for a certain interval of time is equal to the vector increase of momentum during that interval. The case in which the impulse of a force is of most importance is that of what are known as impulsive forces, which arise where actions take place between bodies in such a brief interval that the bodies do not appreciably change their positions during the action, although sensible changes of momenta take place. If in the equa- tions above, the length of the interval ^ — t0 decreases indefinitely, while the force -components X, Y, Z increase indefinitely, the integrals may still approach finite limits Ix=lim In this case we can not investigate the forces in the ordinary manner for the accelerations have been infinite, but the velocities and momenta have received finite changes in the vanishing interval. The work done is in like manner finite, though the distance moved vanish. The impulse and work of all ordinary, that is finite forces acting at the same time may thus be neglected, since the integral of a finite integrand over a vanishing range of integration vanishes. On account of the third law, the action and reaction being equal during the operation, the impulses of the forces on the two bodies are equal and opposite, so that what one gains in momentum the other loses. It is in this manner that the impact of two billiard balls, or the action of a shot on a ballistic pendulum, is to be dealt with. Many instruments used in electrical measurements act on this principle, that the momentum suddenly communicated to a body at 72 HI. GENERAL PRINCIPLES. WORK AND ENERGY. rest, which afterwards proceeds to execute an observed swing, measures the time -integral of an impulsive force.1) In order to find the work done by a given impulse, let us make use of the equation of work and energy, 22), which says that the work done is equal to the increase of the kinetic energy. The latter may be written, bearing in mind the definition of momentum, T = I m (vl + vl + O = ~ (Mxvx + Myvy + M,vJ. Suppose now the particle set in motion by an impulsive force, from rest. The kinetic energy acquired, and accordingly the work done, is then one -half the geometric product of the impulse and the velocity generated, or in other words, the geometric product of the impulse and the average value of the velocity at the beginning and the end of the impulsive action. This may be otherwise shown, whether the particle start from rest or not, by the following considerations.2) Since the interval of time and the distance moved are infinitely small, we may consider the motion as rectilinear. Suppose the initial velocity to be vQ9 and the final value i\, and let s be a parameter which during the interval runs rapidly through all values from 0 to 1, so that at any part of the interval But as the momentum always increases at a rate proportional to the increase of velocity, we have also M = M, + s(Ml- M.] = M0 + tl, 1) Suppose that a body which swings according to the law of the pen- dulum, or equation 8), § 19, receives, when in its position of equilibrium, an impulse I. It swings out according to the equation dx x = asmnt, -j- = ancosnt Ct v during a time t = ?t/%n to a maximum excursion a, at which its velocity vanishes, and it turns back. • If its mass is m, the momentum communicated to it while at rest was T / dx\ I = lm \ = man \ dt/t=o so that if we know m , a , and n = 2 jt/period , we can measure the impulse of the impulsive force. This is the mode of use of the ballistic galvanometer and electrometer, as well as of the ballistic pendulum formerly used in gunnery. The same formula applies (see Chapter X) , to the heeling of a ship when a shot is fired from a cannon. 2) Thomson and Tait, § 308. 27, 28] WORK OF AN IMPULSE. 73 where / is the total impulse. From the equation of motion we have " dt ~ dt so that we obtain for the work *i *i i W=jFds =JFvdt = lj[v, + s (v, - t,0)] d* = * I(Vl + «0). t0 t0 0 Thus we find as before for the work of an impulsive force the product of the impulse by the average velocity at the beginning and end of the action. It is evident that the same is true for the infinitesimal work done by an ordinary, that is finite force, during an infinitesimal interval. This conception of the impulse will be useful to us hereafter, in connection with the following. For a system of particles, we have for the kinetic energy, Now the kinetic energy is known when we know the velocities of every particle of the system, as well as their masses, no matter what their positions. If we consider T as a function of the velocities, we have accordingly ™ dT dT cT Mxr = mrvxr = Q-V ; Myr = ni.rVyr = Q-- > Mzr = mrvzr= ^ ,; or the momentum components of any particle are the partial derivatives of the kinetic energy of the system, considered as a function of all the velocities of the particles, by the respective velocity -components. Thus we may write $T a-;; + which by the theorem of Euler is true for any homogeneous quadratic function. 28. Particular Case of Porce- function. The conditions necessary for the existence of a force -function being 23), we must have, since d*U_ d*U d*U dzU d*U d*U dxdy dydx oydz dzdy dzdx dxdz dYr _ dXr - == ' It will be shown below (§ 31) that these conditions are also sufficient. 74 III. GENERAL PRINCIPLES. WORK AND ENERGY. In the particular case in which the only forces acting on the system are attractions or repulsions by the several particles directed along the lines joining them and depending only on their ^ mutual distances , a force - function always exists. For let the force between two particles mr and ms at a distance apart rrs be Fig. 18. It will be convenient to consider F positive if the force is a repulsion. Consider now the force Fs^ acting on ms and acting in the direction from mr to ms. Its direction cosines are those of the vector rrs. 30) Z(r) Now since r?s = (xs - xr)2 + (ys - yr differentiating partially by xt, 31) and accordingly ^rrs Xs~Xr OA« rs 9 ( rr * ) "*' J and by (p any function of the independent variable, of the dependent variables, and of their derivatives up to the mth order and consider the change in y made by an infinitesimal transformation, where we replace x9 y, 8 by y- where %,vj,£ are arbitrary continuous functions of t. dx dx , d£ , dwx , dkx , d* £ Then — or x is replaced by --- 1- «— and — r by — r + a— v d* d# d# d«* dtk dtk i. e., by x(k) + ggw. Hence p becomes ey ey, s fi, * «, y «, • • which developed by Taylor's theorem for any number of variables, gives on collecting terms according to powers of s £2 £A (p (t, x, y,g, x\ . . .) + stp± + ^(p.2 - - • + fr df 8e* da* a/* Multiplying by «* after having put s = 0 after differentiation, this becomes ^U^= ** = * U) - so that we may write • A (dy\ _ d8y I dx _ d£ dSx /dx\* \dx) dt I ~dt ~di ~dt~ \di) or, once more removing t from explicit appearance, * /dy\ _ ddy dy ddx \dx) dx dx dx If x is the independent variable, dx = 0, so that we have the same formula as before.) Let us now find the variation of the integral , x, y, z, x', y1, /, . . .) dt. Changing x to x + dx, y to y -f dy, x' to x1 + dx', etc., 7+ dl+ (?2/ + - =< -f d 4- and the variations are - =f(

ddx). It may, on occasion, be more convenient to use these more general formulae, not supposing the variation of any variable to vanish.) If the limits are varied, we have, indicating the part of the change in I due to the change in either limit by a suffix, *? * r hfh \I = I cpdt — lcpdt= lcpdt = J J J which are to added to the part already found. In the application of the calculus of variations, we often encounter problems involving a number of independent variables, so that we deal with partial derivatives, and multiple integrals. The principles here given will however suffice for the treatment of all the usual questions. As a celebrated mechanical example of the use of the Calculus of Variations let us consider the question: What is that curve along which a particle must be constrained to descend under the influence of gravity in order to pass from one point to another in the least possible time? Since v = ^ S.> we have for the time of descent t = I - > or dt making use of the equation of energy § 27, 27), = I - J v Let us take for the independent variable corresponding to t above the vertical coordinate 0. We suppose the motion to take place in a vertical plane. We have then If now we make an arbitrary infinitesimal variation of the curve, if t is to be a minimum we mast have the term of the first order in s vanish, dt = 0. WEBSTER, Dynamics. 6 82 HI. GENERAL PRINCIPLES. WORK AND ENERGY. Now t = •o For any particular curve x is a given function of 3. Giving it a variation dx we have § x'dx'dz ot -f Making use of dx' = ^--- and integrating by parts1), Zi Zl x' dx de If the ends of the curve are fixed dx vanishes for both limits #0 and 0i, hence the integrated part vanishes. Consequently for a minimum the integral must vanish. Now since the function da; is purely arbitrary if the other factor of the integrand did not vanish for any points of the curve we might take dx of the same sign as that factor at each point. Thus the integrand would be positive everywhere and the integral would not vanish, consequently the factor multiplying Sx must vanish for each point of the curve, or This is the differential equation of the curve of quickest descent, or brachistochrone. Integrating we have x' - = c, an arbitrary constant. y(l + *")(V-*0[*-*o]) Squaring and solving for xt2 we obtain V 2 1 V 2 Let us put a = ~ -f #0, & = ^— ^ — ^ ^0 (6 is arbitrary, since it involves c), then we have 1) The bar / signifies that we are to subtract the value of the expression before it at the lower limit z0 from the value at the upper limit z:. 29] BRACHISTOCHRONE FOR GRAVITY. If we introduce a new variable # such that — I} a4-b g --- ~ COS ft, we have dz = -s 83 Thus our differential equation becomes dx -i/l -f cos & 1 -f- cos # Consequently Integrating, 1 — cos & sin # where J 70 -f I -5 -TS— ) cos (ny) -f I -g -5-7 cos (nz) \ d S \dz dx) \ox dy] J\ taken over the portion of the surface bounded by the paths 1 and 2 from A to B. Now — I± may be considered the integral from S to A along the path 1, so that I2 — /x is the integral around the closed path which forms the contour of the portion of surface S. We accordingly get the following, known as STOKES'S THEOREM.1) The line integral, around any closed contour; of the tangential component of a vector JR, whose components are X, Y, Z, is equal to the surface integral over any portion of surface bounded by the contour, of the normal component of a vector w, whose components £, 77, £ are related to X, F, Z by the relations t_<^_<>Z *~ dy dz' ax az •J1 = -^ -pr ) 02 OX ar ax 1) The proof here given is from the author's notes on the lectures of Professor von Helmholtz. A similar treatment is given by Picard, Traite d' Analyse, Tom. I, p. 73. 30, 31] STOKES'S THEOREM. CURL. 87 The normal must be drawn toward that side of the surface that shall make the rotation of a right-handed screw advancing along the normal agree with the direction of traversing the closed contour of integration. 37) IE cos (By ds) ds = I Xdx + Ydy + Zdz = I la cos (on) dS = I I { % cos (nx) + ri cos (ny) + g cos (ni)} dS. The vector co related to the vector point -function E by the differ- ential equations above is called the rotation, spin (Clifford), or curl (Maxwell and Heaviside) of JR. Such vectors are of frequent occurrence in mathematical physics. (See Part III.) The significance of the geo- metrical term curl can be seen from the physical example in which the vector E represents the velocity of a point instant- aneously occupying the position x, y, z in a rigid body turning about the ^"-axis with an angular velocity co. Then the vector E= OQ is perpendicular to the radius and its components are (Fig. 22), 0 rig X = B cos (Ex) = — E sin = BGOS(QX) = = XG), where co is constant, and 2co. _ dx dy So that the £- component of the^curl of the linear velocity is twice the angular velocity about the ^-axis. Further examples are presented to us in the theory of fluid motion. 31. Lamellar Vectors. In finding the variation of the integral I in the previous section, since the variations dx, dy, 82 are perfectly arbitrary functions of s, if the integral is to be independent of the path, dl must vanish, which can happen for all possible choices of dx, dy, dz, only if dZ dY_dX dZ_3Y dX _ n ~5i~'9il-~**Ji ~~ d*~ z>* 3y ~ that is if the curl of E vanishes everywhere. In case this condition is satisfied, I depends only on the positions of the limiting points A 88 ni. GENERAL PRINCIPLES. WORK AND ENERGY. and B, and not on the path of integration. Consequently, as stated without proof in § 28, the conditions 38) are sufficient as well as necessary. If A is given, I is a point -function1) of its upper limit _B, let us say (p. If B is displaced a distance s in a given direction to .Z?', the change in the function cp is B' VB' — v dy 7 dx = x> dj==Y> Tz = z' A vector whose components are thus derived from a single scalar function qp is called the vector differential parameter of (p. Accordingly the three equations of condition 38), equivalent to curl R = 0, are simply the conditions that X, Y, Z may be represented as the derivatives of a scalar point -function. In this case the expression Xdx + Ydy + Zdz = d^dx + d2ydy + d^dz = d

mr -f + vmr - = 0. But since A, p, v are perfectly arbitrary this is equivalent to the three equations Since the w's are independent of the time, we may differentiate outside of the summation and write the above d* d* d2 43) -^ 2rmrxr = 0, j^ Zrmryr = 0, -^ 2rmrgr = 0. If we define the coordinates of a point x, y ~, ~z by the equations and if we consider a mass m to consist of m particles of unit mass, being the sum of the x- coordinates of the whole number of unit particles divided by their number is the arithmetical mean of the x- coordinates. If m is not an integer, by the method of limits we extend the ifiotion of the mean in the usual manner. The point x, if, ~s, the mean mass point thus defined is called the center of mass of the system. (The common term center of gravity is poorly adapted to express the idea here involved and had better be avoided. We shall see in the chapter on Newtonian Attractions that bodies in general do not possess centers of gravity.) The equations 43) thus become A A\ d*x ^ d*y „ d* z ^ 44) 5F=°> d^ = 0> 5F-a Therefore the center of mass of a system whose parts exert forces upon each other depending only on their mutual distances moves with constant velocity in a straight line. This is the Principle of Conservation of Motion of the Center of Mass. It evidently applies to the solar system. What the absolute velocity of the center of mass of the solar system is or what its velocity with respect to the so-called fixed stars we do not at present know. 32] MOTION OF CENTER OF MASS. 91 Returning to the equations 39), whether there is a force -function or not, A, [i, v, being the same for each term of the summation, may be taken out from under the summation sign and being arbitrary, the equation 39) is equivalent to the three or as before 46) d^Zrmr = ZrXr, ^Zrmr = ZrYr, ^Zrmr that is: The center of mass of any system of the kind specified moves as if all the forces applied to its various parts were applied at the center of mass to a single particle whose mass is equal to the mass of the whole system. This principle of the motion of the center of mass reduces the problem of the motion of the system to that of finding the motion of a single particle together with that of the motion of the parts of the system with respect to the center of mass. A rigid body is a system of particles coming under the case here treated, since the only constraints are such as render all the mutual distances of individual points constant. Therefore the only new principles required in order to treat the motion of a rigid body are such as determine its motion relatively to its center of mass. If the center of mass is to remain at rest or move uniformly, we must have 47) 2;rxr = o, .zrrr = o, zrzr = o. This will always be the case as shown above for mutually attracting particles, since to. every action there is an equal and opposite reaction. The three equations 47) furnish three necessary con- ditions for the equilibrium of a rigid body. If we introduce the relative coordinates of the particles with respect to the center of mass into the expression for kinetic energy it assumes a remarkable form. Let us put xr = x + |r, yr = y + ijrf sr = ~8 4- £0 then dxr dx di-r ~dt"'=~di^ ~dt' dt ~ dt dt zr dz d£ dt = = ~dt ~^~ ~dt 92 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 48) r= 9— ^T-L 9^ ^!4.0^f ^ dt d< + d< d* + dt dt r\* /Jflr\» / -~i -^> - i[((f )'+(§) + Now in the last three terms we may write if J is the a;- coordinate of the center of mass in the |, 17, g system. But since the center of mass is the origin of the relative coordinates %,ij,£, this is equal to zero. Similarly for the terms in rjr and £r. Thus we have remaining if we write M for the mass of the whole system, The first term is the kinetic energy of a particle whose mass is equal to the total mass of the system placed at the center of mass, while the second is the relative kinetic energy of the system with respect to the center of mass. Thus the absolute kinetic energy is always greater than its relative kinetic energy with respect to the center of mass (unless the center of mass be at rest). The center of mass is the only point for which such a decomposition of the kinetic energy is generally possible. If the principle of the conservation of motion of the center of mass holds we have dx dy , dz _ ~dt~a> ^di — °> 'di=*c> 32] RELATIVE KINETIC ENERGY. 93 and inserting these in the equation of energy for a conservative system, T+W=h, In this case accordingly the principle of conservation of energy holds also for the relative kinetic energy, the constant h heing changed. Inasmuch as we know of no absolutely fixed system of axes of reference it is obvious that the kinetic energy of any system contains an indeterminate part. But in virtue of the above principle if we consider the center of mass of the solar system to be at rest all our conclusions with regard to energy will hold good. The effect in general of referring motions to systems of axes which are not at rest will be dealt with in Chapter VII. As a simple example of the above principle let us consider the case of a rigid sphere or circular cylinder, with axis horizontal, rolling without sliding down an inclined plane under the action of gravity. If the distance that the center of the body has moved parallel to the plane be s, the first part of T is -- -^M ^) • If the angle that a plane through the horizontal axis parallel to the inclined plane makes with the normal to the inclined plane be # (Fig. 23), •j f\ the velocity of a particle with respect to the center is f-^t where r is its distance from the horizontal axis. The relative kinetic energy is thus •7 f\ or since -j-> the angular velocity of rolling is the same for all terms of the summation, The factor 2rmrrl is called the moment of inertia of the system about the horizontal axis through the center of mass and will be denoted by K. Thus we have If the rolling takes place without sliding we have the geometrical condition of constraint, - d& ds where E is the radius of the rolling body. 94 HI. GENERAL PRINCIPLES. WORK AND ENERGY. The loss of potential energy is M g times the vertical distance f alien , ssina, where a is the angle of inclination of the plane to the horizontal. Our equation thus becomes l iTtf/ds\2l K {ds\z\ °4) T \M(di) + a* U) ) - « - const. If — = V when s = 0, determining the constant we have Thus the motion is the same (cf. § 18) as that of a particle falling freely with the acceleration diminished in the ratio Fig. 23. Thus by increasing Ky which may be done by symmetrically attach- ing heavy masses to a bar fastened to the cylinder in such a way as not to interfere with the rolling of the cylinder (Fig. 23), we may make the motion as slow as we please and thus study the laws of constant acceleration. 33. Moment of Momentum. Under the supposition that the equations of constraint were compatible with the displacement of the system parallel to itself and that the force -function was thereby unchanged we obtained the principle of the conservation of motion of the center of mass. We will now suppose that the equations of constraint are compatible with a rotation of the system about the axis of X and that the force -function is thereby unaffected. This will be the case in a rigid system or in a free system left to its own internal forces (if conservative). If we put yr = rr COS C0r, ob) 2r = rr sin «0 such a displacement is obtained by changing all the ror's by the same amount do, leaving the r's unchanged. We have then dxr = 0, dyr = — ^7 ) dzr = rr cos G)rdo = 32, 33] MOMENT OF MOMENTUM. 95 Inserting these values in d'Alembert's equation we obtain 58) then dividing by (b — a) we have that is the definite integral of a function in a given interval divided by the magnitude of the interval represents the limit of the arithmetical mean of all the values of the function taken at equidistant values of that variable throughout the interval when the number of values taken is increased indefinitely. The specification of the variable with respect to which the values are equally distributed is of the first importance. For instance suppose that we change to a new variable such that x = y(y], y = y-i(x) then 6 y jf(x)dx = lf(x)q> The integral may now be interpreted as the mean of the function f(x) tan a = ~ Vx and inserting the two values of t we get two possible elevations. Thus we find that the aim is completely determined (though not uniquely in this case) by the terminal positions and the velocity of projection. For the action we obtain 15) A = = m - *)} dt Using the values of Vz and t found above we obtain two values of the action different for the two paths. Thus there are two possible natural paths, differing from each other by finite distances, for only one of which is the action least. Both however have the property that between two points sufficiently near together the action is less than for any infinitely near path. In case the radical in 14) vanishes, that is the two roots t2 are equal and there is only one course. The terminal point xlf #! then lies on a parabola whose vertex is vertically above the point of projection (Fig. 25). It is easy to see Fig 25 that this parabola is the envelope of all possible paths in this vertical plane starting from the same initial point xQ1 £0 with the same velocity t'0. For it is the locus of the 35] KINETIC FOCI. 105 intersection of courses whose angles of elevation a differ infinitely little. If the second point xlt y^ lie without this envelope it cannot be reached under the given conditions. If upon it it can be reached by one path, and if within it by two paths. In that case the course that reaches xlf y± before touching the envelope has the less action. A point at which two infinitely near courses from a given point with equal energy intersect is called a kinetic focus of the starting point, and if on any course the terminal configuration is reached before the kinetic focus on that course, the action will be a minimum. If the kinetic focus is first reached it will not. Thus in the problem of motion on a sphere under no forces, the point diametrically opposite the initial point is a kinetic focus. Evidently a particle may reach the kinetic focus starting in any direction from the original point, for all great circles through a point intersect in its opposite point. The envelope of all the great circles or courses from a point in this cases reduces to a point, which is the kinetic focus. For the treatment of the difficult subject of kinetic foci, which belongs to the calculus of variations, the reader is referred to Thomson and Tait, Principles of Natural Philosophy, § 358, and Poincare, Les Methodes Nouvelles de la Mecanique Celeste, Tome III, p. 261, also to Kneser^ Lehrbuch der Variationsrechnung. From the principle of least action we may deduce the equations of motion. Of course the principle was itself derived from these equations, therefore, as is always the case, we obtain by mathematical transformations no new facts. It is however instructive to see how by assuming the principle of least action as a general principle we may obtain the equations from it. Let us put in equation 12) fl<3 /7/V.2 I f]»3 l ,7*2 t*or — (*djr -J- ll'i/r \ W6r, *~ - < -£ = y dq ~ dq dq giving 17) If we put since P involves all the coordinates and velocities xr, yr> zr, x'r, yl-, 0'n 106 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 19) iA 'io Now the term /; 2o may be integrated by parts, giving 9 dP . /*. d /oP\ 7 o-l OXr / — I 0 Xr -y- ( 5—? ) »# • 0a?; r/ j r>?3 V'7^/ The terms in d^ and 60'r are to be treated in like manner. Since the variations of the coordinates vanish at the limits the integrated terms disappear, leaving Now in virtue of 18) since N does not contain the coordinates, dP ^ ~" Y M~Wx~r~ M Also since M. does not contain #/., cP 1 T fM cN T fM. ^T == "« I/ -*r a~^ =1/^7 ^^ 2 V N dxr V N and consequently dP d fdP\ -*/~N dW d 5~ -^-(^-i):=--l/irF^ ^a; dcx' M ox d M dxr\ ^F- N dqj The equation of energy, gives or according to 18), from which we get 35] EQUATIONS DEDUCED FROM LEAST ACTION. 107 Inserting this value of dq gives dP _ d / dP \ _ -]/~N_ $W _ _^_ = _ ,/^ fdw Accordingly we have dW In order that this may vanish for arbitrary variations, dxr, dyr, dzrj the coefficient of each variation must vanish, so that we must have dW =0 or «- c d*zr dW d*zr dW mr~w+~Wr= °> mr~dt^~~ ~fWf=*Zr\ which are the ordinary equations of motion for a free system. The variations dxr, dyr, 8zr are arbitrary only if all the particles are free. If there are constraints the variations must be compatible with the equations of condition, that is we must have the & linear relations between the 12, . . . ^, so that k of the factors multiplying the variations vanish identically. Then the 108 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. coefficients of the remaining 3n — k arbitrary variations being put equal to zero with these k give the differential equations From the 3n equations 23) we may eliminate the It multipliers Z1; A2, . . . A* and obtain 3w — k equations of motion , which is the number of degrees of freedom of the system. The equations 23) are known as Lagrange's differential equations in the first form. They can evidently be deduced from equations 16) of Chapter III by d'Alembert's principle , replacing Xr by d*xr Xr — ™r dtz > etc. 36. Generalized Coordinates. Lagrange's Equations. In many investigations in dynamics where constraints are introduced, instead of denoting the positions of particles by rectangular coordinates (not all of which are independent) it is advantageous to specify the positions by means of certain parameters whose number is just equal to the number of degrees of freedom of the system, so that they are all independent variables. For instance if a particle is constrained to move on the surface of a sphere of radius I, we may specify its position by giving its longitude cp and colatitude #, as in § 23. These are two independent variables. The potential energy depending only on position will be expressed in terms of cp and #. The kinetic energy will depend upon the expression for the length of the arc of the path in terms of cp and #. Now we have, if / be the radius of the sphere, Dividing by dt2 and writing #' = -j-> (p' = -~> we have 24) T = y m P (#' 2 + sin2 & y1 2). The parameters # and cp are coordinates of the point, since when they are known the position of the point is fully specified. Their time -derivatives &', &)> y = f* (ft > 0a)» * = /3 (0i, 0a)> from these three equations we can eliminate the two parameters ql9 q2, obtaining a single equation between x, y, s, the equation of the surface. The parameters q± and q2 may be called the coordinates of a point on the surface, for when they are given its position is known. If q^ is constant and q2 is allowed to vary, the point x, yy s describes a certain curve on the surface. This curve changes as we change the constant value q±. In like manner putting q2 constant we obtain a family of curves. The two families of curves, ql = const, q2 = const, may be called parametric or coordinate lines on the surface, any point being determined by the intersection of two lines, for one of which q^ has a given value, for the other, g2. We may obtain the length of the infinitesimal arc of any curve in terms of q1 and g2. We have dx i , dx -. 25) Squaring and adding, 26) ds* = dx* + df + dz* = Edq^ + ^Fdq, dq2 HO IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. Thus the square of the length of any infinitesimal arc is a homo- geneous quadratic function of the differentials of the coordinates q1 and #2 , the coefficients E, F, Gr being functions of the co- ordinates q1} q2 themselves. If the curve is one of the lines q± = const , we have, since dql = 0J ds22=G-dq22, if it is one of the curves q% = const, we have Considering any arc ds as the diagonal of an infinitesimal parallelogram with sides ds^ and ds2 including an angle # (Fig. 26), we have by trigonometry, ds2 = ds±2 -f 2 dsj_ ds2 cos & -f- ds^. Making use of the above values of ds± and ds2 and comparing with the expression 26), we find F cos # = -=• 1/EG If the coordinate lines cut each other everywhere at right angles we shall have cos # = 0, F= 0, so that 28) ds2 = Edql2 + Gdq22. The coordinates qi9 q2 are then said to be orthogonal curvilinear coordinates. In the example above1) & and (p are orthogonal, the lines of constant & and cp being parallels and meridians intersecting at right angles and the product term in d&dcp therefore disappearing. Employing the 'expression 26) for the length of the arc, dividing by dt2 and writing 1) We have the equations of change of coordinates, x = I sin # cos qp , y = I sin O1 sin qp , z = I cos -91, from which - = I cos & cos qp , -- = I cos & sin op , — = — I sin #, <7aT C& Qv — = _ ZsinO-sinqp, ~ = I sin -91 cos op , — = 0, G

(Xrdxr + Yrdyr + Zrder) is a homogeneous linear function in the dq's which we will write 40) d A = P4 dq, + P2 dq2 + • • • + Pmdqm. By analogy with rectangular coordinates we shall call Pr xthe generalized force -component corresponding to the coordinate qr and velocity ql. If the system is conservative, since 41) dW=-dA, Pr=- and in any case dXr + Y 8Vr 4 7 r r Tr+Zr r=l We may now make use of Hamilton's Principle to deduce the equations of motion in terms of the generalized coordinates q. Performing the operation of variation upon the integral occurring in Hamilton's Principle, both the g's and #"s being varied, we obtain to and since dq 36, 37] LAGRANGE'S EQUATIONS. we may integrate the second term by parts. Since the initial and final configuration of the system is supposed given, the dq's vanish at t = t0 and t = t^ so that the integrated part vanishes, and Now if all the dq's are arbitrary, the integral vanishes only if the coefficient of every dqs is equal to zero. Therefore we must have 45) d(T-W) _d_t<) (T-W} dqs di ( dq'g - or if we write L for the Lagrangian function T — W, 46) ±(**L\-?L. ' dt\d^)~dqs Since the potential energy depends only on the coordinates j- = 0, and we may write the equation 45) There are m of these equations, one for each q. These are Lagrange's equations of motion in generalized coordinates, generally referred to by German writers as Lagrange's equations in the second form. Their discovery constitutes one of the principal improvements in dynamical methods and we shall refer to them simply as Lagrange's equations.1) If the system is not conservative, by § 34, 4) we must write fl ft 48) l($¥4 8A)dt = l(dT J J from which we easily obtain 47), except that Ps is not now derived from an energy function. 37. Lagrauge's Equations by direct Transformation. Various Reactions. On account of the very great importance of Lagrange's equations, it is advantageous to consider them carefully, from as many points of view as possible. The deduction from Hamilton's principle is one of the simplest, but does not perhaps appeal as strongly to our physical sense as is desirable. Of course as Hamilton's principle is completely equivalent to d'Alembert's, and that to the equations of motion of Newton, we might have derived 1) Lagrange, Mecanique Analytique, Tom. I, p. 334. 116 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. the equations from either. This we will now do. It is important every time that a new quantity appears in dynamics, to have a clear conception of its physical nature. We should make free use of all analogies that our science may offer us, and here geometry aids us readily. The notion of the geometric product and the terminology of multidimensional geometry here furnish us valuable aid. The geometric product of two vectors in three dimensional space, defined by their components X, Y, Z, X', Y', Z', xx' + rr + zz', is a scalar quantity, symmetrical with respect to both vectors, such that the geometric product of the resultants of two sets of com- ponents is the arithmetical sum of the products of all the pairs of corresponding components. If one of the vectors is an infinitesimal displacement dx, dy, dz, the geometric product is Xdx + Ydy + Zds, and the multiplier of the change dx is called the component of the vector in the direction of the coordinate x. In like manner let us speak of a quantity defined by components P1; P2, . . . Pm as a vector in m- dimensional space. The geometric product of two such, of which the second is an infinitesimal displacement compatible with the constraints, and defined by the quantities dqi} dq%, . . . dqm, may be, by analogy, defined as Pldql -f P2 dq2 -\ h Pmdqm. If now the vector P1? . . . Pm is equivalent to the system of vectors Xr, Yr, Zr, we have equations 39), 40), 42), and the latter, P,= serves to define the component of the vector -system with reference to the coordinate qs. Thus we have spoken of Ps as the force- component of the system for the coordinate qs. It is to be observed that we do not insist here on the idea of direction, and that our terminology is merely a convenient mode of speaking. Nevertheless, the notion of work gives a means of realizing by the senses the meaning of our term component, for, if we move the system in such a way that all the g's except one qs are unchanged the work done in a change of the coordinate dqs will be P,^,.1) Let us now find the component of our velocity -system according to our generalized coordinates. We have, according to our equation 1) For a further elucidation of the nature of the geometric product, in connection with multidimensional geometry, see Note III. 37] GENERALIZED VECTOR -COMPONENT. 117 of definition 42), for the component of the velocity of the rih particle according to qs, 49) *£+*£+«£ Now we have by 32), dividing by dt, -•-//* The derivatives -~ contain only the coordinates q, not the velocities q', which we see enter linearly, accordingly Making use of this relation, the expression 49) becomes Thus we find that the component of the velocity of any particle according to the coordinate qs is equal to one -half the rate of change of the square of its velocity as we change the velocity q'^1). This result is not of itself of great physical importance, but leads us to one that is. Inasmuch as the momentum is the important dynamical quantity, multiplying by the mass of the particle we find m^f! + **;fj + m,*;g - ~ or the component of the momentum of a particle according to any coordinate is the rate of change of its kinetic energy as we change the corresponding velocity. Summing for the whole system, . 52) that is, the component of the momentum of a system according to any generalized coordinate qs is the rate of change of kinetic energy 1) It is to be observed that this "component" is not what we have called the velocity q'g. 118 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. with respect to the corresponding velocity. The equation 38) now says that the kinetic energy is one -half the geometric product of the velocity and momentum systems. Thus we have perfect analogy with the last two equations of § 27. We shall hereafter denote the momentum belonging to qs by ps and effecting the differentiation of 35) we have O Q or every generalized momentum -component is a linear function of the velocities, the coefficients being the inertia -coefficients Qrs. Let us now find the component of the effective forces according to qs, the effective forces being defined by the system of products, for each particle, of mass by acceleration, dx[ dyl del We have to transform which we make use not only of 51), but of a relation obtained as follows. Differentiating 50) by qs) h Using these results in 54), we obtain for the right-hand member, - E «;) - *%] - iA k |* and with similar results for y and 8j summing for all the particles, we have for the component of the effective forces of the system, d dt\dq' Putting the effective force equal to the applied force we have Lagrange's equation 47) by direct transformation. The equation of d'Alembert's principle thus becomes in generalized coordinates 56) s = l 37] . EFFECTIVE FORCE COMPONENTS. 119 If we had begun with d'Alembert's principle we should evidently have gone through precisely the same process that we have here followed, and assuming all the displacements dx, 8y, dz to he virtual, all the dg's would have been independent, so that from the trans- formed equation 56) would have followed the individual equations 47). This was in fact the mode of deduction followed by Lagrange. We have a noteworthy difference between generalized and rectangular coordinates, in that the effective force -component is not dp generally equal to the time -derivative of the momentum -~> but dt dT contains in addition the term — «— • This we may accordingly call the non-momental part of the effective force. Thus in general, even though the momentum ps is unchanging, a force Ps must be impressed dT in order to balance the kinetic reaction -~— - As an example, let us take the case of polar coordinates in a plane. We then have for a single particle, for the coordinates qlf qz the distance r from the origin, and the angle q> subtended by the radius vector and a fixed radius. The kinetic energy is from which we have the momenta, dT = dT , Thus if the momentum pr is constant, which is the case when the radial velocity r' is constant, we still have to impress a radial com- ponent of force The kinetic reaction —Pr = mrcp'2 is called the centrifugal force, a name to which it is as much entitled as any sort of reaction is to the term force. By analogy we might in general call the non-momental parts of the reversed effective forces or forces of inertia the centrifugal forces of the system. These non-momental parts may be absent for some coordinates. For instance in the present example (p does not appear in the kinetic energy, but only its velocity (p'. We have then £\ /j-j - = 0. so that force need be impressed to change

1\ is a linear function of the generalized accelerations $ '. Here again our generalized coordinates differ from rectangular, in that there is a part of the momental force which is independent of the accelerations #", but which is a homogeneous quadratic function of the velocities, r=m t=m o 58) Consequently if at any instant of the motion we can change the signs of all the velocities, and at the same time of all the accelerations, the accelerational part of the momental force F^ will change its sign, while the non- accelerational part F,W will be unchanged. We may thus experimentally discriminate between the two. Effecting the differentiation in the case of the non-momental force, we find which is also a homogeneous quadratic function of the velocities, and thus possesses similar properties to Ff?\ Thus it is difficult to discriminate experimentally between these two, unless we have some experimental means of recognizing when the momentum ps remains constant. In the simple example which we have used above, since the non -accelerational part of the momental force belonging to r disappears, while the centrifugal or non-momental does not, while for cp 37] VARIOUS TYPES OF REACTION. 121 although the non-momental part F^ = ^— disappears, we have the non-accelerational part F^ = 2mr - r' y> . Experimentally this means that, if a particle move with constant radial and angular velocities, we shall have to apply to it not only a radial force F^ = — mry'2 to balance the centrifugal force, hut also a turning force 2mr-r'(p'. This may he done by means of a varying constraint, say by making a particle move upon a rod turning with angular velocity qp'.1) The particle will then react upon the rod, to which the turning moment 2mr - r' cp' must be applied, for if it were not applied, owing to the conservation of angular momentum, as the particle got farther from the center its angular velocity would be less. To keep it constant the particle must be pushed around. We have now carefully analysed the effective forces, when expressed in terms of our generalized coordinates. It is to be care- fully borne in mind that all these parts come from real accelerations impressed on the particles of the system, although the accelerations of the generalized coordinates may disappear. This will depend on our choice of such coordinates. The analysis that we have made is however by no means devoid of physical significance, as we can not usually observe all the bodies with which we have to do so as to find their real motions and determine their accelerations, but are obliged to become acquainted with them in a more or less round- about way, through the reactions that they present to various operations upon them. From this point of view it is of interest to catalogue the various reactions that we meet in dynamics. In our equation of d'Alembert's principle 56), we have called the P's which are 'equated to the effective forces, the impressed forces, or forces of the system. If the system is conservative, the forces of the system are derivable from a potential energy, as we have assumed in 47), while if not, part of the forces may still be derived from such a function. It will be useful to consider not the forces of the system, but the forces which must be impressed from outside in order to counterbalance all the reactions of the system. In other words, if we write Fs^ for the non- conservative part not yet dealt with, 60) Fs = JFS is the force necessary to be impressed on the system from outside under any circumstances whatever, or — Fs is the reaction of the system, exerted through the coordinate qs. 1) The centrifugal force may be balanced by a spring. 122 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. If the system is left to itself, uninfluenced by other systems, then every F9 is zero, and we have equation 47) with If two systems are coupled together, so that any change of the coordinate qs is accompanied by an equal change of the corresponding coordinate of another system, then the Fjs of the two systems are equal and opposite, which is the law of action and reaction. Accord- ing to what happens to the system, the effect of Fs is of different kinds. For instance, if the system is at rest, or moves very slowly, all the jF/r) terms vanish except the last, and we have the static reaction 3W ^ The work that is then done by the external forces, is stored up as potential energy in the system. If there is no possibility of statical storage, and if there is no non- conservative reaction, we have only the kinetic reactions already dealt with. As a simple example of what is meant, suppose the system to consist of a mass attached to a spring tending to draw it to the right. If the mass is at rest, it must be held by a force applied from outside, to keep the spring stretched, and the static reaction of the spring Ps is toward the right. If the mass is let go, it begins to move toward the right, and the kinetic accelerational reaction is toward the left, balancing the static reaction, or internal impressed force of the system, according to d'Alembert's principle. If there is no inertia, so that the effective forces vanish, and no storage, the work done upon the system is not stored, but is said to be dissipated. The reaction — F^ does not, in the cases that exist , in nature, appear except when there is motion, that is, the reaction - jpy4) is a kinetic reaction, though not due to inertia. This work dissipated, is always positive, in other words, non -conservative reactions are always such as to oppose the motion. A case of frequent occurrence is that where there are non -conservative forces proportional to the first powers of the velocities q1, so that any F3W = KsqJ. We may then form a function F which is, like T, a homogeneous quadratic function of the velocities, 37] CLASSIFICATION OF REACTIONS. 123 and since in this case the work dissipated in unit time is F represents one -half the time rate of loss, or dissipation of energy. F is called the Dissipation Function, or the Dissipativity.1) It was introduced by Lord Rayleigh, and is of use in the theory of motions of viscous media, and in the dynamical treatment of electric currents. Beside this case we have dissipative forces not capable of representa- tion o/f by a dissipation function. We will now place our various reactions in a table showing their grouping in various classes and sub -classes. Positional Reactions Inertial Motional or Kinetic ,T , ,( Accelerational Momenta!] Non - accelerational F W Non- momental or Centrifugal JT (8) Non- conservative Having Dissipation -function Others The advantage of this complete classification is as follows. Suppose that a certain system or apparatus is presented to us for dynamical examination. Its parts are concealed from o%r view by coverings or cases, but at certain points there protrude handles, cranks, or other driving points, upon which we may operate v and which will exert certain reactions. All that we can learn of the system will become known to us by a study of the reactions. Maxwell 2) compares such a system to a set of bell -ropes hanging from holes in a roof, which are to be pulled by a number of bell ringers. If when one rope is pulled none of the others are affected, we conclude that that rope has no connection with the others. If however, when one rope is pulled, a number of others are set in motion, we conclude that there is some sort of connection between the corresponding bells. What the connection is we can find out by studying the motions. In general, if when we move one driving point, and let it go, it remains where we put it, we conclude that it is not attached to anything, but is a mere blind member. If when we push it, it 1) A case of perhaps equal importance is one in which the dissipation function contains the squares of differences of the velocities. 2) Maxwell, Scientific Papers, Vol. II, p. 783. 124 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. returns to its former position, we infer that it is connected with something of the nature of a spring, and that the system can store potential energy. If when we push it it keeps on going after we release it, we conclude that it is connected with a system possessing inertia, and capahle of storing kinetic energy. If its motion dies away, we conclude that there is dissipation, and so on. By experi- menting in turn, or simultaneously, on all the driving points, we may conclude how many degrees of freedom the system has, how the inertia is distributed, and how the parts of the system are connected. The means of doing this will we discussed later, and we shall find that in this manner we may learn much of a system, but that our knowledge will not always be complete. This is the nature of the process by which the physicist proceeds in the attempt to explain recondite phenomena, such as those of heat or electricity, by reducing them to the simpler phenomena of motion. The parts of the systems, be they made of molecules of matter, or of the ether, are concealed from him, but he may operate upon them in certain experimental ways, and draw definite conclusions from the results. One of the greatest triumphs of this method was Maxwell's dynamical theory of electricity. Impulsive forces are dealt with by Lagrange's equations in the usual manner. Integrating equations 47) with respect to the time throughout a vanishing interval t± — tQ, since the velocities are finite, the non-momental forces — -«— are by 58) finite, so that the integral of the second term vanishes, and we have =(li-< p ti — toj P.dt. Thus the momentum generated measures the impulse, as in the case of rectangular coordinates, § 27. As a further example of the use of Lagrange's equations let us take the problem of the spherical pendulum, which we used to introduce the subject. We had 24) W =. — mgl cos &. We have for the momenta p$. and p< 62) cT ( uMVVfcfcSITY Of 37,38] ENERGY INTEGRAL 'CGE'S EQUATION. 125 • and our differential equations are d , 72Q,N dT d di(ml®)-d& = ~~ *(mvwt*.9')-. dt^ ^ ^ Off dtp Now since m and I are constant the equation for # becomes 64) ^ — sin#cos# • qp'2== -- ~- sin#, in which the centrifugal force -component according to & is The equation for qp (cp has no centrifugal part), 65) |^(72sin2# V) = 0 may at once be integrated, giving 66) which is the integral equation 50), § 21. Substituting in 64) the value of qp' derived from the integral equation 66), we obtain the differential equation for #, which is the same as the derivative of equation 51), § 21. The remainder of the solution is accordingly the same as in § 21. 38. Equation of Activity. Integral of Energy. Let us multiply each of Lagrange's equations by the corresponding velocity ql, and add the results for all values of r, obtaining The expression on the right, otherwise written ^^ d qr d A. ^Jrl>r~d^==~dt' represents the time -rate at which the applied forces do work on the system. The equation 67) is accordingly the equation of activity, § 27, 20), in generalized coordinates. By means of the property of T expressed in equation 38), § 36, we may transform the left-hand side of the equation, for, since T depends upon both the g's and qns, both of which in an a'ctual motion depend upon t, differentiating totally, dT x^i/dTdtf dT d 126 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. • Now differentiating 38) totally dT dq'r Subtracting equation 68) from 69) the terms ~ — -f -j— cancel and we hav eft dT „ f f d SdT\ dT dqr But this exactly the left-hand member of the equation of activity 67). Thus if the system is conservative, since d^ - _ dw *•£ - dw dt dt dt dt so that the equation of conservation of energy is always an integral of Lagrange's equations. 39. Hamilton's Canonical Equations. Although the equa- tions of Lagrange are by all odds those most frequently used in dynamical problems, yet in many theoretical investigations a trans- formation introduced by Hamilton is of importance. The kinetic energy being a quadratic form in the velocities qf [equation 35)], the momenta pr being the derivatives of T by the q"s are, as we have seen, linear forms in the q^s. dT 53) n n* _L n /•»' _i_ _i_ n ^ Pm = Q- / = = Vml #1 T Vm2^2 ~r ' ' ' ~T ^mrnqm- OCLm These linear equations may be solved for the g^'s, obtaining any as a linear function of the jpr's, say, n i \ . l ~D ., [ ~D ^ i i ~D „ the J?'s being minors of the determinant, I Qml) Qm2> • - • Qmm i divided by D itself. The R's accordingly , like the §'s, are functions of only the coordinates q. Maxwell calls them coefficients of mobility. The solution of the equations assumes that the determinant D does not vanish. This is always the case, being one of the conditions that T is an essentially positive function. 39] HAMILTON'S EQUATIONS. 127 Let us now introduce into T the variables p in place of the variables q', so that T is expressed as a function of all the ^JP r ^ ^r ' #0 1) Webster, Theory of Electricity and Magnetism, § 63, 64. 128 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. The term prdq'r = Pr^. Sqr being integrated by parts and the dq's put equal to 0 at the limits, we have 78) Now since W does not depend upon the momentum prt therefore the coefficients of the dp's all vanish. If the dq's are all arbitrary, their coefficients must accordingly vanish so that we have the first equation being the equation of motion, the second defining q'r = —j-^- These equations 78) were introduced by Hamilton and on account of their peculiarly simple and symmetrical form they are often referred to as the canonical equations of dynamics. In practical problems they are generally not more convenient than Lagrange's equations. We may recapitulate Hamilton's method as follows: Form the Hamiltonian function H, representing the total energy of the system as a function of the 2m independent variables q and p, the coordinates and momenta. Then the time derivative of any co- ordinate q is equal to the partial derivative of H with respect to the corresponding momentum p> while the time derivative of any momentum is equal to minus the partial derivative of H with respect to the corresponding coordinate. A direct deduction of the equations of Hamilton without the use of Hamilton's Principle will be found in the author's Theory of Electricity and Magnetism § 64. The equation of activity is most simply deduced from Hamilton's equations, for by cross multiplication of equations 78), after trans- posing and summing for all the coordinates we get 791 ^ SHdp. r + -~ = But this is equal to the total derivative of If by t, dJS_ dt ~~ u> which being integrated gives H = h, a constant. But since H = T + W, this is the equation of energy. •• 39, 39 a] VARYING CONSTRAINT. 129 If the system is not conservative ; there may be still some forces which are derivable from a potential energy function. In that case the Hamiltonian function is to be formed with that energy, but we must add to the right of equation 78 a) the non- conservative force - Fr^\ Thus our equations become 80) fe. - FV ^-a% dt " " dqr *r > dt~frir' The equation of activity then becomes 0 H dq dH dp. dqr ~ + ~ ~ or dt if there is a dissipation function. iX , 39 a. Varying Constraint. It may happen that the equations of constraint contain the time explicitly, that is xi> yu *!>••• xn> y*, **) = o, 9>2 & Xl9 Vl9 *1?V" **9-y«9 *n) = 0, 82) . . . .......... Such a case is that of a particle constrained to move on a surface which is itself in motion, say a sphere whose center moves with a prescribed motion. The constraint is then said to be variable, and the work done by the constraint no longer vanishes, for the surface has generally a normal component in its motion, which causes the reaction to do work. The variability of the constraint has an important effect on the equations of motion. We can then no longer determine the position of the system by means of a set of in- dependent parameters, but must give not only their values, but also the time. We rnay put 83) from which, by the elimination of the g's, we may obtain equa- tions 82). 1) cf. § 37, 60). , Dynamics. 130 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. Differentiating now totally, we have dy "by dy dyr = -gfdt + ^dq, or on dividing through by dt, i dxr = ~dt 3yr 85) y;=1nr We have now in each x',y',z', beside the linear function of q[, q'2, ... q^, a term independent of the qtJs, but which may be expressed in terms of the coordinates q and t. On squaring there are accord- ingly not only quadratic terms in the g"s, but also terms of the first and zero orders. On forming the kinetic energy s—1 86) T = |VV we accordingly find that instead of being, as before, a homogeneous function of the q"s, it contains not only quadratic terms, but also terms linear in and others independent of the #"s. The effect of these linear terms in the kinetic energy, whatever be their origin, will be discussed in § 50. 40. Hamilton's Principle the most general dynamical principle. We have seen in this chapter how by means of Hamilton's Principle we may deduce the general equations of motion, and from these the principle of Conservation of Energy. As 40,41] HAMILTON'S PRINCIPLE GENERAL. 131 Hamilton's Principle holds whether the system is conservative or not, it is more general than the principle of Conservation of Energy, which it includes. The principle of energy is not sufficient to deduce the equations of motion. If we know the Lagrangian func- tion we can at once form the equations of motion by Hamilton's Principle, and without forming them we may find the energy. For we have L = T - W, E=T+W, Accordingly 87) E-2T-L so that the energy is given in terms of L and its partial derivatives. If on the other hand the energy E is given as a function of the co- ordinates and velocities, the Lagrangian function must be found by integrating the partial differential equation 87), the integration involving an arbitrary function. In fact if F be a homogeneous linear function of the velocities, the equation 87) will be satisfied not only by L but also by L +• F. For, F being homogeneous, of degree one, , cF Consequently a knowledge of the energy is not sufficient to find the motion, while a knowledge of the Lagrangian function is. The attempt has been made by certain writers to found the whole of physics upon the principle of energy. The fact that the principle of energy is but one integral of the differential equations, and is not sufficient to deduce them, should be sufficient to show the futility of this attempt. It is the infinite order of variability of the motion involved in the variations occurring in Hamilton's Principle that makes it embrace what the Principle of Energy does not. 41. Principle of Varying Action. We shall now deal with a principle, likewise due to Hamilton, somewhat broader than that which we have hitherto called Hamilton's Principle or Principle of Least Action, and furnishing a means of integrating the equations of motion. In the principle of least action a certain integral, belong- ing to a motion naturally described by a system under the action of certain forces according to the differential equations of motion, has been compared with the value of the same integral for a slightly different motion between the same terminal configurations, but not a natural motion and therefore violating the equations of motion. Under these circumstances the principle states that the integral is 9* 132 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. less for the natural motion than for the other. The new principle, on the other hand, compares the integrals always taken for a natural motion satisfying the differential equations , but the terminal con- figurations are varied from one motion to another. The principle is therefore known as the Principle of Varying Action. In the process of § 34 equation 2) we cannot now put the integrated part equal to zero,, but instead of 2) we shall have * The integrated part, which is the sum of the geometric products of the momenta and the variations of the corresponding positions at the end of the motion minus the corresponding sum at the begin- ning, may now be transformed into generalized coordinates. The integral S=f(T-W)dt, where T and W are expressed as functions of the time, appropriate to any given motion (whether natural or not) depends upon the terminal configurations, and is called by Hamilton the Principal Function. The terminal configurations being given we had dS = 0. Let us now find an expression for dS in generalized coordinates corresponding to the expression above in rectangular coordinates. Proceeding as in § 3|^ equation 43) we obtain 89 ^.'ll Since the various motions are all natural ones satisfying the differ- ential equations of motion, the factor of every dq in the. integrand vanishes, so that the integral vanishes of itself, and dS is accord- ingly expressed as a linear function of the variations of the initial and terminal coordinates. Since W is independent of the 0"s and cT fi^i =pr, making use of the affixes 0 and 1 for the limits tQ and tv we may write 90) 98 = Sr&1t£-Zr&a£, an equation which could have been obtained from the considerations regarding geometric products at the beginning of § 37. This 41] VARYING ACTION. 133 expression for the variation of S is of great importance , for by means of it we can obtain a method of integrating the equations of motion , and obtaining the coordinates q and momenta p at any time t±. As we are now to consider the upper limit ^ as variable it will be convenient to drop the subscript 1. Suppose we have integrated the differential equations of motion completely so as to obtain every coordinate as a function of the time t, involving 2m arbitrary constants, cly C2, . . . c2m? the number necessarily introduced in integrating the m Lagrangian equations of the second order or the 2m Hamiltonian equations of the first order. Let the integrals be Differentiating these by t we obtain from which by equation 53) we may .find the ^>'s as functions of t, \jfjj pr — (pr \tj C^j C% y • . • C2m)' These equations with 91) constitute 2m integral equations of the system. , Inserting the particular value £0 in our integral equations we have 91') (fr = fr (t , C , C , . 93') VQ. = cp (t c c We accordingly have the 4^m + 1 variables, connected by 2m integral equations. We may thus choose any 2m + 1 of them as variables in terms of which to express the remaining 2m. For instance in the problem of shooting at a target § 35 we saw that the motion was completely determined by the coordinates of the initial and final positions and the initial velocity. The latter determined the time of transit £, so that it together with the initial coordinates, q^0, . . . 0m, and the final coordinates, qlf . . . qm, may be taken as independent variables in terms of which everything may be expressed. 134 IY. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. Thus the integral 94) S is supposed to be expressed in terms of these 2m + 1 variables. Now if the initial and final coordinates are varied without varying the time of transit t — tQ (t the upper limit of the integral) we have We have however proved that under these conditions we have Since these expressions must be equal for arbitrary variations of the #'s and #°'s we must have 96) M= M^ 3S = We may now, if we please, regard the initial coordinates #i°7 • • . qm, and the initial momenta, p^9 . . . p£y as 2m arbitrary constants replacing the c±, C2, . . . c^m of equations 91) and 93). Then the equations 97) will be the general integrals of the equations of motion, for if the form of the function S is known in terms of tj &, . • • qm, #1°, - . . qm, the equations 97) are m equations involving qly . . . qm without their derivatives, which may be solved to obtain the g's as functions of t and 2m arbitrary constants q^9 . . . g£> J^ . . .pH, as in equations 91). It has appeared as if in order to find S it were necessary to integrate the equations of motion, so to obtain T — W as a function of the time, which being integrated would give S. If this were so the statement just made would be of little interest. But this is not necessary, for Hamilton showed that the function S, which he called the Principal Function, satisfies a certain partial differential equation, a solution of which being obtained, the whole problem is solved. The function S is a function of the variables g, the constants (f and the time t, which thus occurs explicitly and implicitly. Differen- tiating by t we have therefore «9 %-% . Differentiating 94) by t, the upper limit, gives however dJl=T-W. at 41] PARTIAL DIFFERENTIAL EQUATION. 135 Equating the two values, by 38). Transposing and writing T + W = H, The function .0", the sum of the energies, depends upon the co- ordinates qr and the momenta, pr = ^ — If the force -function depends upon the time H will also contain t explicitly. Thus we have the partial differential equation oo\ dS . + The equation is of the first order since only first derivatives of S appear, and, from the way in which T contains the momenta [equa- o ci tion 72)], is of the second degree in the derivatives « — Since S appears only through its derivatives an arbitrary constant may be added to it. Thus we have the theorem due to Hamilton: If qlf . . . qm, ex- pressed as integrals of the differential equations in terms of t and 2m arbitrary constants q^, . . . q&, p^, . . .p£, are introduced into the integral 94), and the result is expressed in terms of t, qlf . . . qm, (Zi0, . • • #m, then 8 is a solution of the partial differential equation 99). The converse of the proposition was proved by Jacobi, namely, that if we take any solution of the equation 99) containing m arbi- trary constants, q^, . . . q£ (other than the one which may always be added), the equations 97) obtained by putting the derivatives of S by the m arbitrary constants equal to other arbitrary constants, p^j . . . pm wiU be integrals of the differential equations of motion. For the proof of this the reader is referred to Jacobi, Vorlesungen tiber Dynamik, XX. Before giving examples of the utility of this method we shall show that the arbitrary constants by which we differentiate need not be the ^°'s? but may be any m constants appearing in the integral equations. Suppose that in equations 91) we vary m of the arbitrary constants c1; . . . cm. We then have 136 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION, and putting t = 0, Then equation 90) becomes 100) 68= Zrpr $ P,r = 3j-f> P>r=Wr and equation 105) then is In the case of a single particle comparing equations 110) and 111) we have dA f dA w me ~w In other words if the action A is expressed in terms of the co- ordinates x, y, s, the momentum of a particle describing any path 138 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. under the action of the given forces with the constant energy h is the vector differential parameter of the action A, and therefore, by the properties of lamellar vectors (§ 31), the velocity of a particle moving in this manner is normal to all the surfaces of constant action, and is inversely proportional to the distance between two infinitely near surfaces of constant action. Otherwise expressed, if from all points of any surface particles be projected normally with the same energy h, their paths will always be normal to a set of surfaces, and the action from one surface to another will be the same for all the particles. This theorem is due to Thomson and Tait.1) Suppose first there are no forces acting, then equation 112) becomes which is satisfied by the linear function 115) A = ax + ~by + C89 if 116) a2 -f 62 + c2 = 2wft. In virtue of this last equation only three of the constants a, 6, c, li are arbitrary. Suppose we take a, 6, ft, then we have 117) A = ax + ~by- Then equations 107) or 113) are 118) mx' = a, my' = 1, mz' =-\/2mJi - (a2 which are first integrals of the equations of motion, showing that the motion of the point is uniform. Equations 108) and 109) are dA as = ~ dA lz 119) ~n=y- y%mh-(a*-\- &2) 3 J. m# / / 6 v/i ^^ l/^-mTj — r«2-l-?i^ ° The first two of these equations are the equations of the path, showing it to be a straight line, while the last gives the time. By means of it we may find 2 as a function of the time, and from the first two x and y. Thus 119) are the integral equations of the motion. Corresponding to this solution, a and & being constants, the surfaces of constant action are parallel planes. The path of any 1) Natural Philosophy, § 332. 41] SURFACES OF EQUAL ACTION. 139 particle projected normally to one of these planes with the energy (kinetic) h is a straight line normal to these planes, and the velocity is constant. In order to find solutions suited to surfaces of equal action having other forms, we should require to find other particular solutions of equation 114), which would take us too far into the subject of partial differential equations. Whatever the nature of the surfaces, since the velocity in all the motions considered is constant, the action is proportional to the distance traversed and consequently if we measure off on the normals to a surface of constant action equal distances, the locus of the points thus obtained will be another surface of equal action, or all the surfaces of equal action are so- called parallel surfaces. Next, suppose we have a single particle of mass unity under the action of gravity. Then w=ge, and our equation is TT l[/dA\*. /dA\* , /d-4\2) 7 120) H = ~ + + + g, = ft. We may find a solution A = ax + by + ), where the functions R and F contain only the variables indicated. = = dr~~dr' d» ~ d&' Substituting in equation 135) we have 1q7v 1 Multiplying by r2 and transposing 1QQ\ X 138) On one side of this equation we have functions of r alone, on the other functions of # and

^= /,!y"2-2l+^ jT/tT-% *s V r rz *s f sm2 dA _ r d^ y _ f 148) ^ ~ / ^T/2^8in4^-ysm2^ 2 V7 ~~ ^ ; a^i /* dr Th=h V"- If we put y = 0, necessitating according to the second equation 9 = 0, the first equation becomes 149) v ' ' ^' the equation of the path, which, on performing the integration indicated, takes the form obtained in § 20 equation 23). 144 V. OSCILLATIONS AND CYCLIC MOTIONS. CHAPTEK V. OSCILLATIONS AND CYCLIC MOTIONS. 42. Tautochrone for Gravity. A curve along which a particle will descend under the action of gravity to a fixed point from a variable point in the same time is called a tautochrone curve. If the particle is dropped from rest we have the equation of energy and the time of falling to the level z = 0 is 0 Let the length of the arc s measured from the fixed point be q>(0), then 3) t-- o If the curve is to be a tautochrone this must be independent of #0 or Let us change the variable by putting 3 = 00u, then i o or changing the variable back to z9 o If this is to vanish for all values of the limit #0 the integrand must vanish, or 4) 9/0) + W(*) = °> which is the differential equation of the curve. Writing this y"(«) = !_ g>'(«) 2/ 42] CYCLOID AS TAUTOCHRONE. 145 we may integrate, obtaining log y' (z) = — 2- log z + const. Taking the antilogarithm, r-N . f x C dS 5) ^0) = -= = r> ]/2 d# since s =

