
CONTENTS. CHAPTER I. HYDROKINEMATICS. 1. Introduction............ 1 2. Definition of a fluid........... 3 3. Lagrangian method, and Eulerian or flux method..... 4 4. Velocity and acceleration — Lagrangian method..... 4 56. „ „ „ — flux method....... 5 7. Analytical lemma........... 7 89. The equation of continuity......... 7 10. Definition of the velocity potential, and the forms of Laplace's operator in polar and cylindrical coordinates....... 8 11. Lagrange's equation of continuity........ 9 12. The bounding surface.......... 9 13. Lines of flow and stream lines........ 10 1415. Properties of lines of flow in a liquid....... 11 16. Earnshaw's and Stokes' current function...... 11 17. Molecular rotation........... 12 1819. Formulae of transformation........ 13 Examples............ 15 ^ ON THE GENERAL EQUATIONS OF MOTION OF A PERFECT FLUID. 20. Pressure at every point of a fluid is equal in all directions ... 19 21. Equations of motion of a perfect fluid....... 20 22. Pressure is a function of the density....... 21 23. Equations of motion referred to moving axes...... 21 24. Equations satisfied by the components of molecular rotation ... 22 25. Stokes' proof that a velocity potential always exists, if it exists at any particular instant.......... 22 26. Physical distinction between rotational and irrotational motion .. 23 27. Lagrange's hydrodynamical equations of motion..... 25 28. Weber's transformation.......... 25 29. Proof of theorem that if the pressure is not a function of the density, vortex motion can be generated or destroyed in a perfect fluid .. 26 30. Cauchy's integrals........... 26 31. Integration of equations of motion when a velocity potential exists .. 27 32. Definition of a vortex line......... 27 33. Clebsch's transformation.......... 28 34. Proof thatis a maximum or minimum . 30 3536. Energy and least action......... 31 37. Steady motion — Bernoulli's theorem....... 33 38. Conditions of steady motion which is symmetrical with respect to an axis, or is in two dimensions......... 34 3940. Steady motion of a liquid — Clebsch's method..... 36 41. Maximum and minimum theorem........ 38 42. Impulsive motion........... 38 43. Motion of a liquid surrounding a sphere which is suddenly annihilated . 39 44. Torricelli's theorem........... 40 45. Application of the hypothesis of parallel sections, to the motion of liquid flowing out of a vessel......... 41 Examples............. 43 ^ ON SOURCES, DOUBLETS AND IMAGES. 46. Velocity potential due to a source or sink...... 48 47. do. due to a doublet........ 49 48. do. due to a doublet sheet....... 49 49. do. due to a source and doublet in two dimensions .. 49 50. Theory of images........... 50 51. Image of a source in a liquid bounded by a fixed plane ....51 52. Image of a source in a sphere......... 51 53. Image of a doublet in a sphere, whose axis passes through the centre of the sphere............ 53 54. Image of a doublet in a sphere, whose axis is perpendicular to the line joining it with the centre of the sphere...... 54 55. Image of a source and line sink in a sphere...... 54 56. Image of a source and doublet in a cylinder...... 56 57. Image of a source between two parallel planes...... 56 Examples............. 59 ^ VORTEX MOTION AND CYCLIC IRROTATIONAL MOTION. 58. Statement of problem.......... 62 59. Fundamental properties of vortex filaments...... 62 60. Integration of the equations which determine the components of molecular rotation in terms of the velocities..... 64 61. Velocity due to a vortex.......... 66 62. Velocity potential due to a vortex........ 66 63. Vortex sheets............ 68 64. Surfaces of discontinuity.......... 68 65. Surfaces of discontinuity possess the properties of vortex sheets .. 68 66. Cyclic and acyclic irrotational motion....... 70 67. Flow and circulation.......... 70 68. Stokes' theorem........... 70 69. Circulation due to a vortex filament is equal to twice the product of its angular velocity and its cross section...... 71 70. Polycyclic velocity potentials......... 72 71. Stream lines cannot form closed curves unless the motion is cyclic .. 72 72. Circulation is independent of the time....... 72 73. Irrotational motion which is acyclic cannot become cyclic ... 73 74. Reconcileable and irreconcileable lines....... 73 75. Simply and multiply connected regions....... 73 76. A multiply connected region is reducible to a simply connected one . 74 7778. Reduction of polycyclic velocity potentials to monocyclic functions . 74 79. Vorticity............. 75 80. Green's theorem........... 76 81. Deductions from Green's theorem — Kinetic energy of a liquid .. 77 82. Physical interpretation of Green's theorem...... 78 83. Liquid whose motion is acyclic and irrotational, is reduced to rest if the motion of the bounding surface is destroyed..... 78 84. Extension of Green's theorem to spaces bounded internally by several closed surfaces........... 78 85. Kinetic energy of a liquid occupying such a region..... 78 86. Adaption of Green's theorem to twodimensional space .... 79 87. Stokes' theorem a particular case of Green's theorem .... 80 88. Thomson's extension of Green's theorem....... 81 89. Kinetic energy of an infinite liquid occupying a multiply connected space 82 90. Kinetic energy of a liquid contained within a closed surface, is less when the motion is irrotational and acylic, than if the liquid had any other possible motion......... 83 91. Kinetic energy when the motion is rotational...... 83 92. Kinetic energy due to two vortex filaments is proportional to the electro kinetic energy due to two electric currents..... 84 93. Another expression for the kinetic energy...... 85 94. Kinetic energy in terms of Stokes' current function .... 85 95. Connection between vortex motion and electromagnetism ... 86 Examples............ 89 ^ ON THE MOTION OF A LIQUID IN TWO DIMENSIONS. 96. Statement of problem.......... 90 97. Boundary conditions for a cylinder moving in a liquid .... 90 98101. Conjugate functions, and their properties...... 91 102. Examples of conjugate functions — Circular cylinder moving in an infinite liquid — Initial motion due to a circular cylinder in a liquid bounded by a fixed concentric cylinder — Motion of a liquid contained in a rotating equilateral prism — do. in a rotating elliptic cylinder............ 93 103. Motion of a liquid contained in a rotating rectangular prism .. 96 104105. Motion of a liquid contained in a rotating sector .... 98 106. Further applications of conjugate functions...... 100 107. Motion of an elliptic cylinder in an infinite liquid..... 100 108. Motion of translation of a cylinder whose cross section is the inverse of an ellipse with respect to its centre....... 102 109. Expression of results in terms of elliptic functions .... 104 110. Current function due to the rotation of the cylinder .... 105 111. do. when liquid is contained within a cylindrical cavity of this form............ 105 112. Application of results to the theory of conduction of heat and of electrified cylinders.......... 106 113. Motion of a cylinder whose cross section is the inverse of an ellipse with respect to its focus..106 114. Motion of translation of a cylinder whose cross section is a lemniscate. 106 115. Coefficients of cos nθ in the expansion of (1 + 2c cos θ + c^{2})^{½}^{ }... 107 116. Motion of rotation of a cylinder whose cross section is a lemniscate . 109 117119. Motion of a cylinder whose cross section is a lemniscate of Bernoulli 109 120121. Dipolar coordinates......... 110 122. Motion of two circular cylinders in an infinite liquid .... 112 123. Kinetic energy of an infinite liquid in which two circular cylinders are moving............ 113 124. Expressions for the coefficients of the velocities..... 114 Examples............ 115 ^ DISCONTINUOUS MOTION. 125127. Statement of problems to be solved...... 120 128129. Representation of a vector by means of a complex quantity, and the properties of the latter......... 121 130. Every complex has a differential coefficient...... 122 131. Kirchhoff's method of