L. Calling its roots A1; 12 we have 28) ^ = -1 The general solution is obtained by multiplying the- particular solu- tions e*1* and e^ by arbitrary constants and adding. Thus we obtain 29) s We have to consider two cases, I. K2 > II. %2 < In case I the radical is real, and since its absolute value is less than ;c both Aj_ and ^ are negative and s eventually decreases as the time goes on, vanishing when t = oo. We have This vanishes when or Consequently if 5 and ^i are of opposite signs s will increase to a maximum and then continually die away. If they are of the same sign the motion dies away from the start. Both cases are shown in Fig. 31, where t is the abscissa and s the coordinate. In case II the radical is imaginary and both ^ and A2 are complex. Then writing 150 V. OSCILLATIONS AND CYCLIC MOTIONS. = + v l = — v and making use of the fundamental formula of imaginaries, 31) eivt = cos vt + i sin vt, and the principle that both the real part and the coefficient of i in the imaginary part of a solution are particular solutions, we obtain the two particular solutions eutcosvt and Fig. 31. We thus obtain the general solution 32) s = e>ut(Acosvt + Bsinvf) (A and B being new arbitrary constants), or as in § 19 equation 10), 33) The trigonometric factor represents a simple harmonic oscillation, which on account of the continually decreasing exponential factor dies away as the time increases (Fig. 32). Such a motion is called a damped oscillation, and 7, is a measure of the amount of damping. The extreme elongation occurs when 34) % = a that is when 35) tan 2 J. \ * — X2't—CC] = - / 1/A.lt*-.it* 43] LOGARITHMIC DECREMENT. 151 The smaller the damping ?c, the more nearly does the time of the maximum coincide with that of the maximum of the cosine factor in 33). In any case successive maxima follow each other at intervals equal to the period of the oscillation , 36) T = ^ At two successive maxima on the same side, s1 and S2, the cosine term will have the same value , therefore the ratio of the elongations will be that of the exponential factors , or Fig. 32. The logarithm of the ratio, 37) (5 = ^^ -1 is accordingly constant, and by means of observations on the loga- rithmic decrement we may determine the damping. We see that the decrement depends on and increases with the ratio of the square of the coefficient of damping % to the coefficient of "stiffness" I?. If there were no damping, B = 0, we should have for the period, Introducing these values of T0 and d, we may write 38) T = 152 V. OSCILLATIONS AND CYCLIC MOTIONS. so that if the damping is small, as is usually the case, it affects the period only by small quantities of the second order. As has been shown in § 38 we have here an instance of the use of a dissipation function and the energy is dissipated at a rate proportional to the exponen- tial e~xt. 44. Forced Vibrations. Resonance. The motion considered in the last section being that of a system left to itself is called a free oscillation or vibration. We shall now consider a problem of a different sort from. any yet treated and involving a force depending upon the time, and thus introducing or withdrawing energy from the system. Let us suppose a particle to be subject to the same conditions as above, but in addition to be acted upon by an extraneous force varying according to a harmonic function of the time, 40) F=Eco$pt, so that the differential equation of motion is We may find a particular solution by putting s = acos(pt — a), 42) ds . f , d*s o / j \ -^= -apsin(pt-a), -^ = - aj»8cos (pt — a). Substituting in the differential equation, we have 43) a (h2 —p*) cos (pt — a) — axp sin (pt — a) = Ecospt = E {GOS a cos (pt — a) — sin a sin (pt — a)}. This can be identically true for all values of t only if the coefficients of the sine and cosine of the variable angle (pt — cc) are respectively equal on both sides of the equation, accordingly we must have axp = a(h2- from which eliminating first E and then a, xp 46) tan« = ^, from which we obtain the amplitude 46) a = - =£ 42, 43] PHENOMENON OF RESONANCE. 153 Thus our solution is v 47) s = - ^ cos (pt — a). y(fc»_p»)* + xv The motion represented by this solution is called the forced vibration, for the system is forced to assume the same period as that of the extraneous force F, namely -— > of frequency — ; while the frequency of the free or natural vibration would be — or without damping - — The displacement is not in phase with the force, lagging behind it by less than a quarter -period if tana is positive, that is, if h is greater than p, in other words if the natural frequency is greater than the forced. If on the contrary the natural frequency is less than the forced, tana is negative, and since sin a is positive, the displacement is between a quarter and a half -period behind the force. If the frequencies of the forced and free vibrations coincide, tana becomes infinite, the lag is a quarter period, so that the displacement is a maximum when the force is zero and vice versa. Then 47) becomes 48) s = — sin Jit. p* and if the damping % is small, the amplitude is very large. This is the case in the phenomenon of resonance, of great importance in various parts of physics, including acoustics, electricity, and dispersion in optics. The equation shows how a very small force may produce a very large vibration if the period coincides nearly enough with the natural one, and explains the danger to bridges from the accu- mulated effect of the measured step of soldiers, the heavy rolling of ships caused by waves of proper period, and kindred phenomena. Although in the phenomenon of resonance the excursion and consequently the kinetic energy becomes very large, it is of course not to be supposed that this energy comes from nothing as has been frequently contended by inventive charlatans proposing to obtain vast stores of energy from sound vibrations.1) If we form the equation of activity, by multiplying 41) by ^> inx 'd(T-\-W) . (ds\* 1 d i/ds\2 -dt~ - + *(di) = *di[(di)- -E*p = (— cos a sin pt cospt + sin a coss ^2 1) Of these the United States has produced more than its share. The ignorance of the above mentioned principle enabled John Keely to abstract in the neighborhood of a million dollars from intelligent (!) American shareholders. 154 V. OSCILLATIONS AND CYCLIC MOTIONS. we see that energy is being alternately introduced into and withdrawn from the system by the extraneous force. On the average however, as we find by integrating the trigonometric terms with respect to the time, T /* smptcos£)tdt = 0, — P /• the time average of the activity depends upon the last term containing sin a, and this is always positive, consequently the extraneous force is on the whole continually doing work on the system, which is being dissipated at the rate xl-^} • This work is a maximum when a = — -9 \dt/ 2 when the system is in complete resonance. Thus the mechanical effects producible by resonance are shown to be commensurate with the causes acting, and the impossibility of the common story of the fiddler fiddling down a bridge is demonstrated. The exactness of "tuning", or approach to exact coincidence of period necessary for resonance is shown in Fig. 33, which is the graph of the curve where y = -=?- is the ratio of the actual amplitude of equation 46) to T? the steady statical displacement p produced by a constant force E (that is when p = 0), x = j- is the ratio of the frequencies of forced and free vibration, and «2 = ^-1) The curves are drawn for values of the parameter a2 equal to -01, -05, -10, -15, -20. Thus the magnitude of the resonance for any particular case can be seen by a glance at the figure. The resonance is sharper the smaller a. The maximum amplitude is not for perfect tuning, but for x = 1/1 — — - The value of the maximum is nearly equal to — If there is no friction , for p = h the vibration becomes infinite, which means simply that in this case friction must be taken into account. If there is no friction we have by 44), sinc£ = 0, cos a = 1 1) This parameter a. is not the angle cc above. 44] EFFECT OF TUNING. 155 and the displacement is in the same or opposite phase with the force , according as h is greater than or less than p. In the latter case the excursion is a maximum in one direction when the force is exerting a maximum pull in the opposite direction. This need not appear paradoxical, for consider the limiting case of a system with very little stiffness in proportion to its inertia, that is li very small and the natural period very great. Then the excursion is always opposite in phase to the force on account of the inertia of the system. In the opposite case of a system with very little inertia in proportion to the stiffness, h is very large, and the excursion is in phase with the force. In this case (that of complete agreement) we have what is- called the equilibrium theory of oscillation, the displacement being the same as (~F1\ S = ph except that the force and displacement are varying together. Such a theory was given by Newton for the tides, which consist of a forced vibration of the water covering the earth under the periodic force due to the moon's attraction. The more accurate theory taking account of inertia was given by Lagrange. The relation of the dyna- mical to the equilibrium theory is shown in Fig. 33. The two points of distinction between free and forced oscillations then are, first, that the free vibration has its period determined solely by the nature of the system, while the forced vibration takes the period of the P. force, and secondly, that if there is damping, the free vibration dies away, while the forced vibration persists unchanged. The theory of the forced vibration which we have given does not take account of the gradual production of the motion from a state of rest, but refers only to the motion after the steady state has been reached. We may now complete the treatment and take account of the motion at the start. Our previous solution is merely a particular solution. According to the theory of linear differential equations in order to obtain the general solution we must add to the particular solution just obtained the solution of the equation 41) Fig. 33. 156 V. OSCILLATIONS AND CYCLIC MOTIONS. when the second member is equal to zero, or in physical terms the forced and free vibrations exist superimposed. Accordingly we have If the system starts from rest we must determine A and /3 so that when t = 0, s and -^- are equal to zero. These conditions will be very nearly satisfied, if p and h are nearly equal and x small, by ( i ^ I ~~9y't (-\ / x2 Yl 52) s = a\cos (pt — a) — e cos ( J/ft2 — — • t — a \ r The simultaneous existence of two harmonic vibrations of nearly equal frequencies gives rise to the phenomenon known as beats. Suppose 53) s = a cos (pt— a) + & cos {(p + Ap] t — /3), where Ap is a small quantity, equal to 2 x times the difference of frequencies. We may write the last term & cos {(pt — a) + dp - t -f a — /3) = 6 {cos (pt— a) cos (dp • t + a — /3) - sin (pt — a) sin (dp -t-\- a — /3)}, so that 54) s = {a -f- & cos (Ap • t + a — /5)} cos (pt — a) - & sin (dp - 1 -f a — /3) sin (^^ — a), or if we write a -f & cos (z/# • £ -f- a — /3) = A - & sin (z/p • t -f a — /3) = 5, 55) 5 = Dcos(pt — a — f), where and D = Accordingly the compound vibration may be considered as a harmonic motion of variable amplitude and phase, the amplitude varying from a + ~b to a — &, with the period -^- and frequency -^- equal to the difference of the frequencies of the two constituents. The phenomenon of beats or interferences is represented graphically in Fig. 34. 44, 45] PHENOMENON OF BEATS. 157 In the case of free and forced vibrations coexisting [equation 52)], we have at the beginning beats which gradually die away owing to the factor e 2 in the free vibration, leaving only the forced vibration. Fig. 35. Fig. 34. This is shown in an interesting manner by a tuning fork electrically excited by another fork not quite in unison with it, the phenomenon of a single driven fork apparently producing beats with itself being very striking (Fig. 35). It will be noticed that the first maximum is greater than the steady amplitude. The greater part of this section and the preceding is taken from Rayleigh's Theory of Sound. 45. General Theory of small Oscillations. Having now set forth the general characteristics of vibrations excuted by systems possessing one degree of freedom, we will now treat the problem of the small vibrations of any system about a configuration of equili- brium after the manner of Lagrange, who first investigated it. Suppose a system is defined by n parameters qi9 q2, . . . qn- Its potential energy will depend only on the coordinates q, and developing by Taylor's Theorem, 56) W = TF where the suffix zero denotes the value when all the g's are zero. Suppose that this is a configuration of equilibrium, then W is a . . . (dW\ -, m, TT;r TT;r minimum or maximum and every (-~ — I equals zero. Inus W - rK0 begins with a quadratic function of the #'s. If the motion is small enough we may neglect the terms of higher orders of small quantities. Accordingly, neglecting the constant W0 (for the potential energy always contains an arbitrary constant which does not afPect the motion), we shall put W a homogeneous quadratic function of the #'s with constant coefficients, r — n s = n 57) W= 158 V. OSCILLATIONS AND CYCLIC MOTIONS. If the equilibrium is stable the potential energy must be a minimum so that the constants crs will be such that the quadratic function W is positive for all possible values of the variables q. The kinetic energy will be a quadratic function of the time derivatives, q[, q'2, . . . qi, 58) T= where the a's are functions of the coordinates q alone. We may develop the functions ars in series, thus one term of the sum becomes 59) ctr.qlq,' = *=1 and since the velocities q' are small at the same time as the co- ordinates g, we may neglect all the terms within the braces except that of lowest order aj,, therefore we may consider the a's as constants. If we have besides the conservative forces of restitution, arising from the potential energy W, non- conservative resistances which are linear functions of the velocities, we may make use of a dissipation function F, § 39, such that the dissipative force correspond- ing to the coordinate qr will be - - ~— 7- We thus have the three homogeneous quadratic functions with constant coefficients, r—n « = ! 60) * = -g r=l * = : Each of these has the property of being positive for all possible values of the variables of which it is a function. The a's may be called coefficients of inertia, the c's, coefficients of stiffness, and the jc's, coefficients of viscosity or resistance. We may now form Lagrange's equations for any coordinate qr. dpr dT dW dF ^t~Wr~^= ~Wr~Wr where 45] SMALL OSCILLATIONS. 159 62) dT -— H ----- h ar*qn H h C™_by substitution of any one lr jjn^J^e^ji^tip^^ ratios ^At : A2 : - • - : An. For each value ofjlr we obtain a different set of ratios. We will distinguish the values belonging to Ar by an upper affix r, so that Ars means the coefficient of e*r* in the coordinate qs. The theory of linear differential equations shows us that for the general solution we must take the sum of the particular solutions Arse rt for all the roots Ar, so that we obtain , Dynamics. 11 152 V. OSCILLATIONS AND CYCLIC MOTIONS. 73) It is to be noticed that the ratios of the As in any column have been determined by the linear equations 64), so that there is a factor which is still arbitrary for each column, that is to say, 2n in all. We may now replace the exponentials by trigonometric terms. The appearance of the terms with conjugate imaginaries Are1* + A[ert = 2e^ (ar cos vt - pr sin vt) leads to the disappearance of imaginaries from the result. Changing the notation we will accordingly write = % e^ * cos Vt - + B e^ cos 74) If these be substituted in the differential equations it will be found that the J5's satisfy the same linear equations as the As. Each column then contains an arbitrary constant as before, in the .B's and a second arbitrary constant in the 8 belonging to the column. We may therefore state the general result: - - The motion of any system, possessing n degrees of freedom, slightly displaced from a position of stable equilibrium may be described as follows: Each coordinate performs the resultant of n damped harmonic oscillations of different periods. The phase and damping factor of any simple oscillation of a particular period are the same for all the coordinates. The absolute value of the amplitude for any particular coordinate is arbitrary, but the ratios of the amplitudes for a particular period for the different coordinates are determined solely by the nature of the system, that is, by its inertia, stiffness and resistance coefficients. The 2n arbitrary constants determining the n amplitudes and phases are found from the values of the n coordinates q and velocities q' for a particular instant of time. 45] NORMAL COORDINATES. 163 We further notice that, since the different periods depending upon v are derived from the roots of an algebraic equation, they are not in general commensurable, so that the motion is not as a whole generally periodic. For instance in the case of Lissajous's curves described in § 19, unless the two periods are commensurable the curve will never close. In the case, however, of the spherical pendulum performing small oscillations the periods of the two co- ordinates were equal, so that the path became a closed curve, an ellipse. There is one set of coordinates of peculiar importance. For simplicity let us suppose there is no dissipation, F—Q. Let us make a linear transformation with constant coefficients, putting & = yn 9>i + fia 9>a H f- nn(1 = o , - 1, C , - 1, 0. , . . . = 0; » rows. 0 , 0 , - 1, C ,-!,... 166 V. OSCILLATIONS AND CYCLIC MOTIONS. Expanding the determinant in terms of its first minors we have This equation between three consecutive determinants of the same form suggests a trigonometric relation, namely, making use of the relation sin (a -j- &) -f sin (a — 6) = 2 sin a cos Z>, with 1} = ft, a = n&, we have sin (n + 1) # -f- sin (n — 1) #• = 2 sin w# cos #. Comparing this with the formula 89), we see that they are identical if we put C = 2 cos #, Dn = fc.sin (n -f 1) #, where c is independent of n. To find it put n = 1, sin 2# A 2 cos Accordingly 90) Dn = sin (w -f- 1) •fl- am #• If this is to vanish we must have where Jc is any integer (not a multiple of n -f 1, to prevent sin # in the denominator from vanishing). Introducing the values of # thus found we obtain 91) from which 92) 00 == 2 \ \ Fig. 37. 87) ~ ~ = 2 cos # = 2 cos — COS n+l Letting & = 1, 2, 3, . . . n, we obtain n different frequencies proportional to the abscissae of points dividing a quadrant into (n + 1) equal parts, Fig. 37. Giving ~k other values not multiples of (n + 1), we shall merely repeat these frequencies. There are accordingly n different frequencies for the vibrations. We may arrive at the same result by noticing that the linear equations for the -4's, CAr - = 0, 46] EQUATION FOR PERIODS. 167 are satisfied by As = P smsft, where P is a constant, making use of the same trigonometric for- mula as before. Accordingly let us substitute in the differential equation ma dly 84) - yr_t + ^ -^ + 2yr - yr+1 = 0 the solution 93) yr = Psinr-frcos^ — s). Every term will contain the same cosine, so that dividing out we have - sin(r- 1)# + 2 l -~ - sinr# - sin(r + l)fr = Or which is an identity if giving o a & /^ _ \ v2 = — (1 — cos#), ma v as before, 92). The complete solution is then s = n 94) yr =^PS sin ^ cos (vf« - «,), S=:l with the 2w arbitrary constants Pg, cc5 to be determined by the initial displacements and velocities. Consider the case first in order of simplicity, n equals 2. Then ->-\nr . * -I/IT v = 2 I/ — sin — = I/ — ) . *• \ ma 6 r ma 95) o - 2 I/ - - sm — = ma 3 Thus the frequency of the higher pitched vibration is in the ratio of ~/3 : 1 = 1.732 to that of the lower, — somewhat more than the musical interval of a sixth. In this particular case it is easy to find the normal coordinates. Writing 96) ± ± = 0, f/2 = ~~ 2/i ? an(i the ^wo beads swing in opposite directions with a frequency }/3 times as great as before. The middle point of the string is now at rest, or forms a node. The general case above treated is very interesting when we pass to the limit as the number of beads is increased, giving us the case of a continuous string, of the greatest importance in the theory of musical instruments. Let us introduce in equation 94) the distance of the bead from one end of the string, rl . x = r • a = — ;— - * Accordingly 94) becomes s—n 100) y (x) = x< P* sin — p cos (vs t — as}- 3 = 1 A glance at Fig. 37 shows us that, as we increase n, the ratios at least of the smaller frequencies approach those of the integers, 1, 2, 3, .... By passage to the limit we may demonstrate that this is exactly true for all the frequencies. If Q be the line density of matter of the continuous string, that is, the mass per unit length, we have Accordingly since we have in the limit Ql* Introducing this into the value of vtj 92), •\c\-\\ ^ 46] GRADUAL PASSAGE TO LIMIT. 169 As n increases without limit y preserves its form, while vs approaches the limit 102) ".- We have therefore for the continuous string, y = • STtX s sm ~ cos _ fsTf-l /S \ (r v 7 ' i ~ "•) - The frequencies of the different terms of the series are in the ratios of the integers. Such partial vibrations are called harmonics or overtones of the lowest or fundamental, for which s = 1. Since, if we consider a single term of the series, the excursions of all the particles are in the same ratios throughout the motion, we see that the harmonics are normal vibrations. On account of the factor depending upon x the sth harmonic has nodes for I 2Z (s-l)Z x = —>—,••• 2 - '-> s s s or at any instant the string has the form of a sine curve and is divided by nodes into s segments vibrating oppositely, generally known as ventral segments. In order to show how rapidly the string of beads approximates to the motion of a continuous string, the following table from Rayleigh's Theory of Sound is inserted. It is to be noticed that it does not give exactly the ratios of the frequencies on account of the variable factor s under the sine in vt> but it approximately does so, and for the fundamental, s = 1, it gives exactly the ratio of frequency for n beads to that of the continuous string. n 1 2 3 4 9 19 39 2(n+l) . it -9003 .9549 • 9745 •9836 •9959 -9990 •9997 * ' n2(^+1) By means of an extension of the above method, Pupin has treated the problem of the vibrations of a heavy string loaded with beads, 1) Writing the factor of — I/ — in the form S7C \ : h S7t Q S7C since ,. lim = 1, we obtain the result. 170 V. OSCILLATIONS AND CYCLIC MOTIONS. both for free and forced vibrations, and by an electrical application has solved a very important telephonic problem.1) On account of the importance and typical nature of the problem of the continuous string, we shall also solve it by means of Hamilton's Principle. Replacing the length of a segment a by the differential dx, writing gdx for the mass m, and for yY-of (partial derivative because y depends upon both t and x), and for the sum, the definite integral, we have the kinetic energy 104) r = Similarly in the potential energy the limit of the term 1S so that the potential energy becomes 105) W As the number of degrees of freedom is now infinite we are not able to use Lagrange's equations, but we can use Hamilton's Principle, which includes them. 106) to 0 Integrating the first term partially with respect to t and the second with respect to x, i t, 107) The variation dy is as usual to be put equal to zero at the time limits, and, as the ends of the string are fixed, dy equals zero at 1) Pupin, Wave Propagation over non - uniform Electrical Conductors. Trans. American Mathematical Society, I, p. 259, 1900. 46] PARTIAL DIFFERENTIAL EQUATION. . 171 the limits for x also, consequently we must have the factor of the arbitrary 6y vanish, that is, 108) tting motion of the continuous string, o Putting •— = a2 we have the partial differential equation for the which may also be obtained from the ordinary differential equa- tions 84) by passage to the limit in an obvious manner. The passage from n ordinary differential equations to a single partial differential equation when n is infinite is worth noting as a type of a phenomenon of frequent occurrence. At the same time the notion of normal vibrations gives rise to that of normal functions. To find a normal vibration let us find a particular solution of 109), 110) y = X(x) • x-dw = afd^- Dividing by Xcp we have 1 d*y _ a* d*X ~^ dt* ~X~di*" Since one side depends only on x and the other only on t, which are independent variables, this can hold only if either member is constant, say — v2a2, where v is arbitrary. Thus we have the two equations „•«* = 112) The first of these shows, like 77), that (p is a normal coordinate. Its integral is 113) (p = C cos (vat — a), the integral of the second is 114) X = A cos vx + Bsinvx. The normal vibration is accordingly represented by 115) y = (Acosvx + B sin vx) COB (vat — a), the arbitrary constant C being merged in A and B. 172 V. OSCILLATIONS AND CYCLIC MOTIONS. Since for all values of t, y = 0 for x = 0, we must have J. = 0, and since y = 0 for x = l, we must also have JE?sini'? = 0, that is 116) vl = sit, where s is any integer, accordingly we obtain for the sth normal vibration, 117) v. - ?, and the vibration is given by ., ON -r, . Stt# (STtat \ 118) y = ft sm -j- cos ( — ^ asj . The general solution is therefore represented as an infinite series of normal vibrations, ^ON -n • fSTtat 103) y =2^ J5, sm —— cos s=l the arbitrary constants, Bs, as, being determined by the initial dis- placements and velocities. In order to determine them let us make use of the other fundamental property of normal coordinates, namely, that the energy functions do not contain product terms. Let us write 119) then 120) 0 0 I I = I n A I . snx . ritx -, f(x) sm -y- dx = ^ As I sin — sin ~- dx, o o and by the property just found the integral on the right vanishes in every term except that in which r = s. But i I o Therefore we have the value of the coefficient i 124) Ar = jff(%) sin ^dx. o We are thus led to a particular case of the remarkable trigono- metric series associated with the name of Fourier. Such series were first considered by Daniel Bernoulli in connection with this very problem of a vibrating string. This determination of the coefficients was given by Euler in 1777. The importance of the series in analysis was first brought out by Fourier who insisted that such a series was capable of representing an arbitrary function, as had been maintained by Bernoulli, but doubted by Euler and Lagrange. 47. Forced Vibrations of General System. Let us now briefly consider the question of forced vibrations of the general system of § 45. Suppose that there is impressed upon each coordinate a harmo- nically varying force, Fr = Ercospt, the period and phase being the same for all, the amplitude Er being taken at pleasure. The equations are most easily dealt with if, instead of proceeding as we did in treating equations 41) and 42) we make use of the principle that, in an equation involving complex quantities, the real and the imaginary parts must be equated separately. Let 174 V. OSCILLATIONS AND CYCLIC MOTIONS. us therefore put instead of the above value of Fr the value whose real part agrees with the above, and having found a particular solution of the differential equation, let us retain its real part only. Thus we have instead of equations 63) n equations of which the rth is d22i d*qz d*qn 125) arl i + #r2 2 ~^~ ' l~ arn dq, dq, i dqn +Cr2qz H ----- h crnqn Guided by the result of § 44, assuming these become ( 126) ; If we call the determinant of equation 65) -D(A) and the minor of the element of the yth column and sth row Dr»(X), we have as the solution of 126) 127) Since D(Ji) = 0 is the determinantal equation 65) for the free vibra- tion, whose roots are At, A2, . . . fan, we have 128) where 0 is the proper constant. Accordingly the denominator D(ip) is 129) D(if) = C (V - V = V + G>-vOa, tan «, = -—*. n» Retaining now only the real parts, we have for our solution, — ; Thus if the damping coefficients ^ are small, all the oscillations are in nearly the same phase. If the frequency of the impressed force coincides with that of any one of the free oscillations, p — i>s = 0, and one factor of the denominator reduces to ^is) so that if the damping of that oscillation is small, the amplitude is very large, or infinite if there is no damping. This is the case of resonance. (Resonance may also be defined in a slightly different manner as occurring when ip is one of the roots of the equation D(T) = 0 in which all the «'s have been put equal to zero. This corresponds with our example in § 44. In practical cases the difference is very small.) 48. Cyclic Motions. Igiioratioii of Coordinates. In certain large classes of motions some of the coordinates do not appear in the expression for the kinetic energy, although their velocities may. For instance in the case of rectangular coordinates, the coordinates themselves x, y, z do not appear. In spherical co- ordinates, § 41, 133), (p does not appear while both r and # do. Further examples are furnished in the case of systems in which throughout the motion the place of one particle is immediately taken by another equal particle moving with the same velocity, as for instance in the case of the system of balls in a ball-bearing (bicycle) or better in the case of a continuous chain passing over pulleys, or through a tube of any form, or by the particles of water circulating through a tube. 176 V. OSCILLATIONS AND CYCLIC MOTIONS. In order that this condition may be permanent it is evidently necessary that the path traversed by the successive particles shall be reentrant, or that they shall circulate. Under the conditions supposed it is evident that the absolute position of any particle does not affect the kinetic energy, for throughout the motion at any point on the path of the particles there is always a particle moving with the same definite velocity. On account of the character of these examples the term cyclic coordinates has been applied by Helmholtz to coordinates which do not appear in the kinetic energy. We shall when necessary distinguish cyclic coordinates by a bar, thus 133) |£ = 0 tig. is the condition that q^ is cyclic. This of course involves that every that is the coefficients of inertia do not depend upon the cyclic co- ordinates. Thus a cyclic coordinate is characterized by the fact that the corresponding reaction is wholly momental. Examples of cyclic coordinates are found in x, y, #, qp, above, and cp in the case of plane polar coordinates. Inserting equation 133) in Lagrange's equations we have ia^ d i dt( or the fundamental property of a cyclic coordinate is that the force corresponding goes entirely to increasing the corresponding cyclic momentum. If the cyclic force Pr vanishes, we have and integrating, iw\ VT W,r = Pr = Cr. In this case we may with advantage employ a transformation intro- duced by Routh1) and afterwards by Helmholtz2), which is analogous to that invented by Hamilton and described in § 39. By means of equations 53) and 71) § 39, we have expressed the velocities as linear functions of the momenta with coefficients Brs, which were functions of the coordinates, and have thus introduced the momenta into the kinetic energy in place of the velocities. We have thus been led to use instead of the Lagrangian function L = T — W, 1) Routh, Stability of Motion, 1877. 2) Helmholtz, Studien zur Statik monocydischer Systeme, 1884. Ges. Abh. Ill, p. 119. 48] ELIMINATION OF VELOCITIES. 177 whose variation appears in Hamilton's Principle, the Hamiltonian function H=TJrW. The transformation of Routh and Helmholtz, instead of eliminating all the m velocities q', eliminates a certain number, which we will choose so as to replace those having the suffixes 1, 2, . . . r, by the corresponding momenta, but to retain the velocities with suffixes r + 1, . . . w, in the equations. This trans- formation, while it may be made in the general case, is of particular advantage where the eliminated velocities are cyclic and the corre- sponding momenta constant, as in the case just described. The equations 53) § 39 for the elimination become by trans- position 138) Qriqi + GrSffi +• • '+ Qrrq'r = Pr ~ (fc.r + ltfr'+l +' ' ' + Crmffi). It will be convenient to write the right hand members above, Pi-Si, ...pr-Sr. Let the solutions of equations 138) be &' = -Rll (Pi - $) + R** (P-2 ~ SJ + • • + Rlr (Pr ~ &), 139) qr' = Rrl (Pi ~ St) + Bri (pi - S2) + • • + Err (pr ~ &), where the J^'s are the quotients of the corresponding minors of the determinant Qn> 612? • • • Q Qrl) Qr2, - • • Qrr by the determinant itself, and, like the §'s, are functions of the coordinates only. Introducing the values 139) for the q"s into the kinetic energy, the latter becomes a function of the velocities qr+i, - - • q™ and of the momenta pl9 .. .pr. It is a homogeneous quadratic function of all these variables, but not of the p's or g"s considered separately on account of product terms, such as psqt' which are linear in terms of either the pjs or q"s. The function I thus transformed has lost its utility for Lagrange's equations, but may be replaced by a new function, as follows. Let us call the function T expressed in terms of the new variables T'. We have thus identically 140) T(ql9 q2>... qm, &', &', ...$!»). WEBSTER, Dynamics. 12 178 V. OSCILLATIONS AND CYCLIC MOTIONS. It is to be noticed that since the coordinates q appear in the co- efficients E of equations 139) they are introduced into T' in a way in which they do not appear in T, so that we do not have dT dT' hut since q enters in T' both explicitly and implicitly through equations 139), we have for s = r -f 1, r + 2, . . . w, 141) - = 3 is called by Routh the modified Lagrangian function, and its negative _by Helmholtz the kinetic potential. It is to be understood that 0 is to be expressed in terms of the velocities , 5/4-1, . . . q'n by means of equations 139) in which plf . . . pr have been replaced by C1; . . . cr. The important thing to notice about

like the Q's from which they are derived. The terms of the latter sort in — -rr— cause precisely the same effect as if they were added to the potential energy. The effect of cyclic motions in a system is accordingly partly represented by an apparent change of potential energy, so that a system devoid of potential energy would seem to possess it, if we were in ignorance of the existence of the cyclic motions in it. The effect of the linear terms in 0 is quite different and will be discussed in § 50. A system is said to contain concealed masses, when the coordinates which become known to us by observation do not suffice to define the positions of all the masses of the system. The motions of such bodies are called concealed motions. It is often possible to solve the problem of the motions of the visible bodies of a system, even when there are concealed motions going on. For it may be possible to form the kinetic potential of the system for the visible motions, not containing the concealed coordinates, and in this case we may use Lagrange's equations, as in the case just treated, for all visible coordinates, while the coordinates of the concealed masses may be ignored. Such problems are incomplete, inasmuch as they tell us nothing of the concealed motions, but very often we are concerned only with the visible motions. Such concealed motions enable us to explain the forces acting between visible systems by means of concealed motions of systems connected with them. The process of eliminating the cyclic coordinates of the concealed motions as above described is termed by Thomson and Tait ignoration of coordinates.1) Examples of the process may be obtained in any desired number from the theory of the motion of rigid bodies rotating freely about 1) Thomson and Tait, Natural Philosophy, Part I, § 319, example G. 12* Y- OSCILLATIONS AND CYCLIC MOTIONS. axes pivoted in bearings fastened to bodies themselves in motion. Such motions will be treated in § 94. A very simple case of the above process is encountered in treating the motion of a particle m sliding on a horizontal rod, revolving about a vertical axis, at a distance r from the axis. Let the angle made by the rod with a fixed horizontal line be cp, then the velocity perpendicular to the rod is rep'. The velocity along the rod being rf, the kinetic energy of the body m is 146) T=~ Since (p does not appear in T, (p is a cyclic coordinate. If there is no force tending to change the angle y> we have dT 147) P(p = — = mr* mentioned, ~mr'2 being the quadratic function of the remaining velocity r' and — — ^— g being the quadratic function of the constant c, which contains as a coeffi- cient a function of the coordinate r. We may now, ignoring the coordinate or dr 1R1\ r $ c mW = ^ = ^r*' We accordingly see that the system acts as if, there being no rotation, it possessed an amount of potential energy — C&, producing the force s>2 ^3 directed from the center. This example accordingly illustrates the effect of ignored cyclic motions in producing an apparent potential energy, but it does not illustrate the effect of linear terms in * + &»&" * If qB is the cyclic coordinate, all the Q's are independent of g3, and if the corresponding force P3 vanishes, we have the constant momentum, 153) pB = Cis &' + fts &' + £33 &' - ^3 ; From this we determine the cyclic velocity, .!«) ft,_^-ft.«/-ft.&', inserting which in the kinetic energy gives, on combining terms, 155) T'=± + -«- It is noticeable that the linear terms in #/, q2' have cancelled each other. It will be proved below that this always happens. But when we form the kinetic potential, which is to be used instead, they reappear. We have 156) = T'- ) V33 , csQ13 , csQ, , 1 c32 I O bfl~i/0 b/2 ^/O V33 VSS ? V33 Thus the effect of the cyclic motion, which may itself be concealed from us, is made evident to our observation by the presence of the fourth and fifth terms, which are linear in #/, q2r. The apparent coefficients of inertia, that is the cofficients of qt' 2, q2' 2, #/ g2'; are 182 V. OSCILLATIONS AND CYCLIC MOTIONS. changed from their real values (unless $13 — $23 = 0), while c 2 there appears the term - ~— independent of the velocities, depend- r Vss ing on the coordinates qlf q2. This is, since it gives rise to a conservative positional reaction, undistinguishable in its effect from potential energy. In reality, the reaction to which it gives rise is motional, instead of positional, as it appears to be. If we could explain all potential energy in this manner, namely as due to concealed cyclic motions, we should have solved the chief mystery of dynamics. In his remarkable work on dynamics, Hertz treats all energy from this kinetic point of view. In order to have a successful model for this representation of potential energy, which needs in order to be perfect no linear terms, we must have Q13 = Q23 = 0. We can now see why the simple example of § 48 showed no linear terms, since by putting all the Q's with one suffix 2 equal to zero we pass to the case of a system with two degrees of freedom. If at the same time the coordinates are orthogonal, §13 = 0, so that the single linear term disappears. This was the case above. Let us now pass to the general case. We have for the momenta the equations 53) § 37 and, for the first r, 137) which are written out, Pi = ' 2 m 157) Pr =Qr Pm — Qml ( Let us now form the kinetic energy from the definition, § 36, 38), 158) Multiplying the above equations, the sth line by qa', and adding, we obtain from the first r lines on the right, The terms coming from the last m — r lines, and the first r columns, as marked off by the dotted lines, are found to be, on collecting according to columns, 49] EFFECT OF ELIMINATION. 183 ' s = r 2*'*' «=i on referring to tlie definition of the definitions of the 5s's, 138), 159) S, = Qs,r Finally the terms from the lower right hand square, of m — r rows and columns gives us a quadratic function of the last m — r velocities, namely that part of 2T which originally depended on these velocities and no others. This part we will call 2Ta. We have therefore 160) 2T = 2Ttt + g.' (8. + e,\ 5 = 1 Now if we form the quadratic functions, with the coefficients R from the determinant of equations 139), s =• r t = r = -£•/, /, KitSt we may write equations 139) as lea) m. the equation of activity. These forces are consequently conservative motional forces. They are however perfectly distinguishable by their effects from the conservative motional forces arising from the term C which imitates potential energy, and they in no wise imitate potential energy, as we shall see by an example. A system containing gyrostatic members behaves in such a peculiar manner that their presence is easily inferred. The theory of gyro- stats will be treated in Chapter VII. In the mean time the following simple example will illustrate the theory, and at the same time serve to prepare for the general theory of the gyrostat, of which it con- stitutes a special case. 1) Thomson and Tait, Nat. Phil. § 345^1. 186 V. OSCILLATIONS AND CYCLIC MOTIONS. Let four equal masses, —> be fastened to the ends of two mutu- ally perpendicular arms of negligible mass (Fig. 38), which are fastened rigidly where they cross, at their middle points, to an axis perpendicular to them both, about which they turn. Let the point of crossing of the three arms be fixed while the system can spin about the axis OP, which can move in any manner. We will suppose that during the motion the axis OP makes with the ^-axis a small angle whose square can be neglected in comparison with unity. Let the position of the axis be determined by the coordinates !, 77, of the point in which it intersects a plane perpendicular to the ^-axis at unit distance Fig. as. from the origin. The squares and products of |, y, are con- sequently to be neglected. Let us further specify the position of the system by the angle cp that the projection of the arm OA on the XY- plane makes with the X-axis. Thus the three coordinates |, rj, cp determine the position of the whole system. If the coordinates of the point A are x, y, 8, since it lies in a plane whose normal passes through the point £, ??, 1, we have 174) g + lx 4- yy = 0. But since OA always makes a small angle with the XT- plane, the projection of OA on this plane differs from it in length only by a quantity of the second order, which we neglect. We therefore have Differentiating 174), dy = xdcp. so that we have dx2 + = xdl -f ydri -f (rjx — ly) dcp, + dz* = (Z2 + rfx* + |y - 50] GYEOSCOPIC FORCES. 187 and the part contributed by the particle A to the kinetic energy is The opposite particle C, for which x2, y2, xy have the same values, contributes the same amount. The other pair of particles, for which the values of x2, y2 are respectively those of y2, x2, for the first pair, and the values of xy the negatives of the values for the first pair, consequently contributes an amount of energy which, added to that already found, makes the terms in xy disappear, and replaces each term in x2, y2, by the same term with I2 written in the place of x2 or y2. Neglecting then |2, if, we have finally 176) y=' We accordingly see that cp is a cyclic coordinate for the system, so that if the system is spinning without any force tending to change qp, we are dealing with a case of the example in § 49. We have, pro- ceeding as there, 177) and eliminating qpf, from which we form In order to form the diiferential equations for the motion of |, 77, we have by differentiation 179) and neglecting the squares and products of the small quantities |, and £', ?/, which are small at the same time, w a* ai 188 V. OSCILLATIONS AND CYCLIC MOTIONS. Proceeding in the same manner for 77, we have with the same degree of approximation d$ ml* , c£ W = "~^n "V 181) If W is the potential energy (there being no apparent potential energy due to the cyclic motion, since the part C is here constant), the equations of motion are accordingly, 182) Thus the gyroscopic terms in c have the property proved in 172). If there is no potential energy, the gyroscopic forces cause the motion to be of such a nature that rr + tf'i/ = o, i/i"2+v12 - ^v^+^' that is the acceleration is perpendicular to the velocity, and pro- portional to it. Under these circumstances the motion is uniform circular motion. In fact the equations are satisfied by I = Acospt. vc 183) W = 0, p = ™- t] = A srnpt, ml* Thus the circle, whatever its size, is described in the same time ~> which is inversely proportional to the momentum of the cyclic motion. We may describe the effect of the gyroscopic forces in general for a system with two degrees of freedom by saying that they tend to cause a point to veer out from its path always toward the same side. This effect is characteristic, and cannot be imitated by any arrangement of potential energy whatever. By the aid of this principle all the motions of tops and gyrostats may be explained. 51. Cyclic Systems. A system in which the kinetic energy is represented with sufficient approximation by a homogeneous quadratic function of its cyclic velocities is called a Cyclic System. Of course the rigid expression of the kinetic energy contains the velocities of every coordinate of the system, cyclic or not, for no mass can be moved without adding a certain amount of kinetic energy. Still if certain of the coordinates change so slowly that their velocities may be neglected in comparison with the velocities of the cyclic coordinates, the approximate condition will be fulfilled. These coordinates define the position of the cyclic systems, and may 50, 51] CYCLIC SYSTEMS. 139 be called the positional coordinates or parameters of the system. In the example of § 48 if we suppose the radial motion to be so slow that we may neglect rn in comparison with r2cp'2 we have 184) T=ymrV2, and the system is cyclic, r being the positional, cp the cyclic co- ordinate. In the case of a liquid circulating through an endless rubber tube, the positional coordinates would specify the shape and position of the tube. The positional coordinates will be distinguished from the cyclic coordinates by not being marked with a bar. The analytical conditions for a cyclic system will accordingly be, for all coordinates, either 1Q.s 3T n 3T 18o) ^ = 0 or ^7=^ = 0, or if we use the Hamiltonian equations 78) § 39 with the value of T obtained by replacing the velocities by the momenta, which we shall denote by Tp, since the non- cyclic momenta vanish 186) ^ = 0, and |5? = 0, for the cyclic coordinates, as before. We accordingly have for the external impressed forces tending to increase the positional coordinates, by § 37, 60), § 39, 80) respectively, the first term vanishing, TT) _d(lP+W) i) dT w- P- s~ and for the cyclic coordinates A motion in which there are no forces tending to change the cyclic coordinates is called an adiabatic motion, since in it no energy enters or leaves the system through the cyclic coordinates. (It may do so through the positional coordinates.) Accordingly in such a motion the cyclic momenta remain constant. The case worked out above was such a motion. In adiabatic motions the cyclic velocities do not generally remain constant. In the above example, for instance, the cyclic velocity (p' was given by A motion in which the cyclic velocities remain constant is called isocydie. o rn O rji 1) That - - = -o-^ may be seen by putting r = m in 144) , when the parenthesis becomes T' — 2T=—TP. 190 V. OSCILLATIONS AND CYCLIC MOTIONS. The motion of a particle relatively at rest upon the surface of the earth is isocyelic, taking account of the earth's rotation. In such a motion the cyclic momenta do not generally remain constant, but forces have to be applied. In the example of the bead on the revolving rod if r varied forces would have to be applied to the rod to keep the rotation y the cyclic forces is double the work done by the system against the positional forces. In such motions the energy of the system accordingly increases by one-half the work done by the cyclic forces, the other half being given out against the positional forces. For if we use the energy in the form we have in any change 196) dT = \^l (qj dps + ps dfr'), and in an isocyclic change, every dqs' vanishing, 197) ST- ±2 *•'**• But since 198) ^ = P., dj>. = Psdt, and since qi = -~±> ql dt = dqs, and the above expression for the gain of energy becomes 199) dT = But the work done by the cyclic forces is 200) 8 A =?sPsdqs = 2dT. Therefore the last part of the theorem is proved. Again, in any motion, sol) w and in an isocyclic motion, 202) . 8T But since the work of the positional forces is 203) 8 A = p. 8q, _ - dq. = - ST, the first part of the proposition is also proved. II. In an adiabatic motion, the cyclic velocities will in general be changed. Then they change in such a way that the positional forces caused by the change of cyclic velocities oppose the motion, that is, do a positive amount of work. For since for any positional force 53, 54] WORK DONE BY FORCES. 193 P ar p*~ 3£; the change due to the motion is Of this the part due to the change in the cyclic velocities is 205) • tP. and the work done by these forces is 206) S-, A ~£. 9? P.Sq, - - Now we have for any motion 207) and in an adiabatic motion this is .zero., so that 208) Substituting this in the double sum 206) r we get 209) SA But this expression represents [§..36, 35)] twice the energy of a possible motion in which the velocities would be dqt', and must therefore be positive for all values of dqj, dgr'. Accordingly d- A > 0. The interpretation of this theorem for electrodynamics is known as Lenz's Law1), namely, ato, electrical current being represented by a cyclic velocity, and the shape and relative position of the . circuits by positional coordinates, if in any system of conductors carrying currents, the relative positions of the conductors are changed, the induced currents due to the motion of the conductors are so directed as by their magnetic action to oppose the motion. 54. Examples of Cyclic Systems. Let us consider the example of equation 184) as illustrating the previous theorems. We have for the momenta dT dT 1) These Theorems are all given by Hertz, Prinzipien der MechaniJc, 568 — 583. WEBSTER, Dynamics. 13 194 V. OSCILLATIONS AND CYCLIC MOTIONS. and introducing these instead of the velocities 210) ^ We have for the positional force This being negative denotes that a force Pr toward the axis must be impressed on the mass m in order to maintain the cyclic state. This may be accomplished by means of a geometrical constraint, or by means of a spring. The force or reaction — Pr which the mass m exerts in the direction from the axis in virtue of the rotation is the so-called centrifugal force. We see that if the motion is iso- cyclic, the positional force increases with r, while if it is adiabatic, as in the case worked out above, it decreases when r increases. The verification of the theorems of § 52 is obvious. The cyclic force vanishes when the rotation is uniform, and the radius constant. If, the motion being isocyclic, that is, one of uniform angular velocity, the body moves farther from the axis, Pv> the cyclic force is positive, that is, unless a positive force P9 is applied, the angular velocity will diminish. In moving out from r: to r2 work will be done against the positional force Pr of amount r2 rz 212) - A = -Jprdr = my' *Jrdr = ^ (r22 - r*)9 ^ rj, while the energy increases by the same amount. Thus the first theorem of § 53 is verified. If the motion is adiabatic, o f If the body moves from the axis, cp' will accordingly decrease, so that 213) The change in Pr due to a displacement dr is, by 211), 214) dPr = - m(yndr + 2/V (V), of which the part containing dcp', 215) does the work 54] EXAMPLES OF CYCLIC MOTIONS. 195 216) dv'A = d9'Prdr = or by 213), 217) d(p'A = mr*d we see that the rod carrying the particles will remain at rest relatively to the hori- zontal rod in either a vertical or horizontal position. It is easy to see that the vertical position is one of unstable equilibrium, for, writing the equation 209) 220) we see that if # be slightly different from zero, # will tend to become still greater in absolute value. Writing however # = -y — &' the equation becomes 221) 1) The system is cyclic if we neglect 13* 196 V. OSCILLATIONS AND CYCLIC MOTIONS. If #' is slightly different from zero, it will accordingly tend to approach the value zero, so that the horizontal position is stable. A body moving according to the differential equation 221) is called by Thomson and Tait1) a quadrantal pendulum, since # changes "according to the same law with reference to a quadrant on each side of its position of equilibrium as the common pendulum with reference to a half -circle on each side", or in other words, in the ordinary pendulum the acceleration is proportional to the sine of the angle of deviation from equilibrium, and in the quadrantal to the sine of twice the angle. The small oscillation performed by the bar will be harmonic with the frequency ~- Here we have an ex- cellent example of an apparent potential energy which is really kinetic. 1) Thomson and Tait, Nat. Phil., § 322. PART II DYNAMICS OF RIGID BODIES 55] TRANSLATIONS AND ROTATIONS. 199 CHAPTER VI. SYSTEMS OF VECTORS. DISTRIBUTION OF MASS. INSTANTANEOUS MOTION. 55. Translations and Rotations. A rigid body or system of material particles is one in which the distance of each point of the system from every other is invariable. Its position is known when the positions of any three of its points are known, for every point is determined by its distances from three given points. These three points have each three coordinates, but, since there are three conditions between them, defining their mutual distances, there are only six independent coordinates. Thus, a rigid body has six coordinates. A rigid body may evidently be displaced in such a manner that the displacement of every point is represented by equal vectors, that is equal in length and parallel. Such a dis- placement is called a translation, and, being represented by a free vector, has three coordinates. A rigid body may also evidently be displaced, so that two given points in it, A and IB, remain fixed. Since any point P must move on a sphere of radius BP about B, and also on a sphere of radius AP about A, the locus of its positions is the intersection of the two spheres, that is a circle whose plane is perpendicular to the line AB, and whose radius CP is the perpendicular distance from P to the line AS. If this is zero, the point does not move, therefore all points on the line AB remain fixed. The displacement is called a rotation and the line AB, the axis of rotation. The rotation is specified if we know the situation of the line AB and the magnitude of the angle POP', or the angle of rotation. A line may be specified by giving the two pairs of coordinates of the points in which it intersects two of the coordinate planes. A line has thus four coordinates, and a rotation, five, — the four of the axis together with the magnitude of the angle. Fig. 39. 200 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. Any displacement of a rigid body may be brought about in an indefinite number of ways. Let three points ABC (Fig. 40) be displaced to A'B'C'. We may first give the body a translation defined by the vector A A'. This will bring B to B± and C to Cv Then through A' pass an axis perpendicular to the plane B^B', and rotate the body about this axis through the angle B^A'S1. This brings B± to B' and C± to a new position (72. Finally rotate the body about A'B' until C2 arrives at Cf. We have thus brought about the given dis- placement by means of a succession of translations and rotations. Evidently the order of these may be varied. Accordingly, Any displacement of a rigid body may be reduced to a succession of translations and rotations. We have seen that a translation may be represented by a free vector, a rotation, by a vector that must give the axis and the angle. If we agree to draw the vector in the axis, and make its length numerically equal to the angle of rotation, it will completely specify the rotation, if we adopt a convention about the direction of rotation. This shall be that, if the rotation is in the direction of the hands of a watch, the vector shall point from face to back of the watch. Vector and rotation correspond then to the translation and rotation in the motion of a cork-screw, or any right-handed screw. As the vector may be placed anywhere along the axis, but not out of it, it has five coordinates, and may be characterized as a sliding vector. Translations are compounded by the law of addition of vectors. The resultant of two rotations about the same axis is evidently the algebraic sum of the individual rotations. The resultant of a trans- lation and rotation is evidently independent of the order in which they take place. The resultant of a rotation and a translation perpendicular to its axis is equivalent to a rotation about a parallel axis, for it is evident that all points move in planes perpendicular to the axis, and that the motions of all such planes are alike, or the motion is uniplanar. Now the motions of any two points in a plane determine the motion of the plane parallel to itself. 55, 56] ROTATIONS ABOUT PARALLEL AXES. 201 From 0 (Fig. 41) lay off the translation vector 00' of length, t and find a point C on the perpendicular bisecting 00' which makes the angle OCO' equal to ca, the angle of rotation, and in the right sense. Then if OC be rotated about 0 through the angle to to Cf and then C" be moved by the translation it will return to C. Therefore the point C remains fixed, and is the center of rotation, and thus the rotation co about C is equivalent to the equal rotation about 0 together with the translation, 1) r = 200 sin ~, and if p is the perpendicular from C to 00', a 2 Fig. 41. 56. Rotations about two Parallel Axes. As before the motion is uniplanar and is specified by two points. Let A and B (Fig. 42) be the intersections of the axes with the plane of the paper perpen- dicular to them. Turn about A through the angle co1, bringing B to B'. Then turn about B' through the angle a>2, bringing A to A'. Bisect o1 by AC. B could be brought to B' by rotation about any point of AC, since all such points are equidistant from BB!. Bisect w2 by B'D. A could be brought to A' by rotation about any point in B'D. Therefore the motion of A and B could be produced by a rotation about 0, the intersection of AC and B'D. Triangle AOA' is isosceles. Angle AOD = angle OAB' + angle AB'O = ^ + ^, Angle AOA' =-- 2 - angle AOD = o^ + o>2, that is, two rotations about parallel axes compound into a rotation equal to their algebraic sum about a parallel axis. To find the position of this axis we have OS' OA AB sm— '- —*- sin 2 ~ 2 2 If the order of rotation is changed we obtain a different result. 202 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. If the rotations c^ and o2 are of opposite signs and of equal magnitudes, the intersection of the two bisectors is at infinity and the axis of rotation is thus at infinity. A motion about an infinitely distant axis is a translation. The direct proof is as follows. Let A be the center of rotation a , bringing B to B'. Then rotate about B' through an equal angle in the opposite direction, bringing A to A. Triangles ABB' and AAB' have AB' common, and AB = A'B' and the included angles equal, therefore A A' and BB' are equal and parallel and two points consequently all points — have moved parallel to each other the same distance. The motion is therefore a trans- lation of magnitude, 4) . 43. t = 2AB sin ~ Accordingly every translation may be decomposed into rotations, and we may reduce all displacements to rotations. 57. Rotations about Intersecting Axes. Infinitesimal Rotations. Let OA and OB be two intersecting axes about which we revolve the body through the angles c^ and o>2 respectively. Describe a sphere with the center 0. Let the rotation o1 about A bring B to Bf, and o>2 about B' bring A to A. Pass planes through the vertices bisecting the angles oc^ and «2, then, as in § 56, the displacement just given is equi- valent to a rotation about the line of intersection CO of these planes. The order of the rotations affects the result. Since AC bisects the angle BAB' and the spherical triangle Fig. 44. BAB' is isosceles, angle ABC = angle AB' C = y- Thus the resultant rotation, o = angle ACA' = angle BCB'. Angle ACE= angle B1 CD = angleDOJ5= |- 56, 57] RESULTANT OF INFINITESIMAL ROTATIONS. 203 In the spherical triangle ABC we have . 0>a . 2 are laid off anywhere on their axes, the position of the axis 0 may be found by the following construction. At A a point on the axis of rotation CDI lay off A E = o2 and at I? at a point on the axis of rotation co2 in the opposite direction BS=av Join E and S, and where this straight line ES cuts AB, draw OT parallel to AE, BS equal in length to 04 + «2. For 0 A _ AE _ co2 ~OB~ B~S~~^L' as required by 7). The construction (Fig. 47) shows that if co1 and o>2 have the same sign, the resultant co1 + o>2 has its axis 0 between A and B. If co1 and o2 are of opposite signs the same construction may be used (Fig. 48), but 0 is on AB produced and on the side of the greater rotation. If o^ = — o>2 ,yf\^ evidently 0 is at infinity and o = 0. The resultant is then a translation per- pendicular to the plane of the two axes, and its magnitude t is by 4) equal to CDG31 times the perpen- dicular distance between the axes. Fig. 47. Fig. 48. 58. Vector - couples. A pair of equal, parallel, oppositely directed, sliding vectors will be called a vector -couple. A rotation vector -couple is thus equivalent to a translation perpendicular to its plane, equal to the product of the length of either vector by the 57, 58, 59, 59 a] VECTOR -COUPLES. 205 perpendicular distance between their lines, or the arm of the couple. This product is called the moment of the couple. Two couples whose planes are parallel give rise to parallel translations, and if their moments are equal, to equal translations. Therefore a rotation -couple may be displaced without altering its effect, if its plane is kept parallel to itself and its moment is un- changed. A vector -couple may then be represented by a single vector perpendicular to its plane, whose length is equal to the moment of the couple. Its direction will be governed by the same convention as before, namely, the vector moment is to be drawn in such a direction that rotation in the direction of the couple and translation in that of the moment correspond to the motion of a right-handed screw. Moments will be represented by heavy vectors. The moment of a vector -couple is a free vector, hence the composition of couples is simpler than that of the slide -vectors themselves. We may now state the theorem of the general infinitely small displacement of a body as follows: The infinitely small displacement of a body may be reduced to a translation and a rotation, or in other words to a rotation and a rotation- couple. The choice of components may be made in an infinite number of ways. 59. Statics of a Rigid Body. Two equal, parallel, opposi- tely directed forces applied to a rigid body in the same line are in equilibrium. For otherwise they can produce only distortion or motion. Distortion is excluded according to the definition of a rigid body. They satisfy the conditions of equilibrium, § 32, for if applied at the center of mass they are in equilibrium, and their moments about any point are equal and opposite. Accordingly a force applied to a rigid body may be applied at any point in its line of direction without change of effect. Thus forces applied to a rigid body are not free, but are sliding vectors (five coordinates). (This is not a property of forces, but of rigid bodies.) Forces, whose lines of direction intersect, may be applied at the point of intersection and compounded by the rule of vector addition. AP £$ 59 a. Parallel Forces. Force - couples. Let A&- and •?*$- (Fig. 49) represent two parallel forces applied to a rigid body at A and B. Introduce at A and B two equal and opposite forces AE and BS of any magnitude in the line AB. These being in equili- brium do not affect the system. Find the resultant of AP and AE by the parallelogram, giving AC, also of BQ and BS giving BD. All these forces are coplanar, therefore the lines AC and BD will 206 VI- SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. meet at E, if produced. Slide AC and BD to E, and then resolve into components parallel to the original ones. We get EH and EJ equal and opposite (being equal to AE and JBS), and EK equal to AP and EL to BQ applied at E. Therefore the resultant of two parallel forces is a parallel force equal to their algebraic sum, and applied on a line T EOj whose posi- tion is to be found as follows. From the simi- lar triangles, AO FK BO GL OE ~KE OE LE OE AP' OE BQ By division, since FK=GL, Fig. 49. AO BO BQ AP Thus the position of the resultant of parallel forces is to be found by the same construction as the resultant of two rotations about parallel axes, Fig. 47. If the two forces are oppositely directed (Fig. 50), 0 is on AB produced, and if the forces are equal 0 lies at infinity. Accord- ingly there is no force that can replace two equal, parallel and op- positely directed forces not along the same line, or force- couple. The distance between the lines of direc- tion is the arm, and the product of either force by the arm is the moment of the couple. Fig. 50. We shall prove the following theorems. 59a] THEOREMS ON FORCE -COUPLES. 207 Fig. 51. Theorem I. A couple may be transported parallel to itself either in its own or a parallel plane without changing its effect. Consider the forces Pl and P2 both equal to P, applied perpendi- cularly at the ends of AB (Fig. 51). At the ends of an equal and parallel line A'B' apply four equal and opposite forces P3,P4,P5,P6, each equal to P, which are in equilibrium. The resultant of the equal parallel ^ forces P17 P6 is a force 2P | applied half - way between A and B'. The resultant of P2 and P5 is a force 2P in the opposite direction applied half- way between A and B. Since ABB' A' is a parallelogram these two points of application coincide and the two resultants neutralize each other. We have left the couple P3P4 equivalent in effect to the original couple. Theorem II. A couple may be turned in its plane about its center of symmetry without changing its effect. Let A'B' be a line of the same length and with the same center 0 as AB, the arm of the couple, and in the plane of the couple (Fig. 52). Apply at A' and B' four equal and opposite forces in equilibrium, each equal to P, and perpendicular to A'B' and in the plane of the couple. Consider Pl and P5 applied at (7, their point of inter- section, and by symmetry their re- sultant will be along OC. Similarly the resultant of P2 and P6 is an equal force along OD in the opposite direction. These two resultants neutralize each other, leaving the couple P3P4 which has the same effect as the original couple. 208 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. Theorem III. A couple may be replaced by another in the same plane having equal moment. Let the couple be P^P^ and the arm be A S (Fig. 53). At C on AS produced and at S apply four equal and opposite forces Q of such magnitude that ft B AB The resultant of the parallel forces, P1? QB, is equal to Pl plus Q3 applied at S on account of the above equation. This is counterbalanced by the forces P2 Fig. 53. and Q± applied at B, leaving the couple Q^Qz of moment f\ 7?/T ~p A 7? \cJ • JD\J === JL * J* Ify equivalent to the original couple. A force -couple is determined therefore by its plane and moment, and may be represented by a free vector perpendicular to its plane and of length equal to the moment. Theorem IV. Composition of Couples. Suppose the two couples are in different planes. By turning each in its own plane bring all the four forces into directions perpendicular to the intersection of their planes, and then by varying one of the couples cause them to have the same arm AS. The forces Qjfi applied at A compound by the parallelogram into Ev P2 and Q2 applied at S compound into E% equal and opposite to Ev The arm of all these couples is the same, there- rig. 54. fore their moments are propor- tional to P, Q and E. The vectors representing the moments are perpendicular to AS and to P, Q and E respectively, thus they form the sides and diagonal of a parallelogram similar to that of P, Q, E. Therefore couples are compounded by compounding their moments by the law of addition of vectors. 59 a, 60, 61] KINEMATICAL AND DYNAMICAL DUALISM. 209 6O. Reduction of Groups of Forces. Dualism. Suppose we have any number of forces applied to various points of a rigid body. Let one such be P applied at A. At any point 0 apply two equal and opposite forces equal and parallel to P. One of these P2 forms a couple with P. The other is equal and parallel to P. The moment of the couple is perpendicular to this force. In this manner the points of application of all the forces may be brought to 0, where they can then be compounded into a single resultant E. For each force thus transferred there remains a couple, and all the couples may be compounded into a single one. There- fore all the forces applied to a rigid body may be replaced by a single force and a single couple. We may now state the following dualism existing between infinitesimal rotations and forces: Fig. 55. Infinitesimal rotations are slid- ing vectors. Forces applied to a rigid body are sliding vectors. When their axes intersect they are compounded by the vector law. Parallel infinitesimal rotations I Parallel forces have a resultant parallel and equal the center of mass of their points Two equal and opposite parallel rotations form a rotation - couple represented by its moment, a free vector. Every displacement of a rigid body may be reduced to a rotation and a rotation - couple. The theory of couples is due to their algebraic sum, placed at of application. Two equal and opposite parallel forces form a couple, re- presented by its moment, a free vector. Every combination of forces applied to a rigid body may be reduced to a force and force -couple. to Poinsot. 61. Variation of the Elements of the Reduction. Central Axis. Null - System. We have seen that any system of slide- vectors may be reduced to the resultant of a single vector and a single moment applied at any point whatever. We have now to examine the variation of the pair of elements, vector E and moment S, as we vary the point of application 0. E is invariable. As we move 0 along the line of E there is no change since E may be applied at any point of its axis, and S may be moved parallel to itself. If we WEBSTER, Dynamics. 14 210 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION, ft make the resolution at any other point, 0', the couple to be com- pounded with S at 0', is perpendicular to E and 00'7 so that if S has any component parallel to E it cannot be neutralized by the new couple. Accordingly in order that the couple may vanish for any point Of, the couple S must be perpendicular to E at all other points. As a change of 0 introduces only a component of S perpendicular to E, the com- ponent parallel to E is unchanged. Therefore the projection of S on E is the same for all points 0, Fig. 56. Although in general E and S have different directions, we may find points 0' for which they have the same direction. Let S and E include the angle & at 0. Resolve S into SQ = S cos # parallel to JR, and 8^ = 8 sin # perpendicular to .R. If we take 0' on a line perpendicular to SE at a distance ^ such that d - E = S sin # in the positive direction of translation corresponding to a rotation from E to •$, the component 5t will be neutralized, and we shall have at 0'} E and S' = $0 in the same direction. This property holds for all points on the line of E through 0'. This line is called Poinsot's central axis. In order to consider the resolution at any point 0 we may refer it to the central axis. Drop a per- pendicular from 0 (Fig. 58) on the central axis, and take this perpendicular for the axis of X, the central axis for the axis of Z. Fig. 57. Then as above 9) and if xyz are the coordinates of the end of S, we have -i /^\ a -r) _ E 10) z = /S0, ?/ — *ta> tan ^ = x -« -> and for any point on the line of /S, 11) — = ^~-> or ## = -^ y, y E x Eyj that is the line of 8 lies on a hyperbolic paraboloid. 61] POINSOT'S CENTRAL AXIS. 211 It is evident that if we slide the whole of Fig. 58 along or turn it around the central axis nothing is changed, consequently if we suppose the vector S laid off at every point of space 0, and con- sider the assemblage of couples thus formed, the assemblage re- mains unchanged if we rotate it about or slide it along the cen- tral axis. Every S is tangent to a cer- tain helix, or locus of a point which moves on a circular cylinder , . J Fig. 58. in a path making a constant angle with its generators (Fig. 59). This angle is less as the diameter of the cylinders is less, so that E 10) tan# = # , All these helices have however one constant in common, namely the distance traversed parallel to the central axis for each turn. If dr be the trans- lation for a rotation do, we have xdo *s: & Then 12) I> Fig. 59. is the traverse for each turn, and is called the pitch of the helix. Every helix lies on a ruled screw -surface, made by the revolution of a line perpendicular to the central axis, which slides along it a distance proportional to the angle of rotation, the pitch of the screw o being p = %TC --JJ- The lines of the assemblage of moments have every direction in space — there are a triple infinity of lines of the system (one for each point in space), but only a double infiriity of direc- tions — therefore every plane cutting all these lines has for rjjfcs points (a double infinity), every possible direction for S. Fpj? Qtye point only is this perpendicular to the plane. This point ,-ii 14* 212 VI- SYSTEMS OF VECTORS. DISTBJBUT. OF MASS. INSTANT. MOTION. the focus of the plane. Let the plane cut the central axis in A. Through A draw a plane perpendicular to the central axis, intersecting the given plane in AO. As we go along the line AO, S turns about it, and for one point has the direction of the normal to the given plane. Accordingly to every point in space there corre- sponds one plane, and to every plane one point. The correspondence was discovered by Chasles, and the system of points and planes was called a Null- Fig. 60. System by Mobius. 62. Vector -cross. Besides the reduction to the screw -type we may reduce the system of vectors to two vectors not lying in the same plane, without a couple. This reduction may be made in an infinite number of ways, and the line of one of the vectors may Fig. 61. be given. Let AB (Fig. 61) be the given line. At any point 0 on AB let E be the resultant vector, S the resultant couple. (E and 8 will not in general lie in a plane with AB.) At 0 pass a plane perpendicular to S, intersecting the plane of E and AB in OP. Resolve 8 into the pair of vectors OP and CQ so taken that the resultant of E and OP shall lie in AB. The length of OP is thus determined, and the distance between its line and that of CQ is determined by S. Thus the line AB determines the line CQ. 61, 62] NULL-SYSTEM. VECTOR-CROSS. 213 The lengths OB and CQ are determined as soon as the line AB is given. Two such non- parallel and non-coplanar vectors OB, CQ will be termed a vector -cross. The crossing will degenerate to intersection only when S = 0 and to parallelism when R = 0. As any line may be taken for AB, and as there are a quadruple infinity of lines in space, there are a quadruple infinity of vector- crosses. They all possess a property in common, namely, that the tetrahedron formed by joining the four ends of a vector- cross has a constant volume. Let OB, CQ (Fig. 61) be the vector- cross, and let us reverse the preceding resolution. The volume of a tetrahedron is equal to one -third the product of its altitude by the area of its base. The area of the base OCQ is one-half the moment of CQ about 0, or --- S, while the altitude is the projection of OB on the perpendicular to OCQ, that is, on S. But since BE is parallel to the plane OCQ, OR has the same projection on S as OB, namely .Rcos'fr, consequently v= 4^cos#-4-#=4-^£cos#. O £4 U But by 8), $COS# = SQ, therefore 13) V=±RSQ. This theorem is due to Chasles. Corresponding lines of vector -crosses possess a remarkable relation to the null-system. Let AB and CQ (Fig. 62) be the two lines of the vector- cross. Through CQ pass any plane, cutting AB in 0. The moment of CQ is perpendicular to the plane OCQ, and the other vector has no moment about 0, since it passes through it. Accord- ingly 0 is the focus of the plane OCQ. Thus, if a plane turns about a line, its focus traverses another line, and these two conjugate lines are lines of a vector- cross. We have here shown the intermediate nature of a line between a point and a plane, in the dual role as generated by the motion of a point and by the rotation of a plane. In the first relation the line is spoken of as a ray, in the second as an axis. If two conjugate lines are at right angles, pass a plane through one, AB, perpendicular to the other, CD (Fig. 63). By the preceding 214 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION. theorem, the point of intersection of the plane with CD is the focus of the plane. Resolving at any point P in AB, the moment of OD, being perpendicular to OD E and OP, lies in the plane OAR That line in a plane which has the property that for all its points the resultant moment lies in the plane is called the characteristic of the plane, or of its focus. Its distance OX = d from the focus is such that1) 14) dEsmfr = S. The line OX, of length S Rsinfr' d S is perpendicular to the plane of E anc( S, and drawn toward the side corresponding to the motion of a right-handed screw when rotated in the direction from E to S. If we should go from 0 in the direction OX a distance d' = — ^ — we should reach the central axis, and 15) dd' = W' 63. Complex of Double -lines. If a plane 1 pass through the pole of a plane 2, then the plane 2 passes through the pole of the plane 1. Let P (Fig. 64) be the pole of the plane 1, and let PO be any line in 1 through P. The moment of E about 0 is perpendicular to PO, and so is S, hence so is their resultant. Thus the moment at 0 is per- pendicular to OP, and the polar plane of 0 contains the line OP, that is, if 0, the pole of 2 lies in 1, then P, the pole of 1 lies in 2. In this case the two poles lie in the line of intersection of the planes, and we see that if a plane turns about a line through its pole, its pole traverses that line. Such a double line is conjugate Fig. 64. 1) For the component in AS, E sin #, has the moment S about 0. 62, 63] COMPLEX. FLICKER'S COORDINATES. 215 to itself. The necessary and sufficient condition that a line is self- conjugate is that the pole (focus) of a plane through the line falls in the line. For then as the plane rotates about the line as an axis, the focus describes the line as a ray. Hence the double lines lying in a particular plane all pass through the pole of that plane , and conversely, all the double lines passing through a point lie in the polar plane of the point. Such a system of lines is called by Pliicker a line complex of the first degree. There are in all a double infinity of lines passing through any point in space, but of these only a single infinity belong to the complex. Therefore lines belonging to the complex have one less degrees of freedom than lines in general, or a complex contains a triple infinity of lines. A complex may be represented analytically by a single relation between the four para- meters determining a line. If we mark off on a line any length B, and give its projections on a set of rectangular axes X, Y, Z, and the projections L, M, N of its moment about an origin 0, the line is completely determined. For its direction is given and giving the moment S = T/L2-\- M2 -\- N2 gives the plane through 0 containing R, and the distance from the line, if the length of R is given, but this is given by R = ]/X2 + Y2 + Z2. As the determination of the line is independent of the length of JR, the ratios of the six quantities determine the line. But these five ratios are not independent, for since by § 5, 12), 16) we have the identical relation, 17) LX + MY+ NZ=0, expressing the fundamental property that the moment of a vector is perpendicular to it. The coordinates LMNXYZ are known as Pliicker's line -coordinates. Thus there remain four independent quantities to determine a line. A relation between these denotes a complex, and in particular a linear relation, 18) aX + IY + cZ+dL + eM+fN= 0, denotes a complex of the first degree. Since the double lines of the null- system are the loci of points which are the poles of planes containing the double -lines, at every point of a double -line the resultant moment is perpendicular to it, 216 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. or double lines are lines of no moment. In the kinematical applica- tion, points on a double -line experience no translation along it. If a double -line cuts one of a pair of conjugate lines, it cuts the other. Let PQ be a double -line cutting the line AS. Then the pole of the plane BPQ lies in the line conjugate to AS. But since PQ is a double-line, the pole of BPQ lies on PQ. Hence PQ cuts the conjugate to AB. Conversely, every line cutting two conjugates is a double -line. The complex of double -lines is symmetrical with respect to the central axis. Let AB (Fig. 65) be a line of the complex, and let OX be the common perpendicular to it and the central axis. Now AB is perpen- dicular to the moment S at X, but S is perpendicular to OX, and the distance o OX is d = -jr tan #-. If

* = sin 2a*> cos cc sin2 = ^ cos2 «2 + py sin2 as six equations to determine px, py, sl} s2, aly cc2. We have by elimination £2 — 2*1 = — . (sin 2a2 — sin2aj) or using the first two equations, 27) A = -J^S cos (oj + ai) sin y, P —px 28) Pv—px . = -^ — - sin (cc2 -f KI) cos (K2 — «j ~~ Pi = P* (cos2 a2 — cos2 aj) + py (sin2 a2 — sin = (Px —Py) (cos2 a2 — cos2 ^) = (Px -Py} sin (ag + ai) sin («2 - ^); Pz-Pi = (Px - Py) sin (ag + «0 sin y, + (Px -Py) (cos2 cc2 - sin2 aj = Pz+Py "f (P* -jPy) COS (^ -f C^2) COS y. From 27) and 29) we obtain (_pj -jpj2 = (py -px}2 sm2y; 31) ^-^ smy From 27) and 30), From 29) and 31), - **~Pl COS (^ + Oj) = :?= 32) tan (^ - 220 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. From 32) with a2 — a± = y, Since the cylindroid is thus determined by 31) and 33), a twist about PI can be resolved into a twist about px and one about py. A twist about p2 can be likewise resolved. The two components about px add together, so do those about py, and since the resultant of any twists about px and py lies on the cylindroid , the resultant of p1 and p2 does. Its direction can be found, since the amplitudes E of the two twists about p^p2 compound by the parallelogram law, hence the angle made by the resultant with the axes is known. The pitch is then found from the pitch -conic. 65. Work of Wrench in Producing a Twist. Let us find an expression for the work done during a twist of amplitude Rk about a screw of pitch pk by a wrench of intensity Ef about another screw of pitch pf. We already know the work done by a force in a translation, namely, it is equal to the product of the magnitudes by the cosine of the included angle. If the force is Rf and the translation (rotation -couple) is $#, we have W = EfSkcos(EfSk). Notice that the vector of one system is multiplied by the vector- couple in the other. We can find the work done by the force -couple in a rotation about its axis. Apply the couple so that one of its members P passes through the axis of rotation. In a rotation this member does no work, for its point of application is at rest, while that of the other member Q moves in a rotation a distance da, where d is the arm of the couple. Accordingly the work is W=Pdc3 which is equal to the product of the twist by the moment of the couple. Here again we multiply the vector of one system by the vector- couple of the other. If the axis of rotation is perpendicular to the axis of the couple, the motion is perpendicular to the force, and no work is done. Hence we must take the resolved part of the couple on the vector, as before. We can now find the work of a wrench during a twist. The work of the force in the displacement Sk is jR/^cosa, a being the angle between the two screws. The work of the couple Sf = -^— Ef in the rotation Ek is Pf SfEk cos a = ~ EfEk cos cc. 65, 66] RECIPROCAL SCREWS. But when Ef is changed to the origin of Rk it gives rise to a moment perpendicular to Ef equal to Efd, d being the perpendicular distance between the screws. This moment therefore makes with Ek the angle a -j- -g-> and the work done by it in the rotation Ek is dEfEk cos a -f y = — dEfEk sin «. Thus the whole work is 34) W = EfEk ( cos a - d sin a . It is symmetrical with respect to both screws, hence the wrench and twist might have been interchanged. The geometrical quantity in parentheses is called the virtual coefficient of the two screws, and if it vanishes no work is done, that is, a body free to twist only about a particular screw is in equilibrium under a wrench about another screw if the virtual coef- ficient of the two screws is zero. The two screws are then said to be reciprocal. 66. Analytical Representation. Line Coordinates. In Pliicker's line coordinates referred to any origin, since each component of vector does work on the corresponding component of couple in the other system, 35) W = XfLk + YfMk -f ZfNk + LfXk + MfYk + NfZk. If a screw is reciprocal to two screws on a cylindroid, it is evidently reciprocal to all the screws on it. For two screws to be reciprocal, the condition is, §36) X,L2 + Y,M, + Z,N2 + L,X2 + M,Y2 + N,Z, = 0. If the coordinates of one of the screws be constant, while those of the other be variable, this is the equation 18) of a complex of the first degree, so that all the screws reciprocal to a given screw form such a complex. Since between the six coordinates X1Y1Z1L1M1N1 there is always the identical relation X, A +Y1M1 + Z1N1 = 0, we may always make them satisfy five equations like the above, that is, we may always find a screw reciprocal to five arbitrarily given screws. Suppose the coordinates of the system of vectors for an origin 0 are XYZLMN, being the projections of E and 8 at 0. Let 222 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. XYZ L'M'N' denote the same for a point 0' whose coordinates Then .37) M=M' + N=N' + xY-yX. In order that the point O1 may lie on the central axis, the direction of resultant and couple must coincide , or *L = M- = *[L X~ Y ~ Z ' hence the equations of the central axis in Cartesian coordinates are L-yZ-\-zY_ M-zX + xZ N-xY+yX ~^~ ~Y~ ~W~ The equation of the focal or polar plane to a point x' yr zr is, since it is perpendicular to L'M'N', 39) (x-x')L' + (y-y')M' + (2-z')N' = 0 and inserting the values of L'M'N', or, more symmetrically arranged, 40) L(x-x') + M(y-y') + JV (*-*') + Y(xe' - 0x') + Z(yx' - xy') = 0. This equation is symmetrical with respect to xyg, x'y'z', hence if x' y' z' is fixed, xyg is on its polar plane, or if xyz is considered fixed, x' y' &' is on its polar plane, showing the reciprocal relation of pole and polar. If the vector system is to reduce to a single vector, the resultant and couple at any point must be perpendicular, or 41) LX + MY+NZ = 0. We must have in general, at any point, $cos# = $0 that is, 42) and the pitch p is given by 4QN P 3, LX+MY+NZ 2^t == ~R = ~"X8+ r*-f Z2 The volume of the tetrahedron on a vector -cross is 44) I ES, = \(LX+ MY+ NZ), and this, like the last expression, is independent of the choice of origin or axes, that is, is an invariant. 66] REPRESENTATION BY LINE -COORDINATES. 223 Suppose that the two members of a vector -cross have Pliicker's coordinates X^Z^M^ and X2Y2Z2L2M2N2 with the identical relations, Their resultant has components and the volume of the tetrahedron is one sixth of LX + MY + NZ = which in virtue of the two identities is 45) L,X2 + M,Y2 + N,Z2 4 L2X, + M2Y, + N&. If any two lines are given by their Pliicker's coordinates, the condition that they shall intersect is that the above expression shall vanish. We may now find the equation of the complex of double -lines. We have seen that every line meeting two conjugate lines is a double -line. Let the coordinates of the two conjugate lines be X1 . . . Nlf X2 . . . N2, satisfying the conditions 46) ^ Z,, where Xf^Y^Z^L^M^N^ define the vector -system. Let the coordinates of a double -line be XYZLMN. The condition that it meets the line X^Z^M^ is LtX + MJT+ N,Z -f X,L + Y, M + &N= 0, and that it meets X2Y2Z2L2M2N2) L2X+M2Y+ N2Z+ X2L + Y2M + Z2N = 0. Adding these equations, and using the conditions 46) we obtain, 47) L0X+M0Y+Nl>Z+Xi>L + as the equation of the complex, that is, any linear relation in Pliicker's coordinates represents a linear complex, as stated in § 63. It is to be noticed that the equation 47) does not signify that the line XYZLMN cuts the line XoroZ0L0J!f0JV"0 unless the latter 224 VI. SYSTEMS OF VECTORS. DISTftlBUT. OF MASS. INSTANT. MOTION. are the coordinates of a line (not of a general system of vectors), that is fulfill the relation If they do, then every line of the complex cuts the line X0Y0,ZOJL0Jf0./V0, and the equation may be considered the equation in Pliicker's co- ordinates of the line XQYQZQL0MoNQ (see Clebsch, Geometric, Yol. II, p. 51). For further information on this subject, the reader may consult, Ball, Theory of 67. Momentum Screw. Dynamics. The previous sections have shown how to combine systems of vectors having different points of application, provided they are unchanged if slid along their lines of direction. As one particular system to which the operation is applicable we have had the various rotation - velocities of a rigid body, as another, sets of forces applied to a rigid body. That these vectors are susceptible of such treatment may be considered as due to properties of a rigid body, rather than of the vectors themselves. We have however previously dealt with two other sorts of vectors which may be dealt with in similar fashion, on account of their physical nature, and independently of the nature of the bodies in which their points of application lie. By means of these properties we are able to connect the kinematical aspect of a rigid body, as expressed by its instantaneous screw motion, with its dynamical aspect, as expressed by an applied wrench about another screw. If for each point of the system we consider the momentum, whose six coordinates (one being redundant), in the sense of § 66 are, mvx, mvy, mvz, m(yvz — 2Vy), m(zvx — xvz), m(xvy — yvx), and form the general resultant, we obtain a system whose co- ordinates are Mx = Zmvx, Hx = Zm (yvz — zvy}, 48) My = 2mvy, Hy = 2m (zvx — xvz}, Mz = 2mvZJ Hz = Urn (xvy — yvx), which represent the momentum of the system, the three projections Mx, My, MS, being more particularly characterized as the linear momentum, the others Hx, Hy, Hz, as the angular momentum or moment of momentum with respect to the origin. We have now by the general principles of dynamics, as shown in § 32, 45), § 33, 61), the fact that the time -derivatives of these six components of momentum are equal to the corresponding com- ponents of the resultant wrench, 66, 67, 68] DYNAMICS. IMPULSIVE WRENCH. 225 applied to the system. That is, dM.. dM 49) dH dH Integrating these equations with respect to the time, we may, in the sense of § 27, call the momentum the impulsive ^vrench of the system. Physically, then, the momentum that a system possesses at any instant is equal to the impulsive wrench necessary to suddenly communicate to it when at rest the velocity -system that it actually possesses. As a prelude to the dynamics of a rigid body we must accordingly study the properties of the momentum or impulsive wrench of a body possessing a given instantaneous twist- velocity. All the systems of vectors in question may be reduced to the screw type, and their respective screws are in general all different. Thus we may speak of the instantaneous velocity -screw and instan- taneous axis, the momentum screw, and the force -screw. As the body moves, all these screws change both their pitch and position in the body, describing ruled surfaces both in the body and in space. The integration of the differential equations of motion 49) will enable us to find these surfaces. The kinematical description of the motion will be complete if we know the two ruled surfaces described in space and in the body by the instantaneous axis, together with such data as will give their mutual relations at each instant of time. 68. Momentum of Rigid Body. The properties of the momentum of a rigid body are conveniently investigated by the consideration of the velocity -system as an instantaneous screw -motion. Let F be the velocity of translation, and co of rotation. Then every particle of mass m has one component of momentum parallel to the axis of the instantaneous twist (which we will take for Z-axis), equal to mvz = mV and the resultant for all is 51) M2 = ZmV=VZm = MV, WEBSTER, Dynamics. 15 226 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. where M is the total mass of the body. By the construction of §§ 57, 59 the resultant of parallel vectors P and Q is applied at the center of mass of masses proportional to P and Q placed at their points of application. Consequently the various elements being pro- portional to the masses m, this component of the momentum is applied at the center of mass of the body. There remains the component of momentum perpendicular to the instan- taneous axes. Let OZ (Fig. 68) be the instan- taneous axis, and let r be the perpendicular distance from it of any point ^Pf and let the angle made by r with the X-axis be #. Now P is moving parallel Fig. 68. to the XY- plane with the velocity v = r& perpendic- ular to r, so that the projections of this velocity are vx = — v sin & = — G)r sin # = — v vcos& or cos -9- ox. Thence we obtain the components of momentum Mx = — 2 may = — oZmy = — May, 52) K/r _ where x, y are the coordinates of the center of mass. The resultant momentum is accordingly equal and parallel to the momentum that the body would have if concentrated at the center of mass, but its point of application is different, for the components Mx, My are not applied at the center of mass, inasmuch as their elements are pro- portional, not to m but to my and mx. The magnitude of the resultant momentum being given by Mx, My, M~, we may find its axis by obtaining its three remaining coordinates, representing the angular momentum. We have 53) Hy = Hz = MVy — = — MVx = &2m (x2 -f- y2) = 68] MOMENTS AND PRODUCTS OF INERTIA. 227 Of these the terms in V are the moments of the vector M V in the direction of the Z-axis applied at the center of mass, while the terms in o are applied elsewhere. The equations of the central axis of momentum are, by § 66, 38), x' y! z' being the running coordinates, Hx or inserting the values, •> r\ — (o Zmxz — y' MV -f z' Miax - M.V~x — a Emyz + z' Mo>y -f x' MV M&HC _ co Zm (x* -f y2) — x' Max — y' May MV This does not pass through the center of mass unless, putting x! = x, y' = y, z' =~z, *K\ — aZmxz -f Mazx _ — oEmyz -f M&yz — M.(oy « _ co Zm (xz -f 2/2) — M a) (x MV We see that the resultant momentum involves the various sums ZmXj Zmy, Zmxz, 2myz, Hmr2, the axis of Z being the instantaneous axis. These sums are constants for the rigid body, depending on the distribution of mass in it. The first two represent the mass of the body multiplied by the coordinates of the center of mass. The last represents the sum of the mass of each particle multiplied by the square of its distance from the Z-axis, and is what has been called the moment of inertia of the body with respect to that axis. We are thus led to consider the sums A = Zm (y2 4- A B = 2m (z2 -f #2), C = Zm (x2 + y2), D = Zmyz, E = Zmzx, F = Hmxy. Of these the last three, D, E, F, are termed the products of inertia with respect to the respective pairs of axes. In the case of a continuous distribution of mass, we must divide the body up into infinitesimal elements of volume dt, and if the density is 0, the element of mass is dm = gdt and the six sums become the definite integrals 15 228 VI. SYSTEMS OP VECTORS. DISTBIBUT. OF MASS. INSTANT. MOTION. The determination of these quantities is then, like that of centers of mass, a subject belonging to the integral calculus. The six constants A, B, C, D, E, F together with the mass M and coordinates x, ?7, 0", of the center of mass, completely characterize the body for dynamical purposes, since when we know their values and the instantaneous twist, the momentum or impulsive wrench is completely given. The body may therefore be replaced by any other having the same mass, center of mass, and moments and products of inertia, and the new body will, when acted upon by the same forces, describe the same motion. 69. Centrifugal Forces. As the body moves, its different parts exercise forces of inertia upon each other, so that there is a resultant tending to change the instantaneous screw in the body. Let us suppose the translation to vanish, and examine the kinetic reactions developed by the rotation, or the centrifugal forces. The instantaneous axis being again taken as the axis of Z, a particle P experiences the centripetal acceleration — = ro2 towards the axis, and the centrifugal force is Jtc = mrs? (see p. 119) directed along the radius r from the axis OZ, and having the projections Zc= 0. For the moment of the centrifugal force we have Lc = yZc — zYc = — my #«2, 58) Mc = 2Xc—xZc = rnxsG?) Nc = xYc-yXc = 0 , so that the coordinates of the resultant centrifugal force and couple are Xc= tfZmx = a* MX, 59) c Lc = — & Nc= 0 . Thus the centrifugal force is equal and parallel to that of a mass placed at the center of mass, and moving as the latter point does. It vanishes when the center of mass lies in the axis. The system of centrifugal forces is however, as in the case of the 68, 69, 70, 71] CENTRIFUGAL FORCES. 229 momentum, not to be replaced by a single force placed at the center of mass, for the couple is not equal to what its value would be in that case, unless — = -|-- If the center of mass lies on the axis, although the centrifugal force Rc vanishes, the centrifugal couple 8C does not, unless D = E = 0. The centrifugal forces then in general tend to change the instan- taneous twist, unless the axis of the latter passes through the center of mass, and for it D = E = 0. Such axes are called principal axes of inertia of the body at the center of mass, and are characterized by the property that if the body be moving with an instantaneous twist about such an axis, it will remain twisting about it, unless acted on by external forces. In order to examine the effect of the distribution of mass of the body, we are led to interrupt the con- sideration of dynamics in order to consider the purely geometrical relations among moments and products of inertia. 7O. Moments of Inertia. Parallel Axes. Consider the moments of inertia of a body about two parallel axes. Let the perpendicular distances from a point P on the two axes be p± and p2 and let the distance apart of the axes be d. Let A and B (Fig. 69) be the inter- sections of the axes with the plane of p± and p2. If we take AS for the X-axis, A for origin, we have rig> 69. P^ = Pi* + d2- ^d cos (p^x), 60) Zmppr+2Era + 2Fap = F'(a,p,>y*). The six quantities, A, B, C, A', B\ Cr, being sums of squares, are all positive. We have evidently 67) 71] ELLIPSOID OF INERTIA. so that the sum of any two of the moments A, B, C is greater than the third. If we lay off on the axis a length p and call the coordinates of the point P so determined |, y, g we have If we now make the length of OP vary in such a manner that Q2 Q = 1, we obtain for the coordinates of P the equation 69) *"(!, ,, g) -4'{» + JBV + I > c. The equation now is /y.2 ,,.2 ,.,2 fL i y ___ — — \ a? + 52 c2 ~ The surface is cut by the XY- plane in the ellipse whose semi -axes are a and 5, and whose foci are at distances from the center /a2 — b2 = /% — a2 on the X-axis. The section by the ZX~ plane is the hyperbola with semi- axes a, c, and foci at distances "j/a2 -}- c2 = y^ — aB on the X-axis. The section by the YZ- plane is the hyperbola, with semi -axes fc, c, and foci at distances "J/62 -}- c2 = j/a2 — a3 on . the F-axis. The surface is an hyperboloid of one sheet. 2°. Let two of the constants a19 a2, a3, be negative, say The equation is 72, 73] ELLIPSOIDAL COORDINATES. 235 The sections by the coordinate planes and their focal distances are XY - = 1 Hyperbola, i/o^+T2 = V^~^ on X-axis, ^ - - ~ = 1 Hyperbola, ]/a2T72 = Va^a, on X-axis, * -f = - 1 Imaginary Ellipse, /-(62_c2) = Va^a*. The surface is an hyperboloid of two sheets. 3°. If al9 a2, as are all positive, the sections are all ellipses, and the surface is an ellipsoid. In all three cases, the squares of the focal distances are the differences of the constants %, &2, a3. Con- sequently if we add to the three the same number, we get a surface whose principal sections have the same foci as before, or a surface confocal with the original. Accordingly ^2 _L y* + g2 _ i - represents a quadric confocal with the ellipsoid tf + V + ^ = 1? for any real value of Q. If a > 6 > c and p > — c2, the surface is an ellipsoid. If — c2 > Q > — &2, the surface is an hyperboloid of one sheet, and if - Z>2 > Q > — a?j an hyperboloid of two sheets. If Q < — a2, the surface is imaginary. Suppose we attempt to pass through a given point x, y, s, a quadric confocal with the ellipsoid Its equation is 83), where the parameter Q is to be determined. Clearing of fractions, the equation is 84) ft?) EH (o« + 9) (6* + 9) (o2 + 9) - a? (V + 9) (c2 + a cubic in p. But this is easily shown to have three real roots. Putting successively p equal to oo, — c2, — V, — a2 and observing signs of f(9), P= oo, f(9) = °° 236 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. Call the roots in order of magnitude A, ^, v. The changes of sign above show that I lies in the interval A > — c2 necessary in order that the surface shall be an ellipsoid, ^ in the interval - c2 > ^ > — fr2 that it may be an hyperboloid of one sheet, and v in the interval — fr2 > v > — a? that it may be an hyperboloid of two sheets. There pass therefore through every point in space one surface of each of the three kinds. If we call 85) F(l, x, y, ,) = -^ + -^ + -^ - 1, the equation F=Q defines A as a function of x,y,8. The normal to the surface A (xyz) = const, has direction cosines proportional to til dl dl o — * o — y ^^' ex cy cz Now since identically _F = 0, differentiating totally, 3F , dF , dF , . 0.F , . and we have dF for the required partial derivative of A with respect to #, when and 0 are constant. Therefore 31 2x I i x* * z* 2 a; Similarly 86) dz The sum of the squares of the derivatives being called /^2, we have 4 Now the direction cosines of the normal to the surface A = const, are cos (n*x) = T- o— = ± 88) cos (n^y) = + cos 73, 74] COORDINATE SURFACES ORTHOGONAL. 237 Similarly for the normals to the surface p = const,, cos 89) cos (n^y) = cos (nuz) = The angle between the normals to A and /A is given by f #2 w2 ^ ^ \(a2~F ^) l^2-}- ^) (^24~ ^) (^2~F f1-) i__ Now by subtracting from the equation _x< the equation ,2 W2 -f we obtain / ^1 «2 J_ 1 «2J_ „ f I y \ 13 I 1 7,2 I ., I I or 2/2 f,v,2 »»2 «2 \ (a« + ao(a* + fO + (&2 + ^)(&2 + ^) + (C2 + ^)(c2 + ")J ^ Accordingly, unless h = ii, cos (n^n/^) = 0, and the two normals .are at right angles. Similarly for the other pairs of surfaces. Accord- ingly the three surfaces of the confocal system passing through any point cut each other at right angles. If we give the values of A, ^, v we determine completely the ellipsoid and two hyperboloids, and hence the point of intersection x, y, 8. To be sure there are the seven symmetrical points in the other quadrants which have the same values of A, ft, v, but if we specify which quadrant is to be considered this will cause no ambiguity. Thus the point is specified by the three quantities >L, ^, v, which are called the ellipsoidal or elliptic coordinates of the point. 74. Axes of Inertia at Various Points. Let K = MW be the moment of inertia about an axis whose direction cosines are «, ft y, at a point 0 whose coordinates with respect to the principal axes at the center of mass G- are xyz. Let p be the distance of the axis at 0 from a parallel axis through 6r, and q the distance of the 238 VI. SYSTEMS OF VECTOES. DISTBJBUT. OF MASS. INSTANT. MOTION. foot of the perpendicular from 6r. Then by the two theorems of § 70 and § 71, K = Ace2 + BP* + Cf + fc2 = a2a2 + 62/32 + c2y2 -f p\ Now tf = r2 - (f = r2 - (ax + /3y + r*)2, 94) &2 = ^2 + £202 + c2;;2 _j_ ^2 _ q2 In order to find the principal axes at 0 we must make this a maximum or minimum with respect to a, ft y subject to the condition, a2 + /32 -f r2 = I- Multiplying this by a constant tf, subtracting from 94), and diffe- rentiating 95) {^2-<5( Multiplying these equations respectively by «, ft y and adding, a2 a2 + tfp + cV - g (aa; + fty + ye) - (3 = 0, fc2 _ r2 _ 6 = 0< Thus tf is determined as 97) 6 = W- r2. Inserting this value in 95) we have (a2 + r2 - F) a = qx, 98) (&2 + r2 - F) /3 = 2y, (c2 +ra-fc8)y =g^. Multiplying these equations respectively by x y e and adding, we get, since q divides out, " " 4- z* - 1 -- - If we now put r2 — &2 = ^), this is the same cubic as 83) to determine 9, and gives three real roots for &2, 7. 2 _ 0.2 1 /fcj — T ~ A , 74] DISTRIBUTION OF PRINCIPAL AXES. 239 The direction cosines are then given, according to 95) and 98), by f («2 + r*- V) - f (V + r* - V) = f (c2 + r* - kf), 100) S(^ + ^-V) = (62 + ^-V) = f(c2 + »-2-V), that is etc. Hence the principal axes of inertia at any point 0 are normal to the three surfaces through 0 confocal with the ellipsoid of gyration at the center of mass. This theorem is due to Binet. Since A, > /z, > v, the least moment of inertia is about the normal to the ellipsoid, the greatest about the two -sheeted hyperboloid, and the mean about the normal to the one- sheeted hyperboloid. We have It* + ^ + fa* = 3r2 - (I + 11 + *,). But the sum of the three roots is the negative of the coefficient of 02 in the cubic 83), I -f 11 + v = x* + f ' + z2 - (a2 + V + c2), 101) ^2 + ^22 + ^32 = 2r2 + a2 + 62 + c2. Thus the sum of the principal moments of inertia is the same for all points lying at equal distances from the center of mass. It is now easy to see that any given line is a principal axis for only one of its points, unless it passes through the center of mass, when it is such for all of its points. It is also evident that not every line in space can be a principal axis. If the central ellipsoid of gyration is a sphere, all the ellipsoids of the confocal system are spheres, and all the hyperboloids cones. Every ellipsoid of inertia is a prolate ellipsoid of revolution, with its axis passing through the center of mass. If the central ellipsoid has two equal axes, the ellipsoids of inertia for points on the axis of revolution are also of revolution. If the distance of a point on this line from the center of mass is d> and the moment of inertia about it is M Jc^ 240 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION. If & < & there are two points for which the ellipsoids of inertia are spheres, namely where d = + ]/a2 — fr2. This is the only case, except the above, where there are spheres. If we look for ellipsoids of revolution in the general case when «, &, c are unequal, we must distinguish between prolate and oblate ellipsoids of gyration. 1°. Prolate. The two equal radii of gyration are the two smaller \ and \ . For these to be equal, we must have k = p. But as A and ^ are separated by — c2, if they are equal they must be equal to — c2. In this case the axis of the ellipsoid and one -sheeted hyperboloid are both zero, and the ellipsoid becomes the elliptical disk with axes }/a2 — c2, }/&2 — c2, forming part of the XF-plane, and the hyperboloid all the rest of the XT- plane. Points lying on both surfaces lie on the ellipse whose axes are "/a2 — c2, "J/fr2 — c2, which passes through the four foci of the system lying on the X- and F-axes, and is accordingly called the focal ellipse of the confocal system. (We saw by 92] that if A = fi the two surfaces were not necessarily orthogonal.) All points lying on this ellipse have prolate ellipsoids of gyration, the axes of rotation lying in the plane of the ellipse. 2°. Oblate ellipsoids of gyration. In this case we have The Y"-axes of the two hyperboloids now vanish. That of one sheet becomes the part of the XZ- plane lying within the hyperbola and that of two sheets the remaining parts. The points common to both are those lying on the hyperbola, whose axes are "j/a2 — &2, ~|/ft2 — c2 and which passes through the remaining two foci of the system, and is called the focal hyperbola. The axes of revolution of the ellipsoids of gyration lie in the plane of the hyperbola. 75. Calculation of Moments of Inertia. In the case of a continuous solid, the sums all become definite integrals, as stated in § 68. All the preceding theorems of course are unaltered. If the body is homogeneous all the integrals are proportional to the density. Since the mass is likewise, the radii of gyration are in- dependent of the density. We will therefore put Q = 1. 74, 75] CALCULATION OF MOMENTS OF INERTIA. 241 Rectangular Parallelepiped, of dimensions 2 a, 2b, 2c. a b c A' —'a —b —c a b c '-///• — a —b —c a b c "///• — a — b — c a b c - c r r C' = I I €/ fj *J — a — 6 — c 102) B = Cr + A' --= y abc (c2 + a2), Cr = ^f + i?' = |-a&c Thus the radii of gyration a07 60, c0 are mo\ 103) aQ= Sphere, with radius E. A' = CCCx2dxdydz, Bf =j f fodxdydz, C1 = the limits of integration being given by the inequality #a-f y2+ 02c, B!=*-xb*ca, C' = ~xc*ab. lo lo lo 105) B=C' + A' = 106) Q Thin Circular Disk normal to Z-axis. , C' = 0, 107) A = B = - The moment about the normal to the disk is double that about a diameter. Circular Cylinder of radius JR, length 2L The moment about the axis of rotation, is? as for the disk, C = ~MR2, A = B' = ± C'= fx -i 108) A = B 75, 76] We have MOVING AXES. if 243 Then the cylinder is dynamically equivalent to a sphere, as is also the case for a cube. These examples furnish the means of treating the cases that usually appear in practice. 76. Analytical Treatment of Kinematics of a Rigid System. Moving Axes. In §§ 55 — 57 we have treated the general motion of a rigid system, from the purely geometrical point of view, without analysis. We shall now give the analytical treatment of the same subject. Let us refer the position of a point in the system to two different sets of coordinates. Let %', y\ #' be its co- ordinates with respect to a set of axes fixed in space, and let x, y, z be its coordinates with respect to a set of axes moving in any manner. The position of the moving axes is defined by the position of their origin, whose coordinates referred to the fixed axes are |, iq, g, and by the nine direction cosines of one set of axes with respect to the other. Let these be given by the following table X Y Z The equations for- the transformation of coordinates are then, 109) Since c^, a?, cc3 are the direction cosines of the X-axis with respect to X', Y', Zf, we have 110) and similarly 110) a* + a,2 + ^ = 1, 244 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. Since the axes Y, Z are perpendicular, their direction cosines satisfy the conditions, ill) and similarly, i j . Thus the nine cosines are not independent, but, satisfying six conditions, may be expressed in terms of three parameters. These, with the three |, ^, g, show the six degrees of freedom possessed by a rigid system. By interchanging the roles of the axes, and considering the direction cosines of X', T, Z' with respect to X, Y, Z we find the equivalent conditions "I' + Ai' + yi'-l, 112) «22 + /322 + r22 = i, 113) If we now differentiate the first of equations 109), supposing y> % to be constant, we obtain for the components in the directions of the fixed axes of the velocity of a point fixed to the moving axes. Let us now resolve the velocity in the direction which is at a given instant that of one of the moving axes. To resolve in the direction of the X-axis we have 115) Vx = tfX + C^Vy' + CC3V,' = ^ -^ + CC2 ~ + ttj ^ The coefficient of x in this expression is the derivative of the left- hand member of the first of equations 110), and is accordingly equal 76] TRANSLATION AND ROTATION. 245 to zero. If we now denote the coefficients of y and z by single letters, and compare them with the results of differentiating equa- tions 111), writing we obtain ** = a'§ + «• Tt + «» Ji These equations express the fact that the velocity of a point attached to the moving axes is the resultant of two vectors, one of which, F, is the same for all points of the system, being independent of x, y, 0, and having the components in the direction of xr, y', z' equal *° dt' ~di' dt9 anc* *n ^e Direction °f %> y> z, equal to TT dt- dr\ . d£ r*=tti-dt+a^t + K*dt' 118) ^-A + + ' This part of the motion is accordingly a translation. The other part of the velocity, whose components in the direc- tion of the instantaneous positions of the X, Y, Z-axes are given by vx = qz- ry, being the vector product of a vector o> whose components are p, q, r, and of the position vector p of the point, is perpendicular to both these vectors and is in magnitude equal to co Q sin (ra Q). It accord- ingly represents a motion due to a rotation of the body with angular velocity CD about an axis in the direction of the vector co. Thus we have an analytical demonstration of the vector nature of angular velocity. If we take as a position of the fixed axes one which 246 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. coincides with that of the moving axes at some particular instant of time, the direction cosines vanish with the exception of a1} fl.2, ys, which are equal to unity. We then have -dt d dt --dt- dt But since /33 = cos (?/£'), y2 = cos (#?/), we have on differentiating d&* • f i\ d(iiz') dy« . f ,N d(zy') -/f - - sm(^ ) T^fr' w=~?* ('y ) r&-> and since sin(^) = sin(^)=l, f = Thus it is clearly seen that p, q, r are angular velocities, being the rates of increase of the angles #«/', x#', yx', or in other words, the angular velocities with which the moving axes X, Y, Z are turning about each other. It is to be noticed that py q, r, though angular velocities, are not time -derivatives of any functions of the coordinates, which might be taken for three generalized Lagrangian coordinates q. They are merely linear functions of the derivatives of the nine cosines, which latter may themselves be expressed in terms of three g's. If we seek to find those points of the body whose actual velocity is a minimum, we must differentiate the quantity, 121) v* = (Vx+qz-ryy + (Vy + rx-p^ + (Vz + py-qxy with respect to x9 y, z, and equate the derivatives to zero. We thus obtain r(Vy + rx— pz) — q(Vz+py — qx) = 0, 122) p(Vz +py - qx) -r(Vx + qz- ry} = 0, q ( Vx + qz — r y) — p ( Vy + rx - pg) = 0, which are equivalent to the two independent equations, p q r These are the equations of a line in the body, namely of the central axis, as found in § 66, 38). Calling the value of the common ratio A, clearing of fractions, multiplying by p, q, r, and adding, we obtain the value of ^,, n Y _L « T/ _i_ *. 124) i-*S 76, 77] RELATIVE MOTION. 247 Making use of this value of I with equations 121), 123), we obtain for v for points on the central -axis 125) V = 2r agreeing with 42). If the velocity of points on the central axis is to be zero, we must have 126) pvx + q.Vy + rV, = Q, when the motion reduces to a rotation, as in 41). 77. Relative Motion. In forming equations 114) and the following, we have supposed the point in question fixed in the body, so that x, y, z were constants. If this is not the case we have to add to the right hand members of 114) the quantities dx . a dy . dz ^-st + ^^^dt' IOTN dx . a dy , dz 127) *•-& + &-£ + **' dx , „ dy , dz "•di + hdi + v*di' which, on being multiplied by the proper cosines, will appear in equations 117) as -^j —_> -^> so that we have for the components of the actual velocity in the direction of the axes X, Y, Z at the instant in question, if the origin of the latter is fixed, dx , v*Tjf + &"~ ry> 128) ^g+-«^ dz These equations are of very great importance, for by means of them we may express the velocity components in directions coinciding with the instantaneous direction of the moving axes of the end of any vector x, y, z. If for x, y, 2 we put the components of the velocity v, we obtain the acceleration -components (§ 103), if the components of angular momentum H we have a dynamical result treated in § 84. 248 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. If we apply these equations to a point fixed in space, for which vxf vy, vz vanish, we obtain inc\\ dx dy dz jj = ry-qz, -^=pz-rx, ~ = qx-py. Taking a point on the X'-axis at unit distance from the origin, we have x = cc1} y = ft , z = ylf da* d& and in like manner taking points on the Y' and Z'-axes, da* dS* dy» - - -- da, dp, dy. — == ^r - ~> -j± from 130), we have dp' dp Q dq dr ^-=«^ + Pi-di + Kdt> iq9x dq' dp K dq dr -dt-=a*-di + P*-dt+K^i' dr' dp a dq dr -df = ^dt+^M+^dt- Thus the angular acceleration is obtained by resolving a vector whose components about the axes X, Y. Z are ~ > -£> -.,-- In other words, ctt dt cut the time derivatives of the components p, q, r, of the angular velocity in the directions of the moving axes at any instant are equal to the angular accelerations of the motion about axes fixed in space 77, 78, 79] ENERGY AND MOMENTUM OF ROTATION. 249 coinciding in direction with the position of the moving axes at the given instant. This theorem, which is by no means self-evident, is of great importance, as is the similar property of the angular velocity, of which we have already made use. 79. Kinetic Energy and Momentum due to Rotation. From equations 119) we find for the part of the kinetic energy of the rigid body due to the rotation, supposing -= - = -^ = ~ = 0, dv dv ct v - qrZmyz — rpZmzx — pqZmxy and for the angular momentum, introducing 119) in 48), Hx = Zm [y (py - qx) — z (rx —##)]= Ap — Fq — Er, 134) Hy = 2m\2(qz — ry} — x(py — qx)~] = — Fp -f Bq — Dr, Hz = 2m [x(rx —pz] — y (qz — ry)] = — Ep — Dq + Cr, the last column being what we obtained in § 68, 53). It is evident that o^s „ dT „ dT rr dT Hx=^ ^V^Jq' Hz=frr' so that in this respect p, q, r, Hx, Hy, Hz have the relation of Lagrangian generalized velocities and momenta. Since we have the kinetic energy is one -half the geometric product of the angular velocity and angular momentum. 250 ™. DYNAMICS OF ROTATING BODIES. CHAPTER VII. DYNAMICS OF ROTATING BODIES. 8O. Dynamics of Body moving about a Fixed Axis. The simplest case of motion of a rigid body next to that of translation is a movement of rotation with one degree of freedom, namely a motion about a fixed axis. The centrifugal force exerted by the body on the axis is Md&2 where d is the distance from the axis of the center of mass of the body, and since this is in the direction of d7 which is continually changing, if a body is to run rapidly in bearings the center of mass should be in the axis, otherwise the bearings are subjected to periodically varying forces. At the same time, even if this condition is fulfilled, there will be a centrifugal couple, also tending to tear the body from its bearings, unless the axis is a principal axis of inertia. It is worth noticing that the first condition may be obtained in practice by statical means, by making the axis horizontal, and attaching weights until the body is in equilibrium in any position, but that the second condition is only obtained by experiments on the body in motion. For this reason the former condition is generally fulfilled in such pieces of machinery as the armatures of dynamos, while the latter is not especially provided for. Let us consider the motion of a heavy body about a horizontal axis. The resultant of all the parallel forces acting on its various particles is by § 59 a equal to a single force equal to the weight of the body Mg applied at the center of mass. The position of the body is determined by a single coordinate which we will take as the angle # made with the vertical by the perpendicular from the center of mass on the axis. If the length of the perpendicular is h the work done in turning the body from the position of equilibrium is The kinetic energy is 2) T = a 6A The equation of energy accordingly is 3) — K\-r:\ + Mgh(l — cos -fr) = const. 80] COMPOUND PENDULUM. 251 But this is the equation of motion of a simple pendulum of length 4) I- K --*2 ~m "¥' The body, which is often called a compound pendulum, accord- ingly moves like a simple pendulum of length I. This is called the equivalent simple pendulum. It is to be noticed that in virtue of the constraint of rigidity, points at distances from the axis less than I move more slowly than they would if moving alone in the same paths, while those at greater distances move faster, and those at distance I move just as they would if free to move in the same circular paths. Let kg be the radius of gyration of the body about a parallel axis through the center of gravity. Then by § 70, 4) *» = ftJ + A«, Z = ? + ft, ft so that I is always greater than h. If we take a parallel axis 0' at a distance h' = I — h beyond the center of mass 6r, so that it, 6r, and the original axis are in the same plane, we have 5) hh' = 1$. If now the axis 0' be made the axis of suspension, the equi- valent simple pendulum has a length fc'2 £2 _L h'* 6) V--hT = -~F~ = h + h' = L The axis 0' is called the axis of oscillation, and we have the theorem that the axes of suspension and oscillation are interchangeable and separated by the distance equal to the length of the equivalent simple pendulum. This is the principle of Eater's reversible pen- dulum, used to determine the acceleration of gravity. The pendulum is furnished with two knife edges, so that it may be swung with either end down. Movable masses attached to the pendulum are so adjusted that the time of vibration is the same in both positions, and then the distance between the knife-edges gives the length I from which g = -mr" The present example also includes the metro- nome and the beam of the ordinary balance. The masses of the pans may be regarded as concentrated at the knife-edges. If the fixed axis is not horizontal, the modification in the result is very simple. Suppose the axis makes an angle K with the vertical. Let us take two sets of fixed axes, Z' vertical, Z the axis of rota- tion, T1 horizontal in the plane of Z and Z', Y in the same plane, 252 VII. DYNAMICS OF ROTATING BODIES. and X and X' coincident. Then we have for the transformation of coordinates z1 = — y sin a -f z cos a, and determining the position of the system by the angle & made by the perpendicular from the center of mass on the axis of rotation with the Y-axis, y = r cos #, 0' = — r sin a cos & -\- z cos a. The potential energy is as before W = Mgz' = — Mgli sin a cos # -f const., thus the equation of energy is 7) ' Y K \-TZ\ — Mgh sin a cos # = const. Thus the equation is the same as before, except that the length of the equivalent simple pendulum is increased in the ratio of 1 : sin a. This example includes the case of a swinging gate and of the im- portant physical instrument, the horizontal pendulum of Zollner. The mode of action of the latter depends on the fact that the moment of the force required to produce a given deflection #, 0 = --- = Mgh sin a sin #, may be made as small as we please by decreasing a, which is observed in practice by making the time of vibration long. 81. Motion of a Rigid Body about a Fixed Point. Kinematics. We shall now consider one of the most important and interesting cases of the motion of a rigid body, namely that of a body one of whose points is fixed, and which thus possesses three degrees of freedom. This case was dealt with very fully by Poinsot, in his celebrated memoir "Theorie nouvelle de la rotation des corps", in the Journal de Liouville, torn. XVI, 1851. On account of the instructive nature of his processes, which are entirely geometrical, we shall present his method first. The treatment of the properties of the moment of inertia, which is contained in the same paper, has already been given in §§ 70 — 72. If one point of the body remains fixed, the instantaneous axis must at all times pass through that point. The motion is completely described if we know at all times the position of the instantaneous axis in the body and in space, and the angular velocity about it. 80, 81] BODY WITH FIXED POINT. 253 Fig. 71. Let 0, Fig. 71, be the fixed point, and let OI± be the instan- taneous axis at a given instant. During the time dt suppose a line 0/2 moves to the position OJ2r, and during the next interval /It let the body turn about this line as instantaneous axis. During this interval let another line OI3 move to OJ3' which then becomes the instantaneous axis, and so on. We have thus obtained two pyramids, one Ol^OLOI^ . . . fixed in space, the other O^OZ/OZ^. . . fixed in the body, and we may evidently describe the motion by saying that one pyramid rolls upon the other. As we pass to the limit, making 4t infinitely small, the pyramids evidently become cones, and the generator of tangency is the instantaneous axis at any instant. The rolling cone may be external or internal to the fixed one. In the former case, Fig. 72 a, the instantaneous axis moves around the fixed cone in the same direction in which the body rotates, and the motion is said to be progressive, in the second case, Fig. 72 c it goes in the opposite direction, and the motion is said to be regressive or retograde. It is to be noticed that it makes no difference whether the rolling cone is convex (Fig. 72 a) or concave (Fig. 72 b) toward the fixed cone. (In the figures, in which merely for convenience the cones are shown circular, C± denotes the fixed, C2 the rolling cone.) If one of the cones closes up to a line, upon which the other rolls, it always remains in contact with the same generator, that is, the instantaneous axis does not move. Accordingly if either cone degenerates to a line, the other does also, and the instantaneous axis remains fixed in space and in the body. This case has been already treated. If we lay a plane perpendicular to the instantaneous axis at a distance E from 0, Fig. 73, and if the radii of curvature of its inter- sections with the fixed and rolling cones be Q± and Q2 (taken with the same sign if they lie on the same side of the common tangent), and the angles made by consecutive tangents at the ends of correspond- ing arcs ds1 and ds2 are dri9 drif we have ds1 Fig. 72 a. 254 VII. DYNAMICS OF ROTATING BODIES. The angle turned by the body in rolling the arc ds2 on its equal ds1 is dr2 — drly and the angular velocity > — — (d — fl ^ — — (— ~\ /1 1\ I es Pj/ Fig. 72 c. Now if w denote the angular velocity with which the instantaneous axis is turning about an axis through 0 perpendicular to the common tangent plane to the cones, we have ds which inserted in 8) gives 9) Ql - Q. If the cones have external contact, $2 is negative, and if we consider the absolute values, we must take the sign plus. Fig. 73. or\ 8') o? = 9') ^ Consequently if we give the values at every instant of three of the quantities, co, the angular velocity about the instantaneous axis, w, the angular velocity of change of the instantaneous axis, and ^ 81] FIXED AND ROLLING CONES. 255 Nfi and p2, the radii of curvature of the sections of the fixed and rolling cones, the fourth , and consequently the whole motion, are determined. This corresponds to the fact that the body has. three degrees of freedom. If 01, Fig. 72, is the instantaneous axis, OC± and OC2 the lines of centers of curvature, the point I may be considered to be travelling around the cones with the ds velocity -^-t and about the lines 00X and 002 as axes with the angular velocities v, ^, of which 03 is the resultant. This was proved by Poinsot as follows. Let Fig. 74 represent a section of Fig. 72 a in the plane of the axes OC1; 01, OC2. If r19 r2 are Pig. 74. the perpendiculars from I on 001? 002, and v and u are the angles C^OI, 02OJ, we have, considering absolute values only, 10) dt "*^~' T dt 1 dsl cosu. ~^T = VCO$V, - -rr = II COS U . g^ dt Q2 dt Inserting in 8') also since we have 12) 09 = 4U- COS U + V COS V, r, i\ , -~- = -r^—i and vr, = *ITI 4t sm ^ It sin u = v sin v. sm v sm w That is, since the three axes are in the same plane, and G> is the sum of the components of [i and v in its direction, while their components in the perpendicular direction are equal and opposite, a) is the resultant of ^ and v. Figs. 72 a, 72 b, 72 c show the three cases, where the cones have external contact, where the fixed cone is internal, and the rolling cone is internal, respectively. The parallelogram construction is shown on the figures, and the direction of rotation is shown by the arrows representing the vector rotations. It will be noticed that in each case the arrow on the figure showing the direction in which / is travelling around the rolling cone is opposite to the direction of rotation /i about 02. The rotation about OOj is known as precession. If both fixed and rolling cones are cones of revolution, and /t, v, co constant, the 256 VII. DYNAMICS OF ROTATING BODIES. precession is called regular. If we call # the angle C^OC* between the axes of the cones, we have , fl V CO ~^Tv ~ 53irT^ • im^ 14) Co2 = jr + v2 + 2/iv cos ^. An important case of a regular precession is furnished, us in the motion of the earth, which, disregarding nutation (§ 93), describes a cone with # = 23° 27' 32" in the time 25,868 years, the motion being retrograde, Fig. 72 c. We thus have sin 23° 27' 32" „„ smO , 25.868x365.256 so that the pole of the earth describes a circular cone whose half angle is 0",0087, an angle too small to be perceived by astronomical means, the radius of the circle cut by this cone on the surface of the earth being only 27 centimeters. 82. Dynamics. Motion under no Forces. We have already found, §§ 68, 69; following Poinsot, the expressions for the momentum and the centrifugal forces for the general motion of a rigid body. If the fixed point be the center of mass, both the linear momentum and the centrifugal resultant vanish, and we have to deal only with the angular momentum and the centrifugal couple. At the same time the resultant of the effect of gravity passes through the fixed point, and is neutralized by the reaction of the support. Let us then consider the motion of a body turning about its center of mass, or more generally, the motion of a body under the action of no forces. Such a motion will be called a Poinsot -motion. Let OZ be the instantaneous axis. Then we have from § 68, 53) Hx = - Em, 15) _H?, = Let us call the resultant of Hx and Hy, J?2, Fig 75. We have for the centrifugal couple Sc, from § 69, 59), Lc = - 16) Mc = Nc= 0. Since Nc is zero, the axis of the centrifugal couple is perpendicular to the instantaneous axis. But since 81, 82] 17) CENTRIFUGAL COUPLE. HxLe + HyMc + H2N0 = 0, 257 it is also perpendicular to the angular momentum. Consequently the axis of the centrifugal couple is perpendicular to the plane containing the instantaneous axis and the axis of angular momen- tum, and is drawn in such a direction that if 8C be turned through a right angle in the direction of the body's motion it will coincide in direction with H2. Also since the components of Sc are equal in absolute value to the components of H% multiplied by co, we have Fig. 75. 18) Sc = HZ o = oH sin (Hci). Thus the centrifugal couple is equal to the vector product of the angular momentum by the angular velocity. In case the instan- taneous axis is a principal axis at 0, the direction of H coincides with that of « and the centrifugal couple vanishes. The body will then remain permanently turning about the same axis. This property of a heavy body turning about its center of gravity about a principal axis of maintaining the direction of that axis fixed in space was utilized by Foucault in his gyroscope, the axis of which points in a fixed direction while the earth turns, and thus the motion of the earth is made observable. The same principle is utilized practically in the Obry steering gear contained in the Whitehead automobile torpedo, in which a rapidly rotating gyroscope is made to give the direction to the torpedo, and by acting on the steering gear to make it return to its course if it accidentally leaves it. Suppose on the other hand that the instantaneous axis is not a principal axis. The centrifugal couple then tends to generate an angular momentum whose axis is in its own direction, and this new momentum compounds with that which the body already possesses. Let us consider two successive positions of the body. Suppose that in the time dt the body turns about the instantaneous axis through an angle d(p = &dt. At the end of that time the vector H2 (Fig. 75), would have turned through the angle dcp into the position H2, the length of the infinitesimal vector H2H2 being H2dcp = H2&dt. But during this time the centrifugal couple Sc has given rise to the angular momentum / WEBSTER, Dynamics. 17 258 VII. DYNAMICS OF ROTATING BODIES. This vector, being parallel to Se and thus perpendicular to H2' gives, when compounded with J?2' a resultant exactly equal to H2. The component of H parallel to co being unchanged by the motion, we find, geometrically, that the angular momentum remains constant throughout the motion, as we have found by a general theorem in § 33. As we now wish to follow the motion of the body from one instant to another, it will be convenient to free ourselves from the choice of axes which made the instantaneous axis the Z-axis. Let us take for axes the principal axes of the body at 0. Let the com- ponents of the angular velocity ro on the axes be p, q, r. Then the angular momentum, being the resultant of the three angular momenta due to the three angular velocities p, q, r, are by § 68, 53) or § 79, 134), 19) Hx = Ap, Hy = Bq, H, = Cr. If we draw any radius vector to the ellipsoid of inertia at the fixed center of mass the perpendicular d on the tangent plane at the point #, y, g has direction cosines proportional to Ax^ By, Cg. If we draw the radius vector p in the direction of the instan- taneous axis, so that *>> ! = f = T = f = «> equations 19) give 21) HM or the angular momentum vector bears to the angular velocity vector the relation, as to direction, of the perpendicular on the tangent plane to the radius vector. Otherwise, if the angular momentum is given, the instantaneous axis is the diameter conjugate to the diametral plane of the ellipsoid perpendicular to the angular momentum. The centrifugal couple being per- pendicular to the plane of d and p, lies in the diametral plane conjugate to Q. It produces in the time dt an angular momentum Scdt whose axis is in the same direction. To find the axis of the angular velocity corresponding thereto we must find the diameter conjugate to the plane perpendicular to Sc, that is the plane yd. But the diameter conjugate to a plane is conjugate to all diameters in it, hence the required 82] ROLLING POINSOT ELLIPSOID. 259 diameter is conjugate to Q and lies in the plane conjugate to Q, that is, parallel to the tangent plane at x, y, a. Consequently, if we compound with the velocity co about Q the velocity corresponding to Scdt parallel to the tangent plane, the resultant has the same com- ponent perpendicular to the tangent plane as co. In other words the component to cos (eo, H) is constant throughout the motion. Now we have found that H is constant in magnitude and direc- tion, hence, multiplying by the constant ocos(ca#), 22) Ho cos (G)H) = const. But jffcos (Ho) is that component of the angular momentum which is parallel to the instantaneous axis, and is accordingly equal, by § 68, 53) to the product of the angular velocity by the moment of inertia about the instantaneous axis. 23) H cos (Hm) = Ko. Accordingly 22) becomes 24) K a)2 = const. But this is equal to twice the kinetic energy. Accordingly we obtain geometrically the integral of energy. Thus for a rigid body this principle follows from that of the conservation of angular momentum. In the ellipsoid of inertia we have, § 71, Accordingly 25) and the equation of energy shows that n is constant during the motion, or during the whole motion the angular velocity is propor- tional to the radius vector to the ellipsoid of inertia in the direction of the instantaneous axis. But since cocos(i?(a) is constant, pcos(pd) = d must be constant, and therefore the tangent plane is at a constant distance from the center during the motion. But since the direction of the line d is constant in space, and its length is also constant, the tangent plane must be a fixed plane in space. As the point where it is touched by the ellipsoid of inertia is on the instantaneous axis the ellipsoid must be turning about this radius vector, and hence rolling without sliding on the fixed tangent plane. The motion of the * body is thus completely described, and we see that the problem of a Poinsot- motion is equivalent to the geometrical one of the rolling of an ellipsoid whose center is fixed on a fixed tangent plane, together with the kinematical statement that the angular velocity of rolling is proportional to the radius vector to the point 17* 260 VII. DYNAMICS OF ROTATING BODIES. of tangency. Before taking up the discussion of this result, as given by Poinsot, we will consider the analytical method of establishing the result. 84. Euler's Dynamical Equations. If Hx', HJ, HJ represent the angular momentum about the fixed X', Y', ^'-axes, L', M', N', the moment of the applied couple, the equations of § 67, 49) are dHf dH' dH' <26\ _ £ _ TJ _JL — M1 — — N' dt *"> dt *->. dt ~1V ' where (cf. § 76) Hx= 27) HJ = HJ = a, Differentiating we have, after making use of § 77, 130), dH' dH dH dH 28) -— + A + K + Hx (ft r-yiq) + S, fop -«,»•) + H, (at q - ftp). If we now choose for fixed axes the instantaneous positions of the moving axes, we have ax = /32 = y3 = 1, all other cosines zero, and the equations 28) become simply 29) dH We may obtain the same results by the use of the equations § 77, 128). Let us take for the point x, y, z the end of the vector H. Its coordinates with respect to the moving axes being Hx, Hy, Hz, substituting them in equations § 77, 128) we obtain for their velocities resolved along the X, Y, ^-axes the expression on the left of 29). We must now put for Hx, Hy, Hz the expressions § 79, 134). If now the moving axes are taken at random, the moments and products of inertia of the body with respect to them will vary with the time, so that their time -derivatives enter into the dynamical equations, which are thus too complicated to be of any use. It is therefore immediately suggested that we choose for the moving axes a set of axes fixed in the body, and moving with it. The quantities A9 B7 C, D, E, F are then constants. If in addition we take as axes the principal axes at the origin of the moving axes, D, E, F vanish, 82, 84, 85] EULER'S EQUATIONS. 231 and then since Hx = Ap, Hy = Bq, H2 = Cr, the equations become simply 30) These are Euler's dynamical equations for the rotation of a rigid body. In case the moments of the applied forces about the origin vanish, they become 31) and we see that the quantities on the right, being the vector product of the angular velocity by the angular momentum, represent the centrifugal couple, which alone acts to produce the angular accel- eration, whose components appear on the left. We thus obtain the result obtained geometrically by Poinsot, the quantities on the left denoting the velocity of the end of H in the ~body. The equations 29) may be simplified in another manner, if the ellipsoid of inertia is of revolution. If for one of the moving axes we take the axis of revolution, and for the others, any axes perpen- dicular to it, whether fixed in the body or not, the axes will be principal axes, and the moments of inertia constant, since the moment of inertia about all axes perpendicular to the axis of rotation in the same. Examples of this will be given in §§ 96, 106. 85. Poinsot's Discussion of the Motion. We may now integrate the equations 31) by making use of the fact that the centrifugal couple is perpendicular to the angular velocity and the angular momentum. Multiplying equations 31) respectively by p, q, r and adding wHich is at once integrated as 33) 4 Ap* + 4 Bf + } Cr> = const. This is the equation of energy. 262 VII. DYNAMICS OF ROTATING BODIES. Multiplying now by Ap} Bq, Cr, and adding, 34) A*p% + B.q z = ~- CO 00 0) Now the length of the perpendicular d is, since it is the projection of Q on the direction of the normal, 36) d = x cos (nx) -f y cos (ny) -f z cos (nz) snce 2T -*««-£ Accordingly since T and H are constant, d is constant, and the tangent plane being perpendicular to the invariable line H is fixed in space. Poinsot called the locus of the pole of the instantaneous axis on the ellipsoid, the polhode (rt6ho$ axis, bd6$ path), and its locus on the tangent plane the herpolhode. The ellipsoid of inertia being 37) Ax2 + W + Cz* = 1, the distance of the tangent plane at x, y, z from the center is Since this is to be constant, this equation with that of the ellipsoid define the polhode curve. Combining the equation 39) A*x* + &f + C*s* = -~ with that of the ellipsoid, divided by d2, we obtain by subtraction 40) A -Ax* + B -Bf+ C - CS = 0. 85] POLHODE AND HERPOLHODE. 263 This is the equation of the cone passing through the polhode, with its vertex at the fixed point, that is the rolling or polhode cone. We find then that the rolling cone for a body moving under no forces is of the second order. If it is to be real, we must have 41) A^^C, that is the perpendicular must have a length intermediate between the greatest and least axes of the ellipsoid. If -^ = A the cone is •42) representing a pair of imaginary planes, intersecting in the real line y = 3 = 0, the X-axis. Thus in this case the rolling cone reduces to a line, fixed both in the body and in space. If ^ = C, we have a similar result. If we 43) A(A- B)x* - representing two real planes intersecting in the Y-axis, and making an angle with the XY- plane whose tangent is 44) __ x ~ - C(S-C) These are the planes which separate the polhodes surrounding the end of the major axis from those about the minor axis. The polhodes are twisted curves of the fourth order, whose appearance is shown in perspective in Fig. 77. The separating polhodes are drawn black. Since the polhode is a closed curve, the radius vector of a point on it oscillates between a maximum and a minimum value. If 6 is the distance of a point on the herpolhode from the foot of the perpendicular d, since 62 = g2 — d2, 6 oscillates between two constant values, and the herpolhode is tangent to two circles. Since the polhode is described periodically the various arcs of the herpolhode corresponding to repetitions of the polhode are all alike. The her- polhode is not in general a reentrant curve. The name herpolhode was given by Poinsot from the verb SQXSW, to creep (like a snake) from the supposedly undulating nature of the curve, it has however Fig. 77. 264 VII. DYNAMICS OF ROTATING BODIES. been proved to have no points of inflexion, and is like Fig. 78, which has been calculated for A = 8, B = 5, C = 3, ^ = 4 - 9. 86. Stability of Axes. We have seen that the body if rotating about either of the principal axes of inertia will remain rotating about it. If the instantaneous axis be the axis of either greatest or least inertia, and be displaced a little, as the polhodes encircle the ends of these axes the instantaneous axis will travel around on a small polhode, and the herpolhode will be small, neither ever leaving the original axis by a large amount. These axes are accordingly said to be axes of stable motion. If on the other hand the mean axis be the instantaneous axis, and there is a slight displace- ment, the axis immediately begins to go farther and farther from the original position, and nearly reaches a point diame- trically opposite before return- ing to the original position. pig 78< The mean axis is thus said to be an axis of instability. It is however to be noticed that if either A — B or B — C is small with respect to the other, the separating polhode closes up about either the axis of greatest or least inertia respectively, and thus a small displacement may lead to a considerable departure from the original pole, the rotation is thus less stable. The rotation about either axis is most stable when the wedge of the separating polhode enclosing it is most open. 87. Projections of the Polhode. From the equations of the polhode 37), 39), we may obtain its projections on the coordinate planes by eliminating either of the coordinates. Eliminating x, 45) an ellipse of semi -axes, B(A-B) S the ratio of which is yc(A-c\ 85, 86, 87, 88] PROJECTIONS OF POLHODE. 265 a constant, so that all the projections are similar. The motion about the axis A is most stable when the small polhode is a circle, that is when the above ratio is unity, or B = C. Eliminating g we obtain 46) d2{A(A — C')x2 + B(B—C)y2}= 1 — <7tf2, an ellipse the ratio of whose axes is -i/B(B-C) V A(A-C)' and for maximum stability this is unity, or A = B. These projections are shown in Fig. 79. Fig. 79. Fig. 80. V, Eliminating y, we have 47) 62{A(A — B)x* - an hyperbola the ratio of whose axes is C(B-C) A(A-B)' All the hyperbolas have the separating polhode projections as asymptotes (Fig. 80). 88. Invariable Line. The invariable line describes a cone in the body. Its equation may be simply found from consideration of the reciprocal ellipsoid 4g) Z + F+Z?-1' whose radius in the direction of d is -y and therefore constant. The cone of the invariable axis is accordingly the cone passing through the intersection of the ellipsoid 48) with the sphere 266 VII. DYNAMICS OF ROTATING BODIES. 49) that is 50) The axis of this, like that of the polhode cone, is the axis of greatest or least inertia. Let us find how fast the invariable line revolves around one of the principal axes. Since the invariable axis is fixed in space, its relative motion is equal and opposite to the actual motion of the part of the body in which it lies. If we call A the diedral angle between the plane of the invariable axis and the axis of X and the XY- plane, we may find -^- Projecting H upon the YZ- plane (Fig. 81), the pro jection makes with the F-axis the angle A, given by Fig. 81. 51) from which Differentiating, 52) H Cr C / dr dq\ 1 BC dr ^dq Inserting from Euler's equations 31), dq _ C — A dr _A-B ~dt ~ B r&' ~dt ~ C dl p{B(A-B)q*+C(A-C)r*} 53) dt -H* f\AH*-HP l]i = *\*H?-H P 2AT H* -1 -EL\2 -© sin2 (H x) [88 MOTION OF INVARIABLE AXIS. 267 Similarly for the rotation around the T and Z-axes, dv^ 3*0-1 dt ~ T sin2 (He) Looking at the signs of the numerators, we see that the invariable axis rotates around the axis of greatest moment of inertia in the direction of rotation, about the least axis in the direction opposite to that of rotation, and about the mean axis according to the value of d. If we mark off on the invariable axis a line of unit length, its end describes a sphero-conic, the intersection of the invariable cone 50) with the sphere whose projections on planes perpendicular to the X and Z-axes are ellipses, and perpendicular to the Y"-axis an hyperbola. The radius vector of the X- projection is rx = sm(Hx) and since it turns with the angular velocity ^- it describes area at the rate 55) 2%%£-*(^-i). The time of one revolution of the body turning with the velocity p would be, if p were constant, t = — The equation of the ellipse is obtained by eliminating x from the equations of the sphero-conic as whose axes are and whose area is 57) 'l 2 — : 1 "0 0) - K ^i-c / BC ' (A-B)(A-C) 268 VII. DYNAMICS OF ROTATING BODIES. Now the area described in one revolution about the instantaneous axis would be, if p were constant [see 55)], and the number of turns the body makes for one revolution of the invariable axis about the X-axis is the area 57) divided by this, or 59) BC -B)(A-C) This may be made as large as we please by making A approach I> or C. If B = C or the ellipsoid of inertia is of revolution, about the X-axis, p is constant, and the invariable cone is circular, and described with uniform velocity, the number of revolutions of the body for one circuit of the invariable axis being /^_^\m The motion is direct or inverse, according as the X-axis is that of greatest or least inertia. These properties may all be illustra- ted experimentally by means of Max- well's Dynamical Top1), constructed by Maxwell for the purpose of studying the motion of the earth about its center of mass. An example of this top constructed in the workshop of the Department of Phy- sics of Clark Uni- versity is shown in Fig. 82. The six weights projecting from the bell allow Tig. 82. « . the moments of in- ertia to be changed in a great variety of ways, while at the same time the center of 1) Maxwell, Papers, Vol. I, p. 248. 88] MAXWELL'S TOP. 269 mass is constantly kept at the point of support, a sharp steel point turning in a sapphire cup. Maxwell's ingenious device for the observation of the motion of the invariable axis, is the disk, divided into four colored segments, attached to the axis of figure. The colors chosen, red, blue, yellow and green, combine into a neutral gray when the top is revolving rapidly about the axis of figure. If however the top revolves about a line passing through a point in the red sector, there will be in the center a circle of red, the diameter of which is greater as the axis is farther from the center of the disk and the boundaries of the red sector. Thus the center of the gray disk changes from one color to another as the pole moves about in the body, and by following the changes of color we can study the motion. By noticing the order of the succession of colors we can determine whether the axis of figure coincides most nearly with the axis of greatest or least inertia, and by changing the adjustments we may make it a principal axis, which is known by the disappearance of wabbling, or we may make it deviate by any desired amount from a principal axis. If the deviation is great, and the top spun about the axis of figure, and then left to itself, the top will wabble to a startling amount, but eventually the pole will reach its first position and the wabbling will cease, to be repeated periodically. The recovery of the top from its apparantly lawless gyrations is very striking. If the adjustment is such as to make the axis of figure lie near the mean axis of inertia, the top will not recover, but must be stopped in its motion before striking its support. The path of the invariable axis has been made visible by Mr. G. F. C. Searle, of the Cavendish Laboratory, Cambridge, by attaching to the axis of figure a card, upon which ink was projected from an electrified jet. Acting upon this suggestion, the author attached to the top a disk of smoked paper, upon which a steel stylus, playing easily in a vertical support (shown in Fig. 82 lying on the table) could write with very slight friction. One easily finds by looking at the disk in its gyrations a point which remains fixed, and by applying the stylus to this point, holding it on a proper support, the path of the invariable axis is drawn, and found to be an ellipse or hyperbola. If the stylus is not held exactly on the invariable axis, small loops are formed, which enable us to count the number of turns of the top in going around the polhode, and thus to verify the theory. The results of several spins are shown in Fig. 83, reproduced from actual traces. The loops are turned out if the principal axis at the center of the ellipse is that if greatest inertia, and in if it is the least, for the reason that in the former case the invariable axis and the herpolhode cone lie within the polhode (Fig. 83 a), while in the latter they lie 270 VII. DYNAMICS OF ROTATING BODIES. without (Fig. 83 b) so that if we consider the relative motion, in the former case a point fixed to the herpolhode describes a sort of Fig. 88. Fig. 83. Fig. 88. hypocycloid (loops out) on the card attached to the polhode, in the latter a sort of epicycloid (loops in). Fig. 83 a. Fig. 83 a. The recent astronomical discovery of the motion of the earth's pole is probably due to a sort of variable Poinsot- motion, the moments of inertia of the earth being gradually varied. Cone Fig. 83 b. 88, 89] EXPERIMENTAL VERIFICATION. 271 Fig. 84. 89. Symmetrical Top. Constrained Motion. While we have in the preceding section considered the very interesting and instructive question of the motion of the most general rigid body under the action of no forces, by far the most frequent case under the practical conditions of experiment is that in which the body is dynamically symmetrical about an axis, that is, the ellipsoid of inertia is of revolution. Such a body we shall call a symmetrical top. This will include not only all ordinary tops and gyroscopes, as well as flywheels, rolling hoops, billiard balls, but even the earth and planets. Suppose such a body to be spinning under the action of no forces, about its axis of symmetry. We have seen that it will remain so spinning, and the angular momentum will have the direction of the axis of symmetry. If now the axis of symmetry OF (Fig. 84), is to move to some other position, OF1, which is then to coincide with the new instantaneous axis, the angular momentum HH' must be communi- cated to the body, that is a couple whose axis is parallel to HH' must act on the body. This may be made evident experimentally by placing a loop of string over the axis F of a symmetrical top balanced on its center of mass (Fig. 85) and pulling on the string. The axis of the top, instead of following the direction of the pull P moves off at right angles thereto, although the string can only impart a force in its own direction. The pull of the string, together with the reaction of the point of support constitute a couple, whose moment is perpendicular to the plane of the string and of the point of support, and it is in this direction that the end of the axis, or apex of the top, moves, as is required by the theory. This simple experiment and the theory which it illustrates will make clear most of the apparantly paradoxical phenomena of rotation. We may describe it by saying that the kinetic reaction of a symmetrical Fig. 85. 272 VII. DYNAMICS OF ROTATING BODIES. rotating top is not at in the direction of the motion of the apex, but nearly at right angles thereto. (Exactly at right angles to the motion of OH.} An ingenious application of this principle is found in the Howell automobile torpedo, invented by Admiral Howell of the United States navy. In this the energy necessary for driving the torpedo is stored up in a heavy steel flywheel, weighing one hundred and thirty- five pounds, and turning with a speed of ten thousand turns per minute. The axis of the flywheel lies horizontally perpen- dicular to the axis of the torpedo (Fig. 86), thus steadying the torpedo in its course. If now any force acts tending to deflect the torpedo hori- se. zontally from its course, by means of a moment about a vertical axis, the end of the axis of the disk moves vertically, causing the torpedo to roll instead of yielding to the deflecting force. The rolling is utilized, by means of a vertically hanging pendulum, to bring rudders into action, and to cause the torpedo to roll back to its original position, while main- taining its course. A striking example of the principle enunciated above is found in an inge- nious top (Fig. 87), spinning on its center of mass, with its axis rolling on various curves constructed of metal wire. No matter what the shape of the wire, the axis of the top clings to it as if held by magnetism, no matter how sharply the curve may bend. The passing around sharp corners at a high speed, in apparant defiance of centrifugal force, is extremely remarkable. The explanation of the action is immediate, on the lines just laid down. The instantaneous axis passes though the point of support 0 (Fig. 88) and the point of contact of the axis of the top with the wire. The wire, in fact, Fig. 87. 89] APPLICATIONS OF SYMMETRICAL TOP. 273 constitutes the directrix of the herpolhode cone. Since the ellipsoid of inertia is of rotation, the axis of figure OF, the instantaneous axis 01, and the axis of angular momentum OH, lie in the same plane, which is perpendicular to the tangent plane to the herpolhode cone. During the rolling, all these axes move parallel to this tangent plane, so that the vector HHL, representing the change of angular momentum, is parallel to the tangent plane, and in the direction of advance of the axis of figure. The couple causing the motion accordingly due to the reaction between the wire herpolhode and the top, is always parallel to the tangent plane, and never vanishes, but always tends to press the top against the wire. Or in general, in constrained motion, the motion causes the polhode cone to press against the herpolhode cone. This seems to have been first explicitly stated by Klein and Sommerfeld, Theorie des Kreisels, p. 173. An application of the above principle on a large scale, and the only one known to the author, is found in the Griffin grinding mill. A massive steel disk or roller A (Fig. 89) hangs from a vertical shaft by a uni- versal or Hooke's joint C, in the middle of a steel ring B forming the side of a pan. If now the shaft be set rotat- ing, the roller spins quietly about a fixed axis, with no tendency to move sidewise. If on the contrary it be brought into contact with the ring, it immediately rolls around with great velocity, pressing with great force against the steel ring or herpolhode, and grinding any material placed in the pan with great efficiency. It is interesting to note that a somewhat similar mill, in which the axis, instead of passing through a fixed point, hangs vertically from a revolving arm, and therefore is devoid of the action just described, although both mills possess in common the centrifugal force due to the circular motion of the center of mass of the roller, WEBSTER , Dynamics. 18 Fig. 89. 274 VII. DYNAMICS OF ROTATING BODIES. is much less efficient. The first mill is an excellent example of the centrifugal force and centrifugal couple, while the second lacks the centrifugal couple, the instantaneous axis and the axis of angular momentum being parallel. Let us calculate the couple involved in the con- strained motion involved in a regular precession, as here applicable, in terms of the constants of § 81. If the angular momentum make with the axis of figure the angle a, its end is at the distance from the axis of the fixed cone H sin (a + -91), so that it moves with the velocity vH sin (a + #). This must be equal to the applied couple, 60) K=vH$m(a + &). Now resolving H parallel and perpendicular to the axis of figure, we have 61) H cos a = Ceo cos u, H sin a = AM sinw, so that 62) K= va(Asmucos& + But we have, § 81, -,o\ G> V lo) — — - = —. — j sin -91 sin u 14) a?2 = ^ -f v2 - from which ca sin u = v sin #, & cos u = [i — v cos #, so that finally 63) K = v{Av8w&eo8& 4- (7 sin -9- (^- It is to be noticed that the body will perform a regular precession under no constraint or other applied couple, if putting .ZT=0, n C 64) C-A 9O. Heavy Symmetrical Top. We will now take up one of the most interesting problems of the motion of a rigid body, namely the motion of a body dynamically symmetrical about an axis, on which its center of mass lies, and spinning about some other point of that axis. This is the problem of the common top or gyroscope. In order to determine the position of the top it will be convenient to introduce three coordinate parameters, namely the three angles of Euler. Let these be # the angle between the J^-axis, which we take as the axis of symmetry, and the fixed vertical Zf-axis 89, 90] HEAVY SYMMETRICAL TOP. 275 (Fig. 91). We may call the XT- plane the equator of the top. Let ON (Fig. 91) be the line of nodes, or the line in which the equator intersects the fixed X' F'-plane. Let ij> be the longitude of the line of nodes, or the angle X'ON measured positively from X' to Y'. Let cp be the angle from the line of nodes to the X-axis, the positive direction of increase being from X to Y. By means of the three angles -O1, ^, cp, we may express the nine direction cosines, and the position of the body is completely determined. The meaning of the angles is easily seen on the gyroscope in gimbals (Fig. 92). It will not be necessary for us to express the cosines, as we need only the values of p, q, r in terms of Euler's angles and their velocities, Fig. 91. dt dt dt As these are the angular velocities about ON, OZ' and OZ, respectively, we need only the cosines of the angles made by these lines with the X, Y, ^-axes, which are evidently as given in the following table. z] N Z sin # sin tp sin-O'COsqp cos^ cosqp — sin (p 0 0 • 0 1 Resolving now in the direction of the three axes, we obtain 65) . - . . = ~It Sm m ^ "^ ~dt COS *?' dib . 18' 276 VII. DYNAMICS OF ROTATING BODIES. These are Euler's kinematical equations. They illustrate the statement made in § 76, about p, q, r as not being time derivatives, for it is easily seen that pdt, qdt, rdt do not satisfy the conditions of being exact differentials. The resultant of the weight of all the parts of the body is Mg applied at the center of mass. If this is at a distance I from the fixed point the moment of the applied force is Mglsmft about the axis ON. L= Mglsin&coscp, 66) M = — Mgl sin # sin cp, N= 0, so that Euler's dynamical equations are A -~ = (B — C) qr -f Mgl sin # cos q, Hz = Cr on the vertical OZ', we obtain 70) HJ = Ap sin 0* sin (p -f Bq sin # cos cp -f- Cr cos # == const. If the top is symmetrical about the 2T-axis, we have A = B. Then the third equation 67) is Cdr — 0 G~-°> 71) Cr = const. = H2. 90, 91] EQUATIONS OF TOP. 277 The integral of energy 69) becomes F + f* -jT = «-«cos#, if we introduce the constants 7ox 2ft H* tMgl -A- AC> a= -r- The integral of vertical angular momentum 70) becomes 2 ^4) sin # (p sin cp -f- q cos TT dip P» = H*' 88) and the same force is required for the rotation about the vertical as if there were no spinning, whereas a force is developed tending to turn about the horizontal axis, which must be balanced by the constraint, P$, proportional to -—• Thus the effect of the concealed motion would be made evident, even if the disposition of the concealed rotating parts were unknown. The effect of the gyroscopic term may be described by saying that if the apex of the top be moved in any direction, the spinning tends to move it at right angles to that direction, as shown in § 50. In our present problem, we have Py = 0, ^sin2# -^'4- C(')cos# = const. = H}, or by 83), 89) A sin2 # • ^ + H, cos # = HJ, which is the integral of 70). The , differential equation for # is 90) A — Aty ' 2 sin # cos # + Hz sin # - ty ' = Mgl sin #, which, on replacing ty' by the value from 89), and using the constants of 73), 75) becomes ~^ - -_ a . ~*~ 8~ =S: fj Qi If we now multiply this by 2sin2# • ^r? it becomes an exact derivative, and integrates into 77). Thus our three integrals are immediate integrals of Lagrange's equations. 92. Nature of the Motion. Equation 78) which states that the time -derivative of 2, the cosine of the inclination of the axis to the vertical, is a polynomial of the third degree in 2, shows that 0 is an elliptic function of the time. As we do not here presuppose a knowledge of the elliptic functions, we will discuss the motion without explicitly finding the solution in terms of elliptic functions. We see at once that the solution depends on the four arbitrary constants cc, a, of the dimensions [T~2], which enter equation 78) linearly, and /3, &, of dimensions [l7"1], which enter homogeneously in the second degree, so that if we divide &, /3 by the same number and a, a by its square, while we multiply t by the same number, the two equations 78), 79) are unchanged, that is to any value of -0 280 ™. DYNAMICS OF ROTATING BODIES. corresponds the same value of ^, or the path of the point of the top is the same, but described at a different rate. Thus the shape of the path depends on the three ratios of the constants, or there is a triple infinity of paths. As for the meanings of the constants, a depends simply on the nature of the top, irrespective of the motion, and by comparison with § 80 is seen to be inversely proportional to the square of the time in which the top would describe small oscilla- tions as a pendulum, if supported with its apex downwards, without spinning. If we change the top, we may obtain the same path by suitably changing a, &, /3 as just described. These three constants depend on the circumstances of the motion, b being proportional to the angular momentum about the axis of figure, or to the velocity of spinning, /3 to the angular momentum about the vertical, and a depending on both the velocity of spinning and the energy constant. Expressed in terms of the initial position and velocities they are a=2^Z ^_C_ 92) 0 = With the convention that we have adopted, a is positive. As it is evident that any path may be described in either direction, we shall obtain all the paths if we spin the top always in the same direction. We shall thus suppose b to be positive, while /3 may be positive or negative, according to the sign and magnitude of (-5?) and cos #0. dz \«*/o Since -=- is real, f(si) 78) must be positive throughout the motion, except when it vanishes. Since /"(I) = — (j3 — b)2 and f(— 1) = — (|3 -f b)2 are both negative, f(oo) = oo and f(— oo) = — oo, the course of the function f(z) is as shown in Fig. 93. Thus it is evident that f(z) 'Z has three real roots, two #!, £2, lying between 1 and - 1, while the third, #3, lies outside of that interval on the positive side. Thus the motion is confined to that part of the #-axis between $if z2, and the apex of the top rises and falls between the two values of <#• whose cosines are ^ and #2. The triple infinity of paths may be characterized by giving the three roots all possible real values, instead of giving the constants ft, /3, a. In practice it will be convenient to give the two roots indicating the highest and lowest points reached by the apex, and the value 92] MOTION OF APEX. 281 of -~ the horizontal angular velocity at one qf them, which three data completely characterize the motion. Since g is an elliptic function of the time? the rise and fall is periodic, and after a certain time, g will have attained the same value, and so will -^ and -A accordingly during successive periods the angles ^ and

f(z} - A i (/?-K)2(i-*2) ™ TT?i**(A^:jrH 1_^2 We thus find that 0 — ^ is a factor of the expression on the right, so that, multiplying by 1 — #2, we have f(z) exhibited in the form where {^(z) is the polynomial of order two, so that the other two roots are found by solving the quadratic £(0) = 0. As the roots zl9 s^ approach each other, the rise and fall decreases, and vanishes when f(z) has two equal roots. The condition for this is that /(*) and f (0) = fa (0) + (0 — *i)fi(*) have a common root a., that is that from which 103) a(l-^)2 If % and one of the constants 6, /3 are given, this is a quadratic to determine the other. We find 284 VII. DYNAMICS OF ROTATING BODIES. which is constant, so that the motion is a regular precession, without rise and fall. There are thus, for a given velocity of spinning, and a given angle of inclination with the vertical, two values of the velocity of precession. We may also find these by considering the equation 90), putting & constant in which gives, if sin^ is not zero, 105) -4 a quadratic for ty1 with the roots These values are real if 52 > 2acos#i. If the top be spun so fast that -a?°S is a small quantity whose square may be neglected, we find for one value of ty* which is a large quantity of the order of 6, while the other root is which is a small quantity of the order of y Of these it is the slow precession which is usually observed. It is to be observed that if we put ^' = v, (p* = /A, the first of equations 82) gives for P$ the same result as obtained for K in 63). When we make a vanish, so that the body is under the action of no impressed moment, the root ^ becomes zero, so that the axis of figure stands still, while the root ^ becomes - — that is, the body performs a Poinsot -motion around the vertical as the invariable axis. Thus the effect of the impressed forces may be looked upon as a small perturbation of the Poinsot -motion. We will now consider the motion when the condition 103) is not fulfilled. From equation 78), we have t given by the elliptic integral, 109) t=C- =^= J ya(s-gj(g-sj(e-zj We may easily find two limits within which this value lies, by substituting for the factor ]/# — 03 in the integrand its greatest and least values, as we did in the case of the spherical pendulum. 93] REGULAR PRECESSION. SMALL OSCILLATIONS. 285 Since throughout the motion we have the inequalities, 110) C- dz > t > - - C- By means of a linear substitution we may simplify the integral. Let us put / 1 N Z1~ZZ 111) when the integral becomes 112) C dz - = C dx = p.na— iff 4- J y^-ax*-*,) J yi-x*~ so that we have for t, 113) — cos""1^ > t + const. > cos"1^. If now the difference 8± — z% = x is sufficiently small in comparison with £3 — 0J and #3 — ^2 , we may obtain an approximate result by putting under the radical the mean of the quantities which are too great and too small respectively, so that if ^ -f 22 = 2#0 we have the approximate result 114) 1 4- const. = —cos-1x, from which we obtain x ^ c" " ~c = COS 115) ^ = ^0 + c • cos The arbitrary constant has been taken so that i = 0 when the top is at its highest, and z = £0 -f- c = £r We thus see that when the roots zly ^ are nearly enough equal, the apex of the top rises and falls with a harmonic oscillation g of the small amplitude c = * ~ '• In order to determine when the approximation is justified, we have to consider what will cause the third root £3 to be large. Since #2 and #3 are the roots of the quadratic function /i(#) 102), their sum is the negative of the coeffi- cient of s divided by that of #2, that is a h 286 VII. DYNAMICS OF ROTATING BODIES. Thus we see that by making 1) large enough we may make #3 as large as we please, when gl and #2 are given, so that the approxi- mation is better the faster the top spins. Let us now consider the horizontal motion, or precession. We have 117) ~dt We have already supposed g to be a small quantity, so that if we neglect the square of j— - — ^ we have, after developing the second V1 ~ ZQ ) factor of the denominator, 118) l-fefc, dt (1-V Now inserting the value of g from 115) and integrating, Thus we see that ifj varies with a harmonic oscillation about the value that it would have in the regular precession at the mean height £0, of the same period as the vertical oscillation. If we project the motion of the apex on the tangent plane to the sphere on which it moves, calling f; the horizontal coordinate, and tj the distance moved from the horizontal mean axis, we have, Fig. 99, 120) yi^v Thus we see that the second terms of 115) and 119) represent an elliptic harmonic motion of the apex of the top. This is termed nutation. We thus have a complete description of the motion of a top when differing by a small amount from a regular precession, as a regular precession combined with a nutation in an ellipse about the point which advances with the regular precession. We shall now make an additional supposi: tion with regard to the constants of the motion. We have seen from 108) that in the case of regular precession with rapid spinning, the precession was slow. Let us then suppose that 121) •'=%— P "• is a small quantity of the same order as c, so that their squares and product may be neglected. Since #0 is the cosine of the angle between 93] NUTATION. CYCLOIDAL PATH. 287 the vectors whose magnitudes are b and ft this supposition is equi- valent to saying that the angular momentum makes a small angle with the axis of figure , as we see from Fig. 100, in which the distance DE=p- bz0. Making this supposition, the last term in 116) is negligible, also that in 119). Thus we obtain from 116), Jz «o n and since — is supposed to be large we may neglect — 122) 123) I, so that we have finally, Fig. 100. s* — pi — T~ — ^smbt, 124) | = J^--°-*- ^si Vi-^02 T/i-V yr- It is evident from Fig. 100 that /3 —  is positive, accordingly [cf. 119)] the apex is always moving so that -~ is positive at the bottom of its path, and thus the average motion is in that sense. The motion at the top may be in either direction, according to the magnitude of c. We see that the motion of nutation is opposite to the motion of the clock -hands. Thus the motion of the apex, as given by 124), is that of a point at a distance — from the • yi-V center of a circle which rolls on a line above it with its center advancing at a velocity - — - — • The radius of the rolling circle is - ~V Such a locus is called a cycloid. In the ordinary cycloid, the tracing point is on the circumference of the rolling circle, or /3 — � = be. If the tracing point is an internal one, the cycloid is called prolate. It has no loops, nor vertical tangents, and -^ is never zero, but it has points of inflexion. If the point is external the cycloid is called curtate, and has loops, but no inflexions. It is evident that this curve will be described when the apex is given a push to the left at the top of its motion, while if it be given a push to the right it describes the prolate cycloid, and if it be simply let go, it describes the ordinary cycloid with cusps. (The prolate and curtate cycloids are also called trochoids.) Since the height of a cycloid is to the length of its base as 1 : x, the base being the 288 VII. DYNAMICS OF ROTATING BODIES. distance traversed in one revolution, we see that when the top is spun rapidly, so that the precession is slow, the rise and fall is very / c*\ rapid (for b = r -j-j ? and very small. For this reason it is seldom noticed, and this accounts for the popular opinion, expressed in many text hooks, that the motion of a top is such that its axis describes a circular cone with a constant angular velocity, or a regular precession. Thus the reason of the vertical force of gravity producing a horizontal motion remains a paradox. We have seen that such a motion is the very particular exception, and not the rule, being only exhibited when the necessary horizontal velocity is imparted at the outset, so that the action of gravity is always balanced by the centrifugal couple generated by the precession. If the necessary velocity is not imparted, the top immediately begins to fall in Fig. 101 a. Fig. 101 b. obedience to gravity. The motion which we have just described is called by Klein and Sommerfeld a pseudo- regular precession, and may be called a small oscillation about a regular precession. In Fig. 101 are shown curves of the actual path obtained by photographing a small incandescent lamp attached rig. 101 c. to the axis of a gyroscope, with 0- nearly a right angle. 94. Small Oscillations about the Vertical. In the discussion which has just been given, it has been supposed that 1 — #2 was not a small quantity. If however in the course of the motion the axis of the top becomes nearly vertical this will no longer be true, so that for this case a special investigation is necessary. Let us suppose that & and #' are so small that in the kinetic potential all their powers above the second may be neglected. Let us use for coordinates the rectangular coordinates of the projection of the apex on the horizontal plane, x = r cos ^', y = r sin ^, r = sin #. Using then the expression of 85) for the kinetic potential, with W= Mglcosft, 93, 94] TOP NEARLY VERTICAL. 289 125) ^ = ~ A(&'2+ sin2 » - $' 2) + H, cos «• • ^ - Mgl cos #, we will convert it into terms of #, ?/, #', ?/', neglecting all terms of order higher than the second. In the first term, since , to the order of approximation, r' = cos# •»' = #', we have rt2 -f- ^2^'2, the square of the velocity in polar coordinates, which is in rectangular coordinates x'2 + y'*. Also we have 11 9 , ,• xy' — yx' tan ib = ~> sec2^-^f = — — j2 — > iC iC2 126) ^' = ^#' and since 127) cos a = { 1 - (*2 -f 2/2)}¥ = 1 - ^jp^ we have finally 128) $=i^'H We have then in the term in H, an example of the gyroscopic terms of § 50, in which x = q±, y = g2, Forming the equations of motion, since dy dx we have finally Ax" + H2y' + Mglx = 0, Ay"-H,x' + Mgly = 0, or in terms of our constants, 131) These equations are a particular case of a problem that is interesting enough to he considered in full. If & were zero, they would be the WEBSTER, Dynamics. 290 V11- DYNAMICS OF ROTATING BODIES. equations for the small vibrations of a system of two degrees of freedom , the stiffness and inertia coefficients of which are the same for both freedoms. Let us consider the general system , for which 132) T = ±(Ax"+By'*), W= ±-(Cx* + Df), into which a gyrostat, or rapidly rotating symmetrical solid , is intro- duced ; the direction of whose axis is determined as in the present case by the coordinates x and y. (It is to be noticed that x and y are principal coordinates.) The equations for the small oscillations of the system are then By" - Hzx' + Dy = 0. These may be treated by the general method of § 45 for small oscillations. In order to simplify the notation, it will be convenient to put 134) , A, , when our equations become 8" + &,' + <*-<>, V'-fcl' + cfy^O. Having solved these, we may pass to the case of our vertical top by putting c = d. In accordance with the method of § 45, let us put from which we obtain ^(*2 + C) -4&A The determinantal equation is 137) A4 + (c whose roots are 138) A2 = {-(c + If the solution is to represent oscillations, all the values of A must be pure imaginary, thus both values of ^2 must be real and negative. SMALL OSCILLATIONS ABOUT VERTICAL. 291 If we call them — ^2, — v2, we have for their sum and product the coefficients in 137), 139) ii* + v* = c + d + tf, [i*v* = cd. In order that p, v shall be real it is accordingly necessary that c, d shall be of the same sign, that is our system must be either stable for both freedoms , or unstable for both. Extracting the square root of the second equation 139) , doubling, and adding to and subtracting from the first, * v2 = c d b* 2~/cd = ¥ ~ Extracting the roots, adding and subtracting, 141) The inner double sign is evidently unnecessary. Since fi v = + we have also, From the values of [i and v it is evident that both are real if c and d are positive. If tehy are negative it is necessary in order to have real values that b > ~^c + y^d. Thus we find that even if the system is unstable, sufficiently rapid spinning of the gyrostat makes it stable. This is the case of the top with its center of mass over the point of support. In order to complete the discussion we have to determine the coefficients Alt A^ for the various roots. If we call the roots 19 292 VII. DYNAMICS OF ROTATING BODIES. we have for the general solution , | = 41' e"" + Af e~if" + Af e"" + A(? e'1", n = 41' «"" + Af e-"" + Af e1" + Af e~ivt, where we have by the first of equations 136) , 41' .„_„• 2) ' 3) « > 1 /LA 1 " -— - •> _ ~ >* 1~ ' Introducing the values of the J.2's in terms of the A,? a, and writing 145) 41)+42)=«, 41)-42)=-ift43)+44)=«', 43)-A^=-in', we have, replacing exponentials by trigonometric terms, | = « cos lit + (Ism [it + «' cos v^ 4- /3f sini/tf, 146) e _ u.2 c — ?? = -- (j3 cos ^t — asi with the four arbitrary constants a, ft «', /3', or putting 147) a = ^ cos sl9 /3 = ^ sin £1; a' = A% cos «2; /3' = A2 sin | = Al cos (^^ — fj) -f -^2 cos (v^ ~ ^2)7 148) ft2- sin t - f -- sin v - Accordingly the motion may be described as the resultant of two elliptic harmonic motions of frequencies ^? ^7 the directions of the axes of the ellipses being coincident, and given by the directions in which the system can make a principal oscillation when the gyrostat is not spinning. The absolute sizes of the two ellipses are arbitrary, but the ratios of the axes, and the phases, are determined by the nature of the system and the rapidity of the spinning. Calculating the coefficients in 148) from the values of /it, — > v, — > 149) bv 94] EXPERIMENTAL REALIZATION. 293 If now c = d, as in the case of the top, both these expressions become equal to plus or minus unity, so that both ellipses become circles. The motion of the top making small oscillations about the vertical is accordingly to be described by saying that its apex describes epicycloids (epitrochoids) or hypocycloids (hypotrochoids) upon the horizontal plane. It is to be noticed that according as we take the signs in 149) the relative sense of the rotations in the two circular motions will be alike or different. By considering which way the top tends to fall we may decide whether the cusps are turned inwards or outwards, and it will be found that if the center of mass is above the point of support the cusps or loops are turned inwards, and the curves are epicycloids, while if it is below the cusps or loops are turned outwards and the curves Mg 102 are hypocycloids. An instrument to show these properties of the motion has been constructed by the author, and is shown in Fig. 102. A heavy Fig. 103 a. Fig. 103 b. symmetrical disc hangs by a universal joint (Hooke's or Cardan's suspension) from a shaft which is rotated by an electric motor. 294 VII. DYNAMICS OF ROTATING BODIES. A pointed steel wire slides easily in the end of the axis of the pen- dulum, and draws a curve upon a plate of smoked glass which is Fig. 103 c. Fig. 103d. brought against it by a lifting table: By means of a lantern and a Fig. 103 e. Fig. 103 f. right angled prism the curves are projected upon a wall in the act Fig. 103 g! Fig. 103 h. of being traced. Examples of the curves obtained are shown in Fig. 103. (Figs, g, h, i are hypocycloids drawn geometrically, for 94] COMPARISON OF THEORY AND EXPERIMENT. 295 comparison.) In order to compare theory with experiment, let us calculate how many revolutions in one circle go to one of the other. Let us call this ratio m. We have then from 141) 150) m = ± V Fig. 1031. It is noticeable that this ratio depends only on the constants of the system and the velocity of spinning, hut not on the circum- stances of projection. This is shown in the figures. In each group m is made an integer, by properly adjusting the height of the disc, and the rate of spinning, which is main- tained constant by stroboscopic ob- servation. If the apex is merely drawn aside, and let go, the curves have cusps. If pushed to one side, the curves have loops, and if to the other, there are no loops, but the curve is a sort of curvi- linear polygon, and if the spinning is rapid enough, there are in- flexions. The ratio m is the same for the three types of curve. The slight perturbations no- ticeable in the figures arise from the slight loose- ness in the tracing point, and permit p.g 1Q4 of counting the number of revolution of the top about its axis (thus determining r), 296 VII. DYNAMICS OF ROTATING BODIES. which is found to be the same for the same value of m, as may be verified on the figures. In order to illustrate the more general case above treated, the spinning top is included in a system of two pendulums (Fig. 104), whose frequencies may be made to have any ratio to each other, so that when the top is not spinning the point describes a Lissajous's curve. The influence of the spinning on the curves is shown in Fig. 105. Fig. 105 a. Fig. 105 b. An interesting application of the heavy symmetrical top is the gyroscopic horizon invented by Admiral Fleuriais of the French navy. A small top is spun upon a pivot in vacuo, in a box which is attached to a sextant. The top executes a slow movement of precession Fig. 105 c. Fig. 105 d. about the vertical, and by means of lines ruled on two lenses which it carries, the vertical is observed, so that observations may be made when the horizon is obscured by fog.1) 1) Schwerer. L'horizon gyroscopique dans le vide de M. le Contre-Amiral Flewiais. Annales Hydrographiques. 1896. " ! T 94, 95] APPLICATION OF JACOBFS METHOD. 297 ^^^^•HMfiflW*^^^ 95. Top Equations deduced by Jacotai's Method. We will conclude the treatment of the top by deducing the equations of motion by the method of Jacobi, § 41. Since we have for the kinetic energy, 77) T= and for the momenta p# = Aft', py = J.sin2# • ^' 4- C cos 0- (9' p(p = C(, ty, plus a function & of -9-, which we will determine. We shall obtain the result in the notation of § 90 if we, put 154) S=-ht + A(bg> + ^ + ®). nserting in the differential equation 153), we obtain 155) from which 156) Accordingly we have the solution, 157) S = --ht + A (by + ^ The integrals are obtained by differentiating by the arbitrary constants, *, 6, ft, 298 VII. DYNAMICS OF ROTATING BODIES. dS Bearing in mind that -Fffr) = . vL> and that -- = r, we see that Sill $T O the first equation is the integral of equation 78), the second of 79), and the third of 80). 96. Rotation of the Earth. Precession and Nutation. Since the earth is not an exact sphere, it is not centrobaric, that is the direction of the resultant of the attraction of its various parts on a distant point does not pass through its center of mass. Or, in other words, the attraction of a distant mass -point, not passing through the center of mass of the earth, possesses a moment about it, which tends to tilt the earth's axis. The sun and moon are so nearly spherical that they may ^.^ be considered as concentrated • •< — r~/\ a^ their respective centers of 4<^_J mass. One of them, placed Fig. 106. at M (Fig. 106), attracting the nearer portions of the earth more strongly than the more distant ones, tends to tip the earth's axis more nearly vertical in the figure, and it is seen that this is the same in whichever side of the earth the body lies. Thus the sun always tends to make the earth's axis more nearly perpendicular to the ecliptic, exept when the sun lies on the earth's equator, that is at the equinoxes. The deflecting moment thus always tends to cause a motion of precession in the same direction, the tendency being greatest at the solstices, and disappearing at the equinoxes. The moon, which moves nearly in the plane of the ecliptic, produces a similar effect. It will be shown, in § 148, that the potential of a body at a distant point, x, y, 8 is given very approximately by J, - -- - -2 - -pr- where r2 = x2 -f y2 + £2, and A, B, C, are the principal moments of inertia of the body. If the distant point is the center of the sun, whose mass is m, the force exerted by the earth on the sun is \zr\\ v 2F T/ dV dV X = rm-, Y=ym, Z 95, 96] ROTATION OF THE EARTH. 299 But this is equal and opposite to the action of the sun upon the earth, the moment of which about the earth's center of mass is accordingly 161) M=-(eX-xZ), Differentiating the expression 159), since x appears both explicitly, and implicitly in r, and -^- = — ? J dx r dv^dv^ dx dr r 162) |F = |I^ + oy or r fa — + cV dV z_ r and inserting in 161), L = - 163) We may now insert these in Euler's equations, so that, if x, y, z, the coordinates of the sun, are given as functions of the time, the earth's motion may be found. Considering the earth to be symme- trical about its axis of figure, we put A = _B, so that N = ,0, and the third equation gives r = const., as in the case of the top. It is however more convenient for our purpose to use, instead of Euler's equations a set of equations proposed by Puiseux, Resal, and Slesser, in which we take for axes, as suggested in § 84, the axis of symmetry, and two axes perpendicular to it, that is, lying in the equator, and moving in the earth. We have, since we are dealing with principal axes 164) Hx = Ap, H, = Aq, Hz=Cr, which are to be inserted in equations 29), § 84, where we are to' put the velocities with which the moving axes turn about themselves, which we will call ^0, g0, r0, so that our equations are 165) dH, dt dH dH 300 VII. DYNAMICS OF ROTATING BODIES. If we choose as X-axis the line of nodes, or intersection of the equator with the ecliptic, or plane of the sun's orbit about the earth, we have, in 65), cp = 0, so that Euler's geometric equations become simply, 1 a while r is n°t equal to rQ. Inserting in the third equation 165), we have C— = 0, r = const. = &, where i& is the angular velocity of the earth's daily rotation. We shall content ourselves with an approximate solution of the equations, which may be obtained by neglecting the squares and products of small quantities. Observations show that -^ and -r- are / dib \ small, l-j-j- = 50",37 per year), so that we may neglect r0g0, Thus our equations 165) become 167) If the sun, or other disturbing body, did not move with respect to the axes of X, Y", Z, then Z, M would be constant, and the equations would be satisfied by constant values of j?, #, 168) M In order to ascertain whether these approximations are sufficient when L and M vary, let us differentiate equations 167), substituting in either the value of the first derivative of p or q from the other, obtaining 169) dt* A CO, A ^.T^ 4- M) = dL ~dt"' dM dt Fig. 107. We have now to find the values of L, M in terms of the motion of the sun. If I be the longitude of the sun, that is the angle its radius vector OS makes with the X-axis, we have, passing a plane through the sun perpendicular to the X- axis (Pig. 107), x = r cos I, y = r sin I cos #, £ = r sin I sin -fr, 96] PRECESSION AND NUTATION. 301 so that, inserting in 163), jj = 7 — L_ — 1 sm2 1 sm ft, cos ^ 170) Ti/r 3ym(A— C} . ^ 7 . M = ~ ' sin 6 cos Z sin &. If we suppose the sun's path relative to the earth to be a circle, described with angular velocity n, we have so that 171) d L &yin (C A) \ 3ym'(A—C) f 07 . . 7 ^d& rs" I*'1 COS ^ Sm ^ + sm * cos ^ COS # -TT dt Now if A = C, there would be no motion of the earth's axis, so that C — A is a small quantity of the order of -=J • The angular velocity n, though much larger, is still 365^- times smaller than i&, so that if we neglect its product and that of ^- with C — A, we may neglect the right hand sides of 169). Thus the approximation 168) is justified, for differentiating, it will make ~ > -jrf negligible, so that equations 167) are satisfied. Inserting the values of p, q, L, M, in 168), we have (C~ ^cosfrd -cos2r>, These are the equations for the precession and nutation. In order to integrate them approximately, we may neglect the small difference, on the right, between # and its mean value, so that inserting the value of 21 = 2nt -\- 210, considering & constant, and integrating, 3ym C - A _, * - ifir* "(T COB*' " J 3ym C-J. ,, * = sm * We thus find the motion to be a regular precession, of amount t rr ,j\ i 3ym (7— ^4. 174) * =db^r-cos*' together with a nutation in an ellipse (compare § 93), whose period is one -half that of the revolution of the disturbing body. 302 VH. DYNAMICS OF ROTATING BODIES. By means of observations of the value of the precession, we tQ ^\ may thus obtain the ratio of - — ~n~^' We see that ^ne forces causing precession are proportional to —s- On account of the nearness of the moon, therefore, and in spite of its small mass, the precession produced by the moon is somewhat greater than that due to the sun. Since the moon's orbit departs but little from the plane of the ecliptic the precession due to the moon may be calculated approxi- mately by the above formulae, and compounded with that due to the sun. 97. Top on smooth Table. Having treated in detail the motion of a body with one point fixed, and three degrees of freedom, it remains to consider the motion of bodies which, like the ordinary top, spin upon a table or other surface. We must now consider the reaction between the body and the surface, and we have to distinguish between the ideal case of perfectly smooth, or frictionless bodies, where the reaction is normal, and bodies between which there is friction, so that the reaction is not normal. We will consider the first case. Let us examine the motion of a symmetrical top, spinning on a sharp point resting on a smooth horizontal plane. The top has five degrees of freedom, its position being defined as before by the three angles -O1, ^, (p, and in addition, by the coordinates x, y, of the center of mass, the #- coordinate being given by 0 = I COS ft. Since the only force which we have not already considered is the reaction, which has no horizontal component, the horizontal component of the acceleration of the center of mass vanishes, so that its motion is in a straight line with constant velocity. It therefore remains only to determine the motion of rotation. This being in- dependent of the horizontal motion just found, we may consider the latter to vanish, so that the center of mass will be supposed to move in a vertical line. The motion thus becomes one of three freedoms, and we shall treat it by Lagrange's method as before. By the principle of § 32, 50), the kinetic energy is equal to that which the body would have if concentrated at its center of mass, plus that which it would have if it performed its motion of rotation about the center of mass supposed at rest. If then A and C denote the moments of inertia about the center of mass (in § 90 they were the moments about the fixed point), we have 175) T=~ [M Z 96, 97, 98] TOP ON" TABLE. FRICTION. 303 The potential energy is as before Mglcosfr. Consequently the only difference in the problem from that treated in § 90 is in the extra term in #', Ml* sin* & • &'2 in the kinetic energy. Carrying out the various steps of §§ 90, 91, we find instead of the first equation 76) the equation 176) # and putting s = cos #, d (d*\*— \dtf- where we denote the roots of the denominator by #4, z5. It is to be noticed that they lie outside the interval 1, — 1, for evidently the coefficient of #'2 in 176) cannot vanish for real values of -9-. The square of -j-- being now the quotient of two polynomials in z, s is a hyperettiptic function of t. We may however, without a knowledge of these functions, treat the problem just as we did the former one, and we shall find that the top in general rises and falls between two of the roots of the numerator, and that the motion resembles the motion already discussed. The path of the peg has loops, cusps, or inflexions, according to the initial conditions, as before, while the regular precession and the small oscillations may be investigated as before. Whereas accordingly the functional relations involved are considerably different, physically this motion, which is that of the common top, closely resembles that already studied. 98. Effect of Friction. Rising of Top. We have now to take account of the effect of friction. Here we have in addition to the normal component of the reaction a tangential component called the force of friction, and the ordinary law assumed is that the tangential component is equal to the normal component multiplied by a constant depending on the nature of the two surfaces in contact, called the coefficient of friction. If the friction is less than a certain amount, the two surfaces will slide one upon the other, and the direction of the friction will be such as to oppose the sliding, being in the direction of the relative motion of the points instantaneously in contact. The bodies are then said to be "imperfectly rough". If the friction is greater than a definite amount, it will prevent the sliding, and there is then no relative motion of the points of contact, so that there is a constraint due to the friction, which is expressed by an equation stating that the velocities of the points of the two surfaces in contact are equal. If one of the surfaces is at rest, as is usually the case, the instantaneous axis then always passes through 304 VII. DYNAMICS OF ROTATING BODIES. the point of contact. If it is in the tangent plane, the motion is said to be pure rolling, and the bodies act as if "perfectly rough". If the instantaneous angular velocity has a normal component, this is known as pivoting, and is also resisted by a frictional moment. The pivoting friction is however usually neglected where the surfaces are supposed to touch at a single point. The conception of perfect roughness, involving the absolute prevention of slipping under all circumstances is as far from the truth as that of perfect smoothness, nevertheless slipping may often cease in actual motions, so that motions of perfect rolling, whether or not accompanied by pivoting, are important in practice. For instance, a bicycle wheel under normal circumstances rolls and pivots, if it slips the consequences may be serious. In the following sections we shall consider the methods of treating various cases of friction. We may however, without calculation, consider the effect of imperfect friction on the motion of the top spinning on the table. Let P (Fig. 108) represent the peg, no longer considered as a sharp point. Let OH represent the angular momentum at the center of mass 0. The friction is in the direc- tion Fj opposite to the motion of the point of contact of the peg with the table. The moment of this force with respect to the center of mass is perpendicular to the plane 01$ or K. Thus the end of OH moves in the direction of K, that is rises. Thus the effect of friction is to make the top rise toward a vertical position. When it has reached that, it "sleeps" and the friction has become merely pivoting friction, tending to stop the motion. We have before seen that under conservative forces, the top would never become vertical except instantaneously by oscillation. The effect of friction on the Maxwell top may be most easily seen from the fact that the friction tends to stop the spinning, accordingly it causes a moment which is represented by a vector opposite in direction to ro, Figs. 83 a, b. Compounding this vector with H we see that the moment of momentum vector H tends to move away from the axis of the two cones in Fig. 83 b while it tends towards it in Fig. 83 a, thus the trace of the invarible axis (as it would be but for friction), instead of being an ellipse, is a spiral winding outwards in the former case, and inwards in the latter, as is shown by the arrows in Fig. 83. 99. Motion of a Billiard Ball. We wiU now treat the problem of the motion of a sphere on a horizontal plane, taking 98, 99] MOTION OF BILLIARD BALL. 305 account of friction. The friction of sliding is supposed to be a force of magnitude F= pR where R is the reaction between the ball and the table, and ^ the coefficient of friction. F has the direction opposite to that of the motion of the point of contact of the ball and table. If the axes of X, Y are taken horizontal , Z vertical, we have for the motion of the center of mass the equations ~ = X = Fcos (Fx) = [iR cos (Fx), 178) and since 0 is constant, R = Mg. Euler's dynamical equations are, since A = B=C, 179) a being the radius of the sphere. To determine the direction of JE we have IT V 180) m f-^=, where vx, vy are the velocities of the lowest point of the sphere, Differentiating these equations, and making use of 179), *vx _d*x d^_^J_ai^ 'dt~W a dt~ M" A2 182> S-3+.Jf-i+5'- • Dividing one of these by the other, and using 180), dvx dvy 20 from which WEBSTER, Dynamics. 306 VH. DYNAMICS OF ROTATING BODIES. Integrating we have 184) ^ = const. = ^- Thus we find that F makes a constant angle with the axes of coordinates , and since it has the constant magnitude pMg the center of the sphere experiences a constant acceleration, and describes a parabola. If the center of the sphere starts to move with the velocities Vx, Vy and with a "twist", whose components are ^>0, g0, r0, we have, integrating 179), since X, Y are constant, 185) q = qo-a^t, • r = r0. Integrating the equations for the center of mass Inserting in 181) we find for vx, vy i Xt> ~V 2 187) M V -£ ^"=r = =T Accordingly, X Vx-aq0 188) ^^-g, -/X2+r2 = ^, F^-««o X = - ^M^r 189) Since vX9 vy are linearly decreasing functions of the time, whose ratio is constant, they vanish at the same time t 99, 100] PURE ROLLING. 307 The sliding the ceases. Obviously it cannot change sign, so that the above solution ceases to hold. The ball now rolls without sliding, and we have always, at subsequent instants, the equations of constraint dx dy Y = &y _ n*p_ _ <* V M dt* dt " A From this we obtain so that X = Y = 0, and the ball moves uniformly in a straight line. In reality there is always a certain friction of pivoting, causing a moment about the normal, but this would only affect the rotation component r, which would not affect the motion of the center of the ball. 1OO. Pure Rolling. The preceding problem has illustrated both sliding friction and pure rolling. The treatment of the latter is interesting on account of a peculiarity in the nature of rolling constraint which makes the ordinary treatment of Lagrange's equations require modification. We shall accordingly first present the application of Euler's equations to this subject, but before doing so, we will treat by means of results already obtained one of the most important practical problems, which illustrates the steering of the bicycle, namely the rolling of a hoop or of a coin upon a rough horizontal plane. As the hoop rolls, if its plane is not vertical, it tends to fall, and thus to change the direction of its axis of symmetry. The falling motion developes a gyroscopic action, which causes the hoop to pivot about the point of contact, so that the path described on the table is not straight but curved. The pivoting motion, like the precession of the top, tends to prevent the falling, and to this is added the effect of the centrifugal force due to the curvilinear motion of the center of mass. Thus the hoop automatically steers itself so as to prevent falling, and a bicycle left to itself does the same thing. Let the position of the hoop be defined by the coordinates of its center of mass, and by the angles #, ^,9 of § 90, # being the inclination to the vertical of the axis of symmetry, or normal to the plane of the hoop at its center. We will examine the conditions for a regular precession, in which <$•, qp', ^' being constant, the center of mass and the point of contact of the hoop with the table evidently 20* 308 VII. DYNAMICS OF ROTATING BODIES. describe circles. In this case we have for the moment about the center of mass of the forces tending to increase &, by 82), 191) P^ = The forces which act to change # are, the weight of the hoop, which has zero moment about the center of mass, and the reaction of the table. Let i?, Fig. 109, (an edge view of the hoop) represent the vertical reaction, F the horizontal component due to friction, which is normal to the path of the point of contact, the tangential component disappearing on account of the assumed constancy of the velocity of rolling, as in the case of the rolling sphere. We accord- ingly have, taking moments, 192) P<> = Fa sin # - Ea cos #, F a being the radius of the hoop. But considering rig. 109. the motion of the center of mass, which is uniform circular motion, and supposing all the forces there applied, since there is no vertical motion, the resultant vertical component vanishes, or E = Mg and the horizontal component balances the centrifugal force, so that 193) F = where b is the radius of the circle described by the center of mass. Beside the dynamical equation we have the equation of constraint describing the rolling. Since there is no slipping, the rate at which the center of mass advances in its path is 194) ar = a(y1 + ^' cos#). But this is also, from the circular motion, equal to —biff'. From the equation of constraint, 195) a( 200) A f - (Cr - dr We have finally, as the conditions for rolling and pivoting, the equations stating that the velocity of the point of contact with the plane (whose coordinates are x,y, 8) is at rest. vx 4- qz = 0, 201) vy + rx - pz = 0, vz — qx = 0. The coordinates x, s of the point of contact are obtained as known functions of & from the equation of the meridian of the hody. We have accordingly the eighteen equations 197 — 201) between the eighteen quantities vx, vy, vz, p, q, r, pQ, g0, r0, X, Y, Z, #, ^, qp, Ex, Ey, E2, or just enough to determine them. The differential equations are all of the first order. The reactions may be at once eliminated from the equations 199), 200). By differentiating 201) we may eliminate the derivatives of vX9 vy, vz from 199). In doing this, however, we introduce the derivatives of x, 2, which are functions of #, so that in general the equations become complicated. We shall therefore confine ourselves to the case of a body rolling on a sharp edge, like a circular cylinder with a plane bottom, or a hoop or disk. We then have x, z constants, x = a, z = c, where a is the radius of the circular edge, c the distance of the center of mass from its plane, which is zero in the case of the hoop. 100] GENERAL MOTION OF ROLLING HOOP. 311 The equations of the motion of the center of mass thus become M ( — c TJT ~l~ aq2 ~\- PQ (Q>T — CJP)) = RX ~}~ Mg sin -91, OAON n/r I dp dr \ -n zOZ) Mi c~ — a~j cro giving Aajl + Cc~ + Car + Aapcin& = 0, 207) - Ma*p + Macpctnfr = 0, 312 VII. DYNAMICS OF ROTATING BODIES. as two equations to determine p and r as functions of #. When they are thus determined, the equation of energy 208) M(v2 + v2 + v^+A(p2+q2) + Cr2 = 2{h-Mg(a8m&-ccos&)}, or by 201), 209) (A + Mc2}p2 + (A + Ma2 + Mc2}q2 + (C+ Ma2)r2 - ZMacpr suffices to determine q as a function of #. Thus we see that when- ever & returns to a former value, the circumstances of the rolling are repeated, so that the motion is periodic. Eliminating -~ from 207), we obtain 210) MAa2p = - {AC + M (Aa2 + Cc2}} |£ - MCacr, differentiating which, we may eliminate p, obtaining for r, 0.,-,\ d*r , _ dr . MCa 211) a linear differential equation for r, with variable coefficients. In the case of the disk, where c = 0, by introducing the new variable x = cos2 #, we reduce the equation to the form ,. x d*r . /I 3 \ dr 212) which is the differential equation of Gauss 213) ^(l-^l^ + ^- if we put MCa This differential equation is satisfied by the hypergeometric series 214) F(K,ft,r,x^i + + tt(« + i)(« + a)p(/? + i)(/? + 8) 3 and by the theory of linear differential equations we find that the general integral is -f 100, 101] LAGRANGE'S EQUATIONS AND ROLLING. 313 where ct and c2 are arbitrary constants. From this we obtain Ma* dr * ~ Ma*~ d&' •j f\ ' -t t and from equation 209) we obtain a = -=-> or -=- as a function of #, at d& so that the time is given in terms of # by a quadrature. The explicit completion of the solution is too complicated to be of use in investigating the motion. The equations 207) have been investigated by Carvallo by a development in series, from which the properties of the motion are investigated. 1O1. Lagrange's Equations applied to Rolling. Noii- iutegrable Constraints. In the attempt to apply the method of Lagrange to the problem of rolling we are met with a peculiar difficulty, which has been the subject of researches by Vierkant and Hadamard.1) We shall follow the treatment of the former of the rolling of a disk. Let us characterize the position of the disk by the angles ^, ^, - & -f cos^sin^ • ^'), 220) /n' = y' - a (sin # sin ^ • -9-' — cos ^ cos ^ • ^'), £' = a cos # • #'. Squaring and adding, we obtain for the kinetic energy, 221) T= \ + 2a[— si -h cos # • ^' (— x' sin ijf -f- y' cos ?/>)]} -f |^(#'2+ sin2#- ^'2) -f |C(y + T^'cos^)2. J I Forming now the equation of d'Alembert, adding equations 216) multiplied by A and ^ respectively, and equating to zero the coeffi- cients of dx, dy, d&9 Sty, 4^, we obtain the equations of motion 101] NON-INTEGRABLE CONSTRAINTS. 315 o • e) C(q)' 4- ^'cosd-) - We must observe that if we had taken account of the equations 217) in the expression 221) for the kinetic energy, before differentiating, we should have obtained quite different equations. Having performed the differentiations, however, and introduced in the equations all the reactions belonging to the different coordinates, we may now take account of the equations of constraint, thus introducing, in effect, the statement of the equilibration of some of the reactions, and causing some of the terms to drop out. Now introducing the values of #', y\ from 217) in 222), and eliminating A, \L from 222 a, b, e), we obtain 223) Ma2 1 sin ^ -j- (sin ^ • ') 4- ^sin2# • 4- Jf a2 sin «••«•>' = 0. 316 VII. DYNAMICS OF ROTATING BODIES. Since the last terms of both these equations contains &', it is suggested that we change the independent variable from t to #, which is done by dividing through by #', giving 226a) (Ma2 -f C) ~ (yf -f cos # - ^) - Ma2 sin & • tf = 0, b) ^{(Ma2+ C) cos & dz vz = yp-xq + -ft- The first two terms, representing the vector -product of the angular velocity of the moving axes hy the position -vector of the point, represent the components of the velocity of a point fixed to the moving axes, the last terms represent the velocity relative to the moving axes. We might now, in order to find the components of the actual acceleration along the instantaneous positions of the moving axes, make use of equations 128), § 77, to obtain the velocity of the end of the velocity- vector, that is put for x,y,z the quantities vx,vy,vz, when on the left we should ohtain ax, ay, az, as has been suggested for H in § 84 (after 29) but we shall rather choose for the sake of variety, to proceed by means of Lagrange's method to find the forces tending to increase the relative coordinates x, y, z. Suppose a particle of mass m to have coordinates x, y, z in the moving system. Its kinetic energy is then that is r- !«[($'+ ®'+ -A -=^> the angular velocities Gnf d t (it of the moving axes p, q,r and their derivatives, ^ty ~> ~ u>\ ut Ci/t A point fixed to the moving system at x, y, 8 would have the accelerations • d r * > 236) 0*0 = a - * - y (r2 +^>) + q (rz+px), These may he called the components of acceleration of transportation (entramemenf) or the acceleration of the moving space. They represent the centripetal acceleration of the transported point. (If p, q, r are constant, we have in the last two terms the ordinary expressions for centripetal acceleration, whose resultant is v* divided by the distance from the axis of rotation.) Beside these and the relative accelerations there are terms T <>\ndz vdy -di -rdi These are termed the components of the compound centripetal accel- eration. We accordingly have for the total acceleration 'x, 238) dt that is the actual acceleration of the point is the resultant of the relative acceleration, the acceleration of transportation, and of the compound centripetal acceleration. Accordingly we may consider the axes at rest if we add to the actual forces applied forces capable of producing an acceleration equal and opposite to the acceleration of transportation and the compound centripetal acceleration. This is known as Coriolis's theorem. 320 VII. DYNAMOS OF ROTATING BODIES. The resultant J, often known as the acceleration of Coriolis, is evidently perpendicular to the relative velocity whose components are -jrr> ~> -^ and to the axis of jp, q, r and is equal to twice the vector-product of the angular velocity of the axes and the relative velocity of the particle. It is interesting to notice that the accel- eration of Coriolis arises from the presence of linear terms in the velocities, -^y ^| > -^ in the kinetic energy, the effect of which in introducing gyroscopic terms was explained in § 50. Thus a particle may be arranged to represent hy its motions relatively to a uniformly revolving body, such as the earth, the motions of a system containing a gyrostat. This remark is due to Thomson and Tait. 1O4. Motion relatively to the Earth. Let us suppose the axes chosen are taken fixed in the earth, the origin at the center, the #-axis the axis of rotation. Let the earth rotate with the constant angular velocity <£, which expressed in seconds is and is very small. Then p = q = 0, r = &. The centripetal accel- eration of transportation is then Accordingly for a point at rest on the earth we may consider the earth at rest, provided we add to other applied forces a centrifugal force whose components are m&2x, m&2y. This centrifugal force is 239) ro^yS^Tp = m&2Rco8 y, where E is the radius of the earth and q) is the latitude. This is a subtractive part of g, the acceleration of gravity, which is consequently greatest at the poles, least at the equator. The vertical part of the centrifugal force is m&2Rcos2cp. This acceleration is common to all bodies at rest on the earth, and hence is included with gravity in our ordinary experiments. It need not then be further noticed. There is however to be considered the apparent compound centrifugal force, — m Jx, — mJy, — mJz, which acts on bodies in motion rela- tively to the earth. - mJx= 24°) ; - m J, = 0. 103, 104] EFFECT OF EARTH'S ROTATION. 321 The equations of motion of a body acted on by forces X} F, Z are m dt 241) the terms in & having the usual property of gyroscopic forces. For a falling body we have, if the plane XZ is the vertical plane at the place of observation, X = — mg cos qp, Z = — mg sin -i/8 (£0 - £)3 ^ ~ ~3 V ~ g 104, 105] FOTJCAULT'S PENDULUM EXPERIMENT. 323 The particle falls to the east by an amount proportional to the square root of the cube of the height of fall and to the cosine of the latitude. This has been experimentally verified. 1O5. Motion of a Spherical Pendulum. We have for the pendulum the equation of constraint so that to the previous equations of motion are added terms giving 245) Multiplying by -^i ~t -^ respectively and adding, then integrating, we get the equation of energy, the gyroscopic terms disappearing, as usual. For a second integral we get as in § 23, If we assume that the oscillations are infinitely small, t i- is infinitely small, and the last term above is of the third order and may be neglected. Integrating we have 248) The equation of energy 246) becomes Inserting polar coordinates, | = r cos o, = r sn o, 21 -j. = b — SI sin

r = while if to be the velocity of rotation of the top about the ^-axis, we have for the moment of momentum, the axes being principal axes, though not fixed in the top, 255) Hx = — A& sin #, Hy = A Hz = Co. ~ dt Inserting now in the equations 29), § 84, the constraint producing a couple Z, d& ~dt 256) + n do Q -77 — at . --J-- -f dt ~ -^- = 0. dt From the last of these equations, o is constant, while from the second, neglecting i£2, we have The first equation 256) determines the constraint L. Equation 257) is the equation for the motion of a plane pendulum, § 22, so that the gyroscope will perform oscillations about a line parallel to the earth's axis, or will be in equi- librium when -91 = 0, thus afibrd- ing a means of determining the latitude. The time of a small oscillation will be, 2 which, on account of the small- ness of &, will be very great unless o be made very great. The experiment was performed with success by Foucault. In the second case let us suppose the gyroscope con- strained to move in a horizontal plane. Let us take for IT- axis the vertical, corresponding to the £-axis of Fig. 112, for the Z-axis the axis of figure of the top, making the variable angle (p with the north, towards the east, and for the X-axis a perpendicular to these (Fig. 113). The rotation of the earth gives the components — & sin #, & cos # in the direction of the £, g axes respectively (# being the co- latitude and Fig. 113. 326 Vn. DYNAMICS OF ROTATING BODIES. not variable), which give by the use of the table of direction cosines, X r z i — sinqp 0 — cosqp n — cosqp 0 sing? t 0 i 0 the values of the rotations of the axes p = SI sin # sin op, QCD\ f) CL drp T = £i sin # cos op, and for the angular momenta, o being again the velocity of spinning, Hx = A& sin # sin op, o r" i"\\ ~TT A i 4~\ d ($)\ Hz = Co. Inserting in equations 29), § 84, the constraint producing the couple Z, 4 f~\ * tt CD y^f / f~* (t QP\ yl 5i sin v cos op -T— -j- C o ( 5i cos v" — ~^^ I - J. te cos 0- - ~\ £1 sin ^ cos op = L, d2cp 260) — A -=TJ + ^15i2 sin2 ^ sin op cos op — C&& sin ^ sin op = 0, C-~ -\- A ( 5i cos ^ - ~p ] 5i sin -O1 sin op — A& (£1 cos & — -=?] sin ^ sin op = 0. V «£/ The last equation again shows that o is constant, while from the second, neglecting £i2 we have 261) -v-^ -j~ -j" »&£> sin -&• • sin op = 0. The first equation determines the constraint L. The gyroscope again performs oscillations about the meridian, with the period A which is greater the greater the latitude, being infinite at the poles. The gyroscope in this mounting therefore constitutes a dynamical compass. It is to be noticed in both cases that the equilibrium is stable for # = 0 or qp = 0 if o is positive, and for & = it, cp = it if o is negative, in other words the gyroscope tends to set its axis as nearly as possible parallel with the earth's axis, so that its direction of rotation shall correspond with that of the earth. This was clearly stated by Foucault, although he employed no mathematics. PART III THEORY OF THE POTENTIAL, DYNAMICS OF DEFORMABLE BODIES 107, 108] POINT -FUNCTION. LEVEL SURFACE. 329 CHAPTER VIII. NEWTONIAN POTENTIAL FUNCTION. 107. Point -Function. If for every position of a point in a region of space t a quantity has one or more definite values assigned, it is said to be a function of the point, or point -function. This term was introduced by Lame. If at every point it has a single value, it is a uniform function. Functions of the two or three rectangular coordinates of the point are point -functions. A point -function is continuous at a point A if we can find corresponding to any posi- tive £, however small, a value d such that when _B is any point inside a sphere of radius < d, \f(B)-f(A)\ q3 be three uniform point -functions. Each has a level surface passing through the point M. If these three level surfaces do not coincide or intersect in a common curve, they determine the point M, and we may regard the point -functions qlf q2, q^ as the coordinates of the point M. The level surfaces of qly q2, qB are the coordinate surfaces, and the intersections of pairs (q1q2)) G&fe), (Q'sQ'iX are the coordinate lines. The tangents to the coordinate lines at M are called the coordinate axes at M. If at every point M the co- ordinate axes are mutually perpendicular, the system is said to be an orthogonal system. 110. Differential Parameter. The consideration of point- functions leads to the introduction of a particular sort of derivative. If F is a uniform point -function, continuous at a point M, and possessing there the value F, and at a point M' the value F', in virtue of continuity, when the distance MM' is infinitesimal, F' -V=JV is also. The ratio y, _y ^y is finite, and as MM' — As approaches 0, the direction of MMf being given, the limit 108, 109, 110] DIFFERENTIAL PARAMETER. 331 lim AV _dV As ~ Js is defined as the derivative of V in the direction s. We may lay off x} T7" on a line through M in the direction of s a length M Q = -*— and OS as we give s successively all possible directions, we may find the surface that is the locus of Q. Let M N (Fig. 115) be the direction of the normal to the level surface at M, and let MP7 drawn toward the side on which V is greater, represent in magnitude the derivative in that direction. Let M ' and N be the inter- sections of the same neighboring level sur- face, for which V=V, with MQ and HP. Then AV AV MN MM' MN MM' As MM' approaches zero, we have lim AV dF MM' ,. AV dV lim -v-,= 5—; en MN Hence oV ds that is, the derivative in any direction at any point is equal to the projection on that direction of the derivative in the direction of the normal to the level surface at that point. Accordingly all points Q lie on a sphere whose diameter is M P. The derivative in the direction of the normal to the level surface was called by Lame l) the first differential parameter of the function V, and since it has not only magnitude but direction, we shall call it the vector differential parameter, or where no ambiguity will result, simply the parameter, denoted by P or Py. The above theorem may then be stated by saying that the derivative in any direction is the projection of the vector parameter on that direction. The theorem shows that the parameter gives the direction of the fastest increase of the function V. If V is a function of a point -function #, F= /*(#), its level surfaces are those of g, and and if T~> dV- dVdq /•»/ \$<7 P == __ = *. = f' (Q) _i, ^^ dq on ' *•*? on dn k, P = 1) G. Lame. Lemons sur Us coordonnees curvilignes et leurs diverses appli- cations. Paris, 1859, p. 6. 332 VIII. NEWTONIAN POTENTIAL FUNCTION. where the sign + is to be taken if V and q increase in the same, - if in opposite directions. Suppose now that V = f(ql} q%, q3, • • •) d V d V 0 q^ d V d q% d V 3 qs and if h±, h2, . . . denote the parameters of qly #2, . . . the above theorem gives T> /T^ N 2V 1 /7 \ , 2F7 n \ P cos (Ps) = o — h* cos (his) + o — ^9 cos (h»s) H 0 ql v q% • dV Now + -o— hi is the parameter of V, considered as a function of fa, and we may call it the partial parameter Pl} and since P/ and hi have the same sign if -g— > 0, opposite signs if g—- < 0, we have in either case -o— hi cos (his) = Pi cos (PiS). 0% Pcos (Ps) = P! cos (P^) + P2 cos (P2s) H This formula holds for am/ direction s and therefore shows that the parameter P is the geometrical sum, or resultant, of the partial parameters, T> ~p I -p I Thus we have the rule for finding the parameter of any function of several point -functions. If we know the parameters hlth%, . . . of the functions ql9 q2, . . . and the partial derivatives g — > g — > • • • we lay off the partial parameters Q Y*- in the directions hlf h2, . . . or their opposites, according as ^— > 0, or the opposite, and find the resultant of Plf P2, . . . If the functions qlt q2, ... are three in number, and form an orthogonal system, the equation gives for the modulus, or numerical value of the parameter Examples. (1) in § 108. Let the distance of M in the given direction from the plane be u. 4V=4u = ~- > where a is the cos a angle between the given direction and the given plane. p_ An cos a cos a 110] - DIFFERENTIAL PARAMETER INVARIANT. 333 If the given direction is perpendicular to the given plane P = 1. Accordingly for q{ = x, q2 = y, qB = 2, the rectangular coordinates of a point, we have Px = Py = Pz = I, and for any function f(x, y, e) ,dx) - \dy) " {-$*, The projections of P on the coordinate axes are the partial parameters This agrees with the definition already given in § 31. Consequently, if cos (sx), cos (sy), cos (ss) are the direction cosines of a direction s, the derivative in that direction ~— = P! cos (sx) -f -Pg cos (sy) + -^3 cos (S;r) 0F , v , 01? , v . aF / x = - cos s« + cos $ -f cos s^, which is the same as equation 38 a) of § 31. We have in this section defined the differential parameter in a geometrical manner, not depending on the choice of axes of coordinates. If however we take as the definition of the arithmetical value of the parameter the equation P = and then transform to other coordinates x1, y',0', by equations 109), § 76, we easily find by calculation that w, is equal to P, that is, the parameter is a differential invariant, as is at once evident from its geometrical nature. If f(x, y, z) is a homogeneous function of degree n, by Euler's Theorem, or nf= P{xcos(Px) + ycos(Py) + ^cos(P^)}. Now the ± parenthesis is the distance from the origin of the tangent plane to the level surface at x,y,z. Calling this d, nf=±P*, P = ±*f> 334 VIII. NEWTONIAN POTENTIAL FUNCTION. or the parameter of a homogeneous function is inversely proportional to the perpendicular from the origin to the tangent plane to the level surface. For example, if n = 1, 7 = ax -r by -f cz, The level surfaces are parallel planes, and the parameter is constant, V is proportional to the distance of the level surface from the origin. If n = 2, as P=2y£+£+ii For the surface, F= 1, a familiar result of analytic geometry. 111. Polar Coordinates. If we call the point functions of Examples 2, 3, and 4, of § 108, r, #, ) ' 112. Cylindrical, or Semi -polar Coordinates. If we take the rectangular coordinate s, the perpendicular distance from the Z-axis, Q7 and co the longitude, or angle made hy the plane includ- ing the point M and the .Z'-axis, we have the system of semi-polar, cylindrical, or columnar coordinates, for which we have immediately, The parameter of a function f(z, p, o) is the resultant of the partial parameters = I ' p = + — - _ L , ay ± ' 113. Ellipsoidal Coordinates. Let us now find the value of the parameter in terms of the ellipsoidal coordinates described in § 73, which are defined for a point x, yy z as the three roots of the equation 1) i y "^ M -1 = 0. The three coordinate -surfaces at any point have been proved to be mutually perpendicular at each point x, yy 2. Since the equation 1) is an identity, we have, differentiating totally, that is changing x, y,8, 1, zdz Now if d^ is the perpendicular distance of the tangent plane from the origin, we have by the last formula of § 110, 336 VIII. NEWTONIAN POTENTIAL FUNCTION. A -i so that we may write for the direction cosines of the normal, COS yd i 3) cos (my) = cos Now as we move along the normal, we have *&i dx = dn cos (**#) = ^2 , ^ * dy = ^d; 6?^ = dn cos (WA^) = 2 . C ~j~ Inserting these values in 2), x* , 2/2 -|(^fiy2 + (^ so that i 2 " In order to express this result in terms of the elliptic coordinates alone we may express x, y, 8, in terms of A, p, v. Observe that the function BY N _ 3? y* , z* -i - -* + 2 + 2 - has as roots A, ^, v, and being reduced to the common denominator (e + O(e + &2)(9 + c2) has a numerator of the third degree in Q. As this vanishes for 9 = 1, 9 = P> 9 = v, it can only be - (? - ).) (Q - p) (Q - v). 113] ELLIPSOIDAL COORDINATES. 337 Accordingly we have the identity z\ F(o\ - x* , y" i = - -1 r «" ~ Multiplying this by p + a2 and then putting p = — a2 we get 2 _ (aa + a)(aa + fO(*a + 'Q r (a2-&2)(«2-c2) and in like manner o * ' (&' -<>•)(&•-<»») _ . If A, /i, v are contained in the intervals specified in § 73, these will all be positive, so that the point will be real. If we insert these values in d%, we shall have hi expressed in terms of A, ^, v. This is more easily accomplished as follows. Differentiating the above identity 6) according to p, - ft) (g - y) f 1 1 If we put p = A, all the terms on the right except the first, b.eing multiplied by Q — A, vanish, and we have , x* 22"1" The expression on the left is — Therefore In a similar manner we find and the parameter of any function F(A, ^, v) is WEBSTER, Dynamics. 22 338 VIII. NEWTONIAN POTENTIAL FUNCTION. 114. Infinitesimal Arc, Area and Volume. If we have any three point -functions g1? g2, g3 forming an orthogonal system of co- ordinates j since their parameters are 7 __ V lh ~L __ V */3 cn^ 2 dn^ 3 dns' the normal distance between two consecutive level surfaces q1 and ql + dq± is dn± = -~i consequently if we take six surfaces &*%,& Fig. 118.* the edges of the infinitesimal curvi- linear rectangular parallelepiped whose edges are the intersections of the surfaces are and since the edges are mutually perpendicular, the diagonal, or element of arc is vrft# " dq3 the elements of area of the surfaces qlf q2, qs are respectively and the element of volume is Examples. Rectangular coordinates xf y, z. Polar coordinates rf -9-, cp, r sin -9-' dSr = r^sin&d&dcp, element of area of sphere, 12) dS#=r sin#dr dcp, element of area of cone, dS(p=rdrd& , element of area of plane, dr = 114, 115] INFINITESIMAL SPACE -ELEMENTS. Cylindrical coordinates, 0, 0, co, 339 13) dSz = yd yd a, element of area of plane, dSg = Qd&dz, element of area of cylinder, (a = dQds , element of area of meridian plane, dr Elliptic coordinates, A, ^, v. dpdv >/ Q -v}(\i- 1} (v - 2 + *0 (&1 + fQ (c2 + p) («2 + *) (fc2 + r) (c2+ v) d»» which will be referred to as Green's theorem in its second form. We shall, unless the contrary is stated, always mean by n the internal normal to a closed surface, but if necessary we shall distinguish the normals drawn internally and externally as n{ and n*. If we do not care to distinguish the inside from the outside we shall denote the normals toward opposite sides by % and w2. 1) An Essay on the Application of Mathematical Analysis to the theories of Electricity and Magnetism. Nottingham, 1828. Geo. Green, Reprint of papers, p. 25. 344 VIII. NEWTONIAN POTENTIAL FUNCTION. 116. Second Differential Parameter. If for the function U we take a constant, say 1, du du du -fa~Ty== W= °> ^==0> and we have simply 23) - jyprcos (Pvri) dS = fj d~^dS The function which, following the usage of the majority of writers, we shall denote by z/F', was termed by Lame1) the second differential parameter of V. As it is a scalar quantity it will be sufficiently distinguished from the first parameter if we call it the scalar parameter. We have accordingly the theorem giving the relation between the two: - The volume integral of the scalar differential parameter of a uniform continuous point -function throughout any volume is equal to the surface integral of the vector parameter resolved along the outward normal to the surface S bounding the volume. We may obtain a geometrical notion of the significance of z/F in a number of ways. In the neighborhood of a point 0, let us develop V by Taylor's theorem, calling the coordinates of neighboring points with respect to 0, xy y, z, then where the suffix 0 denotes the value at 0. Integrating the value of V -- F0 throughout the volume of a small sphere with center at 0, we have <£).///•*+! > dr* s W "*" a dx " dy dz and the theorem 23) becomes 35) - ffp cos (Pn) dS = -JJ[X cos (nx) + Fcos (ny) If P is everywhere outward from the surface S, the integral is positive, and /dx . ar, ^^\^n mean ( - — \- -5 — \~ -$— ) > 0. \dx ' dy tiz J Q -yr Q -T^- O ^ Accordingly -=— + ^ -- \- -^— is called the divergence of the vector <7ic ' oy cz point -function whose components are X; Y, Z, and will be denoted by div. R. The theorem as given in equation 35) may be stated as follows, and will be referred to as the DIVERGENCE THEOREM: The mean value 348 VIII. NEWTONIAN POTENTIAL FUNCTION. of the normal component of any vector point -function outward from any closed surface S within which the function is uniform and con- tinuous, multiplied ~by the area of the surface, is equal to the mean value of the divergence of the vector in the space within S multiplied ty its volume. The theorem is here proved for a vector which is the parameter of a scalar point -function V, but it is evident that it may be proved directly whether this is the case or not by putting in equation 17) for W and x successively X, Y, Z and x, y, s respectively. Let us consider the geometrical nature of a vector point -function E whose divergence vanishes in a certain region. In the neighborhood of any point, the vector will at some points be directed toward the point and at others away. We may then draw curves of such a nature that at every point of any curve the tangent is in the direc- tion of the vector point -function E at that point. Such curves will be called tines of the vector function. Their differential equations are o£»\ dx dy dz ~X="Y~-~-~^' Suppose that such lines be drawn through all points of a closed curve, they will generate a tubular surface, which will be called a tube of the vector function. Let us now construct any two surfaces Sl and 8.2 cutting across the vector tube and apply the divergence theorem to the portion of space inclosed by the tube and the two sur- faces or caps S1 and S2. Since at every point on the surface of the tube, E is Fig. 123. tangent to the tube, the normal component vanishes. The only parts contributing any- thing to the surface integral are accordingly the caps, and since the divergence everywhere vanishes in r, we have 37) C CE cos (En±) dS±+ f (*R cos (JR^) dS2 = 0. SL S2 If we draw the normal to S2 in the other direction, so that as we move the cap along the tube the direction of the normal is continuous, the above formula becomes 38) (JE cos (En^ d8± = I I E cos (Rn^ dS2, st s2 or the surface integral of the normal component of E over any cap cutting the same vector tube is constant. 117] SOLENOIDAL VECTORS. 349 Such a vector will be termed solenoidal, or tubular, and the O ~y Q -T7" O f7 condition - - + - + = 0 will be termed the solenoidal condition ex cy oz (Maxwell). We may abbreviate it, div. E = 0. If a vector point- function JR is lamellar as well as solenoidal, the scalar function V of which it is the vector parameter is harmonic, for 0X . dY . dZ A solenoidal vector may be represented by its tubes, its direction being given by the tangent to an infinitesimal tube, and its magnitude being inversely proportional to its cross -section. As an example of a solenoidal vector we may take the velocity of particles of a moving liquid. If the velocity is B, with components X, Y, Z, the amount of liquid flowing through an element of surface dS in unit time is that contained in a prism of slant height E, and base dS, whose volume is E cos (En) dS. The total flux, or quantity flowing in unit time through a sur- face $, is the surface integral I I EGOS (En) dS = I I [X cos (nx) -f Fcos (ny) -f Zcos (w*)] dS. Such a surface integral may accordingly be called the flux of the vector E through S. A tube of the vector E is a tube through whose sides no fluid flows, such as a material rigid tube through which a liquid flows, and the divergence theorem shows that as much liquid flows in through one cross -section as out through another, if the solenoidal condition holds. If the liquid is incompressible, this must of course be true. As a second example of a solenoidal vector we have any vector which is the curl of another vector, for d (dZ 2Y\ _d_l^_<^2\ , JMU_^| =0 J^\dy~ dz\ + dy\ dz dx\^~dz\dx dy identically. The equation is called Laplace's equation, and the operator _a* d^ d^ ~ dx* + dy*~^~ cz^ Laplace's operator. 350 VIII. NEWTONIAN POTENTIAL FUNCTION. The parameter z/F is often called the Laplacian of V. div. P = AV^ 0 Fig. 124 c. div.P=AV = 0 Fig. 124 b. In Fig. 124 a, b, c, are graphically represented regions of divergent, solenoi- dal, and convergent vector s, with the level surfaces of the functions V of which they are the vector parameters. The arrows on the vector lines show the direction of increase of V, and it is evident that Fhas positive concentration (and a maximum value) where P is convergent, negative concentration (and a minimum value) where P is divergent, and no concentration (nor maximum) where P is solenoidal. 118. Reciprocal Distance. Gauss's Theorem. Consider the scalar point -function, F=— > where r is the distance from a fixed point or pole 0. Then the level surfaces are spheres, and the para- meter is and since hr = 1, - dr\r drawn toward 0 (§ 110). Consider the surface integral of the normal component of E directed into the volume bounded by a closed surface S not con- taining 0, or as we have called it, the flux of E into S, 40) C CE cos (En) dS = - fC± cos (rn) dS. 117, 118] GAUSS'S THEOREM. 351 The latter geometrical integral was reduced by Gauss. If to each point in the boundary of an element dS we draw a radius and thus get an infinitesimal cone with vertex 0, and call the part of the surface of a sphere of radius r cut by this cone d2!, dZl is the projection of dS on the sphere, Fig. 125, and as the normal to the sphere is in the direction of r, we have Fig. 125. ± dScos(rn), the upper sign, for r cutting in, the lower for r cutting out. If now we draw about 0 a sphere of radius 1, whose area is 4#, and call the portion of its area cut by the above-mentioned cone do, we have from the similarity of the right sections of the cone da> The ratio d& is called the solid angle subtended by the infinitesimal cone. Accordingly dScoB (nr) 41) . 7 = + - r — -+- do. - r2 Now for every element dm, where r cuts into S, there is another equal one, —do, where r cuts out, and the two annul each other. Hence for 0 outside S, 42) .. If on the contrary, 0 lies inside S, the integral I I dot is to be taken over the whole of the unit sphere with the same sign, and consequently gives the area 4 jr. Hence for 0 within S, 43) rfcos( JJ-1 These two results are known as Gauss's theorem, and the integral will be called Gauss's integral.1) 1) Gauss, Theoria Attractions Corporum Sphaeroidicorum Ellipticorum homogeneorum Methodo nova tractata. Werke, Bd. Y, p. 9. 352 VIII. NEWTONIAN POTENTIAL FUNCTION. These results could have been obtained as direct results of the divergence theorem. For the tubes of the vector function E are cones with vertex 0. If 0 is outside 8, E is continuous in every point within S, and since the area of any two spheres cut out by a cone are proportional to the squares of the radii of the spheres, we have the normal flux of equal for all spherical caps. Consequently E is solenoidal, and the flux through any closed surface is zero. If 0 is within S, E is solenoidal in the space between 8 and any sphere with centre 0 lying entirely within S, and the flux through 8 is the same as the flux through the sphere, which is evidently — 4 it. The fact that E is solenoidal and V harmonic may be directly shown by diiferentiation. If the coordinates of 0 are a, b, c, 44) r2 = (x - a)2 + (y - 6)2 + (* - c)2, AK\ cr x — a Or y — b dr z — c 45) 7j- = -- 9 ~— = - - ? TT- = -- ; ex r oy r oz r —- _ dx\r r*dx~ rs ' _ _ I , 3 (a?-«) ll' = 3(a?-a)2-r8 f ~ f * ~ 5 8. (I) g.(I 48) ^ (i) = W + \r/ dx* dyz and — is harmonic, except where r = 0. 119. Definition and fundamental Properties of Potential. We have seen in § 28, 34) that if we have any number of material particles m repelling or attracting according to the Newtonian Law of the inverse square of the distance, the function = -y^ + ?p + ...+ 'i '2 118, 119, 120] POTENTIAL FUNCTION. 353 where r19 r2, . . . rn are tlie distances from the repelling points, is the force -function for all the forces acting upon the particle ms. If we put the mass ms equal to unity, at a point P whose coordinates are x, y, 2 the function 49) F=^ + ^ + . .+ ^=V- n '» rn ^r is called the potential function at the point P of the field of force due to the actions of the particles mlt m2, ... mn, and y times its negative vector parameter, 50) x = -rdJ, Y--rg, z--&, f cx r dy * dz is the strength of the field, that is, the force experienced hy unit mass concentrated at the point xy y, 0.1) Since any term — - possesses the same properties as the func- i Tr tion -9 § 118, we have for every term, for points where r is not equal to zero, /4 \-\ = 0, and consequently 12O. Potential of Continuous Distribution. Suppose now that the attracting masses, instead of being in discrete points, form a continuously extended body K. Let the limit of the ratio of the mass to the volume of any infinitely small part be o = lim — •> which is called the density. Let ^r=0 4* the coordinates of a point in the attracting body be a, &, c. 1) It is more usual among writers on attracting forces to write the force as the positive parameter of the potential. The convention above adopted in 49) amounts to defining the potential as the work necessary to remove the attracted particle of unit mass from the given point to infinity against the attracting forces, thus keeping the potential function positive, instead of negative as in § 28 (end). It is the usual practice to adopt such units that y is equal to unity. In order to preserve consistency with the units previously employed and at the same time not to be obliged to introduce y throughout all the equations of this chapter, we shall define potential as above 49) and introduce the factor y into those equations which involve the relationship of the force to the potential. If the force is attractive, y will be negative, and putting y = — 1, we get the usual formulae. Putting y = -|- 1, our notation agrees with that customary for electricity and magnetism , for example in the author's Theory of Electricity and Magnetism. WEBSTER, Dynamics. 23 354 VIII. NEWTONIAN POTENTIAL FUNCTION. The potential at any point P, x, y, s, due to the mass dm at Q, a, 6, c, is dF=^, where r is the distance of the point x, y, z from the attracting point at a, b, c. The whole potential at x, y, z is the sum of that due to all parts of the attracting body, or the volume integral Now we have dm = ydr, or in rectangular coordinates rig. 126. dr^dadbdc, dm = gdadbdc. If the body is not homogeneous, p is different in different parts of the body Ky and is a function of «, &, c, continuous or discon- tinuous (e. g. a hole would cause a discontinuity). Since 53) It is For every point x, y, 2, V has a single, definite value, accordingly a uniform function of the point P, x,y, 0. It may be differentiated in any direction, we may find its level surfaces, its first differential parameter, whose negative multiplied by y is equal to the whole action of K on a point of unit mass, and the lines of force, normal to the level, or equipotential surfaces. If for any point x, y, z outside K, r± is the shortest distance to any point of K, and r2 is the greatest distance, we have for any point in K dm dm 120, 121] POTENTIAL OF CONTINUOUS DISTRIBUTION. 355 Since r± and r2 are constant, ' Now since / / / dm = M, the whole mass of the body K, the K above is 54) f '.,r-t ~s cos (rx) > - -^ cos (rx) > - -^ cos (r a?) . rz - rt Multiplying and dividing the outside terms by cos A and integrating, Multiplying by jR2 and letting J5 increase without limit, since ,. JS2 ,. E* ,. cos(ra?) lim -T = I™ — 5- = hm — = 1, = -Jfcos^l, 62) lim [jR2 1?1 = - M cos ^, Therefore the first derivatives of F, and hence the parameter, vanish at infinity to the second order. In like manner for the second derivatives, a2F a« rcr^dr rrr v /i\ , ^ = ^ J J J ~ = J J J ? w (7) dr Every element in all the integrals discussed is finite, unless r = 0, hence all the integrals are finite. We might proceed in this manner, and should thus find that: 121, 122] DERIVATIVES OF POTENTIAL. 357 At points not in the attracting masses, V and all its derivatives are finite and (since their derivatives are finite) continuous, as well as uniform. Also since 63) we have by addition 64) f *j V y U fJ that is, F satisfies Laplace's equation. This is also proved by applying Gauss's theorem [§ 118, 42)] to each element r 122. Points in the Attracting Mass. Let us now examine the potential and its derivatives at points in the substance of the attracting mass. If P is within the mass, the element — at which the point Q, where dm is placed, coincides with P, becomes infinite. It does not however, therefore follow that the integral becomes infinite. Let us separate from the mass K a small sphere of radius s with the centre at P. Call the part of the body within this sphere K' and the rest K1'. Call the part of the integral due to jfiT', Ff, and that due to K", F". Now since P is not in the mass K", F" and its deri- vatives are finite at P, and we have only to examine Ff and its derivatives. Let us insert polar coordinates Fig. 127. 000 so that, integrating first with respect to (p and #, since the absolute value of an integral is never greater than the integral of the absolute value of the integrand, 358 VIII. NEWTONIAN POTENTIAL FUNCTION. 65) \r | if $m is the greatest value of Q in K'. As we make the radius e diminish indefinitely this vanishes, hence the limit is finite. In like manner for the derivative d— = dx~ Separate oft K' from X". The part of the integral from K" is finite. In the other K' introduce polar coordinates, putting & = (rx), 66) dV dx I dr I I \ sin -9- cos # | d&d -^- • oy oz 32V If we attempt this process for the second derivatives -~-^i • • • it fails on account of — > which gives a logarithm becoming infinite in the limit. BV We will give another proof of the finiteness of — • We have which by Green's theorem is equal to This is however only to be applied in case the function -- is everywhere finite and continuous. This ceases to be the case when P is in the attracting mass, hence we must exclude P by drawing a 122, 123] DERIVATIVES WITHIN MASS. 359 small sphere about it. Applying Green's theorem to the rest of the space KUj we have to add to the surface -integral the integral over the surface of the small sphere. Since cos (nx) <^ 1, this is not greater than gm I I — = 4:iteQm, which vanishes with £. Hence the infinite element of the integrand contributes nothing to the integral. In the same way that ^— was proved finite, it may be proved dV 3V" continuous. Dividing it into two parts -5— and -~ — > of which the ox ox second is continuous , we may make, as shown, -~ — as small as we please by making the sphere at P small enough. At a neighbor- ing point P! draw a small sphere, and let the corresponding parts , 3V' , 3V" mi , 3V' be -^ and -5-*— Then we can make ~^±- as small as we please, ox ox ox dV 3V' and hence also the difference -~ --- -^L- Hence by taking P and 3V P! near enough together, we can make the increment of ^- as small o x 3V as we please, or ^— is continuous, and accordingly the second derivatives are finite. 123. Poisson's Equation. By Gauss's theorem [§ 118, 43)], we have when r is drawn from 0, a point within 8. Multiplying by m, a mass concentrated at 0, and calling F=™> 68) ~ cos (>r) dS = - The integral n} ds> where n is the internal normal, is the surface integral of the outivard normal component of the parameter yP, or the inward component of the force. The surface integral of the normal component of force in the inward direction through S is called the flux of force into 8, and we see that it is equal to — 4#y times the element of mass within S. Masses without contribute nothing to the integral. Every mass dm 360 VIII. NEWTONIAN POTENTIAL FUNCTION. thin 8 contributes to the potential at to the flux through the surface S. Therefore the whole situated within 8 contributes to the potential at any point and mass ; when the potential is F>= / / I -— > contributes to the flux K rrr = " **rJJJ ' K and S K Now the surface integral is, by the divergence theorem, equal to 70) The surface 8 may be drawn inside the attracting mass, provid- ing that we take for the potential only that due to matter in the space V within S. Accordingly for r we may take any part whatever of the attract- ing mass, and 71) As the above theorem applies to any field of integration what- ever, we must have everywhere 72) z/F+ 4^0 = 0. This is Poisson's extension of Laplace's equation, and says that at any point the second differential parameter of F is equal to - 4# times the density at that point. Outside the attracting bodies, where Q = 0, this becomes Laplace's equation. In our nomenclature, the concentration of the potential at any point is proportional to the density at that point. A more elementary proof of the same theorem may be given as follows. At a point x,y,2 construct a small rectangular parallel- epiped whose faces have the coordinates x, x -f g, y, y -f vj, 0, z + 6, and find the flux of force through its six faces. At the face normal to the a;- axis whose x coordinate is x let the mean value of the force be - d~ = - Px. dx x 123] POISSON'S EQUATION. 301 The area of the face is 17 £, so that this face contributes to the integral — / / Pco$(Pn)dS the amount — |i^g. o i y- At the opposite face, since o— is continuous, we have for its value terms of hiher order in and therefore, the normal being directed the other way, this side. contributes to the integral the amount and the two together d*V ^y^Wx* ~^~ ^erms °^ higher order. Similarly the faces perpendicular to Y-axis contribute o~ d*V and the others %>n^ i [V Thus the surface integral is and by Gauss's theorem this is equal to where Q is the mean density in the parallelepiped. Now making the parallelepiped infinitely small, and dividing by |^g, we get An important application of Poisson's equation has been made to the attraction of the earth. The acceleration g is made up of the resultant of the attraction of the earth and of the centrifugal accel- eration. Since the latter has the components RPx, &?y along axes perpendicular to the axis of rotation (§ 104), it has the potential function — (x2 -f i/2) , so that if y denote the positive value of the gravitation constant, and n the inward normal to an equipotential surface, we have, putting where 362 VIII. NEWTONIAN POTENTIAL FUNCTION. is the potential of the earth's attraction. But by Poisson's equation, so that we have, 74) Now by the divergence theorem, =-//!>. so that Now if the volume of the earth be v, its mean density gm, the volume integrals are respectively equal to gmv and v, so that, multiplying by this becomes 7Ax &2 76) ^m = Thus if we know the value of g at every point on an equipotential surface, we obtain the value of the product yqm in terms of the angular velocity, and the surface integral of g. Using a formula given by Helmert representing the results of geodetic determinations of g, Woodward1) finds for the value of j>pm = 3.6797 x 10 Richarz and Krigar-Menzel2) obtain, in a similar manner, y$m = 3.680 x 10~ 7 see"2. Combining this result with Boys's value of 7, p. 30 (see erratum), we obtain for the mean density of the earth the value 124. Characteristics of Potential Function. We have now found the following properties of the potential function. 1st. It is everywhere holomorphic, that is, uniform, finite, con- tinuous. 1) Woodward, The Gravitational Constant and the Mean Density of the Earth. Astronomical Journal, Jan. 1898. 2) F. Richarz und 0. Krigar-Menzel,, Gravitationsconstante und mittlere Dichtigkeit der Erde, bestimmt durch Wagungen. Ann. der Phys. u. Chem. 36, p. 177, 1898. 123, 124, 125] CHARACTERISTICS OF POTENTIAL. 363 2nd. Its first partial derivatives are everywhere holomorphic. 3rd. Its second derivatives are finite. 4th. V vanishes at infinity to the first order, dV dV ' ~d~' ~dz vamsn *° second order, lim(E2^} = - R = K\ GX) 5th. F satisfies everywhere Poisson's differential equation and outside of attracting matter, Laplace's equation Any function having all these properties is a Newtonian potential function. The field of force X, Y, Z is a solenoidal vector at all points outside of the attracting bodies, and hence if we construct tubes of force, the flux of force is constant through any cross -section of a given tube. A tube for which the flux is unity will be called a unit tube. The conception of lines of force and of the solenoidal property is due to Faraday. Since F is a harmonic function outside of the attracting bodies, it has neither maximum nor minimum in free space, but its maximum and minimum must lie within the attracting bodies or at infinity. In the attracting bodies the equation — z/F=47tp says that the concentration of the potential at any point, or the divergence of the force from it is proportional to the density at that point, except where Q is discontinuous. 125. Examples. Potential of a homogeneous Sphere. Let the radius of the sphere be B, h the distance of P from its center, Let us put s instead of r, using the latter symbol for the polar coordinate, 364 YE!. NEWTONIAN POTENTIAL FUNCTION. Now Differentiating , keeping r constant, sds = and introducing s as variable instead of -9-, Fig. 128. If P is external we must integrate first with respect to s from h — r to h r. 77) 0 A — r Hence the attraction of a sphere upon an external point is the same as if the whole mass were concentrated at the center. A body having the property that the line of direction of its resultant attraction on a point passes always through a fixed point in the body is called centrobaric. If instead of a whole sphere we consider a spherical shell of internal radius JRt and outer R2, the limits for r being JR1? R2, B, 78) M h' We have dV M dh* h8 If, on the other hand, P is in the spherical cavity, h < JR17 the limits for s are r — h, r -\- h R2 r + h R.2 V= ^ CCrdrds = 4=*$ Crdr 79) which is independent of fc, that is, is constant in the whole cavity. /} V Hence = 0, and we get the theorem due to Newton that a homo- 125] ATTRACTION OF SPHERE. 365 geneous spherical shell exercises no force on a body within. (On account of symmetry the force can be only radial.) If P is in the substance of the shell, we divide the shell into two by a concentric spherical surface passing through P, find the potential due to the part within P, and add it to that without, getting 80) F = 8A dV dh ~ 3 h* dh*~ 3 hs Tabulating these results, 81) F dV dh 3 3 h* Plotting the above results (Fig. 129) shows the continuity of V and its first derivative and the discontinuity of the second derivative at the surfaces of the attracting mass. .We see that the attraction of a solid sphere at a point within it is proportional to the distance from the center, for if fe-o, dV and is independent of the radius of the sphere. Hence experi- ments on the decrease of the force of gravity in mines at known depths might give us the dimensions of the earth, if the earth were homogeneous. Experiment shows, however, that. this is not the case. - 129- 366 VIIL NEWTONIAN POTENTIAL FUNCTION. 126. Disc, Cylinder, Cone. Let us find the attraction of a circular disc of infinitesimal thickness at a point on a line normal to the disc at its center. Let the radius be R, thickness a, distance of P from the center In. 0 0 = 2xeQ{yh2 + R2-h], Attraction of circular cylinder on point in its axis. Let the length be I and let the point be external, at a distance Ji from the center. By the above, a disc of thickness dx at a distance x from the center produces a potential at P -x}2 -(h- x)}. Hence the whole is 84) V- Circular cone on point in axis. 126, 127] DISC, CYLINDER, CONE. 367 Let E be the radius of base, a the altitude, h the height of P above the vertex. A disc at distance x below vertex and radius r causes potential at P, and E E 85) If we have a conical mountain of uniform , density on the earth, and determine the force of gravity at its summit and at the sea level, this gives us the ratio of the attraction of the sphere and cone to that of the sphere alone, and from this we get the ratio of the mass of the earth to the mass of the mountain. Such a determination was carried out by Mendenhall, on Fujiyama, Japan, in 1880, giving 5.77 for the earth's density. m* 1S1- Circular disc on point not on axis. Let the coordinates of P with respect to the center be a, &, 0. Then s2 = a2 -f (b — r cos yt 6. This is Poisson's equation for a surface distribution. If we draw the normal away from the surface on each side, we may write 94) or l cos t WEBSTER, Dynamics. dV , dV •K h o — = •— r cos 24 370 VIII. NEWTONIAN POTENTIAL FUNCTION. 128. Green's Formulae. Let us apply Green's theorem to two functions, of which one, V, is the potential function due to any distribution of matter, and the other, U = — > where r is the distance from a fixed point P, lying in the space x over which we take the integral. Let the space t concerned he that hounded hy a closed surface S, a small sphere H of radius s about P, and, if P is without S, a sphere of infinite radius with center P. Now the theorem was stated in § 115; 22) for the normal drawn in toward T, which means outward from S and 27, and inward from the infinite sphere, as . 184. and since in the whole space r, so that 1) becomes The surface integrals are to be taken over S, over the small sphere, and over the infinite sphere. For a sphere with center at P, -5— = - -5— = __ — , dn ~ dr r r* the upper or lower sign being taken according as the sphere is the inner or outer boundary of T; and for r = oo V vanishes, hence this integral vanishes. Also O T7" j Now at infinity, -- is of order -# and being multiplied by r, still 128] GREEN'S FORMULAE. 371 vanishes. Accordingly the infinite sphere contributes nothing. For the small sphere the case is different. The first integral - / / VdG) *J *J becomes, as the radius s of the sphere diminishes, 4) -^//*— The second part however, since -^ is finite in the sphere, vanishes with s. Hence there remain on the left side of the equation only — 4:7tVP and the integral over 8. We obtain therefore the normal being drawn outward from 8. This formula is due to Green. Therefore we see that any function which is uniform and con- tinuous everywhere outside of a certain closed surface, which vanishes at infinity to the first order, and whose parameter vanishes at infinity to the second order, is determined at every point of space considered if we know at every point of that space the value of the second differential parameter, and in addition the values on the surface 8 of the function and its vector parameter resolved in the direction of the outer normal. In particular, if V is harmonic in all the space considered, we have K\ V b) . VP — — - ' and a harmonic function is determined everywhere by its values and those of its normal component of parameter at all points of the surface 8. Since r == - i | cos (nx) || + cos (ny) ~ry + cos (ne) fz } = cos(nr) a f 24 372 VIII. NEWTONIAN POTENTIAL FUNCTION. we may write 6) 7) Vp=~A Consequently, we may produce at all points outside of a closed surface S the same field of force as is produced by any distribution of masses lying inside of S, whose potential is F, if we distribute over the surface S a surface distribution of surface -density, 1 (Fcos(wr) d_V\ 8) 6 = In the general expression, 5) the surface integral represents the potential due to the masses within S, while the volume integral -mm since everywhere is equal to that is, the potential due to all the masses in the region T, viz., outside S. 129. Equipotential Layers. As a still more particular case of 7), if the surface S is taken as one of the equipotential surfaces of the internal distribution, we have all over the surface V= Vs — const., and the constant may be taken out from the first integral, 9) Fp=_ d8_ ±*J J r ^JJ r $n Now by Gauss's theorem / / C^f^dS = Q, accordingly, so that VP is represented as the potential of a surface distribution of surface -density l dV IF ,-v N +1F G = — - - -5-- = - - cos (Fn) = ~f --- 4:7C on 4:7f y kit y The whole mass of the equivalent surface distribution is 128, 129] EQUIPOTENTIAL LAYERS. 373 which, heing the flux of force outward from S, is by Gauss's theorem, § 123, 68), equal to M, the mass within 8. Accordingly we may enunciate the theorem, due to Chasles and Gauss1): — We may produce outside any equipotential surface of a distribu- tion M the same effect as the distribution itself produces, by dis- tributing over that surface a layer of surface -density equal to - times the outward force at every point of the surface. The mass of the whole layer will be precisely that of the original internal dis- tribution. Such a layer is called an equipotential layer. (Definition — A superficial layer which coincides with one of its own equi- potential surfaces.) Reversing the sign of this density will give us a layer which will, outside, neutralize the effect of the bodies within. The above theorem has an important application in determining the attraction of the earth at outside points. Equation 10) shows that the potential and therefore the attraction is determined at all outside points if F, which is connected with g as in § 123, is known at all points of an equipotential surface. It will be shown later that the surface of the sea is an equipotential surface. Consequently if the value of g is known from pendulum observations at a sufficient number of stations distributed over the surface of the earth the attraction at all points outside the earth can be calculated. Let us now suppose the point P is within S. In this case, we apply Green's theorem to the space within S, and we do not have the integrals over the infinite sphere. The normal in the above formulae is now drawn inward from S, or if we still wish to use the outward normal, we change the sign of the surface integral in 5), 12) vp = - r _ I T A8 _ _L r (P inside 8). Note that both formulae 5) are 12) are identical if the normal is drawn into the space in which P lies. Hence within a closed surface a holomorphic function is deter- mined at every point solely by its values and those of its normally resolved parameter at all points of the surface, and by the values of its second parameter at all points in the space within the surface. A harmonic function may be represented by a potential function of a surface distribution. 1) Chasles, Sur I' attraction d'une couche ellipsoidale infiniment mince, Journ. . Polytec., Cahier 25, p. 266, 1837; Gauss, AUgemeine Lehrsatze, § 36. 374 VIII. NEWTONIAN POTENTIAL FUNCTION. Now if the surface S is equip otential, the function V cannot be harmonic everywhere within unless it is constant. For since two equipotential surfaces cannot cut each other, the potential being a one -valued function, successive equipotential surfaces must surround each other, and the innermost one, which is reduced to a point, will be a point of maximum or minimum. But we have seen (§ 116) that this is impossible for a harmonic function. Accordingly a func- tion constant on a closed surface and harmonic within must be a constant. If however there be matter within and without S, the volume integral, as before, denotes the potential due to the matter in the space" r (within 8), and the surface integral that due to the matter without. If the surface is equipotential, the surface integral &£/#&« **, ±*J J r dne (*^-dS + ~ The first integral is now equal to 4jr, so that Vs being constant contributes nothing to the derivatives of F, so that the outside bodies may be replaced by a surface layer of density 14) tf = - = _ .Pcos (Fne) = + 4tt dng kny - The mass of the surface distribution, 15) ne being the outward normal, is the inward flux of force through S, which is equal to minus the mass of the interim1 matter, and is not, as in the former case, equal to the mass which it replaces. 13O. Gauss's Mean Theorem. As an example of equation 6) let us make the surface S a sphere with center at P. Then in the first term of the integral we have which is constant and may be taken outside the integral. In the second term J being similarly taken outside the integral, we have 129, 130, 131] GAUSS'S MEAN THEOREM. 375 since the function is harmonic in the sphere considered. Accordingly the formula reduces to the first term 16) VdS. The surface integral represents the mean value of F on the surface of the sphere multiplied by the area of the surface, 4^r2. Thus we have the theorem due to Gauss. The value of the potential at any point not situated in attracting matter is equal to the mean value of the potential at points on any sphere with center at the given point and not containing attracting matter. It at once follows from this theorem that a harmonic function cannot have a point of maximum or minimum, for making the sphere about such a point small enough the theorem would be violated. 131. Potential completely determined by its charac- teristic Properties. We have proved that the potential function due to any volume distribution has the following properties: 1. It is, together with its first differential parameter, uniform, finite, and continuous. 2. It vanishes to the first order at oo, and its parameter to the second order. 3. It is harmonic outside the attracting bodies, and in them satisfies The preceding investiga- tion shows that a function having these properties is a potential function, and is completely determined. For we may apply the above formula 5) to all space, and then the only surface integral being that due to the infinite sphere, which vanishes, we have Fig. 135. If however, the above conditions are fulfilled by a function F, except that a certain surfaces S its first parameter is discontinuous, 376 VIII. NEWTONIAN POTENTIAL FUNCTION. let us draw on each side of the surface S surfaces at distances equal to e from S, and exclude that portion of space lying between these, which we will call S± and $2. If the normals are drawn into r we have •> '- The surface integrals are to be taken over both surfaces S1 and $2 and the volume integrals over all space except the thin layer between S1 and 82. This is the only region where there is discon- tinuity, hence in x the theorem applies, and is, 4,r-j£ (i) «, +j r£ (1) Now let us make e infinitesimal, then the surfaces 8lf $2 approach each other and 8. V is continuous at 8, that is, is the same on both sides, hence, since = — (—) = — - — (— ), in the limit the first 3*V\r/ dn^\rj two terms destroy each other. This is not so for the next two. 3 V 7}V for ~— is not equal to -= — because of the discontinuity. In the limit, then The volume integral, as before, denotes the potential / / / -: dr due to the volume distribution, while the surface integral denotes the potential of a surface distribution / / - > where dV - . Hence we get a new proof of Poisson's surface condition, § 127, 94). 132. Kelvin and Dirichlet's Principle. We shall now consider a question known on the continent of Europe as Dirichlet's Problem. Given the values of a function at all points of a closed surface 8 - - is it possible to find a function which, assuming these values on the surface, is, with its parameters, uniform, finite, continuous, and is itself harmonic at all points within 8? 131, 132] PRINCIPLE OF KELVIN -DIRICHLET. 377 This is the internal problem — the external may be stated in like manner, specifying the conditions as to vanishing at infinity. Consider the integral of a function u throughout the space r within 8. J must be positive, for every element is a sum of squares. It cannot vanish, unless everywhere -5- = -^ = -= - = 0. that is u = const. dx dy dz But since u is continuous, unless it is constant on S, this will not be the case. Accordingly J(M) > Q Now of the infinite variety of functions u there must be, according to Dirichlet, at least one which makes J less than for any of the others. Call this function v, and call the difference between this and any other hs, so that u = v -+• hs, h being constant. The condition for a minimum is that J(v) A\ 72 TV N 07 C C Ci^V $S , ^V ds 'd® B 8\ -j n 24) WJ(s} + M- + -r + 3sdr' °' for aZZ values of ft, positive or negative. But as s is as yet un- limited, we may take h so small that the absolute value of the term in li is greater than that of the term in h2, and if we choose the 378 VIII. NEWTONIAN POTENTIAL FUNCTION. sign of Ji opposite to that of the integral, making the product negative, the whole will be negative. The only way to leave the sum always positive is to have the integral vanish. (It will he observed that the above is exactly the process of the calculus of variations. We might put 6 v instead of hs.} The condition for a minimum is then But by Green's theorem, this is equal to Now at the surface the function is given, hence n and v must have the same values, and s = 0. Consequently the surface integral vanishes, and = 0. But since s is arbitrary, the integral can vanish only if everywhere in T, z/v = 0, v is therefore the function which solves the problem. The proof of the so-called Existence -theorem, namely, that there is such a function, depends on the assumption that there is a function which makes the integral J a minimum. This assumption has been declared by Weierstrass, Kronecker, and others, to be faulty. The principle of Lord Kelvin and Dirichlet, which declares that there is a function v, has been rigidly proved for a number of special cases, but the above general proof is no longer admitted. It is given here on account of its historical interest.1) We can however prove that if there is a function v, satisfying the conditions, it is unique. For, if there is another, v\ put 1) Thomson, Theorems with reference to the solution of certain Partial Differential Equations, Cambridge and Dublin Math. Journ., Jan. 1848; Reprint of Papers in Electrostatics and Magnetism, XIII. The name Dirichlet' s Prinzip was given by Riemann (Werke, p. 90). For a historical and critical discussion of this matter the student may consult Burkhardt, Potentialtheorie in the Encyklopadie der Mathematik, Bacharach, Abriss der Greschichte der Potential- theorie, as well as Harkness and Morley, Theory of Functions, Chap. IX, Picard, Traite d' Analyse, Tom. II, p. 38. It has been quite recently shown by Hilbert that Riemann's proof given above can be so modified as to be made rigid. 132, 133] GREEN'S THEOREM IN CURVILINEAR COORDINATES. 379 On the surface, since 0. In T, since and are zero, 4u = 0. Accordingly J(u) = 0. But, as we have shown, this can only be if u = const. But on S, u = 0, hence, throughout r9 u = 0 and v = v1. 133. Green's Theorem in Orthogonal Curvilinear Co- ordinates. We shall now consider Green's theorem in terms of any orthogonal coordinates, beginning with the special case forming the divergence theorem, § 117, 35). 28) -if [X cos (nx) + Tcos + Zco8(n#y\ ~/ff13- dY dz\, ^-\dr. Instead of the components X, Y, Z, let us consider the projec- tions P1; P2, P3 of a vector P along the directions of the tangents to the coordinate lines q_uq%,q% at any point. Then projecting along the normal n to S, we have the integrand in the surface integral 29) If we divide the volume T up into elementary curved prisms bounded by level sur- faces of q2 and g3, as in the case of rectangular coordinates (Fig. 136), we have, at each case of cutting into or out of S respectively, Pt cos (nn±) -f- P2 cos (nn2) + P3 cos dS {, ± dScos(nni) where dS^ is the area of the part out by the prism from the level surface qv Now by § 114, ~ dqz dqs accordingly 30) - Fig. 136. the change from the double to the triple integral involving the same considerations as in the proof given for rectangular coordinates in § 115. 380 vm- NEWTONIAN POTENTIAL FUNCTION. Transforming the other two integrals in like manner, L cos (nn±) -f P2 cos (nn2) + P3 cos ( _ / / { But this is equal to But since multiplying and dividing the last integrand in 31) by \\li^ we find that since the volume integrals are equal for any volume, the integrands must he equal, or If the vector is lamellar, its projections are the partial parameters according to &, &, & of its potential V (§ 110), Pi = hi /S - ? Pj> = fas) Q - f P a = lln Q - ' 18fX 2^?2 3^?8 Equation 32) then becomes 33) This result for the value of z/F was given by Lame, by means of a laborious direct transformation. The method here used is a modification of one given by Jacobi and Somoff.1) In order to prove Green's theorem in its general form, we remark that from the nature of the mixed parameter of the two functions U and F as a geometric product we have 34) A ( U, F) = PfPf + PfP2F + Pf P3r ,, dU dV . 19 dU dV , 72 dU dV = M o— « — h /*; o^ y — h ">\ Forming the volume integral, and integrating the first term partially according to q± we obtain 1) Lame, Journal de I'Ecole Poly technique , Cahier 23, p. 215, 1833; Legons sur les Coordonnees curvilignes, II. Jacobi, Uber eine particular e Losung der partiellen Differentialgleichung JV=Q, Crelle's Journal, Bd. 36, p. 113. Somoff, Theoretische Mechanik, II. Teil, §§ 51, 52. 133, 134] STOKES'S THEOREM IN CURVILINEAR COORDINATES. 381 35) ' which as above is equal to Integrating the other terms in like manner we obtain the general formula, Q*\ CCC\i*dUdV , J93UdV . 19 dU dV\ dq.dq.dq, 36) / / / {W*— js -- h^lo— 5 --- h ---— y ' 8 = -J J Z? in which each integrand is found to correspond to one of those in § 115, 20). 134. Stokes's Theorem in Orthogonal Curvilinear Co- ordinates. The proof of Stokes's theorem given in § 30 can be easily adapted to curvilinear coordinates.1) Let Pt, P2; P3 be the projections of a vector P on the varying directions of the tangents to the coordinate lines at any point. Then, the projections of the arc ds being ds19 ds%, dss, we consider the line -integral 37) I =p cos (P, ds) ds = i dsi + P2 ^ 4- P A B where 1) Webster. Note on Stokes's theorem in Curvilinear Coordinates. Bull. Am. Math. Soc., 2nd Ser., Vol. IV., p. 438, 1898. 382 VIII. NEWTONIAN POTENTIAL FUNCTION. Let us now make an infinitesimal transformation of the curve as in § 30. Then the change in the integral is B 38) dI== The last three terms can be integrated by parts, giving B B 39) f Esddqs = Esdqs - fiqsdEs, (* = 1,2,3), A A and, since the integrated terms vanish at the limits, 40) dI=J(dE1dq±+ dE, dq2 + dE^ dqz -dE1 dq,- dE, dq2- dE3 d &). Performing the operations denoted by d and d, as on p. 85, six of the eighteen terms cancel, and there remain the terms, Now the changes dg9, dq%, d#3, dqs, in the coordinates correspond to distances , , measured along the coordinate lines, and the determinant of these distances, is equal to the area of the projection on the surface q± of the infinitesimal parallelogram swept over by the arc ds during the trans- formation. Calling this area dS, and its normal n, we have dqs - dqB dq^) = cos 0%) dS. If we now continually repeat the transformation, until the curve 1 joining AB is transformed into the curve 2, the total change in 1 is equal to the surface integral over the intervening surface, 42) 134, 135] LAPLACE'S EQUATION IN SPHERICAL COORDINATES. 383 Accordingly the components of as before. Q Q If we had attempted to verify the value 48) of V by direct calculation, we should have found a difficulty in the appearance of a logarithm which would have become infinite when the length of the cylinder became infinite. Nevertheless the attraction is finite, as just shown. It is to be note that all the properties hitherto proved to hold have been for potentials of bodies of finite extent. 136. Logarithmic Potential. We may state the above result in terms of the following two-dimensional problem. Suppose that on a plane there be distributed a layer of mass in such a way that a point of mass m repels a point of unit mass in the plane with a force — where r is their distance apart. The potential due to m is V = — m log r and it satisfies the differential equation a«r a»r 0 + = Similarly, in the case of any mass distributed in the plane, with surface -density /it, an element dm = [jidS produces the potential , and the whole the potential 51) F = -r f fdwlogr = - I CplogrdS, where r is the distance from the repelled point x, y to the repelling dm at a, &, so that We may verify by direct differentiation that, at external points, this V satisfies dv dx == ~~ WxJ J WEBSTER, Dynamics. 25 386 VIII. NEWTONIAN POTENTIAL FUNCTION. 2(i/-&)2 This potential is called the logarithmic potential and is of great importance in the theory of functions of a complex variable. 137. Green's Theorem for a Plane. In exactly the same manner that we proved Green's Theorem for three dimensions , we may prove it when the integral is the double integral in a plane over an area A bounded by any closed contour C. Since we have for a continuous function TF 53) dxdy =[W, -W, +• • . + TF2w - W2n^] dy = — I TFcos (nx) dSj cj where n is the inward normal, ds the element of arc of the contour. dV Applying this to W=U-^--y we obtain 54) Treating the other term in like manner, we obtain C A Interchanging U and V we obtain the second form ds= I I (V4U— U4V)dxdy, c A where we write ^y '•-^ 136, 137, 138] LOGARITHMIC POTENTIAL. 337 138. Application to Logarithmic Potential. If in 56) we put U= 1, we obtain 57) which is the divergence theorem in two dimensions. If the function V is harmonic everywhere within the contour, we have 0- Applying this to the harmonic func- tion logr, where P, the fixed pole from which r is measured, is outside the contour, Cdlogr -, I l..r 4 /*cos(rw) / —5^- ds = I - TT- ds = / - — J dn J r dn J r If the pole P is within the contour, we draw a circle K of any radius about the pole, and apply the theorem to the area outside of this circle and within the contour, obtaining the sum of the integrals around C and K equal to zero, or rf\\ 59) cos (rri) These two results are Gauss's theorem for two dimensions. They may of course be deduced geometrically in the same way as for three dimensions, § 118. We may now deduce Poisson's equation for the logarithmic potential as in § 123 for the Newtonian Potential. The logarithmic potential due to a mass dm being — dmlogr gives rise to the flux of force 2ttdm outward through any closed contour surrounding it, and a total mass m causes the flux = 2it / / pdxdy. Put in terms of the potential this is 60) Cj^ds = - f f^Vdxdy = 2% C A A and since this is true for any area of the plane, we must have 61) JV=-2itii. This is Poisson's equation for the logarithmic potential. 25* 388 VIII. NEWTONIAN POTENTIAL FUNCTION. 139. Green's Formula for Logarithmic Potential. Apply- ing Green's Theorem 56) to the functions logr and any harmonic function V, supposing the pole of P to be within the contour, and extending the integral to the area within the contour and without a circle K of radius £ about the pole, c The third term is (since V is harmonic in K) and the fourth, _ (Y^ds = - f^rd^ = - Cvd», J dn J r J K K K which, when we make e decrease indefinitely, becomes Accordingly we obtain the equation 63) . F, which is the analogue of equation 6), § 128. In a similar way we may find for nearly every theorem on the Newtonian Potential a corresponding theorem for the Logarithmic Potential. A comparison of the corresponding theorems will be found in C. Neumann's work, Untersuchungen uber das logarithmische und das Newtonsche Potential.1) The Kelyin-Dirichlet Problem and Principle may be stated and demonstrated for the logarithmic potential precisely as in § 132. 14O. Dirichlet's Problem for a Circle. Trigonometric Series. We shall call a homogeneous harmonic function of order n of the coordinates x, y of a point in a plane a Circular Harmonic, since it is equal to $n multiplied by a homogeneous function of cos G) and sin 09, and consequently on the circumference of a circle about the origin is simply a trigonometric function of the angular coordinate ra. Any homogeneous function Vn of degree n satisfies the differential equation 1) See also Harnack, Die Grundlagen der Theorie des logarithmischen Poten- tiates; Picard, Traite d} Analyse, torn. II; Poincare, Theorie du Potentiel Newtomen. 139, 140] TRIGONOMETRIC SERIES. 389 so that a circular harmonic is a solution of this and Laplace's Equa- tion simultaneously. The homogeneous function of degree n anxn + dn-ix"-1 H a^xyn~l -f a • yn contains n + 1 terms, the sum of its second derivatives is a homo- geneous function of degree n — 2 containing n — 1 terms, and if this is to vanish identically each of its n — 1 coefficients must vanish, consequently there are n — 1 relations between the n + 1 coefficients of Vny or only two are arbitrary. Accordingly all harmonics of degree n can be expressed in terms of two independent ones. The theory of functions of a complex variable1) tells us that the real functions u(x, y), v(x, y) in the complex variable u -f- ivf which is a function of the complex variable x -\- iy, are harmonic functions of x, y, and making use of Euler's fundamental formula, 65) x -f iy = Q {cos co -f i sin co} = geita, and raising to the nih power, we have 66) (x 4- iy)n = Qnein(a = $n (cos no + i sin no). Accordingly we have the two typical harmonic functions £» rr\ « w * It may be at once shown that these functions are harmonic by substitution in Laplace's equation in polar coordinates, equation 47). Accordingly the general harmonic of degree n is /^ Q \ T 7" 47 ( A | TO * \ m /TT We may call the trigonometric factor Tn, which is the value of the harmonic on the circumference of a circle of radius unity, the peripheral harmonic of degree n. If a function which is harmonic in a circular area can l>e developed in an infinite trigonometric series 69) V(x, y} •• on the circumference of the circle of radius R, the solution of Dirichlet's Problem for the interior of the circle is given by the series For every term is harmonic, and therefore the series, if convergent, is harmonic. At the circumference Q = JR, and the series takes the given values of V. The absolute value of every term is less than the absolute value of the corresponding term in the series 69), in 1) See § 197. 390 VIII. NEWTONIAN POTENTIAL FUNCTION. virtue of the factor — ? therefore if the series 69) converges, the -pn f series 70) does as well. Since the series fulfils all conditions, by Dirichlet's principle it is the only function satisfying them. We may fulfil the outer problem by means of harmonics of negative degree. Taking n negative , the series 71) RT\ Q is convergent , takes the required values on the circumference, and vanishes at infinity. For a ring-shaped area between two concentric circles, we may satisfy the conditions by a series in both positive and negative harmonics, 72) H- ^? Q~*n{A!n cos no 140 a. Development in Circular Harmonics. We may use the formula 63), § 139, to obtain the development of a function in a trigonometric series on the circumference of a circle. Let the polar coordinates of a point on the circumference of the circle be B, o and of a point P within the circumference 0, cp. Then we have for the distance between the two points rig. 139. Removing the factor R2, inserting for cos (CD — Q) its value in exponentials, and separating into factors we obtain 73) r = E Taking the logarithm we may develop and 140, 140 aj DEVELOPMENT IN TRIGONOMETRIC SERIES. 391 by Taylor's Theorem, obtaining 74) logr = logE - 45?- — (en*<»-y> + e-«»-(«-9)) = log B -~e~ cos » (o> - ?). 1 This series is convergent if Q < E, and also if Q = E, unless 03 = (p. Inserting this value of logr in 63), differentiation with respect to the normal being according to —E, we have 75) Expanding the cosines, we may take out from each term of the integral, except the first, a factor Qncosncp or pnsinwgp, so that VP is developed as a function of its coordinates p, cp, in an infinite series of circular harmonics, the coefficients of which are definite integrals around the circumference, involving the peripheral values of V and •*— This does not establish the convergence of the series on the circumference. Admitting the possibility of the development, we may proceed to find it in a more convenient form. In order to do this let us apply the last equation to a function Vm, which is a circular harmonic of degree m. Then at the circumference we have T7" ~Dm T> m -r>m — 1 rrj Vm = ±i JLm, -Q^ = -witf JLm, and 271 76) Fm(P) = ^*- The expression on the right is an infinite series in powers of p, while Vm(P) is simply QmTm. As this equality must hold for all 392 vnl- NEWTONIAN POTENTIAL FUNCTION. values of Q less than JR, the coefficient of every power of Q except the mih must vanish, and we have the important equations 77) / Tmcosn((o — (p)do = 0, o 78) Tm() = V (An cos n o + Bn sin n o) = V Tn (o). 0 0 Multiply both sides by cosm (a — (p)d& and integrate from 0 to 2 it. 81) / F(o) cos m (03 — (p)dc3=^. I Tn((ai) cos m(a) — w}do. *J ^^ *J o oo Every term on the right vanishes except the mth which is equal to nTm((p). Accordingly we find for the circular harmonic Tm the definite integral Sft 82) Tm(y) = — I F(to) cos m (o — cp) do. o For m = 0, we must divide by 2. Writing for Tm(jtp) its value Am cos mcp + Bm sin wqp, expanding the cosine in the integral, and writing the two terms separately, we obtain the coefficients 83) AQ = ~ I F(ra) d&j Am = — I F(o) cos mco do, o o 27T Bm = — I ^C03) sinmodco. 140 a, 141] SPHERICAL HARMONICS. 393 This form for the coefficients was given by Fourier1), who assuming that the development was possible, was able to determine the coefficients. The question of proving that the development thus found actually represents the function, and the determination of the conditions that the development shall be possible, formed one of the most important mathematical questions of this century, which was first satisfactorily treated by Dirichlet.2) For the full and rigid treatment of this important subject, the student should consult Kronecker, Theorie der einfachen und der vielfachen Integrate; Picard, Traite d 'Analyse, Tom. I, Chap. IX; Riemann-Weber, Partielle Differentialgleichungen ; Poincare, Theorie du Potentiel Newtonien.9) 141. Spherical Harmonics. A Spherical Harmonic of degree n is defined as a homogeneous harmonic function of the coordinates x, y, z of a point in space, that is as a solution of the simultaneous equations 84) Q*\ ° i , 85) x -a— + V -o-H- ^ -Q— = n V. dx v dy dz The general homogeneous function of degree n anQxn + an^lt0xn-1y + an-2,<>xn~2y2 h -f an-Litf*-^ + an-2,ixn-2y^ f- contains 1 + 2 + 3 • • • + n + 1 = terms> ^ gum of itg second derivatives is a homogeneous function of degree n — 2 and accordingly contains 2 'n terms. If the function is to vanish identically, these 2 l coefficients must all vanish, so that there (n— 1) n , .. n (n 4- 1) (n 4- 2) „„ . „ , are - — ~— relations among the - ^ - coefficients of a harmonic of the nih degree, leaving 2^ + 1 arbitrary coefficients. The general harmonic of degree n can accordingly be expressed as a linear func- tion of 2n + 1 independent harmonics. 1) Fourier, Theorie analytique de la Chaleur, Chap. IX, 1822. 2) Dirichlet, "Sur la Convergence des Series Trigonometriques1', Crelle's Journal, Bd. 4, 1829. 3) A resume of the literature is given by Sachse, Bulletin des Sciences MatMmatiques , 1880. 394 VIII. NEWTONIAN POTENTIAL FUNCTION. Examples. Differentiating the arbitrary homogeneous function, and determining the coefficients, we find for n = 0, 1, 27 3, the following independent harmonics: n = 0 a constant, n = l x, y, e, n = 2 x2 — y2, f — £, xy, yz, zx, n = 3 3x*y-y3, 3x22-z3, 3y2x-x*, 3y*0-0*, 3z2x-x3, 3*2y-y8, xyz. If we insert spherical coordinates r, ft, a £)/? cy 0 JL ° y I ^ « /? ^ v ' nJ is a harmonic of degree »-(**£+ ^ Since tO VQ = C sponds the harmonic F_i = -> we have If ^ he any constant direction whose direction cosines are cds (\ x) = \ , cos (^ y) = % , cos (^ *) = »!, £^+*$if*l? and J- ^ is a harmonic of degree - 2, and to it corresponds the harmonic, « 93) ^ ®; which is of the first degree. Since ^2+ mf + V = l, the harmonic contains ftw arbitrary constants, and multiplying by a third, 4, we have the general harmonic of degree 1, in the form 94) "i If in like manner ^, hs, . . . hn, denote vectors with direction cosines Z2, m2, W2, . . • lnf mn> nn- d is a spherical harmonic of degree - (n + 1) and to it corresponds 95) ^ = ^+1ii-i(7> a harmonic of degree n, and since every h introduces two arbitrary constants, multiplying by another, A, gives us 2n + 1, and we have the general harmonic of degree n in the form, The directions ^, fc2, . . .'*„ are called the ewes of the harmonic. To illustrate the method of deriving the harmonics we shall find the first two. 143, 144] FORMATION OF HARMONICS. 397 A 5 d d A]~ ^ = [>2 + if 1) Laplace, "Theorie des attractions des sphero'ides et de la figure des planetes." Mem. de I'Acad. de Paris. Annee 1782 (publ. 1785). 145, 146] ZONAL HARMONICS. 399 Considering this as a function of z let us develop by Taylor's Theorem, 103) ^=^_/)= , . f A 1 1 0V /1\ and since tor r = 0, -r = — > — = — l—\> > d r «*> J-7 + <-0 Now multiplying and dividing each term by rn+l, we find 1 1 105) ^ = i where 106) P0=l, P This is the determination of the constant A, adopted by Legendre, for the reason that, since by the binomial theorem, for r' r we find ^Z 0*' I i*' , 7*' ^ In order to find Pn as a polynomial in [i we may write ^ as and develop by the binomial theorem. 109) 5 = 0 Developing the last factor, 400 VIII. NEWTONIAN POTENTIAL FUNCTION. Picking out all the terms for which s -\- 1 = n we get for the coefficient of f— J p __ " 2.4(2n-l)(2w-3) The first polynomials have the values -30^+3), 147. Development in Spherical Harmonics. We may use the formula 6), § 128, for an internal point, to obtain the development of a function of 9, Ym, |I = _ mr»- ir,,, we obtain, since 118) FM(P) - ± 00 If the coordinates of P he rf, -9-', cp', we have, while on the right we have an infinite series in powers of r', with definite integrals as coefficients. Since the equality must hold for all values of rr less than r, we must have, collecting in terms in r's 119) o o *m(&> (P') = o o" that we have for the values of the integral 120) / IYm(&, ) = r0 + r1 + y2+.... Multiply both sides by Pn(p) sinftdftdcp, and integrate over the surface of the sphere and since every term vanishes except the nih we obtain Tt iTt 123) fff(», V) P.GO sin 9 d» dy = ^ Yn (»', «p' 0 0 124) Yn (»', J) = ~f(^, 9) P« W sin * d» d, 0 0 actually represents the function f(&, qp'). This theorem was demon- strated by Laplace, but without sufficient rigor, afterwards by Poisson, and finally in a rigorous manner by Dirichlet. A proof due to Darboux is given by Jordan, Traite d' Analyse, Tom. II, p. 249 (2me ed.). 148. Development of the Potential in Spherical Har- monics. In investigating the action of an attracting body at a distant point, and for many other purposes connected with geodesy and astronomy, it is convenient to develop the potential function in a series of spherical harmonics. \$ x,y,z denote the coordinates of the attracted point P, r its distance from the origin, a, b, c the co- ordinates of the attracting point Q, r1 its distance from the origin, d the distance between them, dr' the element of volume at Q, we have and using the value of -=- from 105), when r > r', .< + which, on removing the powers of r from under the integral signs, is the required development in spherical harmonics, 126) F=£ + 5 + 7* + "- where the surface harmonics Yn are the volume integrals 127) T taken over the space occupied by the attracting body. Since /i enters into the integrand, and, according to 121), it contains the angular coordinates #, qp of P, the surface harmonics Yn are functions of & and (p. 147, 148] DEVELOPMENT OF POTENTIAL. 403 If the body is homogeneous, and is symmetrical about an axis of revolution, since V is independent of #, y, it is evident that all the harmonics are zonal, and we have 128) F=^^ + ^^ + ^i + . r r2 rs where every Pn is the zonal harmonic in cos #. If we know the value of V for every point on the axis of revolution, so that we can develop it in powers of — as 129) F,=0 = ^(r)==|* + i + £*.+... ' then putting cos#=l in 128) and comparing with 129), we find •4* = Bn so that V is completely determined as 130) 7= + + + .... ; ~: If in addition the body has an equatorial plane of symmetry, so that F(cos#) = F(— cosfl-), evidently the development will contain only harmonics of even order. As a case of this we shall develop the potential of a homogeneous ellipsoid of revolution in § 161. Whether the body is homogeneous or not, we may easily obtain the physical significance of the first few terms in 126). For making use of the values in 111) since n = ax++c* we have 131) r'Pii — (a There occur in the first three terms the volume integrals J J J ^dad^dc==My I J I Qadadbdc = Ma, jjJQbdadbdc = Mb, jffycdadbdc = Me, ///< A + B-C gabdadbdc = F, 26* 404 VIII. NEWTONIAN POTENTIAL FUNCTION. where M is the mass, a, l>, c the coordinates of the center of mass, A, B, C, D, E, Fj the moments and products of inertia of the body at the origin. If we choose for origin the center of mass, and for axes the principal axes of inertia at that point, we have a = b = c = D = E=F= 0, so that the second term of the development disappears, and the third simplifies, so that we have * -* -2C)z* In all these developments, it is to be borne in mind that r is greater than the greatest value of rr for any point Q in the body. If the body is a homogeneous sphere, all terms disappear except the first. If the attracted point is at a considerable distance compared with the dimensions of the attracting body, or if the body differs but slightly from a sphere, the terms decrease very rapidly in magnitude, so that the first is by far the most important. Thus under these circumstances bodies attract as if they were concentrated at their centers of mass, or were centrobaric (§ 125). The correction is in any case in which we are dealing with the actions of the planets, given with sufficient accuracy by the second term in 133), from which the moments causing precession were calculated in § 96. In § 161 we shall see how the terms depend upon the ellipticity of an ellipsoid of revolution. 149. Applications to Geodesy. Clairaut's Theorem. Although, as has been stated, the development 125) is not in general convergent inside of a sphere with center at the origin which just encloses the attracting body, on account of the divergence of the series 105) when rr > r, still it may occur that the performance of the integrations in 125) causes the latter series to converge even within this sphere. At any rate for a body having the properties of the earth, it has been shown by Clairaut1), Stokes2), and Helmert3), that the series 125) converges at all points on the surface of the body, and also that for the earth the two terms in 133) represent the attraction with quite sufficient approximation for applications to the figure of the earth. In order to exhibit the surface harmonics 1) Clairaut, Theorie de la Figure de la Terre, tiree des Principes de I'Hydrostatique. Paris, 1743. 2) Stokes, "On the Variation of Gravity at the Surface of the Earth.' Trans. Cambridge Phil. Soc., Vol. VHI, 1849. 3) Helmert, Geoddsie. 1884. 148, 149] CLAIRAUT'S THEOREM. 405 in terms of angular coordinates, let us introduce the geocentric latitude ty = — — Q- and longitude qp, in terms of which x = r cos i[> cos qp, y = r cos ^ sin cos2 = 1 — sin reduces to 134) C-(l- 1-1- cos 2 op .9 1 — cos2op 2 - — 2 In order to deal with the apparent gravity g, we have to add to Vj the potential of the attraction that of the centrifugal force, as in § 123, 73), putting 135) y Vc = i co2 (>2 + y*) = { G32r2 sin2 #. It is to he noticed that by writing 136) Fc is itself exhibited as — o2r2 plus a spherical harmonic. If we now write ^ 2 K= -M~ we have the approximate expression for the potential of terrestrial gravity 137) U= with If the surface of the earth is an ellipsoid whose radius vector differs at every point from that of a sphere by a small quantity of the first order, the angle between the normal and the radius vector 406 VIII. NEWTONIAN POTENTIAL FUNCTION. will be small, its cosine will differ from unity by a small quantity of the second order, neglecting which we may put -i on\ 139) # = -y^ = . 9 B- A , COS COS Determinations with the pendulum show that g varies very slightly with the longitude, we may therefore put B=A, so that 140) U- 1A1\ On a level surface, such as the surface of the ocean will be shown in § 179 to be, U= const. = U0. ^M. For such a surface, equation 140) gives, putting a = ^r and in the ^0 parenthesis substituting a for r, we obtain the equation of the surface, 142) r = al + (l The substitution of o, for r is permitted in the higher powers because of the assumption that — differs from unity only by a small quantity 77, where square is neglected, and thus rm = am(l -\-mrf). Inserting the value of r from 142) in 141), and approximating in like manner, 143) .9 = or 4A\ r^ 144) g=sr-r n 145) g = g0 (1 + n sin2 ^), AZ\ 2co2a8 ZK 146) ^ = — ^ — .r-^- ylf 2a2 The equation 142) is easily seen to be that of an ellipsoid of revo lution, and putting ^ = 0, ^ = — its semi -axes are found to be M7\ 147) 149] CLAIRAUT'S THEOREM. 407 Accordingly the ellipticity, or flattening (aplatissement, Abplattung), denned as the ratio of the difference of axes to the greater, is re~rp ZK 148) e= = « The quantity c = °^ is equal to the ratio of the centrifugal accel- eration G) 2 a at the equator to the acceleration of gravity ^—^ at the same place, while n is equal to the ratio of the excess of polar over equatorial gravity to the latter. Thus equation 148) gives us Clairaut's celebrated theorem, 149) e + n = ~c. , Polar gravity — equatorial gravity Elhpt^c^ty of Sea -level -\ -- Equatorial gravity 5 Centrifugal acceleration at equator 2 Gravity at equator The values of the constants in 145) adopted by Helmert as best representing the large number of pendulum observations that had been made up to 1884 are given by 150) g = 978.00 (1 + 0.005310 sin2 9), agreeing closely with the formula given on p. 33. The value of the centrifugal acceleration is known from the length of the sidereal day, the time of the earth's rotation, giving 86,164.09 sec. and the earth's equatorial radius, given by Bessel as 6,377,397 meters. From this is found c = 0.0034672 = ^^ giving by 149) e = ~ x 0.0034672 - 0.005310 = 0.0033580 = — ^- & u - ' ox dq ox (W_dVd*qdqj)_ (dV\ dx* ~ dq dx* + dxdx(dq) In like manner g2F _ " ~" ~ * dq dz* "+" \dz) dq*' dV _ dV Accordingly Q\ ^2 3) T# = - dV dq\™*dq) dq Now since F is a function of q only, the expression on the right must be a function of q only, say ^gt = h*h* d \ ^ V " 153. Application to Elliptic Coordinates. Applying this to elliptic coordinates gives 412 VIII. NEWTONIAN POTENTIAL FUNCTION. Al _ d L 1 -I/ (gt+l^ "Sir^i K Ql)i^-f)(« 1 I 1 1 1 \ _ m 2 a'-V 2- 2 ~ which is independent of /i and i>, and therefore the system of ellipsoids A can represent a family of equipotential surfaces. We have 9) w «ji - + + 10) V=AC- J V(«° J5 must be such a constant that when A = oo, which gives the infinite sphere, F= 0. This is obtained by taking the definite integral between I and oo. ' 00 11) V-A C- =ds A being taken for the lower limit, so that A may be positive, making V decrease as A increases. V is an elliptic integral in terms of A, or A is an elliptic function of F. For J dl ia\ A*(d A \d a differential equation which is satisfied by an elliptic function. We may determine the constant A by the property that lim(VF) = M, or that ~v lim (r2 -*rA = — M cos (rx). r= > i , . ^ We have ^F ^ra^ [by § 73, 86)] 153, 154J ELLIPSOIDAL EQUIPOTENTIALS. 413 From the geometrical definition of A, lim A = 1. Now consider, for simplicity, a point on the X-axis, where $1 = x = r. The denominator becomes infinite in A2, that is, r5, and so does the numerator. Hence so that 154. Chasles's Theorem. We have now found the potential due to a mass M. of such nature that its equipotential surfaces are confocal ellipsoids, but it remains to determine the nature of the mass. This may be varied in an infinite number of ways; we will attempt to find an equipotential surface layer. By Green's theorem, § 129, 11), this will have the same mass as that of a body within it which would have the same potential outside. If we find the required layer on an equipotential surface S, since the potential is constant on S, it must be constant at all points within, or the layer does not affect internal bodies. The surface density must be given by 10), § 129, (3 = — — «--> where m is the outward normal to A, and 3V _ dV dl __, dV d^i ~ 111 fh^ ~ l ~dl' Now since 15) (? = - — di — %TC d'k Since V is a function of A alone, the same is true of -=T-> which for a constant value of A is constant. Hence tf varies on the ellipsoid S as d%. Therefore if we distribute on the given ellipsoid S a surface layer with surface density proportional at every point to the perpen- dicular from the origin on the tangent plane at the point, this layer is equipotential, "and all its equipotential surfaces are ellipsoids confocal with it. Consequently if we distribute on any one of a set of confocal ellipsoids a layer of given mass whose surface density is proportional to d the attraction of such various layers at given 414 VIII. NEWTONIAN POTENTIAL FUNCTION. external points is the same, or if the masses differ, is proportional simply to the masses of the layers. For it depends only on A, which depends only on the position of the point where we calculate the potential. Since by the definition of a homceoid, the normal thickness of an infinitely thin homceoid is proportional at any point to the perpendicular on the tangent plane, we may replace the words surface layer, etc., above by the words homogeneous infinitely thin homceoid. The theorem was given in this form by Chasles.1) 155. Maclaurin's Theorem. Consider two confocal ellipsoids, 1, Fig. 142, with semi -axes e^, @lt ylf and 2, with semi -axes «2, /32, y2. The condition of confocality is 16) Fig. 142. If we now construct two ellipsoids 3 and 4 similar respectively to 1 and 2, and whose axes are in the same ratio & to those of 1 and 3, these two ellipsoids 3 and 4 are confocal (with each other, though not with 1 and 2). For the semi -axes of 3 are wcc-i* v'Lj-i, vV-\ « and 01 4 are /ucc^» i/p^. v^o. and hence the condition of confocality, 17) is satisfied. Now if on 3 we distribute one infinitely thin homceoidal layer between 3 and another ellipsoid for which # is increased by d&, and on 4 a homoeoidal layer given by the same values of & and d&, and furthermore choose the densities such that these two homoeoidal layers have the same mass, then (since these homceoids are confocal) their attractions at external points will be identical. Now the volume of an ellipsoid with axes a, 6, c, is —stabc, o that of the inner ellipsoid of the shell 3 is accordingly and that of the shell is the increment of this on increasing # by (vol. 3) = (vol. 4) = or Similarly 1) Chasles, "Nouvelle solution du probleme de 1'attraction d'un ellipsoide h^terogene sur un point exterieur. Journal de Liouville, t. V. 1840. 154, 155, 156] ATTRACTION OF CONFOCAL ELLIPSOIDS. 415 Consequently, if we suppose the ellipsoids 1 and 2 filled with matter of uniform density ^ and p2, the condition of equal masses of the thin layers 3 and 4, is simply 18) that is, equality of masses of the two ellipsoids. And since for any two corresponding homoeoids such as 3 and 4 (#• and # -f dti) the attraction on outside points is the same, the attraction of the entire ellipsoids on external points is the same. This is Maclaurin's celebrated theorem: Confocal homogeneous solid ellipsoids of equal masses attract external points identically, or the attractions of confocal homogeneous ellipsoids at external points are proportional to their masses.1) 156. Potential of Ellipsoid. The potential due to any homoeoidal layer of semi -axes a, /3, y is found to be from our preced- ing expression for F, 14), 14) M r * V vw where A is the greatest root of ^^ Now if the semi -axes of the solid ellipsoid are a, &, c, those of the shell a = &a, /3 = #&, y = frc, we have M = k-st^d&abc, if the density is unity, and 20) d,V **9***dbcJ- where A is defined by 21) To get the potential of the whole ellipsoid, we must integrate for all the shells, and 22) V^Zxalfi fad ft C ds J J V(aW + 1) Maclaurin, A Treatise on Fluxions. 1742. 416 VIII. NEWTONIAN POTENTIAL FUNCTION. For every value of # there is one value of A, given by the cubic 21). Let us now change the variable 5 to t, where, # being constant, s = &2t, ds = &2dt, and put A = &2u. Then 1 oo /"* /* dt 23) V= 2xabc I &d& I — g 9 > €x €X 0 w where u is defined by 24) ^ - Since &* is thus given as a uniform function of u, we will now change the variable from # to u. Differentiating 24) by #, 25) When ^ = 0, w = oo, and when ^ = 1, u has a value which we will call 6j defined by X2 ?/2 22 ^rG + ^f^ + ^qr^ = Accordingly, changing the variable, 27) F=^a&c / \^^ + ^^ + ^^^du C-^=^=^^ The three double integrals above are of the form 28) where This may be integrated by parts. Call 156] POTENTIAL OF ELLIPSOID. 417 Now

W= ff(t}dt, ?>'(«) = - /"(«)• Inserting' these values in 29), or the variable of integration being indifferent, we may put u for t in the first integral. Applying this to our integral 27), by putting C successively equal to a2, IP, c2, multiplying by #2, i/2, #2, and adding, 31) V= Now the first three terms of the integrand are, by definition, equal to 1, so that 32) V This form was given by Dirichlet.1) If the point x, y, s lies on the surface of the ellipsoid, . then 0 = 0 and 33) V=itabcf {l_-j£-_--j£ rr-l-: ~^~ =• ^y I az+% o*-}-u c -fu} y^^.uj^^.uj^^_u^ o We find for the derivatives of F, oo fly /* du ^ = CXJ (a2 + W)V(^R)(&2 + ^)(c2-H*) a -, dai* x* y* z* \ 1 — xabc^l 1 — asj—Q— pV^ — ^2jr^| /— g 2 8 — — • 1) Dirichlet, "Uber eine neue Methode zur Bestimmung vielfacher Integrate." Abh. der Berliner Akad., 1839. Translated in Journ. de Liouville, t. iv., 1839. WEBSTER, Dynamics. 27 418 VIII. NEWTONIAN POTENTIAL FUNCTION. By definition of (3, the parenthesis in the last term vanishes, and 00 = — 2nabcx C du I - J (a8 + w)l/(a8 + w)(&8 + w)(c8-|-« a 34) ~ = -2xabcy f — V9 J (62 + ^)V(a2 + ^)(i a co 0 , = — ZTtaocs 157. Internal Point. In the case of an internal point, we pass through it an ellipsoid similar to the given ellipsoid, then by Newton's theorem it is unattracted hy the homoeoidal shell without, and we may use the above formulae for the attraction, putting for a, &, c, the values for the ellipsoid through #, yy 0, say &a, #&, &c. Since the point is on the surface of this, 6 = 0. 35) o Now let us insert a variable u' proportional to u, u = co dV 36") -TT— = — 2rt& abcx v$ ft, 0 The # divides out, and writing u for the variable of integration 00 dV C du I - . J (a2 + ^)]/(^ + ^)(62 + ^)(c2 + ^) 0 So that for any internal point, we put yLx, Therefore, since for two points on the same radius vector, -17- - T7- - ^ - X, F, Zt rt The forces are parallel and proportional to the distance from the center, though not directed toward the center. 158. Verification by Differentiation. For an outside point we have, differentiating 34) a 27' 420 VIII. NEWTONIAN POTENTIAL FUNCTION. Now by § 73, 86), 06 2x /I x* y* z* \ dx a* + 6/ j(a2+<>)2 ~*~ (62+<02 (c2-f<02j Forming •«— , and -^ and adding, QO C\ 1 1 1 I a The integration may be at once effected. Since 7/ s (du , dv , div] d(UVW)=UVW\- ->; , 1 u v w I we nave 45) d\ , l = =} (V(a* + u)(b*-\-u)(c* + u)l _ _ _ du _ f ya*+u } ~ y(a* 4- u] (&2 + u} (c2 -f u) I 2 |/(a2 -f u)s I M 1 i 1 I 1 1 du 2 |a2+tt 62 + ^ c2 + wjy^2^w)(&2_^w) The integral becomes then which cancels the second term, and z/F=0. For an internal point /* I - J ( At infinity tf = oo, and V and its derivatives accordingly vanish. Therefore the value of V found satisfies all the conditions. 159. Ivory's Theorem. If x, y, z is a point on the ellipsoid (1) 4T) $ + £ + # = 1, the point (%i Oj Cj lies on the ellipsoid (2) 158, 159, 160] IVORY'S THEOREM. 421 These will be called corresponding points. We shall now assume that these two ellipsoids are confocal, and (2) the smaller. Then 49) V = a22 + A, V = V + *, V = <*' + *• The action of (2) on the external point x, y, z is 50) X2 = - 2 v + v + we must have (5 = 'k. If now we substitute 51) X2 = — 2y7taJ)<,Cc>x I - u J («i 2 + «') V(oi 2 + «') o Now the attraction of the ellipsoid (1) on the interior point «« 5« c2 . > y^> 2 is ax f-.^ q cc 52) Zj = - 277ca1b1cLx^ C - du so that 57) r~,aVf- -W-W -- ^7 ss(62- a!)V(6s- a8) (1 - s8) ds no\ TT- = 59) X = 2*aVxT ___ J S8(ft2 4:7cab2x / s*ds ~A / " ~ (&2-a2)2,y (l-s2) Now /s2ds s ^l==W 160] so that 60) 61) ELLIPSOIDS OF REVOLUTION. X = s8(62-a2)2 '/ 4«a68y / s2^s = (»._a,v '|7 No w s*ds so that 62) Y= for if then -,/&«_ a* -, /62-a Jbor sm~~] I/ 2 we may write tan~] I/ 2 423 These formulae all serve for an oblate spheroid, where a < 6. For a prolate spheroid, "b > a, they introduce imaginaries, from which they may be cleared as follows. Call sin— 1 (iu} = #, then iu = sin &, 1/1 -}- u2 = cos -91, therefore Put cos # — i sin # = ]l + u2 -f w, sin—1 (iu) = & = i log { u. 424 VIII. NEWTONIAN POTENTIAL FUNCTION. Therefore 64) 65) y = (a'-&2)2 ..! In all these formulae, 6 is the larger root of the quadratic 66) _-L + _£_!, for an outside point, and 6 = 0 for an inside point. In the latter case, we have functions only of the ratio -v--1) 161. Development of Potential of Ellipsoid of Revolution. We may develop the expression 58) for the potential of an oblate ellipsoid of revolution in a series of spherical harmonics. Considering first a point on the axis of revolution, let us put x = r > a, y = 0, so that we have by 66), 61), 67) -^- = 1, J = 0 «2-|-(7 and using the tan"1 instead of sin"1 in 58) and 60), we obtain , 2 tan~] 2 ~Qx Jr 68) Vk = Remarking that M the mass of the ellipsoid is -nab2, and developing the antitangent, we have, if r > a > ]/62 — a2, 62 < 3M = 3 M I ^, (- 1)" (fe2 - ay ^, (- 1)* (&2 - a2)" 2 }^/2w + l r2w+A ^-/2n + 3 1) Thomson and Tait, Natural Philosophy, Part H, § 527. 160, 161, 162] ENERGY OF DISTRIBUTIONS. 425 from which, by § 148, 130), -we obtain finally, 70) F.g and this series is convergent for points on the surface of the ellipsoid itself, if &2 < 2 a2. The series converges extremely rapidly if ^ differs little from unity. 162. Energy of Distributions. Gauss's theorem. If a particle of unit mass be at P, (x, y, g) at a distance r from a particle of mass m(J, the work necessary to bring the unit particle from an infinite distance against the repulsion of the particle mq will be 71) W=7mJ=yV(X,y,z)=rrp. If, instead of a particle of unit mass, we have one of mass mp the work necessary will be mp times as great, m 72) Wpq = r-?mp = ympVp = ymqVq, where _ % **-% In other words, this is the amount of loss of the potential energy of the system on being allowed to disperse to an infinite distance from a distance apart r. Similarly, for any two systems of particles mp, mq, 73) wft = Vp being the potential at any point p due to all the particles q and Vq being the potential at any point q due to all the particles ^?. This sum is called the mutual potential energy of the systems p and q. If however we consider all the particles to belong to one system, we must write where every particle appears both as p and q, the — being put in because every pair would thus appear twice. This expression has been given in § 28, 36). If the systems are continuously distributed over volumes t, t1 we have 75) Wpq = 426 VIII. NEWTONIAN POTENTIAL FUNCTION. The theorem expressed by the equality of the two integrals is known as Gauss's theorem on mutual energy, where Vp' represents the potential at p due to the whole mass Mq, Vqj that at q due to the whole mass Mp}) The above equality may be also proved as follows. Since 76) Qf--{ and the triple integrals in 75) become respectively, 77) ~-4 and , /»/»/» Vq^Vq'dlq. Now since outside of r, 4V = 0 and outside of T', z/Ff = 0 the integrals may be extended to all space. But by Green's theorem, both these integrals are equal to dV dV dV dV 3V cV dx dx dy dy dz dz since the surface integrals on vanish at infinity. Gauss's theorem accordingly follows from Green's theorem and Poisson's equation. 163. Energy in terms of Field. For the energy of any distribution consisting of both volume and surface distributions, the sum 74) becomes the integrals 78) W = Now at a surface distribution Poisson's equation is 1 f dV cV\ •"1 1) Gauss, uAllgemeine Lehrsatze in Beziehung auf die im verkehrten Ver- haltnisse der Entferming wirkenden Anziehungs- und Abstossungskrafte." Werke, T» J TT _ + r\m Bd. Y, p. 197. 162, 163, 164] ENERGY OF FIELD. 427 If, as in § 131, we draw surfaces close to the surface distribu- tions, and exclude the space between them, we may, as above, extend the integrals to all other space, so that the normals being from the surfaces S toward the space r. But by Green's theorem, as before, this is equal to the integral »> "- £///!«©'+ ©'+©><• 00 Thus the energy is expressed in terms of the strength of the field at all points in space. This integral is of fundamental importance in the modern theory of electricity and magnetism. It is at once seen that this always has the sign of y, so that it is positive for electrical or magnetic, negative for gravitational dis- tributions. CHAPTER IX. DYNAMICS OF DEFORMABLE BODIES. 164. Kinematics. Homogeneous Strain. We have now to consider the kinematics of a body that is not rigid, that is, one whose various points are capable of displacements relatively to each other. In the general displacement of such a body every point x, y, z moves to a new position x\ y', z1 , so that xf, y', 0' are uniform func- tions of x, y, 0. The functions must also be continuous, that is, two points infinitely near together remain infinitely near together, unless ruptures occur in the body. The assemblage of relative displacements of all the points is called a strain. The simplest sort of strain is given when the func- tions are linear, that is, x1 = a^x -f a2y -j- a^z, 1) y' = \x + \y + M, 0' = W + %y -f c3z, where the a's, Vs and c's are nine constants. 428 IX. DYNAMICS OF DEFORMABLE BODIES. No constant terms are included because a displacement represented by x' = a, y' --=~b, #' = c, would denote a translation of the body as if rigid, which is unaccompanied by relative displacement or strain. Let the solutions of the equations 1), which we shall term the direct substitution, be 2) where ABx' . . , — z/ etc. — z/', etc. 4ii ft* I .' C2 j CS A strain represented by the equations 1) is said to be homogeneous. If the accented letters denote initial positions, and the unaccented letters final positions, the strain represented by equations 2) is said to be inverse to the first strain. In virtue of equations 1) or 2) a linear relation between x, y, z becomes a linear relation between x1, y', z\ Accordingly in a homo- geneous strain a plane remains a plane, and a straight line, being the intersection of two planes, remains a straight line. Finite points remain finite, since the coefficients are finite, accordingly parallel lines, intersecting at infinity remain parallel. Parallelograms remain parallelograms (their angles being in general changed), and therefore the changes of length experienced by equal parallel lines are equal, and for unequal parallel lines proportional to their lengths. Thus any portion of the body experiences the same change of size and shape as any equal and similarly placed portion at any other part of the body. This is the meaning of the term homogeneous, which signifies alike all over. When two vectors OP of length r and OP' of length r' drawn from the same origin are so related that their respective components x, y, 2, x\ y\ z' are connected by the equations 1) or their equi- valents 2) either vector is said to be a linear vector function of the other. The properties of such linear functions are of great importance in mathematical physics, and will now be taken up before their application to strain. 164] HOMOGENEOUS STRAIN. 429 Let us examine the conditions that the two vectors OP and OP' shall have the same direction. The condition for this is where A is to be determined. Introducing the values x' = &x, y' = ky, %] = fig into equations 1) we obtain (% — A) x 4- a2y 4- a3 # = 0, 4) \x + &-A)y 4-M = 0, c^ + c2# 4- (cs — A) £ = 0, a set of linear equations to determine x, y, z. The condition that these shall be compatible is that the determinant of the coefficients vanishes. 5) , 6, 0. This is a cubic in A. Let its roots be Alt A2, A3. Inserting any one of these in 4) we may find the ratios of x, y, z giving the direction of the vectors in question. Supposing that A1; A2; A3 are real, let us find the condition that the three directions are mutually perpendicular. Substituting first I = ^ and then I = A2 in 1) and 3), we have, denoting the values of x, y, z by corresponding subscripts, + Vi^ i + &2«/i -+ ^^ = ^i + -f Multiplying the first three respectively by #2,2/2,'#2 and adding, and subtracting the sum of the last three multiplied respectively by x # we 7) fe - &8) (2/1^2 - *i 2/2) 4- (OB ~ ci) (^i ^2 - %^2) 4- (&i- «a = (A! - A2) fe^ + 2/^2 4- ^^2). The condition for perpendicularity of rlf r2 is % 4- /«/ + ^^ = 0. 430 IX. DYNAMICS OF DEFORMABLE BODIES. Accordingly the left-hand member of 7) must vanish. If rs is perpendicular to t\r2 its coordinates must be proportional to their vector -product. Thus we may write 7) as Inasmuch as the order of suffixes 1, 2, 3 is indifferent, if the three vectors r19 r2, rs are to be mutually perpendicular, equation 8) must be satisfied by the components of all three. This can be true only if we have that is, the determinants of the substitutions 1) and 2) are symmetrical. In this case the linear vector -function is said to be self - conjugate, and a strain represented by such a function is called a pure strain. 165. Self -conjugate Functions. Pure Strain. We will consider this important case in detail. Adopting a symmetrical notation, let us write . 7 0 = ax + hy + gs, 9) y' = hx + l)y -f fz9 z1 = gx 4- fy+ cz. If by cp we denote the homogeneous quadratic function 10) fp = axz -\- by* -f £#2 + %fy& + 2g2x + 2hxy, equations 9) may be written so that the vector OP' (Fig. 143) is parallel to the normal at the point P, whose co- ordinates are x,y,z, lying on the quadric cp = + E2} where E is a constant introduced merely for the sake of homogeneity. In like manner calling Fig. 143. 11) (p' = -f equations 2) are -f 2 X 12) so that OP is parallel to the normal at P', whose coordinates are x', y', z', a point on the quadric 166] PURE STRAIN. 431 By means of either of the quadrics 9 and

, if/ coincide in direction. Multiplying together 16) and 22) we obtain 24) <.9'=S2, and, since the directions of (> and p' coincide for the axes of either ellipsoid , we see that the ellipsoids are reciprocal with respect to a sphere of radius S. Multiplying 15) by 16) and 22) respectively we get for the axial directions , 165] STRAIN- QUADRICS. 433 that is, the axes of the ellipsoids i(j, ij> are proportional to the squares of the axes of the quadrics tp and g>'. By means of either pair of quadrics * _ ~ 1 Off X m^^ii 1 dcp' y' 2 "dy b 1 d(p' z' ¥ U7 = ~c In the cubic for the axes of The determinant of the coefficients of the substitution 9) accordingly represents the ratio of expansion, and since parallel lines are stretched V in the same ratio, the ratio of expansion of volume -=• is everywhere the same. 165, 166] SIMPLE ROTATION. 435 166. Rotation. Let us return to the case of the general homogeneous strain given by equations 1) and let us find the condi- tion that all points situated before the strain on a sphere with center at the origin remain on the same sphere after the strain. The condition x'* + y'a + e'* = x* + f + £ gives 31) fax -f a.2y + a3#)2 -f (\x + bzy -f &3#)2 which being true for all values of x, y, s necessitates the equality of the coefficients of corresponding sqares and products on both sides of the equation , that is, V -I- If + c,2 = 1, 32) aS + V + cS-*!, a^a^ -f &i&2 -f 33) V3 + &2&3 + Equations 32) show that a, &, c with the same suffix are direction- cosines of a line, equations 33) show that the three lines are mutually perpendicular, in other words, the equations of strain are merely those of transformation of coordinates, and the result of the strain is merely a rotation of the body as if rigid. Let us obtain the analytical expression for an infinitesimal rotation about an axis. Let the direction -cosines of the axis be X, [i, v and the angle of rotation be do. Since we have proved in § 57 that infinitesimal rotations may be resolved like vectors and treated like angular velocities, we have the components^ of rotation, 34) cox = ^do3, G)y = /idea, oz = vd&, from which by equations 119), § 76, we obtain the infinitesimal displacements, x1 — x = dx = z&y — ycoz = (0fi — yv) do, 35) y1 — y = dy = x&z — ZG)X = (xv — gfy do, #f -- 8 = dz = ox — xc3 = k — %i do. From this we obtain the substitution for the rotation considered as a strain, 28 , x = 1 • x — voo - y + 36) y1 = vdo -x+l-y- Udo 436 IX. DYNAMICS OF DEFORMABLE BODIES. The determinant of the substitution is skew symmetrical. The ratio of expansion is 37) V ~V Ado , Ado , 1 which is equal to 1 plus terms involving do2 which are negligible, agreeing with the result that rotation as a rigid body is unaccompanied by change of volume. 167. General Small Strain. We shall now consider small strains in general, that is, strains in which the displacements of all points are small quantities whose squares and products may be neglected. Let the components of the displacements or shifts be 38) x — x = u, y — y = v, s — z = w, so that if we now write the coefficients of the strain as 39) y' = \x 4- (1 - we have the shifts given by the substitution a.2y 40) w= and if the a's, &'s and c's are small, u, v, w will be small quantities of the same order. The ratio of dilatation is by 30) 41) = The quantity 42) terms of higher order, 1 4- &i 4- &2 + cs + terms of higher order. IT! -\r is the increment of volume per unit volume, and will be referred to simply as the dilatation and denoted by (5. 166, 167] COMPOSITION OF SMALL STRAINS. 437 Suppose two small strains take place successively according to the equations 39) for the first, and X' 43) for the second. Substituting the values of x\ y'} #r from 39) in 43) we obtain *4(l+< -f -f Neglecting terms of the second order we obtain the equations of the resultant strain -f- -f 44) ?" = & and for the resultant shifts 45) (a2 -f that is, successive small strains are compounded by adding their shifts. This important proposition enables us conveniently to resolve small strains into types already studied. Every small strain represented by equations 40) can be written by addition and subtraction of equal terms u = a^x + (aa -f &x)y + (a3 + cJ0 46) -f 438 IX- DYNAMICS OF DEFORMABLE BODIES. Accordingly we may write the strain as the resultant of two, U = U ! + u-2 9 V = vi + ^2 J W = W-L + W2, where denoting a pure strain, and 48) v2 = — (^ — a2) x — — (c2 - denoting a rotation CD whose components are 49) OIT = -9(03 — Thus every small strain may be resolved into a pure strain and a rotation. In order to bring out the symmetry let us write the pure strain % = sxx + gzy + gyz, 50) v1 = gzx + syy + gx0, MI = 9y% + 9xy + 5,-er, where Thus the six quantities g and co are respectively the half sums and half differences of shift -coefficients symmetrical about the main diagonal.1) 1) In the usual notation the #'s are defined as the above sums without the coefficient — i as stated by Todhunter and Pearson, A History of Elasticity and Strength of Materials, Vol. I, p. 882, "The advantage which would arise from 167, 168J TYPICAL SIMPLE STRAINS. 439 The general small strain is accordingly completely defined by the nine small coefficients, sx, sy, ss, gxj gy, g,, a*, &y, G>Z. 168. Simple Strains. Stretches and Shears. The pure strain 50) may be resolved into two parts (a) (b) 52) vl = syy, <=^-f 0 + gx0, wj = szs, Wi" = gyX + gxy + 0. A strain whose equations contain but a single constant is called a simple strain. Thus we may resolve the strain (a) into three simple strains of which the first is given by u = sxx} v = 0, w = 0. This represents a displacement in which each point is shifted parallel to the x -axis through a distance proportional to its x coordinate. Such a displacement is called a stretch. The constant sx represents the distance moved by a plane at unit distance from the YZ- plane and measures the magnitude of the stretch or the linear expansion per unit length. If s is negative the stretch becomes a squeeze. The strain (a) accordingly represents the resultant of three simple strains, namely stretches, of different amounts in the directions of the coordinate axes, which are evidently the axes of the strain. The semi- axes of the strain -ellipsoid are 1 + sx> 1 + sy, 1 -f sz and its equation ? "• 7i i 0 N2 === •'J or neglecting squares of small quantities, (1 - 2s*) x* + (1 - 2sy) ^ -f (1 - 2s,) £ = 1. The dilatation is by 42) 53) 6 = sx + sy + sz. Obviously we can have 6 = 0 if at least one of the stretches is replaced by a squeeze. If the three s's are equal we have a simple introducing the — into the slides is thus obvious", and we have therefore so introduced it, although to them "it seemed too great an interference with the nearly general custom.'1 We have also introduced a single suffix, gx, instead of the more usual double suffix notation, g z, feeling that the brevity and analogy with a>x thus gained justifies the change. 440 IX. DYNAMICS OF DEFORMABLE BODIES. strain known as a uniform expansion for which the strain -ellipsoid is a sphere and the dilatation \ r' — r 05) Sr = -^— is a small quantity of the first order. But, since the angle between r and r' is infinitesimal, we have to the first order, if q is the dis- placement PP', 56) r' = r -f gc -~ _ gcos(gr) _ r r2 Now if UjVjW are given by equations 50) the numerator becomes 58) r*sr = sxx2 + syy* -f szz* + 2^^ + 2^£# + 2g,xy = #. If we put this equal to unity we have 59) ^ = ~2> where r is the radius vector of the quadric 60) z-l. This is called the elongation and compression quadric, and it is to be noticed that the displacement of any of its points is in the direction of the normal, for /»w\ l 3% l d% 1 d% 61) M = -^, #= *, ^= * % ox % dy 2 cz Since any one of the six coefficients may be positive or negative, the quadric may be an ellipsoid or an hyperboloid. In the latter case not all the lines drawn from the origin will meet the surface, and for those which do not r is imaginary and sr is negative. If we construct the conjugate hyperboloid, % = — 1, those rays which do not meet the first hyperboloid meet this, and the magnitude of the compression is given by 62) *<.—.£• Lines that meet both hyperboloids at infinity and therefore have a zero stretch or compression lie on the cone % = 0, asymptotic to the two hyperboloids, and known as the cone of no elongation. All lines which are equally elongated with the stretch S, where 63) S = i {sxx* + syif + s,s* + 2gxyg + 2gysx + 2gzxy], lie on the cone 168 a] ELONGATION QUADRIC. 443 64) (sx — £) x* + (s9 — S)y2 + ($y — S)z*+ 2gxyz + %gyxz + 2ggxy = 0, which may be called a cone of equal elongation S, of which the cone of no elongation is a particular case. Let us form the elongation quadrics for expansions and shears. If the slides vanish we have 65) x = SxX* + s,y2 + s,s* = 1, and for a simple stretch in the X- direction 66) sxx* = 1, the elongation quadric breaks up into the two parallel planes, l^SxX —1=0 and y ' sxx +1=0, at distances + — from the origin. Since for any line making the angle # with the X-axis we have r = — = 9 ysx cos # the stretch is given by 67) sr = -g = sx cos2 #. The cone of no elongation is therefore the plane & = ^ parallel to the above pair of planes. In equation 65) if sx) sy, sz are of the same sign the quadric is an ellipsoid and the cone of no expansion is imaginary. If one s has a sign different from that of the others we have two hyperboloids and the cone of no expansion is real and separates the stretched from the squeezed lines. In the general shear sx = Sy = sz = 0 we have 68) x = 2 (gxyz + gyex + g»xy) = 1, and the cone of no elongation 69) gxyz 4 gysx + gexy = 0, contains the three coordinate -axes as generators. These are therefore unstretched. In a simple shear parallel to the XY- plane we have 70) % = 2gzxy = ± 1 which represents equilateral hyperbolic cylinders with axes bisecting the angles between the x and y axes. The cone of no elongation, xy = 0, breaks up into two coordinate -planes, x = 0 and y = 0. These two planes are undistorted, and are the planes of circular section of the strain -ellipsoid. 444 IX. DYNAMICS OF DEFORMABLE BODIES. A combination of two simple shears in planes at right angles obtained from 52 b) by putting gz = 0, has the elongation quadric 71) gxyg -f gyxg = 0, which breaks up into the two planes 72) # = 0 and gxy at right angles to each other. It is to be noticed that the cone of the resultant of three simple shears in mutually perpendicular planes does not so break up. We have seen that we require nine constants to specify the general homogeneous strain, of which three belong to the rotation, six to the pure strain. Let us consider the number of data required to specify a simple pure strain. To specify a uniform dilatation we require only the constant of dilatation tf; for a simple stretch, the direction of the axis, involving two data, and the magnitude of the stretch, making three in all; for a simple shear, four data, the magnitude of the shear, two to fix the plane of the shear and one additional for an axis. Consequently we may always represent a general strain as the resultant of three simple expansions, or of two simple shears and a uniform dilatation. 169. Heterogeneous Strain. If the displacements are not given by linear functions of the coordinates, the strain is said to be heterogeneous. In this case we may examine the relative displacements of two neighboring points. Let the coordinates of the first point P be before the strain x, y, z, and after it x -f u, y + v, z + w, and those of the second, §, be before x + f, y + g, 0 + h, and after % + f -\- u', y 4- g -4- v\ z + h + w1. If the point Q be referred to P as an origin both before and after the strain, it has as relative co- ordinates before f, g, h, and after f -f u' — u, g + vf — v, h -f w' — w, so that the relative displacements are u' — u, v' — v, wf — w. Now u, v, w may be any functions of the coordinates x, y, 3 of P, but they must be continuous, otherwise the body would be split at surfaces of discontinuity. Accordingly u\ v',w* being the values of u, v, w for x -\- ff y -\- g, % -\- h may be developed by Taylor's theorem, so that, neglecting terms of order higher than the first in /*, #, li i ~c-u . du . ^ du u — u = f-x — \- g- — h » -o-> 1 dx ' y oy ' dz rroN f r ^V , fiv . •* dv 73) v ' — v = fjr- + g^r -f h -5-9 1 dx ' y dy ' dz 3w , 7 dw 168 a, 169] HETEROGENEOUS STRAIN. 445 Thus the relative displacements are given as linear functions of the relative coordinates f, g, li whose coefficients are the values of the nine first derivatives at the point P, that is to say, constants for all points Q in the neighborhood of P, consequently the relative strain of the portion of the body in the neighborhood of P is homogeneous. Thus we say that any continuous heterogeneous strain is homogeneous in its smallest parts. Comparing with equations 49) and 51) we find the stretches, dilatation, slides and rotations at any point to be respectively „ A\ du dv dw 74) sx = -, Sy = -, S, = -, 75) ' ox 1 dw . dv\ 1 du dw\ l dv . du 1 (w . v\ 1 (u , w\ l (dv . du ?.-¥W + J^ ^ = ¥\^ + W 9*=*(dx+Jy 1 /dw dv\ 1 (du dw\ l (dv du X== ~ \-~ -- -x— ? 2 V^i/ W = — ^ -- -o— > G) * == — ^ -- -K— 2 \^^ &ar/ 2 Thus the volume dilatation is equal to the divergence of the dis- placement, while the rotation is equal to one half its curl. We might have obtained the value of 0 by the divergence theorem. Consider any closed surface S fixed in space so that por- tions of the deformable body flow through it daring the strain, and let us find the volume of the matter which passes outward through S. Through an element dS at which the displacement is q there passes out a quantity filling a prism of slant- height q and base dS whose volume is therefore qcos(nq)dS, where n is the outward normal to S. Through the whole surface there accordingly issues an amount whose volume is 78) Q = I I qcos(nq)dS = I I [u cos (nx) + v cos (ny) -f w cos (n&)} dS by the divergence theorem. This is accordingly the increase in volume of the portion of substance originally included by the surface S. The ratio of this to the original volume is accordingly the mean value of the divergence in the volume in question, and making the volume infinitesimal, this becomes the dilatation 6. In order that a strain shall be everywhere irrotational we must have the curl components of the displacement vanish everywhere. 446 IX- DYNAMICS OF DEFORMABLE BODIES. But by § 31 this is the condition that the displacement is a lamellar vector and ' dxy dy' ~ dz Then (p is called the strain -potential. Only when the strain is ir- rotational can a strain -potential exist. The line integral along any curve AS of the tangential com- ponent of the displacement B B 80} / q cos (q, ds) ds = I (udx + vdy + wdi) A A is called the circulation along the path, and for irrotational strain is independent of the path, equal to cpB — yAj and vanishes for a closed path. Surfaces for which

-^=r> -=-• The normal component Fn Fn Fn 83) Fnn = In cos (Fnri) = Xn cos(nx) -\- Yn cos (ny) -\- Zn cos (ne). If we draw the normal in either direction from the element dS, and if we understand by Fn the force exerted through dS by the portion of the body lying on the side toward which n is drawn on the portion lying on the other side, then if the normal component Fnn = Fn cos (Fnri) is positive it is called a traction, if negative, a pressure. In other words it is a traction if its effect is to cause the portions of the body to approach each other, a pressure if it is to make them recede. The force upon any element dS can be expressed in terms of the forces upon three mutually perpendicular plane elements at the same point. Construct, enclosing the point P, an infinitesimal tetrahedron bounded by the element dS and three planes parallel to the coordinate planes (Fig. 146). Let the areas of the four triangular faces be dS, dSx, dSy, dS~, the suffix in each case denoting the direc- tion of the normal to the face. Further denote the stress -vector for any face by a suffix giving the normal to that face, and let the stress -vectors be those for the portion of the body within the tetrahedron. Suppose that forces are applied to every portion of matter in proportion to its mass, such, for instance, as gravity, the components being X, Y, Z per unit mass. If d-c denote the volume of the tetrahedron the X- component of these external forces is accordingly Xgdt. Let us now form the equations for equilibrium of the matter contained in the tetrahedron under the influence of the external forces and the stresses developed. The first of these is 84) Xgdr + XndS - XxdSx - XydSy - X3dSz = 0. Fig. 146. 448 IX- DYNAMICS OF DEFORMABLE BODIES. But since the three other sides are the projections of dS, we have 85) dSx = dS cos (nx), dSy = dS cos (ny), dSz = dS cos (nz). Inserting these in the equation 84), dividing through by dS, and taking the limit, as the edges of the tetrahedron become infinitely small the ratio of the volume to the surface disappears, so that we have finally 86) Xn = Xx cos (nx) -f X.y cos (ny) -f Xz cos (nz) , and similarly Yn = Yx cos (nx) -\- Yy cos (ny) -f Y, cos (nz), Zn = Zx cos (nx) -f- Zy cos (ny) -f Zz cos (nz). Let us now consider the equilibrium of any portion of the body bounded by a closed surface S. Resolving in the X- direction, we have as the condition for equilibrium, considering both the stresses on the surface and the volume -forces, 87) Making use of equations 86) for Xn, 88) / / { Xx cos (nx) + Xy cos (ny) + Xz cos (ne)} dS and by the divergence theorem, n being the outward normal, r r ridx sx ax 89) JJJ y + w + ^ Since this must hold for every portion of the substance which is in equilibrium, the integrand must vanish, and we have consequently together with the result of resolving in the two other directions, dX dX dX These are but three of the six equations for equilibrium. The other three are obtained by taking moments, the first being 91) 170] NATURE OF STRESS. 449 Introducing the values of Ynj Zn from 86) this becomes , $Z from equations 90) and of 92) / / {y [Zx cos (nx) -f Zy cos (ny) -f Z* cos (w0)] - 0[YX cos (w#) -f Yj, cos (ny) -f F, cos (ws)]} d$ *SL (^ \0aj +-*+ - Writing the term and and applying the divergence theorem, all the surface integrals cancel each other and there remains only the volume integral 93) As before, the field of integration being arbitrary, the integrand must vanish, and we obtain, after applying the same process to the remaining two equations, QA\ v 7 7 v ~y v y±) JL z = Zyy, ZJX = jC^Zj J^y =F Lx. We may also obtain these equations by considering the stresses on the faces of an infinitesimal cube (Fig. 147). We shall denote the tangential components or shearing stresses 94) by Tx, Ty, Tz, the normal components or trac- tions by PXJ Py, Pz. The stress at any point is determined in terms of these six components, for we may find the stress -vector Fn, whose direction - cosines . are u'j P'> ?' f°r anJ stress plane whose normal has the direction cosines a, fl, y by equations 86), which in our present notation Fig. 147. become 95) Xn = Fna' = Yn = Fnp = TyK WEBSTER, Dynamics. 29 450 IX. DYNAMICS OF DEFORMABLE BODIES. These are the exact analogues of equations 17). In other words, the stress -vector is a self - conjugate linear vector -function of the normal to the stress -plane. The stress -vector Fn occupies the place of * in 17). Accordingly the whole geometry of the linear vector func- tion may be applied to the consideration of stress as follows. 171. Geometrical Representation of Stress. If we construct the quadric 96) cp EEE Pxx2 + Pyy* + Pzs* + 2Txyz + 2Tyzx + 2Tzxy = ± E2 any stress -vector Fn is perpendicular to the tangent plane drawn at the point where the normal to the stress- plane cuts the quadric cp (Fig. 148). This is known as Cauchy's stress -quadric. Let its equation, referred to its principal axes, which are known as the axes of the stress, be 97) cp = P±x2 + P2y2 + P3*2 = ± E2. P1; P2, P3 are called the principal tractions, being the normal stresses on the planes perpendicular to the axes, these planes being subject to no tangential stresses. Thus, as for any strain we may find three planes for which the slides vanish, so for stress we may find three planes for which the shearing stress vanishes. In the reciprocal quadric, «.2 »,2 *2 QQ\ ' — _|_ " _|_ i 7)2 the stress -vector is conjugate to its stress -plane, for the normal to the stress -plane is parallel to the normal to cp' where it is cut by the stress -vector. The quadric cp' is known as Lame's stress -director quadric. In equations 17) and 14) putting Fn for -- we obtain Fig. 148. 99) p == _}_ ^L __ _j ±i — pr — r2cos(?*r') or 100) So that the traction or component of the stress -vector normal to its stress -plane is inversely proportional to the square of the radius- vector of the quadric (p in the direction of that normal, or is directly proportional to the square of the perpendicular upon the tangent plane to the quadric cpr parallel to the stress -plane. 170, 171, 171 a] GEOMETRY OF STRESS. 451 If P±, P2, P3 are all of the same sign the quadrics cp and 9?' are ellipsoids. If they are positive we must take the positive sign with R?, and the normal stress on every plane is a traction. If they are negative, we must take the negative sign, and the normal stress is always a pressure. If one of the P's has a different sign from the two others, we use both signs and have pairs of conjugate hyperboloids. In this case for directions parallel to the generators of the asymptotic cone cp = 0 to the stress quadric, we have r infinite and Fnn = 0. Accordingly for stress -planes perpendicular to these generators, the normal stress vanishes or the stress is a shearing stress. These planes envelop a cone called Lame's shear -cone, which divides the directions for which the normal stress is a traction from those for which it is a pressure. In the reciprocal quadric qp', when the radius vector is infinite, it lies in its conjugate plane, the stress -plane. But the radius vector to this quadric has the direction of the stress -vector, so that the shear -cone is the asymptotic cone to this quadric that the stress -vector for any plane is directly proportional to the radius vector in its own direction. This ellipsoid is called Lame's stress -ellipsoid, or ellipsoid of elasticity. 171 a. Simple Stresses. A simple stress is one that contains but a single constant in its specification. These are: 1°. Uniform traction or pressure. P~P ~P T> i = -L a = J-n - JC . All the quadrics are spheres and every stress is normal to its plane and of invariable amount P. Such a stress is physically realized by a body subjected to hydrostatic pressure. 29* 452 IX. DYNAMICS OF DEFOBMABLE BODIES. 2°. Simple traction , PX=P, Py = P. = 0, 105) T T T J-x— J-y= -Lz — M- The stress quadric is 106) y = Px2 = ± 1, a pair of planes perpendicular to the X-axis at a distance from the origin. The stress on any plane is parallel to the X-axis. The stress -director quadric and the shear -cone reduce to the axis of X, all planes tangent to which experience only shear. Cauchy's ellipsoid, 107) P V + 0 • f + 0 - £ = 1, with axes, p» oo, oo, is a pair of planes perpendicular to the X-axis, and Lame's ellipsoid with axes, P, 0, 0, becomes simply that part of the axis of X from x = — P to x = P. From the property of this ellipsoid the stress -vector is proportional to the perpendicular on the tangent plane parallel to the stress plane. Since the tangent plane here always passes through one of the extremities we have 108) , Fn = Pcos(nx) as is indeed evident from equations 95). 3°. Simple shearing stress. p _ p — p — o , y * ' Tz = T, Tx = Ty = 0. Equations 95) become 110) The stress quadric is 111) which represents a pair of rectangular hyperbolic cylinders with the semi -axes — =• The stress -director quadric is yr 112) The shear cone xy = 0 represents the coordinate planes of X. and YZ. 171 a] SIMPLE STRESSES. 453 A shearing stress may also be written, referred to its principal axes, 113) / = ~ 'T ' " J.X= -Ly= J-Z = V, when the stress quadric becomes the pair of hyperbolic cylinders referred to their axes and Xn = Fna' = Pa, -J -J £ \ T7* 77* /? ' ~P ft np ~J71 f f\ We accordingly have = or /?' that is, all stresses are parallel to the XY- plane, and the stress- vector and the projection on the XY- plane of the normal to the stress plane make equal angles with the X-axis on opposite sides (Fig. 149). Squaring equations 115) and adding, 116) Fn* = P2(a2 + /32)=P2(1-^2). If y = 0, that is, if the plane is tangent to the Z-axis Fig. 149. the normal stress being a traction if the normal to the stress -plane falls nearer to the X-axis, a pressure if nearer to the F-axis. The shear cone x2 — y* = 0 is composed of the two planes bisecting the dihedral angle between the XZ- and YZ- planes. From this manner of representing the stress it is evident that a simple shearing stress is equivalent to an equal traction and pressure in two directions perpendicular to each other. Compare the repre- sentation of a shearing strain as an equal stretch and squeeze. For this case Lame's ellipsoid -=, (x2 -f y2) + — = 1 has the axes P, P, 0 and reduces to a circular disc normal to the ^-axis. Since all tangent planes pass through its edge, 117) as abore in 116). Fn = P sin (nz) = 454 IX. DYNAMICS OF DEFORMABLE BODIES. 172. Work of Stress in producing1 Strain. If every point in a body move a distance dq, whose components are 8ti, dv, dw, and if there act upon every unit of mass of the body the external forces X, Y, Z, and upon each unit of surface the forces Xn, Yn, Zn, the work done by all the forces in the displacement is 118) dW= I UXndu + Yndv + Zndw}d8 + / / I Q[Xdu + Ydv + Zdw}dr, which becomes by equations 86), 119) d W = I j {[_XX cos (nx) + Xy cos (ny) + Xz cos (ngj] du + [Yx cos (nx) + Yy cos (ny) + Yz cos (nz)\ dv + [Zx cos (wfl?) 4- Zy cos (wy) + ^ cos (mi)] dw} dS and transforming surface integrals into volume integrals by differentia- tion in the manner of the divergence theorem and making use of equations 94), YydV w + F^i; + Zdw) •"» az 0Zf, dZ ' ' + ^ + X^+Y1r^ + Z, TT /^^'^ _j_ ^V\ , ^ (W<* . ~\ J-z \ "iaTT ' Q * / ~t~ ^a; I o« 172, 173J WORK OF STRESS. 455 By equations 90) the coefficients of du, dv, dw vanish identically, so that, interchanging the order of differentiation and variation, 121) *W-{x.*& + Y,* + Z.9% + Y, or in our later terminology, 122) 8W= j i C{Px8sx + Pydsy + P,8s, + 2Txdgx + ZTydgy + 2T2dg2}d>t. Thus each of the six components of the stress does work on the corresponding component of the strain, and the work per unit volume in any infinitesimal strain is the sum of each stress component by the corresponding strain produced, except that with our terminology the shearing stresses are multiplied by twice the shearing strains or slides. 173. Relations between Stress and Strain. If a body is perfectly elastic the stresses at any point at any time depend simply upon the strain at the point at the time in question, so that if the elastic properties of the body are known at every point the stress components will be known functions of the strain components, which may differ from one point of the body to another. The stresses will be uniform and continuous functions of the strains and may be developed by Taylor's theorem. If then the strains are small, the terms of the lowest orders will be the most important. The strains dealt with by the ordinary theory of elasticity are so small that it is customary to neglect all terms above those of the first order. The results thus obtained are in good accordance with those obtained by experiment under the proper limitations. The law that for small strains the stresses are linear functions of the strains may be regarded as an extension of the law announced in 1676 by Hooke in the form of an anagram, ce^^^nosssttuu Ut Tensio sic Vis. The force varies as the stretch, or in our terminology the stress varies as the strain. Making this assumption we accordingly have X = 9>oi 456 IX. DYNAMICS OF DEFORMABLE BODIES. The gp's will in general be functions of the coordinates of the point, but if the body is homogeneous, that is alike at all points, they will be constants. We shall assume this to be true. If there be a natural state of the body or one in which the body is in equilibrium under the action of external forces, so that the stresses vanish for this state, it is convenient to measure the strains from the natural state. Then the stresses and strains vanish together, so that the terms ^P01, . . . qpog vanish. For such a body there are accordingly thirty -six constants qp, the so-called coefficients of elasticity. In the case of a gas there is no natural state, for a gas is never in equilibrium, unless kept so by an envelope, so that every portion of the gas always experiences pressure, consequently we cannot measure the strains from any natural state. We have now the theory of elasticity as it was left by its founders, Navier and Cauchy. The idea is due to Green1) of supposing the elastic forces to be conservative and accordingly due to an energy function of the strains. If we call the function &(sx> sy, sz, gx, gy, gz] we have for the total potential energy due to any strain 124) The work done in changing the strain is then 125) Comparing this with equations 121 — 122) we find If then the stresses are to be linear functions of the strains, £> must be a quadratic function, and, if we measure from the natural state, a homogeneous quadratic function. A homogeneous quadratic function of six variables contains twenty -one terms, so that instead of thirty -six elastic constants for the general homogeneous body we have only twenty -one, that is, the determinant of the qp's in equa- tions 123) is symmetrical, fifteen coefficients on one side of the 1) Green, Mathematical Papers, p. 243. 173, 174] STRESS -POTENTIAL. 457 principal diagonal being equal to the corresponding fifteen on the other side. If the body besides being homogeneous is isotropic, that is, at any point its properties are the same with respect to all directions, there are many relations between the coefficients, so that the number of independent constants is much reduced. In an anisotropic or eolo- tropic body there are generally certain directions (the same for all parts of the body) with reference to which there is a certain symmetry, so that there are various relations involving a reduction in the number of constants. Such bodies are known as crystals. We shall deal here only with isotropic bodies. 174. Energy Function for Isotropic Bodies. In isotropic bodies the stresses developed depend only on the magnitude of the strains, not on their absolute directions with respect to the body. Accordingly if we change the axes of coordinates the expression for the energy must remain unchanged, or the energy function is an invariant for a change of axes. The cubic for the axes of the elongation quadric 58) belonging to the shift -equations 50) is the determinant — A, g, , gy g, , sy - I, gx 9y > 9* , S* — or expanding the determinant, 128) 43 - (sx + sy + s.) tf 4- (sysz + szsx + sxsy — gl — g} — gl) I + sxgl + sygl + s,gl - sxsys, - 2gxg1Jgz = 0. If the roots are A17 Z2? A3, the equation is 129) tf - (^ + A2 + *3) tf + (M2 + Vs + Mi) * - M2*3 = 0. If we transform to another set of axes X'Y'Z' with the same origin, so that the strain components are sx', sy', sz; gxr, gy; gz', since the elongation quadric is a definite surface, the equation for its axes must have the same roots as before. Accordingly its coefficients are invariants. The roots A1? A2, A3 are the stretches for the directions of the principal axes of the strain. Therefore we have the three strain invariants, symmetrical functions of the roots, /! = ^ + ^2 + Ag = Sx + Sy + 8,, 130) I2 = A^2 + M3 + Vi = svsz + B*SX + sxsy -gl-gl — gl, J3 = ^A^g = 2gxgyg, -f sxsys, - sxgl - svg*y — s,g*,. The invariant Jx represents the cubical dilatation (?, which by its geometrical definition is evidently independent of the choice of axes. 127) 0, 458 IX- DYNAMICS OF DEFORMABLE BODIES. The energy function for an isotropic body, being unchanged when we change the axes, can contain the strains only in the combinations I,, J2, I3, but these are of the first, second and third degrees respectively, and since <& is of only the secoud degree it cannot contain J3. Since it is homogeneous (except for a gas) it can contain Jt only through its square. We therefore have 131) 0 = - PIi + AI* + #/2, where P, A, B are constants. P is zero, except for gases, and is then positive, for if the gas expands it loses energy. The constant A refers to a property common to all bodies, namely, resistance to compression, and is positive, for work must be done to compress a body. The constant B is peculiar to solids. All symmetrical functions of the roots may be expressed in terms of the invariants, for example: 132) &-*,)« + ft -*»)' + &- AJ1 = 2 ( v + v + V) - 2 ft A2 + 1, ^ -t- A3 ;g A2 + A3)2 - 6 ft;, + A2*3 + A,^) Also 2&Jt + J,J, + J,i1)~(i1 + l1 + ^'-.(V+V + V), or r2 We may accordingly write A 1^ -f BI2 as a linear function, of J,2 and of either (^ - A2)2 + (A2 - A3)2 + (X - Aj2, or of V + V + A32. Suppose we write the quadratic terms 134) y2 + (is - Itf + (i, - which is the form given by Helmholtz. The constant H, being multiplied by tf2, refers to changes of volume without changes of form, representing in this case the whole energy, for if there is no change of form the stretches of the principal axes, ^, A2, A3 are equal. The term in C on the other hand refers to changes of form without change in volume, for it vanishes when A, = A2 = ^3, and represents the whole energy if L and p are involved in changes of volume. We thus see that isotropic bodies possess two elastic constants. By means of certain assumptions as to the nature of elastic stresses, making them depend upon actions between molecules, Cauchy and the earlier writers on elasticity reduced the energy function to a form depending on a single elastic constant, the same theory reducing the number of constants for an eolotropic body from twenty- one to fifteen. For this theory the reader may consult Neumann, Theorie der Elastizitat, Todhunter and Pearson, History of the Theory of Elasticity. Experiments have not however confirmed this theory, and it is no longer generally held to be sound. Thomson and Tait inveigh against it with particular emphasis. We shall accordingly assume that an isotropic body has two independent constants of elasticity A and ^. 460 IX. DYNAMICS OF DEFORMABLE BODIES. 175. Stresses in Isotropic Bodies. We may now calculate the stresses by means of equations 126), inserting the values of Sx, sy, ss, gx, gy, gz. . v ' 1 d$ ftw dv\ J-2 = AV = IT o — = V* o -- h -o— 2 C9X \dy o*J 1 fo du . dw The first equation of equilibrium 90) becomes i^o\ 143) or, considering the value of tf, we may write the equation with its two companions, 144) The equations at the surface of the body are by 86), using the above values of the stresses, Xn = (it + 2{*|| - P) cos (na?) f ^ (du . dw\ f N cos (w^) ^. ffJiw-t W cos (w^' c°s (nx^ + (ie + ^ ll - p) cos ^ 146) . dw , *=~-tl- + cos v + cos -r P cos (ne). 175, 176] STRESSES IN ISOTROPIC BODIES. 461 176. Physical Meaning of the Constants. Let us consider a few simple cases of equilibrium with homogeneous strain under stress, there being no impressed bodily forces X, Y, Z, and putting P = 0. 1°. A simple dilatation. u = ax, v = ay, w = as, sx = a, sy = a, sz = a, gx = gy = gz = 0, & = sx + sy -f sz = 3a, 146) Xx = Yy = Zz The surface forces become simply Xn = pcos(nx), 147) Yn=pco*(ny), Zn =pQos (n0), or the surface force is normal to the surface, and 148) F,, = P The ratio of normal traction to cubical dilatation, or of normal pressure to cubical compression, 149) ^, = n±^ = fi is called the bulk -modulus of elasticity, the term modulus being applied in general to the ratio of the stress to the strain thereby produced. 2°. A simple shear. u = ay, v = w = 0, sx = sy = sz = (3 = gx = gy = 0, 150) 6 = we have Xn = 2 11 (a + 6) cos (nx) = ~ • a cos (nx). If the body is a cylinder, with generators parallel to the X-axis; bounded by perpendicular ends, experiencing a normal traction p, there is no force on the cylindrical surface, for which cos (nx) = 0, and on the ends 157) The ratio of the tractive force to the -stretch a, E=^- a may be called the stretch or elongation modulus, and is generally known as Young's modulus. The ratio of the lateral contraction to the longitudinal extension 159) r, = ± l 1 a 2 IL 1) Thomson and Tait use the notation: bulk -modulus =jfe = ^, -[--—, simple rigidity =w, m = ^-(-/i = — -J-^- 8 176, 177] ELASTIC MODULI. 463 is called Poisson's ratio. According to Poisson and the older writers A = ^, so that t] = — - We must certainly have - 1< — y ^ < 0, making the rigidity negative. If 17 < — 1, making the bulk -modulus negative. No known bodies have 77 < 0, and in experiments on isotropic bodies 77 has generally been found nearly equal to —> the value assumed by Poisson, the value being found to approach more nearly to Poisson's value the more pains were taken the secure isotropic specimens. The bulk-, shear- and stretch -moduli and Poisson's ratio are the important elastic constants for an isotropic body, any two of which being known, all are known. CHAPTER X. STATICS OF DEFORMABLE BODIES. 177. Hydrostatics. Let us now consider the statics of a perfect fluid, that is, a body for which p = 0. If each element of the fluid is subjected to forces whose components are X, Yf Z per unit mass, equations 144), § 175 reduce to while the equations for the surface forces 145) become Xn = (JL6 — P) cos (nx), 2) rn The surface force is accordingly normal and equal to 3) ll times the derivative of 6 by the corresponding derivative of (— p), our equations of ^equilibrium 1) are ^ ~ dx' ^ dy' ^ dz Thus the fluid can be in equilibrium only under the action of bodily forces of such a nature that Q times the resultant force per unit mass, that is to say, the force per unit volume, is a lamellar vector. If the pressure at any point depends only on the density, and conversely, and we put — = -j— '» 5) «/ * so that dP _ dP dp _ 1 dp dx dp dx Q dx 6) d^=dPdp==^3p) dy dp dy Q dy' dP _ dP dp _ 1 dp dz dp dz Q dz 7, Our equations 4) are dP dP dP Accordingly in this case the bodily forces per unit mass must be conservative. If V is their potential, multiplying equations 4) by dXj dy, dz respectively and adding, we have 8) Q (Xdx + Ydy + Zde) =*-gdV If we have two fluids of different densities in contact we have at their common surface 9) so that 10) (ft - therefore dV and dp are each equal to zero and the surface of separation is a surface of constant potential and constant pressure. 1) Not the constant P in 3). 177, 178] EQUATIONS OF HYDROSTATICS. 465 Also, since, by 8) V differs from — P only by a constant, the sur- faces of equal pressure are equipotential or level surfaces. If the fluid is incompressible y is constant, so that we have 11) -V= const. + -•' For gravity we have, if the axis of Z is measured vertically upward so that 12) P~tt(C If, neglecting the atmospheric pressure, we measure z from the level surface of no pressure 13) P = -9Q*, which is the fundamental theorem for liquids, namely, that the pressure is proportional to the depth. 178. Height of the Atmosphere. If we consider a gas whose temperature is constant throughout, the relation between the pressure and volume is given by the law of. Boyle and Mariotte p = a$, accordingly 14) P = f*2 = r°±!L = a log e + const. and 15) V=gs = c — a 16) ? = 9o«", where p0 is the density when 0 = 0. Thus as we ascend to heights which are in arithmetical pro- gression, the density decreases in geometrical progression, vanishing only for # = oo. If on the other hand, we consider the relation between pressure and density to be that pertaining to adiabatic compression, that is compression in such a manner that the heat generated remains in the portion of the gas where it is generated, we shall obtain a law of equilibrium corresponding to what is known as convective equilibrium. The temperature then varies as we go upward in such a way that, if a portion of air is hotter than the stratum in which it lies, it will rise expanding and cooling at the same time until its temperature and density are the same as those of a higher layer. When there is no tendency for any portion of WEBSTER, Dynamics. 30 466 X. STATICS OF DEFORMABLE BODIES. air to change its place convective equilibrium is established. The principles of thermodynamics give us the relation for adiabatic com- pression 17) p = !)(>*, where % is the ratio of the specific heat at constant pressure to that at constant volume, whose numerical value is about 1.4. We then have 18) Since x > 1, p diminishes as 8 increases and is equal to zero when gz = cf so that on this hypothesis the atmosphere has an upper limit, which may be calculated when the value of Q for a single value of 8 is known. It is obviously improper to consider the equilibrium of the atmosphere to an infinite distance without taking account of the variation of gravity as the distance above the surface of the earth increases. Considering the earth to be a sphere with a density that is a function only of the distance from the center, we have^ with 7 positive1), as in §§ 123, 149, instead of equation 8), Xdx+ so that on the hypothesis of equal temperature 19) - - — = const. — a log p, 20) 9 = 9,e^. On this hypothesis the density decreases as we leave the earth, but not so fast on account of the diminution of gravity, so that at infinity the density is not zero but equal to the constant In this example we have neglected the attraction on the gas of those layers lying below. From equation 20) the barometric formu is obtained. Proceeding in the same manner for convective equilibrium, we have 21) - - Here again Q decreases as r increases, giving an upper limit to the atmosphere for Q = 0 for a finite value of r. 1) It is to be observed that in equation 8) the forces are taken as the negative derivatives of the potential, but as in the following examples involving the earth's attraction they are obtained by multiplying ilae positive derivative byy, we must in the integral equation change the sign of V and multiply by y. 178, 179] ATMOSPHERE. ROTATING FLUID. 467 179. Rotating Mass of Fluid. If a mass of fluid rotates about an axis with a constant angular velocity ro, we may by the principle of § 104 treat the problem of motion like a statical problem, provided we apply to each particle a force equal to the centrifugal force. If we take the axis of rotation for the ^-axis the centrifugal force may be derived from the potential, If an incompressible liquid rotates about a vertical axis and is under the influence of gravity, we have by 11), 23) V=gz- y02 + r% 24) p = o(c-gs-\- ^(a^ + y*)). V * / Consequently the surfaces of equal pressure are paraboloids of revolution. Measuring 0 from the vertex the equation of the free surface, for which p = 0, is O ^ \ ^ / 2 i 2\ The latus rectum is \> On this principle centrifugal speed indicators are constructed. An important case which we have already treated by this method in § 149 is the shape of the surface of the ocean. If we seek an approximation, assuming the earth to be centrobaric, the potential due to the attraction of the earth and centrifugal force will be, as we find either directly, or by putting K=Q in § 149, 140), and the equation to the surface of the sea will be, U= const., which may be written, writing — for the constant, where ty is the geocentric latitude, and a is the polar radius. In the case of the earth ,.. = noo . . which is so small that the second ym. 288.41 term may be considered small with respect to the first, and its square neglected. Accordingly putting in this term r = a the equation of the surface is f w2as 1 1 ' 2yM ' ^J 30! 468 X. STATICS OF DEFORMABLE BODIES. which is the equation of an oblate ellipsoid of revolution. The ellipticity is > + ££)- Sir' The difference between this and the value — - given in § 149 is due to the fact that we have neglected the attraction of the water for itself and that the nucleus is not exactly centrobaric. 18O. Gravitating, rotating Fluid. A problem of great importance in connection with the figure of the earth and other planets is the form of the bounding surface of a mass of homogeneous rotating liquid under the action of its own gravitation. If V is the potential of the mass of fluid at any internal point, and we take the X-axis for the axis of rotation, we have The form of the function V depends upon the shape of the bounding surface of the liquid, which is to be determined by the problem itself. The complete problem is thus one of very great difficulty and has been only partially solved.1) We will examine whether an ellipsoid is a possible figure of equilibrium. We have found in § 157, 37) for the potential of a homogeneous ellipsoid QO C '( T2 it2 ?"*• \ fiii 30) V=x<>al>c J |l--^-&^--^| ™ — o = const. -^{Lx2 + Mf + where du L = 2itoal>c I - J (a^-\-u}-[/(a^-\-u)(l^-^ 31) M=2it(>abc C = I (h% _j_ -jA I/Yd2 -4- u ^/ \ J r \ 0 I - J ( 1) Poincare, "Sur Tequilibre (Tune masse fluide animee d'un mouvement de rotation." Acta math., t. VII, 1885. Also, Figures d'equilibre d'une masse fluide. Paris, 1903. 179, 180] GRAVITATING ROTATING FLUID. 469 Inserting in the integral equation 11) (see footnote p. 466), 32) ~ + i{ Lx* + W + N^ - 7 (f + *2) } = c°™t., the surfaces of equal pressure are similar to the ellipsoid 33) L, and ^0) = 0 for I = 2.5293, for which value 470 x STATICS OF DEFORMABLE BODIES. The course of the function is shown in Fig. 150, from which it is evident that if < 0.22467 Fig. 150. there are two values of I satisfying equation 38) and accordingly two possible ellipsoids of rotation. If on the contrary > 0.22467 " no possible ellipsoid of rotation is a figure of equilibrium. When co is very small one of the values of 'L tends to zero and the other to infinity, that is, one of the ellipsoids is a sphere, the other a thin disc of infinite radius. In the case of the earth using the value of yQ of § 123 and of co of § 149. - - = .00230. and the smaller of the two values 7 %icyg of A coincides most nearly with the actual facts1), giving A2 =0.008688, e = ^- The actual ellipticity being however ^^ we can only conclude that ^jy y the earth when in its fluid state was not homogeneous. The transcendental equation 36) written out is 40) HD -e2) C\-*- >J la'+« or otherwise 41) (V - du {<*»*+ =0. Besides the solution 6 = c there is another given by putting the integral equal to zero. When a = 0, the integrand and consequently the definite integral is negative, when a = — - - the integral is j/&2-t-e2 positive. There is accordingly a real value of a which satisfies the equation and there is an ellipsoid with three unequal axes which is a possible figure of equilibrium, if co lies below a certain limit. This result was given by Jacobi in 1834. For further information on this subject the reader is referred to Thomson and Tait, Natural Philosophy, §§ 771—778. 1) Tisserand, Traite de Mecanique Celeste, Tom. II, p. 91. 180, 181] POSSIBLE ELLIPSOIDS. 471 181. Equilibrium of Floating Body. Let us apply the equa- tions of equilibrium to a solid body immersed in a fluid under the action of any forces. Let us find the resultant force and moment of the pressure exerted by the liquid on the surface of the body. If we call the components of the resultant H9 H, Z, and of the moment L} M, N, we have S = / / p cos (nx) dS, 42) H = I I p cos (ny) dS, Z= I I pcoB(ne)dS, L = I I [yp cos (ns) — zp cos (ny)} dS, 43) M = / / {#p cos (nx) — xp cos (nz)} dS, N = I I {xp cos (ny) — yp cos (nx)} dS. If the body is in equilibrium it is evident that we may replace it by the fluid which it displaces, which would then be in equilibrium according to equations 4), and might then be solidified without disturb- ing the equilibrium. If the body is only partly immersed we must apply the integration to the volume bounded by the wet surface and a horizontal plane forming a continuation of the free surface of the liquid and called the plane of flotation. Over this plane p = 0, consequently the surface integral is taken only over the wet surface, while the volume integral is as before taken over the volume of the fluid displaced. With this understanding we may couvert the surface integrals into volume integrals taken throughout the space occupied by the displaced liquid, that is, within the surface of the solid body below the plane of flotation. We thus have 441 472 X. STATICS OF DEFORMABLE BODIES. 45) If the force acting on the liquid be gravity we have X = Y=0, Z=-g, accordingly where m is the mass of the fluid displaced by the body. This is the Principle of Archimedes: A body immersed in a liquid has its weight diminished by the weight of the displaced liquid. For the moments we have L 46) M = 9 i I I Qydt = gmy, = — g I I I Qxdr = — gmx, N = 0, where x, y denote the coordinates of the center of mass of the displaced liquid. If the body is in equilibrium, by Archimedes' Principle the weight and therefore the mass of the body is equal to that of the displaced liquid. Consequently the resultant of the forces acting on the body is equivalent to a couple whose members are forces mg exerted downward at the center of mass of the body and upward at the center of the mass of the displaced liquid. If the couple is to vanish one of these must be vertically above the other. The center of mass of the displaced liquid is called the center of buoyancy of the body. If the floating body is slightly displaced through a small angle deo from the position of equilibrium by the application of a couple, the mass of the displaced fluid must remain unchanged, but the position of the center of buoyancy is slightly altered. Let us take the origin in the intersection of the old and new planes of flotation (Fig. 151). For the new position the figure is to be tilted in the direction of the arrow until the new position of the water line W'L' is horizontal. The old center of buoyancy JB is now no longer under the center of mass Gr and consequently, if the same portion were immersed, 181] FLOATING BODY. 473 the body would be acted upon by a couple equal to mg times the horizontal distance between the verticals through G and B, and tending to increase the angular dis- placement. If 1) denote the ^ M length GB which now makes an angle d co with the vertical, this horizontal distance is w , and the couple mgbdco. \ j- Ju 7 _ . j.-, — i>— _L-^" \ """^-r' But this is not the only couple, \ B ) for the immersed part is now \^__ ^ different from that formerly Fig 151 immersed by the volume of the two wedges of small angle tfco, the wedge of immersion EOE1 and the wedge of emersion DOD'. The buoyancy added by the wedge of immersion and that lost by the wedge of emersion both produce moments in the direction tending to decrease the displacement. These moments must accordingly be subtracted from that previously found, to obtain the whole moment tending to overset the body. It is evident that if no vertical force is to be generated, the volumes of the wedges of immersion and emersion must be equal. Since the wedges are infinitely thin we may take for the element of volume at = zdxdy = yo&dxdy. The condition for equal volumes is then / I zdxdy = dG> I I 47) the integral being taken over the area of the plane of flotation. This will be the case if the axis taken through the center of mass of the area of flotation. The moment due to the wedges is 48) L'=g$ I I j ydx = gQ I j sydxdy / / y*dxdy = where KX is the radius of gyration of the area of flotation about the X-axis. In like manner 49) M' = — gg I I I xdt = — g$ I I xzdxdy = — ggdco I I xydxdy. 474 X- STATICS OF DEFOEMABLE BODIES. If we take for the axes of x and y the principal axes of the area of notation the integral / / xydxdy vanishes. Accordingly a rota- tion about a principal axis through the center of mass of the plane of notation developes a couple about that axis of magnitude ggdoSnl tending to right the body. We have accordingly for the whole moment of the righting couple 50) L = gdco (p£x| — mV). On account of the change in the immersed part the center of buoyancy has moved from B to B'. If we draw a vertical in the new position through B\ the point M in which it cuts the line BG is called the metacenter and the distance MG = h, the metacentric height. Since the couple acting on the body is composed of the two forces, mg acting downward at G and upward at B', it is evident that if the equilibrium is stable or the righting couple is positive M must be above G. The arm of the couple being the horizontal projection of MG is equal to hx-d(o9 and L = mghxd(D. We accordingly have, inserting this value of L in equation 50) for the metacentric height 51) mhx dividing by m and writing — = V, the volume of the displaced liquid, 52) k_£j?_6. The equilibrium is stable or unstable according as this is positive or negative. For the displacement about the F-axis we have in like manner a couple proportional to the displacement, with a new metacentric 53) ft*=y-&, where Ky is the radius of gyration of the plane of notation about the F-axis. It is evident that the metacentric height is greater for the displacement about the shorter principal axis of the section. Thus it is easier to roll a ship than to tip it endwise. Since the rotation about either axis is resisted by a couple proportional to the angular displacement, the body will perform small harmonic oscillations about the principal axes with the periodic times ««« and ! where Kx and Ky are respectively the radii of gyration of the solid about the principal axes in the plane of notation. 181, 182] METACENTER. 475 Introducing the values of hx and hy we may write 54) '£ Since for a body of the shape of a ship 5T and # increase together, we see that the larger x corresponds to the shorter time. The pitching of a ship takes place more rapidly than the rolling. The locus of the center of huoyancy for all possible displacements is called the surface of buoyancy, and the two metacenters are the centers of curvature of its principal sections. Evidently the body moves as if its surface of buoyancy rolled without friction on a horizontal plane, for it would then be acted on by the same couple as under the actual circumstances. 182. Solid hollow Sphere and Cylinder under Pressure. We have dealt in § 176 with a few cases of equilibrium of solid bodies under stress, where the strain produced was homogeneous. We shall now treat a number of cases in which the strain is not homogeneous. If there are no bodily forces the equations 144), § 175, become 55) Forming the divergence, by differentiating respectively by x, y, 2, and adding, and interchanging the order of the operations d and A, 56) (I + 2ii) 4 = ^, ™ = 87; *-^ = «» and (p is equal to the potential of a mass of density - — > coinciding 4 7t with the body under investigation. If this be a sphere or spherical shell, we find, as in §§ 125, 135, er>\ 58) 59) 9^^+, where & is a second arbitrary constant, from which we have 476 x- STATICS OF DEFORMABLE BODIES. 60) c_? dy flz la*" ,,.2) „. ; a b dw _ 36^o? du- _ Sbzy div _ a b giving (? = a. The values of the surface forces are by 145), § 1757 I~« ^ /a b . 3&^2\~| f ^ Xn = [id + 2/i (¥ - p + -p-)J cos (nx) r / N /NT / cos * + ^ cos ( # cos Wj8f + ^ cos H — ~ [^ cos (Mic) 4- y cos (wt/)]. Collecting the terms bx Ix 11 / \ z / \\ fttibx Qiib p- (7 cos (ns) + f cos (ny) -f 7 cos (»*)] = -^- = ^- cos and writing 63) these become 64) Xn=pcos(nx), Tn = pcos(ny^ Zn=pcos(nz), so that p is a normal traction or pressure. If E± and E2 are the internal and external radii of the shell, pi = Ha + jfsr' which determine a and 6. _ 66) ^8 s 182] HOLLOW SPHERE AND CYLINDEE. 477 In Oersted's piezometer the internal and external pressures are equal, so that 6 = 0, and the sphere receives a homogeneous strain, which is of the same magnitude, 6 = ^~> as if the sphere were solid. A second practical application is found in the correction of thermometers due to the pressure of the mercury causing the bulb to expand, the amount of expansion being found from 66). In treating the case of a very long hollow cylinder, we proceed in precisely the same manner, except that the problem is a two dimensional one. We will number the equations in the same manner, with the addition of an accent. The formulae have an application in finding the pressure able to be borne by tubes and boilers. 57') w = ' V==-' w==0> 6 59') d fp (a . , b \ x u = -^~ = ( — r + - ) - dx \2 r/r dm /a , b\ y v = -2- = ( — r + — I— dy \2 r) r du a b 2bxz du 2bxy d^Y + r*" ~r^' J~y = ~^~ dv _ 2bxy dv a b Jx = ~7^"; ~d = ¥ + T2 •v fi o (a . b 2bx*\~\ , x 4:bxy Xn = \la + 2fi (- + T2 -- -H cos (nx) -- -f- cos (ny), 62'") ' ^ ±bxy , r, 0 la . b 2bx*\~] , ^ Yn = - -^ cos nx + \la + 2^ (- + -2 - -^jj cos (ny), 63')' l^Ol+fOa-^' 64f) Xw = p cos (wa?), TJ, =._p cos (ny), 66') «-,- _2 f~— —\ 478 X. STATICS OF DEFORMABLE BODIES. 183. Problem of de Saint -Veiiant. We shall now treat a problem dealt with by the distinguished elastician Barre de Saint- Venant, in two celebrated memoirs on the torsion and flexion of prims, published in 1855 and 1856.1) For the full treatment of this subject the reader is referred to Clebsch, Theorie der Elasticitdt, or especially to the French translation of the same work, edited with notes by de Saint -Venant himself. The problem is thus defined by Clebsch, whose analysis is here followed. What are the circumstances of equilibrium of a cylindrical body, on whose cylindrical surface no forces act, and whose interior is not subjected to external forces, under which the longitudinal fibres composing the body experience no sidewise pressure?2) What forces must act on the free end surfaces, in order to bring about such circumstances? We have already treated a special case of this problem in 3°, § 176. If the ^-axis be taken parallel to the generators of the cylindrical surface of the body, the conditions that adjacent fibres exercise no stress on each other perpendicular to their length are 67) The conditions at the cylindrical surface are Xn = Xx cos (nx) H- Xy cos (ny) + Xz cos (nz) = 0, 68) Yn = Tx cos (nx) + Yy cos (ny) + Yz cos (ne) = 0, Zn = Zx cos (nx) + Zy cos (ny) + Zz cos (nz) = 0, of which the first two are satisfied identically, since cos nz = 0, and the last is simply, 69) . Zx cos (nx) + Zy cos (ny) = 0. In order to remove the possibility of a displacement of the cylinder like a rigid body involving six freedoms we will suppose fixed: a point, a line, and an infinitesimal element of surface of one orthogonal end or cross -section. Take the fixed point as origin, so that UQ '= v0 = WQ = 0. Take the fixed element of surface for the XT- plane, and the fixed line for the X-axis. For a point near the origin, the shifts are 1) Memoire sur la torsion des prismes, avec des considerations sur leur flexion. Memoires des Savants etr -angers, 1855. Memoire sur la flexion des prismes. Journ. de Math, de Liouville, 2me Serie, T. I, 1856. 2) That is, the stress on a plane parallel to a generator is only tangentij and in the direction of the generator. 183] ST. VENANT'S PROBLEM. 479 ^ dx If the point is in the XY- plane, dz = 0, and as this plane is fixed, w must be 0, necessitating If the point is on the X-axis, dy = dz = 0, and as it must remain on the X-axis, v and w must vanish, necessitating (TT) = o- \#SB/tl The six conditions at the origin are accordingly, while the conditions 67) everywhere satisfied are 72) Of these the first two give x du dv I, fciu dv 3w\ k / du 3w fa===jy= ~2^\a^ + ^+ aij= ~2 ^. ^w? dw ~ Z (I -\-ti~dz = ~ Vjz' with the third b) |^+|f = 0. 0^ r ^o; The equations 144) of equilibrium are 73) 480 x- STATICS OF DEFORMABLE BODIES. From the first of a) and b), by differentiation, 74) l**v gfj ^ 0** From a), n~\ Bu , Bv , dw ,+ The equations 73) become c) ; ^ B*w 3*W . ^ V and the conditions 71) at the origin, constitute the mathematical statement of the problem. Differentiating e) by #, and subtracting the derivative of c) by x and of d) by y, 77) Inserting the values of -«- and ^- in terms of -«- from a) gives 78) |^ = 0. Adding the derivative of c) by y and of d) by x, 79) + (l- *2 cos (ny) , 184] TORSION OF PRISMS. 485 V, F15 F2 are uniquely determined, each vanishing at the origin. The origin is any point on the line of centers of mass of the cross- sections. The most general solution of de Saint -Venant's problem now contains six constants, a, a1? a2, /3, &1? &2. We may examine separately the corresponding simple strains, by putting in each case all the constants but one equal to zero. I. a 4= 0. u = Z2 = Ea, This is a stretch -squeeze of ratio ?y, which has been already treated in § 176. II. /3+0. Ill) u = pyz, v = — pxs, w = pV. These equations represent a torsion, whose rotation is proportional to the distance from the origin, or fixed section. We have u = v = w = 0 for x = y = 0 the line of centers. For the stresses we find 112) Z.-E%-0, Zx = so that the stress on any cross -section is completely tangential. If the contour of the bar is circular, we have cos (nx) : cos (ny) = x : y, dV so that ^n = 0 and V = tv = 0, and accordingly we see that the plane cross -sections remain planes. For any other form of cross- section than a circle, V does not vanish, so that the cross -sections buckle into non- plane surfaces. This buckling was neglected in the old Bernoulli -Euler theory of beams, and constitutes, as was shown by de Saint -Venant, a serious defect in that theory, which is still largely used by engineers. In order to produce this torsion, we should apply at the end cross -section the stresses 113) Zx=X, = iipy, Zy = Yz = -iipx, that is at every point a tangential stress perpendicular to the radius vector of the point in the plane and proportional to its distance from the line of centers. As in practice it would be impossible to 486 X. STATICS OF DEFORMABLE BODIES. apply stresses varying in this manner, we make use, in this and the other cases, of de Saint -Tenant's principle of equipollent loads, viz: If the cross- section is small with respect to the length, stresses applied to a body near the ends produce approximately the same effect if their statical resultants and moments are the same. Consequently we may apply to the end faces or to the convex surface near them the forces and couples, X = I I X,dS = iifi I JydS = 0, Y = / / F, dS = - [ift I i xdS = 0, Z= I J Z2dS = 0, 114) L =(VZ> ~ *Y') dS = M= C f(zXz- xZz}dS = pp C iyzdS = 0, 0 0 The twisting couple divided by the rotation per unit of length, 115) ^ (oz is proportional to the fourth power of the radius. This is the law announced by Coulomb, in his work on the torsion -balance. This factor of the applied couple multiplying the twist per unit length is called the torsional rigidity of the prism. Thus the shear -modulus p may be determined by experiments on torsion. For other contours than the circle it is convenient to introduce into the problem instead of V its so-called conjugate function ^, defined by the equations, 3V rOW cV dW 116) - = -jr-J H— = — -7f-> ex cy cy ex 184] TORSION OF PRISMS. 487 If we now form the line - integral around the contour of the cross- dV section, of -? from 109) , on dV ll7) /ferfs=/(Ecos(^)+ = / [— y cos (nx) -\- x cos 7 ds cos (nx) — — cos (nyy\ ds, iayv'-vw"V $x since if we circulate anti- clockwise (Fig. 152), ds cos (w#) = dy, ds cos (ny) = — dx the integrals become 118) = - / (ydy + xdx), *J and since both differentials are perfect, integrating, 119) gr^C-ite'+tf2). Fig. 152. Accordingly if we can determine a harmonic function *P which on the contour shall have the value 119), the problem is solved. For example take the functions, giving equal to a constant on the ellipse tfz , y' a2 -t- -ftl if we take giving ~ 2 488 X. STATICS OF DEFORMABLE BODIES. We thus obtain u**fl9*, v= -fxs, w The curves of equal longitudinal displacement, or contour-lines of the buckled sections, are equi- lateral hyperbolas (Fig. 153). The stresses are, by 112), Fig. 153. The twisting couple is f* f* f* r* 120) N= I I (xYz - yX,) dS = - ^t. / / (a*y* + tftf) dx dy> In a similar we may deal with an equilateral triangular prism. If a be the radius of the inscribed circle, the equation of the boundary may be written, (x -d)(x- i//3 + 2 a) (x + y V 3 + 2 a) = 0, or The function J.(.^3~3^2) is a circular harmonic, and is constant on the boundary if we take Fig. 154. The curves of equal distortion are shown in Fig. 154. For any contour we have for the couple, from 112) 121) N (xY2 - 184, 185] TORSION OF PRISMS. 489 where I is the moment of inertia of the section about the If we call 122) ff(X |T - y fl) dS = (1 - q)ff(*> + f) dS then 123) N = and the moment per unit twist per unit length qpl is called the torsional rigidity of the bar, and q is de Saint -Tenant's torsion -factor. For the circle q = 1, for the ellipse q = 2^fc2 2 , since I = ^-ab(a2+ &2). If S = nab is the area of the ellipse, the rigidity may be written, ii S*. and by a generalization of this formula, de Saint -Venant writes Having solved the problem for a great variety of sections, he found that, when the section is not very elongated, and has no reentrant angles, K varies only between .0228 and .026, its value for the ellipse being .02533. We may thus put in practice obtaining a most valuable engineering formula. Considering the dimensions of S and I, we see that for similar cross -sections, the rigidity varies as the square of the area of the section, as stated by Coulomb, but for different sections the results differ much from those of the old theory, in which q was supposed to be unity. . 185. Flexion. For the third case we put III. «,+(), 124) u = - y{£2-M02-r% v 125) Zz = Ed^ = Ealx) The force on any section, Z= I i ZzdS = Ea± I I since the origin is the center of mass of the section. In this case the couples L and N vanish, while 490 X. STATICS OF DEFORMABLE BODIES. = I C(gX,-xZ,}dS= -Ec a parabola, or, since the displacement is supposed small, a circle of radius — • This displacement is called uniform flexure, for the curvature of the central line is constant. It is produced by the action of no forces, but of a couple applied at the ends. The couple is the same for all cross -sections, and is equal to the product of Young's modulus by Iy and the curvature of the central line 127) M This is the theorem of the bending moment. Such a strain is produced in a bar when we take it in our hands and bend it by turning them outwards. If the bar has a length I from the fixed section, the deflection of the end is 128) M = _ and the flexural rigidity, or moment per unit displacement per unit of length is 2EIy. For a rectangular section of breadth 6 and height h, I= =J J _A _A ¥ 2 For a circular section of radius R, For a circular and rectangular beam of equal cross -sections, since = bh the ratio of stiffnesses is ^rectangle _ bhs and if & = 7& Circle ' 12 •^circle ^ = 1.0472. Since w = a^xz, a plane cross -section at a distance 0 from the origin remains a plane which is rotated through the angle — = a^z, 185] UNIFORM FLEXURE. 491 cutting the XY- plane at a distance - below the origin. Such a plane remains normal to the line of centers, and the flexure is circular (Fig. 155). If the section is rectangular, the sides y = + — become, snce v = — while the sides x = + -= become, since or since g is constant, circles, so that the cross -section becomes like Fig. 156. . Fig. 155. Pig. 156. In an experiment by Cornu a bar of glass was thus bent, and points of equal vertical displacement were observed by Newton's fringes produced by a parallel plate of glass placed over the strained plate. The curves u = const, are #2 — ny^ = const. a set of hyperbolas whose asymptotes make an angle a with the Z-axis given by ctn2« = y. By photographing the fringes and measuring the angle Cornu obtained a value of ^ very near to —> the value given by Poisson. The case a2 4= 0 *s precisely like that just treated, except that the roles of the X and F-axes are interchanged. We pass therefore to: 492 X- STATICS OF DEFORMABLE BODIES. IV. ^ 4= 0. 129) « = -^ «; = ^ where Fj is defined by cos (w*) + (2 + ^) *y cos (»»)• If the cross -section is symmetrical with respect to both axes of X and Y, evidently the boundary condition is satisfied by a func- 3V tion Fj_ evm in y, and o -j-i ~ are the velocity components of the particle, u, v, w. Accordingly we have KX dF 2F , dF , dF , dF 5) •^T~==^T + ^O h v Q h^^~- dt dt dx ' dy dz We shall call this mode of differentiation particle differentiation.1) Introducing this terminology, dividing by Q and transposing, our equations of motion 3) become If + du U o p ' ^^ . du Vd-y + W8z = X- 1 dp o — ' Q dx ^\ dv . 6) 8t + W ~ f- ' ^iC ' v^- + w — dy z = r- Q dy dw 3«0 , div dw # 1 dp ~dt + ^iC dy dz ^Jz' DF 1) In most English books the symbol •> used by Stokes, is used for jj t particle differentiation because of the very objectionable practice of making no distinction in the symbol for ordinary and partial differentiation. , Dynamics. 32 498 XT- HYDRODYNAMICS. If we consider any closed surface fixed in space , expressing the fact that the increase of mass of the fluid contained therein is represented by the mass of fluid which flows into the surface, we shall obtain an additional equation. The velocity being q} the volume of fluid entering through an area dS in unit time is, as in § 169, equation 78), qcos(q,n)dS, and the mass, pgcos (q, n) dS. We have therefore for the total amount entering in unit time 7) I / gq cos (q,ri)dS =1 I p{iecos(w#)-f- vcos(ny)-\-wcos(n0)}dS But this is equal to the increase of mass per unit of time, for the volume of integration is fixed, that is, independent of the time, consequently we may differentiate under the integral sign. Writing this equal to the volume integral in 7) and transposing, Since this holds true for any volume whatever the integrand must vanish, so that we have _ A ^t~ ~^r + ~W~ ~0^~ which is known as the Equation of Continuity. Performing the differentiations we have 00 , 00 , 00 , 00 , (0** , 0^ . or in the notation of particle differentiation, If now we fix our attention upon a small portion of the fluid of volume V as it moves, its mass will be constant, say, m = $ V= const. By logarithmic differentiation, I^ + ^-^-O g dt H F dt ~ so that the expression ... du.dv . div _ 1 dQ _ 1 dV ^^J^^ Tz~ ~^~di~=T^dt' that is, the divergence of the velocity is the time rate of increase of volume per unit volume. This corresponds with the expression 186, 187] EQUATION OF CONTINUITY. 499 for the dilatation found in equation 75) § 169, the divergence, which we shall still call '2 \aa; ds// ~ S, which represent the components of a vector co, which by comparison with the expressions for &x, &y, &z in § 169, 77) is seen to be the angular velocity of rotation of a particle or the vorticity. Let us accordingly write, putting v, ^w; and adding, the terms in 1, 17, £ disappear and integrating over any volume, we have - / / [« cos (wa;) + » cos (ny) + «c cos (»«)] ( UQ —p) dS ^ -L ?^") -L ~ ' Introducing the value of U and of its multiplier in the last integrand, o -ST* 29) o which by equation 10) is equal to - • -ST* an^ transposing, we obtain, If the applied forces are independent of the time, -^r = 0, and we may write the left-hand member, 188, 189] EQUATION OF ACTIVITY. 503 30) since the volume of integration is fixed. The first term of the integral represents the kinetic energy and the second term, the potential energy due to the applied forces. The term on the left in 29) is accordingly the rate of increase of the energy, kinetic and potential. Of the terms on the right, the amount of matter Qqcos(qn)dS flowing through dS in unit time brings with it the energy so that the first part of the surface integral represents the total in- flow of energy. The remaining surface integral and volume integral containing p represent the work done by the pressure, for at the surface the velocity q and the force pdS give the activity pqcos (qri)dS, so that the surface integral represents the activity of the pressure at the surface. If we consider a small element of volume F, the work done in compressing it by an amount dV is as above —pdV, and the activity Q1\ dV du . dv 31) -r- = Putting V=dr and integrating, we find that rrr isu . a. is the activity of the pressure in producing changes of density in the whole mass. Transposing this term we find that equation 29) expresses the following: The rate of increase of energy of the fluid, both kinetic and potential, due to the external forces plus the activity of compression (production of intrinsic energy) is equal to the rate of inflow of energy plus the activity of pressure at the surface. Equation 29) is therefore the equation of activity or conservation of energy. 189. Steady Motion. Steady motion is defined as a motion which is the same at all times. Assuming that not only X, T9 Z, V but^f, v,w,p, Q are independent of t, equations 27) for steady motion become 32) 504 XL HYDRODYNAMICS. If the motion be non- vortical, the left-hand members vanish, and we immediately obtain the integral 33) F + P -f 4 a = -S0 which determines q± in terms of the difference of pressures. The flux in unit time is then The theorem expressed by equation 33) is known as Daniel Bernoulli's theorem. For gases expanding isothermally P = a log p = a logp -f const. Consequently equation 33) becomes 39) a logp -}- -— q2 = const. This formula may be used to calculate the velocity of efflux through an orifice from a vessel containing gas under pressure. If the pressure in the vessel at a point so remote from the orifice that the air may be considered at rest is p and if the pressure of the atmosphere at the orifice where the velocity is q is p0, we have alogp = alogpQ + y£2, \^ 40) (f = 2 a log — • If the efflux is adiabatic, as in practice it nearly is, by § 178, 18) Accordingly 41) q2 = \ which is the usual formula for the efflux of gases. If the external force is gravity V=g0, so that equation 33) becomes for an incompressible fluid, 42) — -f qz + -^ q2 = const. Q 2 If we consider efflux from a reservoir whose upper free surface is so large that q is negligible, the pressure being that of the atmo- 506 XI. HYDRODYNAMICS. sphere, the £- coordinate zl9 the velocity of efflux q at a point where z = #2 ig given "by 43) a' or the velocity of efflux is equal to that acquired by a body falling freely from a height equal to that of the free surface to the orifice. This is Torricelli's theorem. 190. Circulation. We define the circulation along any path as the line integral of the resolved tangential velocity, B B 44) v = -^-9 w = -£-> ox £y dz and qp is called the velocity potential, a term introduced by Lagrange. (When there is vorticity there is no velocity potential.) Before 1858 only cases of motion had been treated in which a velocity potential existed. In that year appeared the remarkable paper by Helmholtz 1) on Yortex Motion. Let us now find the change of circulation along a path moving with the fluid, that is, composed of the same particles, the forces being conservative. Our equations of motion 3) may be written, putting U' = — (F-f P) 49) du ~dt dv dw ~dt dU' vdz dU' Fig. 161. The change of circulation along the path AB is 50) dt d dt A B I (udx -f vdy + wdz), in which dx9 dy, dz vary with the time, being the projections of an arc ds composed of parts which move about. If after a time dt the arc ds assumes a length ds' whose components are dx', dy', dz* we have (Fig. 161) 1) Vber Integrate der hydrodynamischen Gleichungen, welche den Wirbel- bewegungen entsprechen. Wissenschaftliche Abhandkmgen I, p. 101. 508 XL HYDRODYNAMICS. ' = x + dx + dt [u + <^-dx + ^dy + — - (x + udf) cy z and therefore the change per unit time in the projections are du 7 K.I\ d /? \ dv -, . dv ^ , cv 51) Thus we have B KON ^w , dv dw 52) and substituting from equations 49) and 51) B -ON dffAB C[3U' . cU' . 8U' 53) -dr=Jb^dx + i^dy + -wd* A du . 3v . 3w ^ + Vd-x + WZx du . 3v , dw ^ + v8j + w du . dv . A which vanishes for a closed curve. Therefore if the forces are conservative, the circulation around any closed path moving with the fluid is independent of the time. Thus if the circulation around any closed path is zero at one time, it is always zero, or in other words if a velocity potential once exists, it always exists. This theorem is due to Lagrange. 190, 191] VORTEX MOTION. 509 191. Vortex Motion. We will now consider the case in which no velocity potential exists , that is, the case of vortex -motion, according to the methods of Helmholtz. From the equations 27), whose right-hand members are the derivatives of — ( V + P + ^ en , this quantity may be eliminated by differentiation. Differentiating the last equation by y, the second by #, and subtracting, we obtain *Wis U dz ~*~ * dz W dz ' or otherwise -•I!-: On the right the coefficient of w vanishes identically by 47), and that of | is by the e equation 55) becomes that of | is by the equation of continuity 12) equal to - ~> thus Now we have £ M ^ *| _ I *i ® dt\Q/ dt Q dt and accordingly we may write our equation 56) and its two companions i ^ , 1 ^ . 1 ^ 9 d# 9 ay 9 2z' rrrx _ | ^V 7] ^V ^V ^i-Si"^-??? • e 27' I fiw r\ diu g Bw 7 ^ 7 3y "^7 a« ' Thus the time derivatives of — > — > — for a given particle are homo- geneous linear functions of these quantities. By continued differentiation with respect to t and substitution of the derivatives from these equa- tions, we see that all the time derivatives are homogeneous linear functions of the three quantities themselves. Consequently if at a certain instant a particle does not rotate, it never acquires a rotation. This we find by developing —t—>— as functions of t by Taylor's theorem, for if the derivatives of every order vanish for a certain 510 XL HYDRODYNAMICS. instant, the function always vanishes. Stokes1) objects to this method of proof as not rigorous inasmuch as it is not evident that the func- tions I, rjf £ can be developed by Taylor's theorem, and replaces it by the following demonstration. Let L be a superior limit to the numerical values of the coef- ficients of — 9 — i — in the second member of equations 57). Then Q Q Q evidently £, 17, £ cannot increase faster than if their numerical or absolute values satisfied the equations 58) A dt _d dt d dt \ Q instead of 57), |, ??, g vanishing in this case also when # = these three equations and writing Adding we obtain 59) ^ The integral of this equation is and since 52 = 0 when t = 0, c must be zero, and £1 is always zero. Since the sum of the absolute values cannot vanish unless the separate values vanish, the theorem is proved. Let us now consider two points A and B lying on the same vortex line at a distance apart ds = s — > where s is a small constant. Since the particles lie on a vortex -line we have £xvx dx dy dz ds e We have for the difference of velocity at A and B £1\ ^U J %U 7 VU J bl ) Un — MA == o — u>% ~~r ^r~ dy ~r ~^~ u>% or by equations 57), 62) J_ Q dx^ Q 1) Stokes, Math, and Phys. Papers, Vol. II, p. 36. 191, 192] VORTEX -MOTION. 511 Now at an instant later by dt, when the particles are at A' and Bf, we have dx' = dx + (UB - O dt = 8 + 63) dy' = dy + (u* - UA} dt = sQ+± g) dt\ de' = dz + (UB - UA) dt = e[± + ± g ) dt] Therefore the projections of the arc ds' in the new position are proportional to the new values of — > — ? — > as they originally were, so that the particles still lie on a vortex -line. Accordingly a vortex- line is always composed of the same particles of fluid. Also since the components of ds have increased or have changed so as to be always proportional to the components of — ? if the liquid is in- compressible the rotation is proportional to the distance between the particles. And whether Q vary or not, if S be the area of the cross- section of a vortex -filament, gSds, the mass of a length ds remaining constant, so does So, the strength of the filament. It is easy to see that this is equivalent to a statement of the conservation of angular momentum for each portion of the fluid. Evidently in a perfect fluid no moment can be exerted on any portion, since the tangential forces vanish. Accordingly the strength of a vortex -filament is constant, not only at all points in the filament but at all times, consequently a vortex existing in a perfect fluid is indestructible, however it may move. It is from this remarkable property of vortices discovered by Helmholtz that Lord Kelvin was lead to imagine atoms as COD- sisting of vortices in a perfect fluid. 192. Vector Potential. Helmholtz's Theorem. We have seen that any curl is a solenoidal vector. We may naturally ask whether conversely any solenoidal vector can be replaced by the curl of another vector. It was shown by Helmholtz that any uniform continuous vector point -function vanishing at infinity can be expressed as the sum of a lamellar and a solenoidal part, and the solenoidal part may be expressed as the curl of a vector point-function. A vector point -function is completely determined if its divergence and curl are everywhere given. Let q be the given vector, which in our case is the velocity of the fluid. Let us suppose that it is possible to express it as the sum of the vector -parameter of a scalar func- 512 XI. HYDRODYNAMICS. tion (p and the curl of a vector function Q, whose components are U, V, W. Accordingly let us put _ - 64) dt dx dy Finding first the divergence of q we have f>*\ j- du . dv . dw bo) div. q — % ho "~ -Q— = the divergence of the curl part vanishing. But by § 128, 5) we know that if cp and its first derivatives are everywhere finite and continuous, we have ST. a 7 — dr. Since q is continuous by hypothesis, div. q is finite. Consequently the lamellar part of q is determined by its divergence. Secondly finding the curl of q, dy All 4- dU 4- dV 4- dW -^U + + + 1Z_ zu\ _ —^ > — respectively, is called the vector potential of the vector — -• We may thus abbreviate our results in the vector _ Tt equations, 71) q = vector parameter

(u* + v* + w^ dr dlJ cW V\ - - Ti) = 2 / / K*7 w - w F) cos (WflO + (wU—uW) cos (ny) -f If the integral be taken over all space, since the motion is supposed to vanish at infinity the surface integrals vanish, and 81) T= <>fff[ Vt + Vrt + Wf\ dr, or inserting the values of U, V, W from 70) and the integration may now be restricted to the vortices. If again we integrate by filaments, we find where the integration is expressed as over the length of each of the double infinity of vortex -filaments constituting the vortices. This is the form obtained for the energy of two electric currents by Franz Neumann. 195. Straight parallel Vortices. Let us now consider the case in which the vorticity is everywhere parallel to a single direc- tion, that of the axis of g. Let the motion be uniplanar, that is parallel to a single plane, the X3^-plane, and the same in all planes parallel to it. All quantities are therefore independent of g. The vortices are columnar and either of infinite length or end at the free surface of the liquid. Such vortices may be produced standing vertically in a tank with a horizontal bottom. Under the conditions imposed we have QA\ du dv CM c. rr rr 84) 0 = w = TT- = Q- = ^ = $ = 7? = c7=F, dz dz dz and QKX dW dW 85) u = -5—> v = -- ;—> cy Ox 33* 516 XI. HYDRODYNAMICS. 86) 2? = ^-|' and by equations 57) so that £ is independent of the time for any given vortex -filament. The function W is not a velocity potential, but is said to be conjugate to a velocity potential (p for which 87) — fe — If- The function TF has a simple physical meaning. If we find the amount of liquid which flows across a cylindrical surface with generators parallel to the #-axis of height unity in unit of time, we have B B . 88) iff = I q cos (qn) ds = I [u cos (nx) + v cos (ny)\ ds, A A the line integral being taken around any orthogonal section of the cylinder. Now we have ds cos (nx) = dy, ds cos (ny) = — dx, so that 89) * •-{ ~ vdx) -dx + dy -W,- WA. A function, the difference of whose values at two points A and J5 gives the quantity flowing in unit time across a cylinder of unit height drawn on any curve with ends at A and jE?, is called a flux or current function. The quantity crossing is independent of the curve because the fluid is incompressible. In the present case the vector potential "FT is a current function. The stream lines being lines across which no current flows are given by the equation ^ = const. Substituting the values of u and v from 85) in 86), we have But this is the equation for a logarithmic potential with density — § 138, 61), so that we have as the integral 91) W=- as may also be found from equation 70) by integrating over the infinite cylinder as in § 135, subject to the difficulty mentioned on p. 385. The value of W given in equation 91) satisfies the equation 195] COLUMNAR VORTICES. 517 outside of the vortices and equation 90) at points within them, as shown in § 138. If we have a single vortex filament of cross-section dS and strength n = £'dS, 93) W = - ^-logrdS=- -logr, 7T 7C and the lines of flow are circles, r = const. Then = =^ y-y dy 'XT' r cW x x-x1 V = ex Ttr r 95) ,2 the velocity is perpendicular to the radius joining the point x, y with the vortex and inversely proportional to its length. It is to be observed that although the motion is whirling, every point describing a circle about the center, the motion is irrotational except at the center, where the vortex -filament is situated, each particle describing its path without turning about itself, like a body of soldiers obliquing or changing direction while each man faces in the same unchanging direction. The motion in the vortex on the contrary is similar to that of a body of soldiers wheeling or changing direction like a rigid body rotating. If we have a number of vortices of strengths ^, 3«2, . . . %„, and form the linear functions of the velocities of each, Z7 r, where u,9 vsy is the velocity at the vortex s both vanish. For any pair of vortices r and s we have where ur is the part of the velocity at xr, yr due to the vortex of strength x, situated at xsys. Thus while similarly xr-xs XSUS = KsXr -^— »—> so that the terms of the sum destroy each other in pairs. 1) U and V have nothing here to do with the components of the vector- potential. 518 XL HYDRODYNAMICS. Similarly for vortices continuously distributed, the strength of any elementary filament being 97) U which again vanishes, since every point is covered by both dS and dS'. If we define the center of the vortex as x where then since g does not depend upon t, if we follow the particle differentiating, 99) the integrals being taken over areas moving with the liquid. Therefore or the center of all columnar vortices present remains at rest. If we have a single vortex filament of infinitesimal cross- section S, for which 101) the velocity depends on the current function W = - — log r. In the vortex and close to it, if x is finite, g, W, u, v are infinite. But at the center u = v = 0, the vortex stands still and the fluid moves about it in circles with velocity — The angular velocity and the area of the cross -section remain constant, although the shape of the latter may vary. If we have two such vortex- filaments each urges the other in a direc- tion perpendicular to the line joining them, they accordingly revolve about their center, maintaining a constant distance from each other. If they are whirling in the same direction the center is between them (Fig. 162), but if in opposite direc- tions, it is outside, and if they are equal it lies at infinity. Such a pair of vortices may be called a vortex-couple or doublet, jijg 162 and they advance at a constant velocity, keeping symmetrical with respect to the plane bisecting perpendicularly the line joining them. This plane is a stream -plane and may accordingly be taken as a boundary of the 195] VORTEX PAIRS. 519 fluid. Since either vortex moves with a velocity - - and half way between them the velocity being due to both is - — = —^ we find ~¥~ that a single vortex near a plane wall moves parallel to it with a velocity one fourth that of the water at the wall. This is an illustration of the method of images, of frequent application in hydrodynamics. As another illustration con- sider the motion of a single vortex-filament in a square corner inclosed by two infinite walls. The motion is evidently the same as if we had a pair of vortex- couples formed by the given Q A O vortex and its images in the two walls, turning as shown in Fig. 163 and forming what may p. 16g be called a vortex kaleidoscope. From the symmetry it is evident that the planes of the walls are stream -planes, so that we may consider the motion in one corner alone. If x and y be the coordinates of the vortex considered, we have as due to the others, Y. K y Y. x* Zn x*-\-y* 2 it y (x2 -f 102) xx x Y, y* ' ' *~* = ~ * Since u and v are the velocities -jr>. -~ of the vortex, we have for at at the equation of its path dx irvoN dt u xs dx 103) -j— = — = -- - = —, or ay v y ay dt dx _ dy !c*~ ~~y*' whose integral is 111 T12?/2 104) + = ' ** + f = -' = -, sin2 & cos2 ^, n" ' a and in polar coordinates, 105) the equation of a Cotes's spiral, having one of the axes as an asymptote. 520 XI. HYDRODYNAMICS. The same problem gives us the motion of two equal vortex- couples approaching each other head on, or a single vortex- couple approaching a plane boundary, showing how as they are stopped they spread out. The beharior of vortex -couples will serve to illustrate that of circular vortex rings, for the theory of which the reader is referred to Helmholtz's original paper. The two opposite parts of a circular vortex appear to be rotating in opposite directions if viewed on their intersection by a diametral plane normal to the circle, thus resembling a vortex -couple. It is found that the circular vortex advances with a constant velocity in the direction of the fluid in the center, maintaining its diameter, but that when approaching a wall head on it spreads out like the vortex -couple. Two circular vortices approaching each other do the same thing, but if moving in the same direction the forward one spreads out, the following one contracts and is sucked through the foremost vortex, when it in turn spreads out and the one which is now behind passes through it, and so on in turn, as may also be shown for two columnar vortex-couples traveling in the same direction. Most of these properties of circular vortices may be realized with smoke rings made by causing smoke to puff out through a circular hole in a box, or mouth of a smoker, or smoke-stack of a locomotive. The friction at the edge of the hole holds the outside of the smoke back, while the inside goes forward, establishing thereby the vortical rotation. As previously stated no vortex could be formed if there were no friction. It is to be noticed that the direction of the fluid on the inside of the vortex gives the direction of advance. 196. Irrotational Motion. We shall now consider the non- vortical motion of an incompressible fluid. We then have a velocity potential qp and -t r\r>\ d

v = ^-i w = ^-- ox oy dz The equation of continuity becomes 107) 4y = 0, and the potential is harmonic at all points except where liquid is being created (sources) or withdrawn (sinks). The volume of flow per unit time outward from any closed surface S is 108) - / / [u cos (nx) -f v cos (ny) -f- w cos (nzj] dS so that if this is not equal to zero, it is equal to the quantity created in the space considered in unit time, 195, 196, 197] IRROTATIONAL MOTION. 521 so that if we put z/qp = tf , 6 is the amount of liquid produced per unit volume per unit of time. The total amount dt — i i i Gdr is called the strength of the source. If (3 is given as a function of the point we have Accordingly the velocity potential has the properties of a force potential, the density of attracting matter being represented by L times the strength of source per unit volume. The negative sign occurs here from the different convention employed, it being customary to define the force as the negative parameter, the velocity as the positive parameter of its potential. • In particular a point source of strength m produces a radial velocity of magnitude -^—^ This system is called by Clifford a squirt. 197. Uniplanar Motion. A simple and interesting case is that of uniplanar flow as defined above. We then have all quantities independent of s, so that Laplace's equation reduces to nn A powerful method of treatment of such problems is furnished by the method of functions of a complex variable. The complex number a 4- ib, where a and & are real numbers and i is a unit defined by the equation »»=-i, (the same root being always taken) is subject to all the laws of algebra, and vanishes only when a and ~b both vanish separately. Any function of the complex number obtained by algebraic operations, after substituting for every factor i2 its value — 1, becomes the sum of a real number plus a pure imaginary, that is a real number multiplied by i. Any equation between complex numbers is equivalent to two equations between real numbers, being satisfied only when the real parts in both numbers are equal as well as the real coeffi- cients of i in both members. If z denote the complex variable x + iy, any function of & may be written w = f(z) ==• u + iVj 522 XI. HYDRODYNAMICS. where u and v are real functions of the two real variables x and y. For instance z* = C» + ty)8 = x* - i w = #2 — ?/2, v 1 _ 1 x — iy z U = Let us examine the relation between an infinitesimal change in z and the corresponding change in /"(#)• We have, x and y being real variables capable of independent variation, 114) dz = dx + i dy, 115) Consequently by division, • du , d«* . /#« , dv \ 7 i . -, o — dx •+• -^— dy -+- 1 1 75 — dx -+- -^ — a w I ^.,,N die du -\-idv dx dy \o x cy } dyl dx The ratio of the differentials of w and z accordingly depends in general on the ratio of dy to dx, that is, if x and y represent the coordinates of a point in a plane, on the direction of leaving the point. If the ratio of dw to dz is to be independent of this direc- tion and to depend only on the position of the point x, y, the numerator must be a multiple of the denominator, so that the expression containing ~ divides out. In order that this may be true we must have du . . dv that is 117) \dx n dx/ dy dy Putting real and imaginary parts on both sides equal, -,^Q^ du cv dv du HO) O— = ^~ > 7T~ = a" > dx cy ex cy and 11Q. dw du . dv dv .du = + * = -' 197] COMPLEX VARIABLE. 523 In this case the function w is said to have a definite derivative defined by j,f , N j. dw dy = 0 and it is only when the functions u and v satisfy these conditions 118) that u + iv is said to be an analytic function of g. This is Riemann's definition of a function of a complex variable.1) The real functions u and v are said to be conjugate functions of the real variables x, y. It is obvious that if w is given as an analytic expression involving 0, w = f(&), then w always satisfies this condition. For dw df(z] dz f'C\ dw df(z] dz . ~, / x dx dz dx I \ )i fly d% fly I \ )' Accordingly .dw . (du . dv\ dw cu . dv cu _ dv dv _ du dx dy' ex dy If we differentiate the equations 118), the first by x, and the second by y and add, since d*v dzv dxdy ~ dydx' we obtain 120^ — 4-— — 0 dx*^ dy*-"' Differentiating the second by x and the first by y and subtracting, we find that v satisfies the same equation Thus every function of a complex variable gives a pair of solutions of Laplace's equation, either one of which may be taken for the velocity potential, representing two different states of flow. It is to be noticed that the question here dealt with is simply one of kinematics, since Laplace's equation is simply the equation of continuity and there is no reference to the dynamical equations. The question arises whether any two solutions of Laplace's equation will conversely give us the function of a complex variable. It obviously will not answer to take any two harmonic functions, for they must be related so as to satisfy the equations 118) or be mutually conjugate. In order to avoid confusion with the velocity 1) Riemann, Mathematische Werke, p. 5. 524 XL HYDRODYNAMICS. components u and v, let us call the two conjugate functions (p and ^7 satisfying the equations It is evident that qp and ^ have the relation of the velocity potential and stream function denned in § 195. If one function is given we can find the conjugate, for we must have which by equations 122) is Now if we call this Xdx + Ydy it satisfies the condition for a perfect differential dX __ dY dy dx that is, in this case, dy* dx* Consequently the line integral dtp -. . dm -. } — dx + 7T2- dy \ dy dx y) from a given point #0, yQ to a variable point x, y, is a function only of its upper limit and represents ^. Similarly if ^ is given 123) , Furthermore the first of the equations 122) is the condition that V dx + ? ^F which in virtue of the equations d$ dW ib = 75— = -- o— 9 dx dy 125) = a* = ^ ^ 3y dx' are conjugate to each other and give a new analytic function of z, 197] CONJUGATE FUNCTIONS. 525 whose derivative is gp + iif>. From these by new integrations we may obtain any number. The method of the complex variable accord- ingly gives us the solution of an unlimited number of uniplanar problems. The equations 122) are geometrically the condition that the lines (p = const., iff = const, intersect each other everywhere at right angles. If ijj is the stream function the lines ijj = const, are the lines of flow, which we know intersect the equipotential surfaces at right angles. As examples consider the cases worked above, w = £2, (p = x2 — 2/2, $ = %xy. The equipotential lines are sets of equilateral hyperbolas, intersected at right angles by the system of equilateral hyperbolas forming the stream lines (Fig. 164). The stream line iff = 0 consists of the X and Y axes, which may accordingly be a boundary, so that one quarter Fig. 164. of the figure represents the flow in a square corner of a stream of infinite extent. The function w = — gives x .1. _ y 526 XL HYDRODYNAMICS. The equipotential lines give a set of circles all tangent to the Y-axis at the origin , while the lines of flow are a similar set all tangent to the X-axis (Fig. 165). The water flows in on one side of the Fig. 165. origin and out at the other as if there were a source on one side and an equal sink on the other close together. The function zn, of which the two examples just treated are particular cases, gives an interesting case which is most simply worked out by the introduction of polar coordinates. x = r cos G), y = r sin o>, z = x + iy = r (cos co + i sin CD) = rei(a, -f from which we obtain the two conjugate functions 126) u = If we multiply these two harmonic functions by constants and add, the sum 127) rn [An cos (wo) + -#« sin n& is the circular harmonic function treated in § 140. We may accord- ingly develop the velocity potential in a series of circular harmonics, 197] FLOW AROUND CORNER. 527 128) y and if we know the values of qp on the circumference of a circle with center at the origin, we may find the coefficients by the method of Fourier as in § 140 a, 83). Let us examine the motion in a segment between two walls making an angle 2 a at the origin and reaching to infinity. If we use the value of cp given by equation 128), the coefficients and the values of n admissible are to be determined by the condition along each wall. But since dn = rda), we have 129) - ^ which must vanish for to = + a. If a*-f-i na = — ~ — n, and if ncc = nit, sin (+ no) — 0, 3c being any integer. Therefore if we put when n is an odd multiple of — 9 An = 0 and Bn = C^x+i and for even multiples, Bn = 0 and An = C^x, we shall have as a solution of the problem n 2 a • /2x-fl n V-.--AT « /»* • C2y. + ir sm (-— ^— • — '») + ft^ cos ^— o The tangential velocity at the wall is given by 131) The exponent of the lowest power of r is ^— -1. If this is negative, that is if a > --> the velocity is infinite for r = 0, that is at the corner, unless C^ = 0. The pressure is given by the equation p = const. — -| £2, so that at a sharp projecting edge around which the water flows there would be an infinite negative pressure. This being impossible, around such an edge the motion is discontinuous, so that instead of 528 XL HYDRODYNAMICS. flowing as in a) Fig. 166, the water flows as in b), the flow being discontinuous at the dotted line. In actual fluids such surfaces of discontinuity give rise to vortex motion, so that we see eddies formed at projecting corners. a) The function Fig. 166. with = logr 4- gives us radial stream lines forming a uniplanar squirt, while y> = co, ijf = log r Fig. 167. and for a free surface, p = 0, 133) C - gives us flow in circles with a velocity dcp The velocity at the center is infinite. This flow is exactly what we found in § 195 to be produced by a vortex filament at the center. If the fluid is under the influence of gravity, we have 42) p = const. — gqz — — 197, 198] WAVE MOTION. 529 If z is zero when r = , 0 = 0, and the equation of the sur- face is The form of the surface is shown in Fig. 167. This is approximately the form taken by the water running out of a circular orifice in the bottom of a tank, although the above investigation takes no account of the vertical motion. 198. Wave Motion. The case of uniplanar water waves may be dealt with by the method of the preceding section.1) Let us take the XY- plane vertical, the Y-axis pointing vertically upward and the motion as before independent of the 0 coordinate, so that we may use e to denote the complex variable. We shall find that the waves travel with a constant velocity and iib will therefore simplify the problem if we impress upon the whole mass of liquid an equal and opposite velocity so that the waves stand still and the motion is steady. Such still waves are actually seen on the surface of a running stream. Let us first consider waves in very deep water. At a great depth the vertical motion will disappear and we shall have only the constant horizontal velocity that we have impressed, so that u = — a, v = 0, from" which cp = — ax. The function f(z) = — a z + Ae~ik* = - a (x + iy) -f Ae-^+W gives (p + ^ = — a (% + iy) 4- Aeky (cos kx — i sin kx), 134) (p = — ax + Aekv cos kx, if/ = — ay — Aeky sin kx. When y = — cx> this makes (p = — ax, as required. The free surface of the water being composed of stream lines is represented by one of the lines ^ = const, and if we take the origin in the surface its equation is consequently 135) ay -f- Ae^sinkx = 0, which shows that y is a periodic function of x with the wave-length I = -^- The longer the wave-length, that is the smaller k, the more 1) Rayleigh, On Waves. Phil. Mag. I, pp. 257— 279, 1876. Scientific Papers, Vol. I, p. 261. WEBSTER, Dynamics. 34 530 XI. HYDRODYNAMICS. nearly does the exponential reduce to unity and the more nearly is the profile a curve of sines. The velocity is given hy 136) £2 = u2 + v*, where u = fl = v = ~ a-Ake^smJsx, 137) a * I? = = - > = Ake**coakx, 138) #2 = a2 + ^fcV** + 2^afce** sin lex. So far all our work has been kinematical. The relation to dynamics is given by introducing the equation 33) for steady motion, and at the surface putting jp = 0, and making use of the equation 135), 140) gy + i {a2 + A*k*#*y - 2a*ky} = C. Since the surface passes through the origin, putting y = 0 we obtain C=1-{a* + A*V}, inserting which gives 141) (g - a2fy y + ± AW (e^ - 1) = 0. This equation can be only approximately fulfilled, but if the height of the waves is small compared with the wave-length, so that 2 ky is small, developing the exponential and neglecting terms of higher order than the first in ky we have giving the equation connecting the velocity and wave-length 142) g - a2k + AW = 0. If ky is small the equation of the surface 135) is approximately 143) y = -- sin kx so that the maximum height of the waves above the origin is T$ = —- Inserting the values of the height and wave-length in equation 142) it becomes 8f2«/. 4:*2.BM 144) a {TV --^1 ~ * an equation connecting the wave-length, height and velocity. For 198] WAVES IN DEEP WATER. 531 waves long enough, in comparison with their height to neglect — ,2 -> we have 145) . «»=.£ If s is the height from which a body must fall to acquire a velocity equal to the wave - velocity, since a2 = 2gs, the equation becomes 146) I = ±xs, accordingly the velocity of propagation of long waves in deep water is equal to the velocity acquired by a body falling freely from a height equal to one -half the radius of a circle whose circumference is the wave-length. In order to study the motions of individual particles of water let us now impress upon the motion given by 137) a uniform velocity a in the X- direction. Equations 137) now give the motion with respect to moving axes travelling with the waves, so that in order to obtain the motion with respect to fixed axes we have to add a to the u of 137) and replace x by x — at, obtaining u = — A~ke*y sin k (x — at], 147) ij = Akeky cosk(x — at), for the equations of the unsteady motion of the actual wave -propa- gation. For the velocity of a particle we have 148) q = -\/u2 + v2 = AJce*y showing that the velocity decreases rapidly as we go below the surface, so that for every increase of depth of one wave-length it is reduced in the ratio e~27t = .001867. If the displacement of a particle which when at rest was at x, y is |, 17 we have -FT = — A~kekv sin~k(x — at), 149) if we neglect the small change of velocity from x,y to x -\-%,y -\- r}, so that we obtain by integration | = 1 F\A\ il = Thus each particle performs a uniform revolution in a circle of 2 it, >L radius Beky in the periodic time y— = — • We thus see how the ka a motion is confined to the surface layers. The direction of the motion in the orbit is such that particles at the crest of the wave move in the direction of the wave -propagation, those at a trough in the opposite direction. 34* 532 XI HYDRODYNAMICS. Let us now discuss the form of the wave -profile 135) when the restriction that the height of the waves is small in comparison with the wave-length is removed. The equation of the surface is This may be conveniently done by means of a graphical construction, Fig. 168. Let us construct two curves, with the coordi- the runnng nates X, Y, first the logarith- mic curve Fig. 168. X = e*Y which must be and the second the straight line X = — p . 7 B smkx separately constructed for each value of x. At the intersection of the line and curve , we have Y+Be*Y sin 7^ = 0, so that the value of Y thus obtained may be taken for the y coordinate of the wave -profile with the abscissa x. As x varies, the line swings back and forth about the X-axis, and we see that when sin&# is positive there is one intersection of the line and curve, while if sin ~kx is negative there are two, giving two values of y, both positive. Any positive y is greater in absolute value than the corresponding < / negative for the \ / / symmetrical posi- \ I \ / tion of the line. / / Thus the unsym- \ / metrical nature of "'V_..X trough and crest is made evident. Beginning with _^> -\^^ ^ ^\^^ x = 0, the two 'f^^ ^^"-~~^_ _—- — ^^ ^~ values of y are Fig- 169. one zero, the other infinity, and as x increases, y has a single negative value. When x=-^- = —> y is again zero and infinity, and as x increases the two values of y, both Q positive approach each other until y = — A, then recede until y = L The form of the curve as constructed in this manner is shown in Fig. 169, the lower branch representing the wave -profile. If B is greater than a certain quantity the values of y between certain limits 198] HIGHEST WAVES IN WATER. 533 are imaginary. This limiting value of B is that which makes the highest position of the straight line, for which sin kx = — 1, tangent to the exponential curve. We then have j -y V -y -| jy = kX for the curve, equal to yr= ^- for the line, from which The upper and lower branches of the curve 151) then come together, and the wave -profile has an angle. Waves cannot be higher than this without breaking. By differentiation of 151) we find for the summit, -^- = + 1, so that the angle between the two sides of the wave is a right angle (Fig. 170). As a matter of fact, before the waves are as high as this, the \ equation 141) is \ / no longer satisfied \ / with sufficient \ / approximation for the waves to have the form in question. By an elaborate system of approximation, Michell1) has shown that the highest waves have a height .142 A, while the equa- tion 151) gives .2031. It was shown by Stokes2) that at the crest the angle was not 90°, but 120°, as follows. In the stationary wave, in order to have an edge, the velocities u and v for a particle at the surface must both vanish together, for if v alone vanishes, there will be a horizontal tangent. Consequently, if we place the origin at the crest, equation 139) becomes Fig. 170. gy o. But if we represent the surface by a development of the form of equation 128), on account of symmetry there will be only sine terms, and if in the neighborhood of the origin we retain only the most important term, we may put 1) Michell, The highest Waves in Water. Phil. Mag. 36, p. 430, 1893. 2) Stokes, On the Theory of Oscillatory Waves. Trans. Cambridge Philo- sophical Society, Vol. VIII, p. 441, 1847. Math, and Phys. Papers, Vol. I, p. 227. 534 XL HYDRODYNAMICS. 152) gp = Arn sin no, ty = Arn cos ncc>, o being the angle measured from the vertical. We have for the radial velocity G OP . qr = -£- = Anrn~-1smn(o, and if a is the inclination of the surface to the vertical at the crest q = Anrn~1smn a. But we have g2 = — 2gy = 2gr cos a and accord- o ingly 2 (n — 1) = 1, n = —- Also as in 129), cos na = 0. Thus The problem of waves in water of finite depth may be treated in a similar manner, by putting instead of 134), cp -f ty = — az + Ae~ikz -f jBe*'**, qp = — «# -f (^le*y -f Se~ky) coshx, 153) il> = — ay — (Aekv - Be~*y) sin A; a;, w = — a - v= If the depth is h, we must have v = 0 for y = — h9 giving Ae~kh = Bekh. Calling this value C, we have 154) ^ = -ay- <7(e *(*+*) - e-*(*+y)) sinA;^ = 0, as the equation for the wave -profile. For the first approximation, for waves whose height is small compared to their length, replacing eky,e~ky by unity, we have 155) ay = — C(ekh — e-kh} • sin&#, and neglecting (GY&)2, 156) u2 + v2 = a2 + 2CW (ekh -f e~k/<) sin kx. Thus the surface equation 139) becomes 157) const = ^--^ (ekh - e~kh) sin Jcx ^ a -f a2 which is satisfied by 158) a~k(e] giving the velocity 159) 198, 199] WAVES IN SHALLOW WATER. 535 If h is infinite this reduces to 145), while if the depth is very small with respect to the wave-length, it reduces to a? = gh. Accordingly long waves in shallow water travel with a velocity independent of their length, being the velocity acquired by a body falling through a distance equal to one -half the depth of the water. Consequently the resultant of such waves of different wave-lengths is propagated without change, contrary to what is the case in deep water. Changing to fixed axes, we have for the running wave - *(A0 - -*(A in & (x _ af), and by comparison with 147), 150), we find that the particles describe ellipses with semi -axes equal to 161) G (e*(A+y)-|- e-*(A+?)), C (e*(A+y)-_ g- *(*+?)). If we consider the resultant of two equal wave -trains running in opposite directions, we have sfc(# — at) -f cos ~k (x -f at)] = 2 (7 («*<*+*>+ e-^+y^cosJcxcosJcat, sin k (x - at} -I- sin k(x + at)] The equation of the profile is now of the form, y is equal to a function of x multiplied by a function of tf, so that the profile is always of the same shape, with a varying vertical scale. Such waves are called standing waves, and we see them in a chop sea. The difference between them and the stationary wave in a running stream, with which we began, is very marked, as here every point on the surface oscillates up and down, while there the water -profile was invariable both as to time and place. 199. Equilibrium Theory of the Tides. We shall now briefly consider some aspects of the phenomena of the tides, the general theory of which is far too complicated to be dealt with here. The earliest theory historically is that proposed by Newton, which supposes that the water covering the earth assumes, under the attraction of a disturbing body, the form that it would have if at rest under the action of the forces in question. This so-called equilibrium theory, which neglects the inertia of the water, belongs logically to the subject of hydrostatics, but will be now treated. If U denote the potential of gravity, including the centrifugal force, as in § 149, we have, as there, for the undisturbed surface of the ocean, 536 XI. HYDRODYNAMICS. 163) U (r0, if>, ,,g>)-U (r0, ^ 9) + F = const, = C. But if we put h = r — r0, h is the height of the tide, and being small with respect to the radius, we may put 166) U(r,^ri-U(rQ,^,ri = h™, giving 167) v-c=-hj?- But g = — ?-*-> as in § 149, so that we obtain for the height of the tide 168) h = r°. We may determine the constant in 168) by the consideration that the total volume of the water is constant. If dS is the area of an element of the earth's surface, the total volume of the tide above the surface of equilibrium must vanish, giving 169) 0=hdS, VdS = cdS, V=C, where F is the mean value of the disturbing potential over the earth's surface. Now we have found in § 150, equation 154), the value of the potential of the tide - generating forces, 170) F-£g(3cos»Z-l), where Z is the zenith-distance of the heavenly body at the point in question. If we refer other points on the earth's surface to polar coordinates with respect to this point and any plane through it, with coordinates Z, <&, we have 2rt TI C CvdS = ™ Cd® f@ cos2 Z - 1) sin Z dZ = 0, 0 0 so that the mean of F vanishes. Accordingly we have 171) fc-.(3oo.»Z-l). 199] EQUILIBRIUM THEORY OF TIDES. 537 This equation shows that the tidal surface is a prolate ellipsoid of revolution, with its axis pointing at the disturbing body. Let us now express cos Z in terms of the latitude iff of the point of observation and of the declination d and hour -angle H of the disturbing body, which for brevity we shall call the moon. If we take axes in the earth as usual, with the XZ- plane passing through the point of observation and measure H from this plane, we have for its coordinates and those of the moon respectively r cos iff, D cos d cos H, 0 ; D cos d sin H, r sin ijt, D sin d , from which we obtain the cosine of the angle included by their radii cos Z = cos iff cos d cos H -f sin if> sin d. Squaring this, replacing cos2 H by -_ - (1 4- cos 2-ET), cos2 iff cos2 d by (1 — sin2 i{i)([ — sin2 d), we easily obtain 3 cos2 Z — 1 = -~ [cos2 d cos2 ty cos 2H + sin 2 d sin 2^cosH (l-3sin2(?)(l-3sin2ij/n 8^ — J- Inserting this in 171), replacing g by its approximate value -Tr and, as we have already done, neglecting the attraction of the disturbed water, we have the equation for the tide, 172) * = ^^ [cos2 d cos2 ^ cos 2H + sin 2 d sin 2 # cos H (1-3 sin2 d) (1 - 3 sin2i|;)-| ~~3~~ J The first term in the brackets, containing the factor cos 2 JET, where H is the moon's hour -angle at the point of the earth in question, is periodic in one -half a lunar day, consequently this term has a maximum when the moon is on the meridian, both above and below, low water when the moon is rising or setting. The effect of this term is the semi-diurnal tide, which is the most familiar, with two high and two low waters each day. This tide is a maximum for points on the equator, where cos2 ty = 1, and for those times of the month when cos2 d = 1, that is when the moon is crossing the equator. These are the so-called equinoctial tides. The second term, containing the factor cosJS, is periodic in a lunar day, and gives the diurnal tide. This gives high water under the moon, and low water on the opposite side of the earth. On the side toward the moon, these two tides are therefore added, while on 538 XL HYDRODYNAMICS. the opposite side we have their difference. Consequently, at any point, the difference of two consecutive high waters is twice the diurnal tide. This difference is generally small, showing that the latter tide is small. It vanishes for points on the equator, and at the times of the equinoctial tides. The third term, which vanishes for latitude 35° 16', does not depend on the moon's hour -angle, but only on its declination. This declinational tide, depending on the square of sind, has a period of one -half a lunar month. Beside the tides due to the moon, we must add those due to the sun, for which the factor outside the brackets in 172) is some- what less than one -half that due to the moon. The highest tides therefore occur at those times .in the month when the sun and the moon are on the meridian together, namely at new and full moon. These are known as spring -tides. The lowest occur when the moon is in quadrature with the sun, and the lunar and solar tides are in opposition. These are known as neap-tides, and occording to this theory would be only one -third the height of the spring -tides. The greatest spring- tides would be those in which the moon was on the equator, or the equinoctial spring- tides. Now it is found that, instead of this, the high tides come about a day and a half later. Consequently, although the equilibrium theory indicates to us the general nature of the different tides to be expected, it does not give us an accurate expression for their values. Roughly speaking we may say that the tides act as if they were produced as described by the action of the sun and moon, but that the time of arrival of the effects produced was delayed. A correction was introduced into the equilibrium theory by Lord Kelvin, to take account of the effect of the continents. For if the height of the tide were given by the equation 171), removing the various volumes of water in the space actually occupied by land would subtract an amount of water now positive, now negative, so that the condition of constant volume would not be fulfilled. In order that it still may do so, the integral 169) is to be taken only over those parts of the earth's surface covered by the sea, The value of V is then not zero. The effect of this is to introduce at each point on the earth's surface change of time of the arrival of each tide, varying from point to point. The practical effect of this correction is not large. 20O. Tidal Waves in Canals. In the dynamical theory of the tides, taking account of the inertia of the water, we have the problem of the forced oscillations of the sea under periodic forces. As a simple example illustrating this method we shall consider waves 199, 200J CANAL THEORY OF TIDES. 539 in straight canals. Let the motion be in the plane of XY, as in § 198, and let Ji, the depth of the canal, be small in comparison with the wave-length. We shall suppose the displacements of all the particles, with their velocities and their space -derivatives, to be small quantities whose squares and products may be neglected. We shall also neglect the vertical acceleration, so that the equation for y is that of hydrostatics, giving the pressure proportional to the distance below the surface. If the ordinate of the free surface is h -\- y, this gives 173) P 174) = ' ox y * ox while the equation of motion, the first of equations 6), is 175) |» x_i|£. Ot Q OX Combining these two equations, we have 17«\ cu v dr] 176) w=x~^' Q and if X is independent of yy since ^ is also, this shows that u depends only on x and t, or vertical planes perpendicular to the XY plane remain such during the motion. Integrating the equation of continuity du cv A Wx + Wy="(}> with respect to y from the bottom to the surface, 1 nn\ tdu ^ /7 . N du 1<7) v- -J^dy- -(h + r,)^, or approximately, at the surface, -, rro\ $n 7 ^U 178) v = Tt=-]lTTx O £ Now putting u = 3! ? the equation of continuity 178) becomes, ot 1 7Q^ ^^ = ~ and on integration with respect to the time, 180) 1~-*H- Substituting in 176) we have for the horizontal displacement 540 XI. HYDRODYNAMICS. If there is no disturbing force, X = 0, and we have the equation for the propagation of free waves, which we might have used in order to obtain the results of § 198, for instance it is satisfied by equations 150) if we put a2 = gh. This is the same equation as we had in § 46, equation 109), for the motion of a stretched string, and the standing waves of 162), § 198, putting y = 0, are the normal vibrations of equation 115), § 46. The general solution of equation 182) is obtained in the next section, for the present it is sufficient to consider the wave already obtained which advances unchanged in form with the velocity a. We have then, in the case of an endless canal encircling the earth, the curvature of which we may neglect, the case of a free wave, running around and around, without change, so that at any point, the motion is periodic in the time — ; where I is the length of the endless canal. We thus have a system with free periods, and when we consider the action upon it of periodic disturbing forces, we may expect the phenomena of resonance, as described in Chapter Y. Let us now suppose the canal coincides with a parallel of latitude, and that x is measured to the westward from a certain meridian. We then have for the horizontal component of the disturbing force where V is given by 170), and H, the hour -angle of the moon at the point x, is x dV cV 1 183) H=at— —> so that — ~— = ^^ - —> rcosi|) ex dHrcoaip 03 being the angular velocity of the moon with respect to the earth. We accordingly find X to be composed of two terms each of the form where for the semi-diurnal part 184) A = — -jjj- cos2 ^ cos tyj m — 2&, fr = — Introducing this into the equation 181), 185) r4 = &2 o-| — ^.sin(w£ — Jcx). Ot VX we may find a solution 200] .DIRECT AND INVERTED TIDES. 541 186) % = Bs where by insertion in 185) we find From 180) we obtain 188) 77 = ^irr£i cos (m t - Jcx). The coefficient of the cosine is positive or negative according as ok is greater or less than m, so that we have, according to circumstances, high or low water under the moon. In the former case, the tides are said to be direct, as in the equilibrium theory, in the latter they are inverted. But — '- is the ratio of the time period of the force, m or half a lunar day, to the time required for a free wave to travel half around the earth, and the tide is direct or inverted according as this is greater or less than unity. Equation 188) is the analogue of equation 50), § 44. Inserting the values of the constants in 188) we find that the canal theory gives the height of the tide as given by the equilibrium theory in 172) (which we also obtain by putting m = 0), multiplied by the factor -f-Y (ak) exactly as described for the system with one degree of freedom on page 155. If we introduced into our equations a term giving the effect of friction we should obtain a change of phase, as in § 44, of amount other than a half -period, or inversion. In order to determine the directness or inversion of the tides, let us insert the values of m, k from 184) in 188), by which we find that the tides are direct or inverted according as we have the upper or lower sign in the inequality 189) 0&^r8a>2cos2V>. Supposing the lunar day to be 24 hours, 50 minutes, the earth's circumference forty million meters, we find at the equator the critical depth, determining the inversion, to be 20.46 kilometers, or 12.7 miles. As the depth is less than this, the tides are inverted. For any depth less than the critical depth, there will be a latitude beyond which the tides will be direct. Accordingly we see that even if we consider the ocean to be composed of parallel canals separated by partitions, the tides will be very different in different latitudes, so that if the partitions be removed, water will flow north and south. We thus obtain an idea of the complication of the actual motion of the tides. 542 XL HYDRODYNAMICS. By introducing the complete expressions for the accelerations with respect to revolving axes, given in § 104, and applying the principles of forced oscillations, we obtain the more complete theory given by Laplace. 2O1. Sound -Waves. Let us now consider the motion of a compressible fluid which takes place in the propagation of sound. In the production of all ordinary sounds, except those violent ones produced by explosions, the motion of each particle of air is extremely minute. We shall therefore suppose that the velocity components u, v, w and their space derivatives are so small that their squares and products may be neglected. Let us put 190) p = p0(l-M), where QO is a constant and s is a small quantity, of the same order as the velocities, called the compression. From the equation of •continuity we have 1Q1x _ du cv dw _ 1 d$ _ 1 ds -fa^d^^fa- ~~Q~di~ ~1+10*' or neglecting the product of s and its derivative, -t f\c\\ ds 192) "=-**• In order to calculate P, we have, since the changes in Q are small 193) dp = a2dQ = a2QQds, where a2 is a constant representing the value of the derivative J- for Q = QO, the density of the air at atmospheric pressure. We therefore have 194) P _ =-«• log (1 +.) = ««,, to the same degree of approximation. Neglecting small quantities the equations of hydrodynamics 6) become, when there are no applied forces, o o CU 9 08 -or = — a 3— 7 dt dx dv 9 ds 195) - = — a?2-> dt dy dw o ds —- = — a2-^-) dt cz with 192) «=-f, Differentiating the equations respectively by x, y, z, adding and observing the definition of 6, we obtain 200, 201, 202] SOUND-WAVES. 543 196) = and differentiating 192) by t and combining with this 197) *J Since the motion is assumed to be irrotational, introducing the velocity potential into equations 195) they become the derivatives by x, y, z respectively of the equation 198) Si—''- Differentiating by t, making use of equation 192), 199) j£ = - - a2 d~ = a2 6 = a*4plf the motion is sometimes called a pulse. A pulse is none the less a wave. Thus the general solution of equation 201) represents two plane waves propagated in opposite directions with the same velocity a. The velocity of sound a = y -JJT depends upon the elasticity of the air and was calculated by Newton, assuming that the process was isothermal, using Boyle's law. As this was found to give results not agreeing with experiment Laplace suggested that the compression was adiabatic, the vibrations being so rapid that the heat generated did not have time to .flow from the heated to the cooled parts. Thus the constant K, equation 17) § 178 representing the ratio of the two specific heats of the air is introduced. The velocity of sound gives one of the most accurate ways of determining this ratio sc. The velocity of the particle of air is obtained by C)f\Q\ d

= i{F1(r-aO + F,(r + at)\, of which the first term represents a wave proceeding outwards, the second one proceeding inwards, the magnitude however varying according to the factor — • For a periodic solution representing a simple tone proceeding from a single point -source we may take 218) (p = — -- — cosJc(at — r). * The physical meaning of the constant A is obtained as follows. Let us find the volume of air flowing in unit of time through the surface of a sphere with center at the source. We will call this the total current, 219) I = A {cos ~k(at — r) — kr sin ~k (at — r)}. Accordingly when r = 0 we have I = Aeoshat and A, the maximum rate of emission of air per unit of time, is called the strength of the source, agreeing with the definition of § 196. In order to obtain the activity of the source, that is the rate of emission of energy per unit of time, we may find the rate of working of the pressure at the surface of a sphere, as explained in § 188, 220) P = In order to find p, we have, if pQ is the undisturbed atmospheric pressure, by integration of 193), and by 198), 35* 548 XL HYDRODYNAMICS. &£\. j p — _2^Q == ^ ^o^ == — ?o ~ from which we obtain 222) P = This contains a part which is alternately positive and negative, and also one which is always positive. If we seek the mean value of P throughout the period, that is T P = — f Pdt, T=~, o we easily find, since the mean of cos#, sin#, cos # sin #, is zero, while the mean of sin2^ is —> 223) P = A*f**a which is independent of the radius, as it should be. The mean energy -flow per unit of time and per unit of area of the sphere is which is a measure of the intensity of the sound. Tihs decreases as the inverse square of the distance. In order to give an idea of the extremely small dynamical magnitudes involved in musical sounds, it may be stated that measurements made by the author1) showed that the energy emitted by a cornet, playing with an average loudness, was 770 ergs per second, or about one ten -millionth of a horse- power, while a steam -whistle that could under favorable circumstances be heard twenty miles away emitted but - — ? or one -sixtieth of a horse -power (see note, p. 153). 2O5. Waves in a Solid. The equations of motion for an elastic solid are obtained from the equations of equilibrium 144), § 175 by the application of d'Alembert's principle in the same manner as the equations of hydrodynamics were deduced from those of hydro- statics. It will be convenient here to revert to the notation of Chapter IX where u, v, w and 6 refer to displacements rather than to velocities. Applying d'Alembert's principle we thus obtain 225) p (X - g) + (I + ,*) fl + ^u = 0, etc. 1) Webster, On the Mechanical Efficiency of the Production of Sound. Boltzmann- Festschrift, p. 866, 1904. 204, 205, 206] WAVES IN SOLID. 549 If there are no bodily forces we have the equations of motion da 226) e=(* + d*w ,. . Ndff . V ~W = ( + **' Tz + P w' Differentiating respectively by x, y, z and adding we obtain 227) e = (i which is the equation for the propagation of wave -motion, the dilatation being propagated with a velocity b = y- — -• Taking the curl of equations 226) we have 228) Thus the components of the curl are propagated independently, each with a velocity a—y— - The velocity of the compressional wave which is unaccompanied by rotation depends upon the bulk modulus and the modulus of shear. The velocity of the torsional wave which is unaccompanied by change of density depends only upon the mo- dulus of shear. The general motion of an elastic body is a com- bination of waves of compression and of torsion. The wave of torsion is that upon which the dynamical theory of light is founded. Inasmuch as p vanishes for a perfect fluid no wave of torsion is propagated, so that the luminiferous ether must have the properties of a solid and not those of a fluid. 2O6. Viscous Fluids. We have now to consider a class of bodies intermediate in their properties between solids and perfect fluids, namely the viscous fluids. By definition a perfect fluid is one in which no tangential stresses exist. We have then 229) Xx = Y, = Z, = -p, X, = Y2 = ZX = 0. In a fluid which is not perfect no tangential stresses can exist in a state of rest, but during motion such stresses can exist. While in a solid the stresses depend on the change of size and shape of the small portions of the solid, in the case of a viscous fluid the stresses 550 XL HYDRODYNAMICS. depend on the time -rates of change, that is on the velocities of the shears, stretches, and dilatations. The simplest assumption that we can make is that the stress -components are linear functions of the strain -velocities. The fluid being isotropic, considerations regarding invariance bring us to precisely similar conclusions to those we reached in § 175, so that to the stresses of equations 229) for a perfect fluid are added stresses given by equations 142), § 175, A and p being constants for the fluid, and u, v, tv, 6 now denoting velocities, instead of displacements, returning to the notation of this chapter. (We put P = 0, since these additional terms vanish with the velocities.) We thus obtain Z2 = -p + ^ + 2ti~z 230) which are of the same form, with a different meaning, as 142), § 175. If the fluid is incompressible we find, putting 6 = 0, Xx + Yy + Z,= - 3p, and assuming that this holds also for compressible fluids we must have 231) 3^ + 2^ = 0. a Replacing A by its value - — jt, we find for the forces, as in § 175, 144), v dp . 1 232) r, 3p . 1 eZ-U+ 3 which are to be introduced into the equations of hydrodynamics 6). Thus we obtain the general equations, putting — = v, du , du , du , cu v ds ^ 1 dp ^rr-h^K -- h^o -- h^o -- -^ o -- V^U = X -- K^-J dt dx ' dy ' dz 3 0x Q dx rtoo\ ^^ dv dv t dv v da T;r 1 dp 233) - H-MO — h^^-f^o- - vdv = Y -- 7p-> ' ot dx dy dz 3 dy Q dy dw . dw . dw . dw v ds ^,1 dp -r-\-U~ -- h^o -- h^o -- ^ - —vAw = Z -- r~? dt ' dx ' dy ] dz 3 dz Q dz 206] VISCOUS FLUIDS. 551 which reduce to 6) when ^ = 0. The coefficient p is called the viscosity of the fluid, and its quotient by the density, v, is called by Maxwell the kinematical coefficient of viscosity. The equations 233) are too complicated to be used in all their generality. We shall here consider only the case of incompressible fluids, for which the terms in 6 vanish. If we form the equation of activity as in § 188, we obtain beside the terms in the first integral of 29) the additional terms ~ P I I I 4- which by Green's theorem may be converted into dv dw u\2, /2»\2 , /3sA3, /M2 , w + u + y + u + If the integration be extended to a region where the liquid is at rest, say the surface of a containing solid, where the liquid does not slip, the surface integrals vanish, and the volume integrals give a positive addition. That is to say, the applied forces have to do an amount of work over and above that going into kinetic and potential energy, and this work is dissipated into heat. If there are no applied forces, the energy of the fluid is dissipated, and it will eventually come to rest. In order to find simple solutions of our equations, we may deal either with steady motion, or with motions so slow that we may neglect the terms of the second order in u, v, w and their derivatives. Let us first consider steady motion. The simplest case is uniplanar flow parallel to a single direction, or as we may call it, laminar flow. If we take noA\ du dp 234) 0 = v = w= -75- = -^-9 } dz dz the equation of continuity gives 235) *£ = 0. If there are no applied forces, equations 233) reduce to OQ£\ d*u dp dp A 236) ^Q-2=:r^ ^ = ^ ^ cy* ox oy Since u depends only on y and p only on x, this equation cannot hold unless each side is constant. Accordingly dzu dp 1 a 9 ^72==a==^' M = c + &«/-f-¥ -y2, p = d + ax, 552 XI. HYDRODYNAMICS. where a, Z>, c, d are constants. If we determine them so that the velocity vanishes for planes at a distance +h from the X-axis, we have 238) u = ^(f-V). The amount of liquid that flows through such a laminar tube per unit of width parallel to the Z-axis is accordingly 239) — h and for a length of tube I the difference of pressures at the ends is 240) • A-ft-aJ, 3=££, so that the flow is proportional to the difference of pressures at the ends and inversely to the viscosity. For the practical determination of viscosity, we may take the almost equally simple case of cylindrical flow, where the velocity has everywhere the same direction, and depends upon the distance r from the axis of a circular tube, at the surface of which it is at rest. If we put u = v = 0 we have the equations of motion and of continuity o^i\ dp 241) and since w depends only on r the first becomes C*AO\ , dp 242) where a is a constant as before. This equation is integrated as in § 182, 58'), 243) w = ^-r2 + fclogr -f c. Since w is finite when r = 0 we must have b = 0 and if w vanishes for r = R we obtain 244) „„£(,. For the flow we find ctAc.\ s\ 245) Q = 0 This method was invented by Poiseuille1) for the measurement of 1) Poiseuille, Eecherches experimentales sur le mouvement des liquides dans les tubes de tres petits diametres. Comptes Rendus, 1840 — 41; Mem. des Savants Strangers, t. 9, 1846. 206] DETERMINATION OF VISCOSITY. 553 the viscosity of fluids. His verification of the proportionality of the flow to the fourth power of the radius of the tube has been taken as a proof that the liquid does not slide when in contact with a solid. As another example of steady flow let us consider uniplanar cylindrical flow, in which each particle moves in a circle with velocity depending only on the distance from the axis, as in the case of the lubricant between a journal and its bearing. Each cylindrical stratum then revolves like a rigid body, which requires 246) u = — &y, v = ox, where co depends only on r — "J/#2+ y2. We then find du xy do 3u y* do — = -- -=—y ~ — = — O --- -3—? cx r dr cy r dr ' dv . X* dco dv xy dm x . = <*> H --- -5—' o~ = — ~^- r dr oy r dr and, most easily by the application of equation 86), § 141, and by the expression of z/co in terms of r, d(o\ ~ y ~ - -=- — 2- -=-t r drj r dr dco\ . rt x dco o) . 1 dco\ . rt x r + - -J-) + 2- * ' r dr) ' r dr Thus the first two of equations 233) become /d2a . 3 d&A 1 x dp -f vyl-j-j- H — -3-1 — -- - i » \dr* r dr) Q r dr 249) 2 ( I ^ \ * 2/ **.P \c?r2 r dr/ Q r dr Multiplying the first by y} the second by x and subtracting, 250) 4^ + - ?- = 0, a?*2 r ar a differential equation whose solution is 251) « = £ + &. Determining the constants so that CD = 0 for r = R± and o> = & for r = R*, 252) „,= f-V Jj^-'.-i|. 1 2 \ J Multiplying equations 249) by x and y respectively, and adding, we have to determine PJ 253) £eo2r = ^- For the stresses we obtain, using equations 230), v I 0 xydo)\ (x*-y* Xn = ( — p — 2p -^ ^-J cos (nx) H- it I - •* -rr I cos 554 XI. HYDRODYNAMICS. n*A\ T^ /x*—y*da\ / \ , / «r» xti d(o\ , N 254) Yn = ii (—^L ^j cos (nx) + (- # + 2jt -^ ^J cos (wy), Zn = — p cos (w#) = 0. This shows that there is a normal pressure p, together with a tangential stress which we obtain by resolving along the tangent, 255) T = Yn cos (nx) - Xn cos (ny) dofo;2— w2, 9 9 \ , ±xy / \ / >1 ~ ^dr \ — r~ (COS "~ COS ^' "^ -- COS v1*) cos (w^) I and since cos (nx) = —^ cos (ny) = —^ The moment of the tangential stress on the cylinder of radius r and unit length, is accordingly 257) We may accordingly use this method to determine the viscosity, as is in fact done in apparatus for the testing of lubricants. We see that if the linear dimensions are multiplied in a certain ratio, the moment is increased in the square of that ratio. We also see that the moment of the force required to twist the cylinder is independent of the pressure p, which contains an arbitrary constant, not given by the equation 253), but depending on the hydrostatic pressure applied at the ends. Let us now consider some simple cases where the flow is not steady, limiting ourselves to the case of small velocities, so that the terms in 233) involving the first space derivatives, being of the second order, are negligible. Let us once more consider laminar flow, defined by equations 234), 235). Let us also put p = const. Instead of 236) we now have for the first of equations 233), 258) ^==vd^. dt fly* This equation is the same as that which represents the conduction of heat in one direction. Let us first consider a solution periodic in the time, such as may be realized physically by the harmonic small oscillation in its own plane of a material lamina constituting the plane y = 0, along which the liquid does not slip. We may take as a particular solution which inserted in 258) gives n = vm 206] SLOW MOTIONS. 555 If this is to be periodic in t, n must be pure imaginary, say n = ip. Then we have and of which complex quantity both the real and the imaginary parts must separately satisfy the equation 258), when multiplied by arbitrary constants. Let us accordingly take 259) u = | A cos (pt — y -j- • y) + J5sin (pt — \ \ i A r / \ This represents a wave of frequency ~^ and wave-length 2jrl/- travelling with velocity y%vp, which as we see varies as the square root of the frequency. Unlike our waves in perfect fluids however it falls off in amplitude, being rapidly damped as we go into the fluid, being reduced in the ratio e~ ** = — in each wave-length. Thus such motions are propagated but a short distance into a fluid. In a similar manner the absorption of light by non- transparent media is explained, the ether there having the properties of a viscous solid. If we treat the equations 233) in the same way as we did 27) in obtaining equation 57), § 191, we obtain instead the following, d /|\ &du ndu £ du <. di (j) = i fa + 7 ^ + -Q dJ + v * «> 260) dt \Q/ Q dx g dy d £\ g dw r 3w Under the circumstances of slow motions these also reduce to 261) Thus we see that the three components of the vorticity are propagated independently, each according to the equation for the conduction of 556 XL HYDRODYNAMICS. heat. The example just treated is an example of this, for we find at once 2 f) QJ and the vorticity is propagated like the velocity. As a final example, let us consider a case of laminar motion in which u, as a function of y, has a discontinuity, this having an important application to the theory of thin plane jets and flames, including sensitive flames.1) We will suppose that at the time t = 0 for y < 0 u has a certain constant value, and that for y > 0 it has a different constant value. It is easy to see that this is equivalent to supposing that there is no vorticity except in an infinitely thin lamina at y = 0. For we have s s /-* /-* 263) /£^ = --l / ^dy = ~(u±-u,) / i ••> i/ 2 / c i/ 2 ^ * •' t/ t/ y — e — c where u± is the velocity on one side, u2 that on the other of the layer of thickness 2s. Now if the thickness decrease without limit, while g increases without limit, the integral may still be finite, as we shall suppose. We have then to find two solutions u and f of equation 258), so related that g = — — -^- • Let us put s = — '*— > and try to find a particular solution that is a function of s alone. We have du du ds _ 1 du y dt ~~ ds dt 2 ds }/i* du du ds du I d^==~ds^~'ds'y^) d*u _ 1 d*u ds _ £ d*u dy9 yt ds* dy t ds* so that our equation becomes the ordinary differential equation, 1) Rayleigh, On the Stability, or Instability, of certain Fluid Motions Proc. London Math. Soc., xi., pp. 57— 70, 1880. Scientific Papers, Vol.1, p. 474. 206] SHEET OF DISCONTINUITY. 557 The integral of this equation is given by 267) los<^ = -£- + const., £ = ««'£, » The last indicated quadrature cannot be effected except by development of the integrand in series, but if we take for the lower limit the value zero, we may express u in terms of the so-called error -function, occurring in the theory of probability, 268) Tables of the values of- Erf(x) have been calculated, and are found in treatises on probability. (Lord Kelvin reprints one such on p. 434 of Vol. 3 of his collected papers.) Since the integrand is an even function of x, it is evident that Erf(x) is an odd function of its upper limit x. It may be easily shown that the definite integral between zero and infinity has the value -^p so that putting x* = -j—> and adding a constant, we have 269) This determination of the constants makes, for all positive values of y and for t = 0, u = u^ (the upper limit being -f oo), and for all negative values u = u2, thus giving the dis- continuity required at y = 0. For all other values of t however, no matter how small, the values from the negative side run smoothly into those on the positive, showing how the discontinuity is instantly lost. This is shown in Fig. 171, in which successive curves show values of u at times equal to 1, 2, 3, 4, 5, 6 times — Differentiating by the limit, we find 270) fs»_4.|«. 558 XL HYDRODYNAMICS. which is infinite when t = 0, - = = 0, as we supposed, but which immediately drops to a finite value, and, no matter what the value of y, immediately acquires values different from zero. Thus the Fig. 172. vorticity, originally confined within the infinitely thin sheet of dis- continuity, is instantaneously distributed throughout the liquid, as shown in Fig. 172, for the times -^9 — > —> 1, 4 times — Thus we see how discontinuities of the sort shown in Fig. 166 are impossible in nature, being replaced by the formation of eddies. NOTES, NOTE I. DIFFERENTIAL EQUATIONS. The differential equations of mechanics are of the type known as ordinary, as opposed to partial, that is they involve a number of functions of a single variable, the time, and the derivatives of these functions with respect to that variable. Suppose for simplicity that we have three func- tions #, «/, 0 of the variable ty and that instead of being given explicitly, they are defined by the equations If we now differentiate these equations, bearing in mind that #, «/, 0 are dependent on t, we obtain dF, dx dF, dy dF, dz dF, _ dt "1" dy dt "*" ds dt 1' dt ~ dF, dx dj\ dy dF, dz dj\ _ 3x dt "t" dy dt "l" dz dt "*" dt ~ ' dj\d^ dj\dy_ dj\d^ dj\ = dx dt "•" dy dt ~* dz dt ^ dt Suppose now that the functions F contained, besides the variables indi- cated, certain constants, c1? C2 . . . Each time that we obtain an equation by differentiation, we may utilize it in order to eliminate from the equations 1) one of the constants c. Thus we obtain (since the partial derivatives are given functions of x, y, z, f), instead of the equations 1), the following, 3) which, since they contain the derivatives -^i -j^i -^t are differential equa- tions, of which equations l) are said to be integrals. If we again differentiate equations 2), we obtain , dF, d*x d*F, /day d*Ft dxdy _ ~fa ~w + ~d^ (~dt) + 2 --^ 560 NOTES. which we may again use to eliminate constants c from 3), so that instead of l) or 3) we now have the system . dx dy dz d*x d*y d* z\ y, M, -, -> , f , = 0, ~) =0, These differential equations, since the order of the derivatives of the highest order contained in them is the second, are said to be of the second order. In like manner we may continue, and successively eliminate all the constants cx, C2 . . ., obtaining differential equations of successively higher orders. Reversing the process, each set of a given order is said to be the integral of the set of order next higher. Any of the sets of differential equations represents the functions x, y, #, but with the following distinction. If the equations 1) contain constants, to which different values may be assigned, 6) Ft(x, y, 0, t, CJL, c2 . . . c^ = 0, F9(x, y, z, t, c1? c2 . . . cn) = 0, .F30,2/, *, t, ct, c2 . . . cB) = 0, for every set of values that may be assigned to the constants, a different set of functions is represented, so that we have an infinity of different functions, the order of the infinity being the number of constants contained in the equations. Now the differential equations obtained by eliminating the arbitrary constants represent all the functions obtained by giving the constants any set of values whatever. Thus the information contained in the differential equations is in a sense more general than that contained in the equations 6), in which we give the constants any particular values. If we reverse the process which we have here followed to form the differential equations, we see that every time that we succeed, by 'inte- gration, in making derivatives of a certain order disappear, we introduce at the same time a number of arbitrary constants equal to the number of derivatives which disappear. Thus the integral equations of a set of differential equations of any order will contain a number of arbitrary constants equal to the order of the differential equations multiplied by the number of dependent variables. As an example consider the very simple case of equations 38), § 13. 38") d ~ » - ' dt* ~ ' Integrating these we obtain 39) x = c±t + d±1 y = c2t+d2, z = c3t + d3, containing the six arbitrary constants c,, c2, c3, d1? d%, c?3. The meaning of these integral equations is that the point x, y, z describes a straight line with a constant velocity. But the differential equations 38) represent I. DIFFERENTIAL EQUATIONS. 561 the motion of a point describing any line in space with any velocity. Now there are a four -fold infinity of lines in space, and a single infinity of velocities. We therefore see the very general nature of the information contained in the differential equations. So in the example of § 13 the statement that all the planets experience an acceleration toward the sun which is proportional to the inverse square of the distance expresses a very general and simple truth, in the form of a set of differential equations, while the integral states that the planets describe some conic section in some plane through the sun, in some periodic time, all the particulars of which statement are arbitrary. The characteristic property of the differential equations of mechanics, for the phenomena furnished us by Nature, is apparently that they are of the second order. This , although leaving possibilities of great generality, suffices to limit natural phenomena to a certain class, in contrast to what would be conceivable. For the consequences of the removal of this limitation, the student is referred to the very interesting work by Konigsberger, Die Principien der Mechanic. In order to determine the particular values of the arbitrary constants applicable to any particular problem, some data must be given in addition to the differential equations. It is customary to furnish these by stating for a particular instant of time, the values of the coordinates of each point of the system, and of their first time -derivatives, which amounts to specifying for each point its position and its vector velocity for the particular instant in question. This furnishes six data for each independent point, which is just sufficient to determine the constants. Thus if we are dealing with a system of n points free to move in any manner, under the action of any forces, the statement of the problem will consist in the giving of the differential equations dt ' dt ' dt*' d together with the so - called initial conditions , that for t = tQ , % = V, 2/i = */i° - • • • zn = %n dxi _ iy no fyi _ r.J no . . ^ _ rJ 10 ~dt ~ I*1-1 >~di ~ Ll/lJ ' ~dt " L*wJ ' From these it is required to find the integrals % = /i(0» 2/i = /i(0» *i = /s(0» • • • *« = fi-W- Cases involving the motion of points whose freedom of motion is limited are dealt with in subsequent chapters. WEBSTER, Dynamics. 36 562 NOTES. NOTE II. ALGEBRA OF INDETERMINATE MULTIPLIERS. On page 61 we have an example of the use of indeterminate multipliers in elimination. It may be somewhat more clear if we examine in just what the process involved consists Equation 12) is a linear equation involving the 3n quantities d^, ... dzn, each multiplied by a coefficient which is independent of the d's. Besides this equation the quantities d satisfy the equations 14), which are of the same form, that is, linear in all the $'s, with coefficients independent of them. Aside from this the d's may have any values whatever. It is for the purposes of this discussion quite immaterial that the d's are small quantities, we are concerned simply with a question of elimination. Let us accordingly represent them by the letters x^ X2, ... #m, between which we have the linear equation 1) A^ 4- A2x2 4- ---- h Amxm = 0. The x's are however not independent, but must in addition satisfy the equations B^XI 4- B12x2 - - - 4- BimXm = 0, B^ 4- #22^2 ---- H • 4- The number of these equations, &, is less than »z, the number of the x's. The question is now, what relations are involved among the A's and J5's when the x's have any values whatever compatible with the equations 2). We may evidently proceed as follows. Transposing m — k terms in 2), say the last, we may solve the equations for the quantities #1? X2 • • • xki as linear functions of the remaining #A_J_I, . . . xm. These m — k quantities are now perfectly arbitrary. Inserting the values of x± . . . Xk in equation l), this becomes linear in the m — k quantities afc+i, . . • xm, which being purely arbitrary, in order for equation l) to hold for all values of the x's, the coefficient of each must vanish, giving us the required m — k relations between the A's and B's. Instead of proceeding in the manner described, the method of Lagrange is to multiply the equations 2) respectively by multipliers Aj, yl2, . . A*, to which any convenient value may be given, and then to add them to equation l). We thus obtain (Al 4- *! -Bn 4- A2 -Bai • • • 4- -f (A2 + A! 512 + A2 £22 • • • 4- 4- (4»4- *i-Blm+ AS •»»»»• ' ' 4- III. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 563 In this equation the afs are not all arbitrary, but as before & may be determined as linear functions of the remainder, say a?1, . . xk in terms of #£_f.i, . . . flJ/n, which are arbitrary. But the multipliers A are as yet arbitrary. Let us determine them so that they satisfy the equations A + *i #11 + *2 #21 ' ' • + A* JBti = 0, v ^-2 ~l~ ^1 #12 ~ #22 * ' * -f ^/fc#&2 = 0, J-yfc -f ^i #1 A + Aj #2* ' ' * ~ which are just sufficient to determine them We thus have + At -Bl, yfc-fl + • • • + 5) + (Am 4 A! 5lm + • - - + A4 Skm)xm = 0, in which the x's are all arbitrary, so that the m — k coefficients must vanish, giving the m — If equations. = 0 =0. Inserting in these the values of the I's already found, we have the m — Jc required relations between the A' 8 and J5's. Obviously the result of the elimination may be expressed in the form obtained by writing equal to zero each of the determinants of order fc + 1 obtained from the array of A's and J3's in equations 4) and 5) by omitting m — ft — 1 rows, only m — & of the determinants thus obtained being independent. NOTE III. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. The method of transforming the equations of motion used in § 37 and the application of hyperspace there occurring render a somewhat more detailed treatment of the question desirable. In order to elucidate matters, we will begin with the very simple case of a space which is included in ordinary space, namely the space of two dimensions forming the surface characterized by two coordinates ^n #»» as on Pa£e HO- We have seen that this space is completely characterized by the expression for the arc as the quadratic differential form A point lying on this surface may be displaced in any manner, in or out of the surface. If it is displaced in the surface, its displacement is a vector belonging to the two-dimensional space considered. We will 36* 564 NOTES. call the changes dql,dq2 the coordinates of the displacement. We have found that when the displacement is made so as to change only one of the coordinates of the point ql or q2. the arcs are respectively ds^ = y'Edql, ds% = yrGrdq2, and that the angle included by them is given by F cos # = If now we have any displacement ds, whose coordinates are dq1, dq2, and project it orthogonally upon the directions of ds1, ds2, we easily see (Fig. 26) that the projections daL, d62 are *„, = ds, + ^ cos & = da, = ds, + ds, cos 4> = ya dq2 + We shall now, following Hertz, introduce the reduced component of the displacement along either coordinate -line, defined as the orthogoneal pro- jection divided by the rate of change of the coordinate with respect to the distance traveled in its own direction. These reduced components we shall denote by a bar, so that 3) The fundamental property of these reduced components is found in the equation giving the magnitude of the displacement 4) ds2 = dq^dq^ + dq2dq2, that is the square of an infinitesimal displacement is the sum of products of each coordinate of the displacement multiplied by the respective reduced component. In like manner the geometric product of two different displacements ds, ds', whose coordinates are dq^, dq2, dq±\ dq% is found to be ' cos (ds, ds') = dxdx' -f dydy' + dzdz' dx , dx -. \ fdx , , , dx ( 6) q2'-l-dq2dql') + Gdq2dq2' dq2dq2' = dq^dq^ + dq2'dq2. HI. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 565 The geometric product of two displacements is equal to the sum of products of the coordinates of either vector by the reduced components of the other. Thus the geometric product is denned by means of the quadratic differential form l) denning the space in question. Solving the equations 3) for dqv dq%, we obtain 6) from which we obtain 7) ds2 = En dq^ -f 2 J212 dq± dq2 -f E22 dq22. The expression 7) is called the reciprocal form to l). Corresponding to it we obtain the form of the geometric product We may now define any vector belonging to the space considered, as one whose components have the same properties as those possessed by those of an infinitesimal displacement. Suppose that X, Y, Z are the rectangular components of a vector E, it does not belong to the space l) unless it is tangent to the surface in question. If so, we have a displace- ment such that ds dx dy dz dql dq^ s Then Q^ Q2 are the coordinates of the vector in the system gA, (a, 568 NOTES. If we multiply this equation by an arbitrary constant — /I and add it to _F(a, |3, y), we obtain the condition by writing the derivatives of F — lq> equal to zero Thus we obtain 4) Now the direction cosines of the normal to the quadric at a point #, y, & are proportional to ^E. dJL dJL. dx dy dz At points where the normal is in the direction of the radius vector we have dF dF dF dx _ dy dz x y z But dF(x, y, z) dF(cc, ft y) dx da etc., so that the equations 4) show that at the ends of the principal axes the tangent plane is perpendicular to the radius vector. Effecting the differentiations the equations 4) become (A - 5) F0 Dfi 7 =0, y =0, -h(<7-/l)y=0. The condition that these equations, linear in a, |3, y, shall be compatible for values of a, |3, y, other than zero is that the determinant of the coefficients shall vanish. — I, F , F , B - A, E D E D (7 This is a cubic in A, which being expanded is IV. AXES OF CENTRAL QUADRIC. 569 We shall show that this always has three real roots. Put A — X = u -\- q, 8) B - I = v -f r, C — I = w + S, where #, r, 5 are to be determined later. Then 0* or arranged according to powers of u, v,'w, uvw 4- qvw + rww + 9) + u(rs - D2) + v(sq - iJ2) + gr5 + 2DEF - D2q - E2r - F2s = 0. Let us now determine q, r, s, so as to make the terms of first order in Uj v, w vanish. rs = D2, sq = E2, qr = F2, from which by multiplication and division 10) qrs = Thus there remains , EF FD . DE 11J uvw -\ — =r- v w H — ^- ivu H — =- uv = 0. Now from 8) u = A — H — q = A — EF/D - K = a — A, 12) v = B -X-r = B - FD/E - I = b - A, w=C -l-s = C - DE/F - I = c - A, if we write A - EF/D = a, B-FDjE=l, C-DE/F=c. Also since from 10) #, r, 5 are all of the same sign, let us call them + Z2, w2, w2, so that we have from 11) /•(T) - (a - A) (6 - A)(c - A) ± [Z2(fc - A) (c - A) + M2(c - A) (a - A) + n\a - I) (b - A)J. Substituting for A in turn the values — oo, c, ft, a, +00, we obtain /•(-oo)=oo, />(c) =±^2(«-C)(&-C) 14) /"(a) = /•(oo) - - oo. 570 NOTES. Let us suppose a > b > c. and take the upper sign in 14). Then for I = — oo f(l] is 4- c 4- & o + 00 and the function f(l) behaves as shown in Fig. 13. As there are three changes of sign, there are three real roots. It is to be noticed that the reality of all the roots depends on #, r, s being of the same sign. Let us call the roots A15 A2, A3. Either one of these being inserted in the equations 5), the equations become compatible, and suffice to determine the ratios of the direction cosines. There are therefore always three principal axes to a central quadric surface. If we call the cosines belonging to the roots Ax, a1? /31? y1? those belonging to A2, «2, /32, y2, equations 5) become 15) Multiplying the first three respectively by a2, /32, yt and adding, 4- , 4- D(ftyi 4- ft72 4- ^y2^ + yi«2 4- If we multiply the second three equations respectively by a1? jSj, 7l and add we obtain for the same expression. Accordingly we have 17) (^ - A2) (X a, 4- ft ft + 7l 72) = 0, so that if the roots /Lj, ^2 are unequal the corresponding axes are perpendicular. In like manner if the determinantal cubic has three unequal roots, the quadric has three mutually perpendicular principal axes. If two roots are equal the position of the corresponding axes becomes indeterminate, and it may be shown that all radii perpendicular to the direction given by the third root are principal axes of the same length. The surface is then one of revolution about the determinate axis. If all three roots are equal, the surface is a sphere, and any axis is a principal axis. IV. AXES OF CENTRAL QUADRIC. 571 We will now transform the equation of the quadric l) to a new set of axes coinciding in direction with its principal axes. Let the new coordinates be x\ y\ #', and let the direction cosines of the angles made by the new with the old axes be given in the table below. The equations of transformation of coordinates are then x = a^x + ft?/ -f yis, 18) y1 --= a^x -f ft?/ -f 722, x = a^x' 4- 19) y = (llX' + ' -f Now using equations 19), we obtain ft which in virtue of equations 15) is equal to *!«!#' + ^22/' In like manner Multiplying respectively by #, «/, ^ and adding, we obtain Ax2 + By* + n\i - - - Q>nn Pnlj ' • • Pnn ?(«,....*.) In this notation 20) becomes / ~WT^i ^~\ = fti. • • • If the functions Mlt . . . ttn in a Jacobian are the partial derivatives o /» of the same function /", wr = -jr^-* so that the element in the rth row gsr ^^r and sth column is -~ — 5 — the determinant is called the Hessian of the dxrdxs function. Thus the determinant H of 2) is the Hessian of /", and will be denoted by Hxf. If now we transform the quadratic form 5) by the substitution 17), so that 22) 576 NOTES. we may find a relation between the Hessians of f with respect to the x's and that with respect to the y's. Using the notation for Jacobians, by 21), 23) • • • fi But in every derivative 8V ay Consequently the Jacobian on the right of 23) is the same as that on the left of 21). Thus we find 3(u u] fci---Pi» 2 > for which does not vanish, since icn is independent of these variables, we may by suitably choosing the value of xn make the form have either sign, it is therefore indefinite. V. TRANSFORMATION OF QUADRATIC FORMS. 577 (If the sum ^J r=l *=1 must have arscryis is zero for all values of y1? . . . yn-\, we for 5=1, 2, , ... n — 1, but since f(c^ . . . cn) is zero we must have r=n also ^ arncr, and these n equations require the determinant of the form r=l to vanish, and the form is singular.) As a result of this theorem we see that ifji form is to be definite, no coefficient arr of a square xr2 must be absent, and all must have the same sign*. For if arr = 0, putting all the variables equal to zero except xr would make the form vanish, and if arr is not zero, the same assumption would make the form have the sign of arr. Consequently all these coefficients must be of the same sign. Let us now consider two ordinary quadratic forms of the same variables, with real coefficients1) r=l s = l r = l s=l from which with an arbitrary multiplier A we construct the form 28) Ay -f i/;. As we give A an infinite set of real values, we obtain an infinite sheaf of forms. Let us examine whether they are definite or not. The determinant of the form Ago -f i/; 28) -f- -f -f -f cnn = f« is identical with Lagrange's determinant, page 159, when the %'& are zero. (We here have written I for the A2 on p. 159.) We shall now prove that if the equation f(k) = 0 has a complex root, all the forms of the sheaf Ago -f- i/; are indefinite. Let A = a -f if be a complex root of the determinantal equation /"(A) = 0. Then since the form (a 4- if) y -f ^ is singular, it may be represented as a sum of less than n squares, and since it is complex, these may be squares of complex variables, so that we have 29) (a 1) Kronecker, Uber Schaaren quadratischer Formen. Monatsber. der Konigl. PreuB. Akad. d. Wiss. zu Berlin, 1868. pp. 339 — 346. Werke, Bd. I, p. 165. WEBSTER, Dynamics. 37 578 NOTES. the «/'s and g'a being real linear forms in the #'s, any of which, but not all, may be zero. Separating the real and the imaginary terms. we obtain r==n — \ r = n — 1 30) «9 + ^ From the two forms the whole sheaf may be obtained. Solving 30) for g> and 32) A cp + y which we may write r=^~ 1 / ! v 33) I Cp -f ib = >Yyr — ^#r) [9r H #r)» ^_ \ f* ./ r=l where ^ ^ 0 a quadratic in ft, giving for every real value of A, a real value of jii. Now each of the forms 33) vanishes for values of #15. . . xn other than zero, which satisfy the n — 1 linear equations and is accordingly indefinite. Conversely if there is in the sheaf a single definite form, the roots of /"(A) = 0 are all real. Now in the mechanical application, the form go, which is proportional to that given by the value A = oo, is the kinetic energy, a definite positive form, consequently the reality of the roots is proved. If Ax is one of the real roots of the equation /"(A) = 0 the form A*9> + ty being singular, can be expressed in terms of less than n linear functions of a^, ...#„, say y^...yn—\. Let yn be any other linear function of the #'s, such that the determinant of the functions «/1? . . . yn is not zero, then we can express the function

, a definite positive form. V. TRANSFORMATION OF QUADRATIC FORMS. 579 A linear divisor such as A — Kr of the determinant of the form A g/ -|- tyr is also a divisor of the determinant of A g> -f- i/;, for on writing out the determinant of the form 35) in terms of y^ . . . yn^\tsy.^ we find 36) a^ to . . . ,,„_,(* ?> + *) = (x - i.) tf* . . . r._,(V + *0, so that the vanishing of the determinant of order n — 1 on the right makes the determinant on the left vanish. But this equal to the deter- minant of A (p -f- ty in the variables x± , . . . xn multiplied by a constant. We may now treat the form hep' -\- i^f in the same manner, and so on, so that finally we obtain 37) lg> + y = (I - Aj^2 + (I - A2>22 + - - - + (A- Aw)^2, where A15 . . . Kn are the roots of the determinantal equation f(k) = 0. Since this is true for all values of A we have which is the simultaneous transformation of two quadratic forms required in the treatment of principal coordinates. It is obvious according to this method that it makes no difference whether the determinant has equal roots or not. Absolute system, 31. Acceleration, definition of, 13. angular, 248 centripetal, 14. components of, 13,14. compound centripe- tal, 319. constant, 33. of Coriolis, 320. — moment of, 17. normal, 17. radial, 16. tangential, 14. transverse, 16. Actio, 66. Action, definition of, 101. least, 101. examples of, 140, 141. surface of equal, 139. varying, 131. Activity, equation of, 66. in Lagrange's coor- dinates, 125. — in hydrodynamics, 501. Addition of vectors, 5. Adiabatic motion, 189. d'Alembert's Principle, 63. — statement in words, 65. — in generalized coordinates, 118. Amplitude, 35. Analytic function, 523. Angle, solid, 351. Angular acceleration, 248. — velocity of moving axes, 246. Anisotropic body, 457. Aplatissement , 407. Appell, 54, 61, 309, 316. Archimedes' principle, 472. Areas, law of, 18. Atmosphere, height of, 465. Attracting forces, 76. Atwood's machine, 23. Axis, central, 209. fixed, body moving about, 250. INDEX. (The numbers refer to pages.) Axis, of suspension and oscillation, 251. Axes, moving, motion with respect to, 316. rotating, 317. parallel, 201. — theorem of in- ertia, 229. of spherical harmo- nic, 396. — — strain, 433. Axioms, physical, 20. Bacharach, 378. Balance, 33. Ball, 224. Ballistic pendulum , gal- vanometer, electro- meter, 72. Base-, golf, tennis ball, 34. Basset, 499. Beats, 156. Beads, string of, vibration of, 164. frequencies of, 167. Beltrami, 113. Bending moment, 490. Bernoulli, 58, 84, 173. Bernoulli-Euler theory, 485. Bernoulli's theorem, 505. Bessel, 407. Billiard ball, motion of, 305. Binet, 231, 239. Bodies, three, problem of, 31. Bourlet, 309. Boussinesq, 309, 346. Boyle's Law, 544. Boys, 30, 362. Brachistrochrone, 77, 82. Brahe, Tycho, 18. British Association, 31. Buckling of sections, 485. Bulk-modulus of elasticity, 461. Bullets, toy, 23. Bunsen pump, 504. Cadmium, wave-length of, 27. Calculus of variations, 77. Cambridge, 269. Carvallo, 309, 313. Cauchy, 451, 456, 459. Cavendish, 30. — Laboratory, 269. Center of mass, motion of, 89. Central axis, 209. forces, 38. Centripetal acceleration , compound, 319. Centrifugal force, 119. — — on earth, 320. — in rigid body, 228. Centripetal acceleration, 15. — — compound, 319. Centrobaric body, 364, 404. C. G. S. system, 33. Chasles, 212, 213. Chasles's theorem, 373, 413. Characteristic function, 136. — of plane, 214. Circle, Dirichlet's problem for, 388. Circular Harmonic, 388. — — development in. 390. Circulation, 506. Clairaut, 33. Clairaut's theorem , 404, 407. Clebsch, 224, 478. Clifford, 87, 521. Clark University, 268. Coefficients of inertia, stiff- ness, resistance, 158. Coffin, 52. Complex, 215. — equation in line coor- dinates, 221, 223. — variable, 521. Component, of vector, 116, — — generalized,! 16, — of momentum, 117, Composition of screws, 216 Concealed motions, 179. Concentration, 345. — proportional to den- sity, 360. INDEX. 581 Condition for equipotential family, 410. Cone, rolling, 253. of equal elongation, 443. Confocal quadrics, 235. Conjugate functions, 523. Conical pendulum, 55. Conic section orbit of pla- net, 18. Conservation of motion of center of mass, 90. Connectivity of space, 339. Conservation of energy, 68. — — integral of La- grange's equations, 126. Conservative system, 65, 68. Constant of gravitation, 30. Constants of elasticity, phy- sical meaning of, 461. Constraint, equations of, 57. non-integrable, equations of, 313. varying, 129. Continuity, equation of, 498. Convective equilibrium, 466. Coordinates, 5. curvilinear, 110, 330. cyclic, 176. cylindrical, 335. ellipsoidal, 234, 335. generalized, 109. line, 215. normal, 163. orthogonal, 110, 330. polar, 11. positional, 189. principal, 163. ignoration of, 179. Coriolis, 317. — theorem of, 319. Corner, flow around, 528. Cornu, 491. Correction for finite arcs, 48. Cotes1 s spiral, 519. Coulomb, law of. Couple, 204. arm, 205. causing precession, 299. — composition of, 208. — in regular preces- sion, 274. Couple, of forces, 205. — moment of, 205. — righting, 473. — theorems on, 207. Curl, 87. — in curvilinear coor- dinates, 383. Curvature, 16. Curve, expression for arc of, 111, 113. — parametric represen- tation of, 10. — tautochrone, 144. Curvilinear coordinates, Green's theorem in 380. Cusps on curves of equal action, 140, 141. Cyclic coordinates, 176. — systems, 188. — — examples of,193. reciprocal rela- tions in, 191. — — work done on, 192. Cycloid, 84. as tautochrone, 145. — drawn by point of top, 287. Cycloidal pendulum, 148. Cylinder under pressure, 475. — moment of inertia of, 242. Cylindrical coordinates,335. flow, 653. Cylindroid, 218. Damped oscillations, 148. Damping, coefficient of, 151. Darboux, 113. Decrement, logarithmic, 151. Deformable bodies , kine- matics of, 427. — — statics of, 463. Density, 353. Derivative, directional, 88, 331. — of analytic function, 523. — particle, 497. Development in circular harmonics, 390. Development of reciprocal distance, 398. — in spherical harmo- nics, 400. — of potential in har- monics, 402. — of potential of ellip- soid of revolution, 424. Differential, perfect, 88. — equation for forced vibration, 152. — — of Legendre, 398. — — of particle un- der Newtonian law, 39. — parameter, 88, 330. — — arithmetical va- lue of, 333. invariant, 333. — — mixed, 343. — — second, 344. Dilatation, 436. Dimensions of units, 27, 28, 29. Dimensional, two, poten- - tial, 385. Dirichlet, 69, 376, 377, 378, 393, 402, 417. Dirichlet's problem, 376. — for circle, 388. — — for sphere, 395 Disc, moment of inertia of, 242. Displacement, infinitesimal, 59. virtual, 58. — lines of, 446. Distributions , energy of, 425. — surface, 367. Dissipation, 122. — function, 123. Divergence, 347. — theorem, 347. Double-lines , complex of, 214. Driving points, 123. Dualism, 209. Dyne, 29. Earth, motion relative to, 320. 582 INDEX. Echo, 545. Ellipse , equation relative to focus, 18. Ellipsoid of elasticity, 451. of gyration, 233. — inverse, 233. — Jacobi's, 470. — Maclaurin's, 469. — moment of inertia of, 241. rolling, in Poinsot- motion, 259. potential of, 415. — — at internal point, 418. differentiation, 419. of revolution, poten- tial of, 421. Ellipsoidal coordinates, 234, 335. — as equipotential family, 412. Elliptic function, 45. — integral, 45, 47. Ellipticity, 407. Elongation and compres- sion quadric, 441. Energy, 65. — conservation of, 68. — equation of, 67. — of distributions, 425. in terms of field, 426. — emitted by sound- source, 548. — function for isotro- pic bodies, 457. — invariant, 457. integral of, in top, 277. — kinetic, 66. — general form of, 112. maximum or mini- mum for equilibri- um, 69. potential, 68. — — exhaustion of,76. — all due to mo- tion, 182. — of normal vibrations, 164. — relative kinetic, 92. — of vortex, 515. Eolotropic body, 457. Epicycloid, on polhode, 270. Epitrochoids described by heavy top, 293. Equation of activity, 66. — — in hydrodyna- mics, 501. — — in generalized coordinates, 125. — of continuity, 498. — differential, of mo- tion, 24. — — of forced vibra- tions, 152. — of equilibrium for de- formable body, 448. — of hydrodynamics by Hamilton's principle, 500. — Euler's, for rotation, 261. — Laplace's, 349. — of motion from Ha- milton's principle, 114. — — from least ac- tion, 106. — — Lagrange's first form. — — — second form, 115. — — Hamilton's ca- nonical, 128. — Poisson's. — — applied to earth, 361. — — for two dimen- sions, 387. Equilibrium, 25. — conditions for, 61,69. — stable and unstable, 69. — convective, 466. — equations of for de- formable body, 448. — theory, 155. of tides, 535. Equipollent loads, 486. Equipotential surfaces, 354. — — condition for fa- mily of, 410. — layers, 372. — surface of strain, 446 . Erg, 56. Erg, unit of energy, 70. Ether, luminiferous , 65. Euler, 73, 173, 499. — equations of hydro- dynamics, 499. — angles of, 274. — dynamical equations of, 260. — kinematical equa- tions of, 275. — theorem of 114, 127. Everett, 29, 33. Existence -theorem, 378. Experiment, comparison with theory, 51, 52. Expansion -ratio, 434. Falling body affected by earth's motion, 323. Faraday, 363. Field, energy in terms of, 426. — strength of, 353. Fixed point, motion about 252. Fleuriais, 296. ,, Flexion, 489. Flexure, uniform, 490. — non- uniform, 494. Floating body, equilibrium of, 471. Flow around corner, 528. Fluid, perfect, 458. — rotating, 467. — gravitating rotating, 468. Flux of vector, 349. Flux -function, 516. Focus, kinetic, 105. — of plane in null- system, 212. Formula, Green's, 370. — — for logarithmic potential, 388. Force, definition of, 24. — accelerational, 120. — central, 38. — centrifugal, 119. — — in rigid body, 228. — component, genera- lized, 114. — effective, 64. INDEX. 583 Force, effective, generalized component of, 118. function, 68. — — particular case of, 73. — — for Newtonian law, 75. containing time, 69. — of inertia, 64. — gyroscopic, 185, 278. — momental, 120. motional, 26. — n on -momental, 119. — parallel, 205. — positional, 26. — reduction of groups of, 299. tidal, 408. Forced vibrations, 152. Formula of Green, 370. Foucault, 257, 324, 325, 326. pendulum of, 324. gyroscope of, 324. Fourier, 173, 393. — coefficients in series of, 392. Freedom, degrees of, 58. Free vector, 199. Frequency, 35. Friction, effect on top, 303. Fujiyama, 367. Function, analytic, 523. characteristic, 136. flux, 516. — Hamiltonian, 117. — Lagrangian, 115. linear vector, 428. normal, 171. — fundamental property of, 172. of point, 88. principal, 132. self - conjugate , 430. — of St. Venant, deter- mination of, 483. g, formula for, 406. g, value of, 32, 33. Galileo, 3, 32, 58 Gauss, 32, 312, 350, 351, 373, 374, 387, 425, 426. — differential equation of, 312. Gauss, theorem of, 350, 387. — — on energy, 425. — — of mean, 374. Generalized coordinates, 109, 111. — velocities, 112. Geodesic line, 103. Geodesy, application of spherical harmonics to, 404. Geometric product, 7, 116. Geometrical representation of stress, 450. Geometry of motion, 3. Gleitung, 441. Glissement, 441. Gravitating rotating fluid, 468. Gravitation, constant of, 30. — kinematical state- ment of, 20. — universal, 29. Gravity, center of, 90. — terrestrial, potential of, 405. Green, 340, 343, 370, 379, 386, 388, 456. — formula of, 370. — — for logarithmic potential, 388. — theorem of, 340. — — in curvilinear coordinates , — — for plane, 386. Griffin mill, 273. Gyration, ellipsoid of, 232. Gyroscope, 274. — in torpedo, 257. — as compass, 326. — latitude by, 325. Gyroscopic forces, 185, 278. terms, 185. — pendulum, 282. — horizon, 296. — system, curves drawn by, 293, 294. Gyrostat, 186. — stability of spinning, 291. Hadamard, 313. Hamilton, 7, 21, 97, 126, 128, 131, 176. — equations of, 126. Hamilton, equations from Hamilton's principle, 127. — method of, 136. — principle of, 97. — — equations of hy- drodynamics by, 499. — — — of string by, 170. — partial differential equation of,; 135. — theorem of, 135. Hamiltonian function, 127, 128. Harmonics, circular, 388. — — development in, 390. — of pipe, 546. — spherical, 393. — — examples of,394. — — forms of, 395. — — axes of, 396. — — in spherical coordinates, 398. — — zonal, 397. — of string, 169. Harmonic function, 345. — ~ motions, 35. — — elliptic, 36. Harkness and Morley, 378. Harnack, 388. Hay ward, 324. Heat, dynamical theory of, 64. Heaviside, 87. Height, metacentric, 474. — of atmosphere, 404, 408, 465. Helmert, 33, 362, 407. Helmholtz, 3, 86, 176, 179, 458, 507, 509, 511, 512. — energy form, 458. — theorem of, 512. Herpolhode, 262, 263. Hertz, 21, 113, 182, 193. Hilbert, 378. Heterogeneous strain, 444. Hodograph, 19. Homoeoids, ellipsoidal, 409. Homogeneous strain, 428. Horse-chestnut, toy, 23. Hooke, 455. Hoop, rolling, of, 308. 584 INDEX. Howell torpedo, 272. Huygens, 148. Hydrodynamics, 496. equations of, 497. Hydrostatics, 463. Hyperellyptic function for top,. 303. Hypergeometric series, 312. Hyperspace, 113. Hypocycloid on polhode, 270. Hypotrochoids described by heavy top, 293. Ignoration of coordinates, 179. — example of, 181. Impulse, 70. — and velocity, geome- tric product of, 72. Impulsive forces, 71. — — in Lagrange's equations, 134. Impulsive wrench, 225. Indeterminate multipliers, 62. Inertia, 21. axes of, distribution in space, 239. coefficients of, 112. ellipsoid of, 231. — force of, 64. moments of, calcu- lation of, 241. principal axes of,229. — moments of, 231. — products of, 227. Infinitesimal arc, area, vo- lume, 338. Integral of function of com- plex variable, 524. Interferences, 156. Invariable axis, motion of in body, 266. — — and plane, 95. Invariants of strain, 457. Invariant, second differen- tial parameter, 347. Irrotational motion, 520. Isochronous vibration har- monic, 146. Isocyclic motion, 189. Isotropic body, stresses in 460. Isotropic body, energy function for, 457. Ivory's theorem, 420. Jacobi, 135, 297, 380, 470. — ellipsoid of, 470. — method of, top equa- tions by, 297. Jordan, 402. Kater, 251. Keely, 153. Kelvin, 3, 376, 378, 511, 538, 557. — and Dirichlet's prin- ciple, 376. Kepler, 18, 38. laws of, 31. Kilogram, weight of, 32. Kinematics, 3. — of deformable bo- dies, 427. — of rigid system, 243. Kinematical equations of Euler, 276. Kinetic energy, general form of, 112. — — due to rotation 249. focus, 105. potential, 179. linear terms in, 184. — reaction, 64, 65, 119, 120. Kinetoscope, 23. Kirchhoff, 69, 499. — energy function of, 459. Klein and Sommerfeld, 273. Kneser, 105. Korteweg, 316. Krigar-Menzel, 362. Kronecker, 378, 393. to Lag, in forced vibration, 153. Lagrange, 155, 157, 164, 173, 507, 508. equations of motion, first form, 108. equations of motion, 115. by direct transformation, 115. Lagrange, equations for small oscillations, 158. — — of equilibrium, 62. — — for pure rolling, 313. — determinantal equa- tion, 159. — — roots of, 160. — method of, top equa- tions by, 279. Lagrangian function, 115. — modified, 179. method in hydro- dynamics, 499. Lamb, 499. Lame, 329, 331, 344, 380. Lame's shear -cone, 451. — stress-ellipsoid, 451. Lamellar vectors, 87. Laminar flow, 554. Laplace, 3, 398, 401, 402r 542, 544. — equation of, 349. in spherical and cylindrical coordina- tes, 383 satisfied by po- tential, 357. — operator, 349. Laplacian, 350. Law of areas, 18, 38. — Coulomb, 486. inverse squares, 20. Kepler, 18. Lenz, 193. motion, 20. Layers, equipotential, 372. Least action, 99. Legendre, 47, 399. — differential equation of, 398. polynomials of, 397. Lenz's law, 193. Level surface, 88, 89. of potential function, 329. sheet, 89. Line-coordinates, 215, 221. Line-integral, 84. Line -integral, independent of path, 87. Lines of force, 354. INDEX. 585 Lines of vector -function, 348. Linear terms in kinetic energy, 130. — — — potential, 184. — vector function, 428. Lines of displacement, 446. Liquids, fundamental the- orem for, 465. Logarithmic decrement, 151. — potential, 385. Loops, 546. Maclaurin, 415. Maclaurin's ellipsoid, 469. — theorem, 414. Maupertuis, 97. Mass, 23. — dynamical compari- son of, 23. center of, 90. Material point, 21. Matter, 3. — definition of, 64. Maxwell, 123, 126, 268, 345, 349, 551. Maxwell's theory of elec- tricity, 124. — top, 268. — — effect of friction on, 304. Mean, integral as, 70, 90. theorem , Gauss's, 374. Mechanics, 3. problem of, 26. Mechanical powers, 60. Mendenhall, 367. Metacenter, 474. Metacentric height, 474. Metre prototype, 26. Michell, 533. Modulus, bulk, 461. — shear, 462. — Young's, 462. Momentum, 26. generalized compo- nent of, 117. Mobility, coefficients of, 126. Mobius, 212. Moment, axis of, 7. — bending, 490. , Dynamics. Moment of inertia, 93, 227. — of momentum, 95. — of velocity, 12. Momentum, 26. — moment of, 95. — — conservation of, 96. — screw, 224. — of rigid body, 225. Moon, motion of, 20. Motion, 3. adiabatic, 189. — change of, 22. — concealed, 179. — constrained, 41. — differential equa- tions of, 24. — geometry, os, 3. — harmonic, 35. — — elliptic, 36. irrotational, 520. — isocyclic, 189. — laws of, 520. periodic, 35. relative, 247. relatively to earth, 320. — steady, in hydrody- namics , — uniform, 21. — — circular, 37. vortex, 509. — of waves, 529. Motus, 26. Moving axes, 243. — — motion with re- spect to, 316. Multipliers , indeterminate, 62. Navier, 456. Neumann, 388, 459, 515. Newton, 3, 20, 29, 31, 66, 155, 365, 535, 544. Newton's theorem, 4P9. Newtonian constant, 30. — law, force - function for, 75. — — motion under,39. Node, 168, 169, 546. Non- conservative system, 69. Normal coordinates, 163. — functions, 171. Normal functions, series of, 173. Null -system, 209. Numeric, 26. Nutation of top, 283. Obry, 257. Oersted's piezometer, 477. Oscillation, axis of, 251. — damped, 148. Oscillations, small, 157. Overtones, 169. Pantheon, 324. Parabola, path of projec- tile, 34. Parallelepiped, moment of inertia of, 241. Parameters of Lagrange, 111. — differential, 88, 330, 344. Parametric representation of curve, 10. Particle derivative, 497. Path, 9. Pendulum , affected by earth's rotation, 323. — compound, 251. — conical, 55. — cycloidal, 148. — gyroscopic, 282. — horizontal, 252. ideal, 42. — Eater's, 251. — plane, 45. — quadrantal, 196. — small vibrations of, 54. — spherical, 48. — — by Lagrange's equations. path of, 50. Perfect differential, 88. — fluid, 458. Period of pendulum, 45, 47. Perpendicularity, condition of, 7. Phase, difference of, 37. Picard, 86, 378, 388. Piezometer, 477. Pitch -conic, 220. Pitch of helix, 211. Pivoting, friction of, 304. 37* 586 INDEX. Planet, motion of by Ha- milton's method, 142. * orbit of, 18. period of, 19. Plucker, 215. Poincare, 105, 388, 393, 468. Poinsot, 252, 255, 256, 261, 263. — central axis, 210. Point -function, 88, 329. Point, material, 21. Poiseuille, 552. Poisson, 402, 491. equation of, 359. — for two dimen- sions, 387. ratio, 463. Polar coordinates, 334. Pole, motion of earth's, 270. Polhode, 262. cone, 263. figure of, 263. — projections of, 265. Polygon of vectors, 5. Polynomial of Legendre, 39. Potential, definition of, 352. characteristics of, 362. of continuous distri- bution, 353. determined by pro- perties, 375. derivatives of, 355. — development in spherical harmonics, 402. of disc, cylinder, and cone, 366. due to cylinder, 384. — of earth's attraction, 298. energy, apparent, 180. of ellipsoid, 415. — for internal point, 418. — — of revolution, 421. — — — development of, 424. — differentiation of, 419. — of gravity, 405. Potential, kinetic, 179. — logarithmic, 385. of sphere, 363. of strain, 446. — of tidal forces, 408. vector, 511. of vector, 89. — velocity, 507. Positional coordinates of cyclic system 189. Precession, 255. — couple in regular, 274. aijd nutation of top, 283. — of earth, 298. Pressure, 447. Principal axes, of inertia, 229. — coordinates, 163. — function, 132. Principle of Archimedes, 472. of center of mass, 91. of energy, 67. of Hamilton, 97, 98. — broader than that of energy, 99. — most general principle, 130. of least action, 97, 99, 101. — of moment of mo- mentum, 96. - of varying action,! 31. Problem of Dirichlet, 376. — for circle, 388. — for sphere, 395. — of three bodies, 31. Product, geometric, 7. of inertia, 227. scalar, 7. — vector, 8. Projections, 4. of vector, 6. Projectile, path of, 34. Puiseux, 54, 299. Pulley, 60. Pump, Bunsen's, 504. Pupin, 169, 170. Pure strain, 430. Quadrantal pendulum, 196. Quadrics, confocal, 235. Quadrics, elongation and compression, 441. — reciprocal, 431. Ratio of Poisson, 463. Rayleigh,123, 157, 169,529, 556. .Reaction, 22. accelerational , 122. — of constraint, 42. — — does no work, 65. kinetic, 64, 119, 120. — non- conservative, 122. — static, 25. Reciprocal distance, deve- lopment of, 398. — relations in cyclic systems, 191. quadrics, 431. Reduction of groups of forces, 209. Reflection of wave, 545. Relations between stress and strain, 455. Relative motion, 247. Resal, 299. Resistance of air, 26. Resonance, 152. — general theory of,175. Resultant, 5 Richarz, 362. Riemann, 378, 523. Riemann -Weber, 393. Righting couple, 473. Rigid body, displacement of, 200. statics of, 205. Rigidity, 462. Rolling, 307. — ellipsoid, 259. — treated by Lagran- ge's equations, 313. Rotation, energy due to, 249. — of earth, 298. — momentum due to, 249. of rigid body, 199. — — — about in- tersecting axes, 202. — — about pa- rallel axes, 201. of rigid body, infini- tesimal, 203. INDEX. 587 Rotation, as vector, 203. Roating axes,, 317. — fluid, 467. Routh, 176, 179. Routh and Helmholtz, trans- formation of, 177. Sachse, 393. de Saint Tenant, 478, 485. — problem of, 478. Scalars, 4. Screw, momentum, 224. Screws, composition of, 216. — reciprocal, 221. Schwerer, 296. Searle, 269. Second, mean solar, 27. — differential parame- ter, 344. Self-conjugate function,430. Sense, muscular, 24. Series, hypergeometric, 312. — trigonometric, 173, 388. Sevres, 26. Shear, 439. amount of, 441. general, 441. — modulus of, 462. Shifts, 436. Ship, heeling of, 72. Simple strains, 439. stresses, 451. Sleeping top, 304. Slesser, 299. Slides, 441. Small oscillations, 157. Solenoidal condition, 349. vector, 347. Solid angle, 351. Somoff, 380. Sound-waves, 542. Sources and sinks, 520. — strength of, 521. Source of Sound, strength of, 547. Space, connectivity of, 339. of m dimensions, 113. Sphere, moment of inertia of, 241. — potential of, 365. under pressure, 475. Spherical harmonics, 393. — — axes of, 396. Spherical harmonics, deve- lopment in, 400. — — — of potential in, 402. — — forms of, 395. — zonal, 397. — waves, 547. .Squeezes, 439. Squirt, 521. Statics, 57. foundation of, 58. — of rigid body, 205. Steps, 4. Steady motion in hydro- dynamics, 503. Stiffness, 151, 158. Stokes, 84, 86, 345, 404, 497, 510, 533. — theorem, 86. — — in curvilinear coordinates, 381. Strain, axes of, 433. ellipsoid, 432. — general small, 436, — heterogeneous, 444. — homogeneous, 428. — inverse, 432. — irrotational, 433. pure, 430. — potential, 446. relatively homoge- neous, 445. — composition of, 437. Strength of field, 353. Stress, 24, 446. — ellipsoid, 451. Cauchy's 451. Lamp's, 451. — geometrical represen- tation of, 450. — in isotropic bodies, 460. — simple, 451. — vector, 447. work of, 454. — and strain, relations between, 455. String, vibrations of, 164. Surface distribution, 367. — of equal action, 139. level, 88. — parallel, 139. Sum, geometric, 5. System, conservative, 65. System, cyclic, 188. — non-conservative, 69. Tait, 141. Target, problem of shoot- ing, 103. Tautochrone, 144. Taylor's theorem, 36, 78, 79, 157, 346, 391, 399, 444, 455, 509. Tensor of vector, 6. Terrestrial gravity, poten- tial of, 405. Theorem, of bending mo- ment, 490. — Bernoulli's, 505. — Chasles's, 413. Clairaut's, 404. of Coriolis, 319. divergence, 347. — Euler's, 114, 127. — Gauss's, 350. — for two dimen- sions, 387. — on energy, 425. Green's, 340, 343. — — for plane, 386. — — in curvilinear coordinates, 381. — Hamilton's, 135. — Helmholtz's, 612. — Ivory's, 420. for liquids, 465. — Maclaurin's, 414. of mean, Gauss's, 374. — Newton's, 409. of parallel axes, 229. Taylor's, 36, 78, 79, 157, 346, 391, 399, 444, 455, 509. Stokes's, 86. — — in curvilinear coordinates, 381. — Thomson and Tait's, 138. Torricelli's , 506. Thomson and Tait, 21, 29, 72, 76, 105, 138, 148, 179, 196, 278, 378, 424, 459, 462, 470. Tides, canal theory of, 539. — equilibrium theory of, 535. — inversion of, 541. 588 INDEX. Tide-generating forces, po- tential of, 408. Time, 3, 4. measurement of, 22. Tisserand, 470. Todhunter and Pearson, 438, 459. Tone, pure, 546. Top, curve, 272. heavy symmetrical, 274. — Maxwell's, 268. — kinetic reaction of, 271. — rising of, 303. rise and fall of, 285. on smooth table, 302. — nearly vertical, 289. symmetrical, 271. — equations by Jacobi's method, 297. Lagrange's me- thod, 277. Torricelli's theorem, 506. Torsion, 483. Traction, 447. Translation of rigid body, 199. - decomposed into rotations, 202. Trigonometric series, 388. Trochoids drawn by top,288. Torpedo, Whitehead, 257. Howell, 272. Triangle of vectors, 5. Tube of vector-function, 348. Tuning, 154. Tuning-fork, 36. Twist of billiard-ball, 306. Uniform motion, 21. circular motion, 37. Units, absolute system of, 31 . derived, 27. — dimensions of, 27. — C. G. S., 29. Variable, complex, 521. Variations, calculus of, 77. differentiation of, 79. — of integral, 80. Varying action, 131. — constraint, 129. Vectors, 4. addition of, 5. couples of, 204. free, 199. lamellar, 87. — product, 8. — polygon of, 5. sliding, 200. tensor of, 6. Vector -cross, 212. — — conjugate lines in, 213. — function, linear, 428. — potential, 511. Velocityv-definition of, 9. — angular, 11. — components of, 10. composition of, 10. generalized, 112. moment of, 12. — sector, 12. — transverse and radial, 11. potential, 507. — due to vortex, 514. critical, for planet,40. Venturi water-meter, 504. Vibrations, energy of, 164. forced, 152. — free, 153. — forced and free coex- isting, 157. isochronous necess- arily harmonic, 146. — normal, 163. small, 36, 46. Vierkant, 313, 314. Virtual work, 57. — — principle of, 61. — displacement, 58. Viscosity, 551. Viscous fluids, 549. — solid, 555. Vortex, 506. — couple, 520. — energy of, 515. — in corner, 519. — ring, 620. — straight, 515. — strenght of, 507 — velocity due to, Vortex -motion, 509. — — conservati 511. Vorticity, 502. Water-meter, Ventu Wave -motion, 529. Waves in deep water, — differential equat of, 543. — highest, 533. — plane, 543. — in shallow *• 535. — in solid, 548. — standing, 535 — of sound, 54. Webster, 127, 381, Weierstrass, 378. Weight of kilogram Whitehead torpedo Wien, 499. Woodward, 362. Work, 56. — of stress, 454. — of wrench, 22 — unit of, 56. — virtual, 57. Wrench, 216. — impulsive, 225. — work of, 220. Young's modulus, 462 Zollner, 252. Zonal harmonics, 397. ERRATA. p. 30, line 21, for y = 6.576 read y = 6.6576. „ 205, line 7 from bottom, for AB and PQ read AP and BQ. Dresden. — ^WAtSONU^KO^^^, "-^saaSffSt-i 901 U.C. BERKELEY LIBRARIES ' CD5DflS3im i t /cr f UNIVERSITY OF CALIFORNIA LIBRARY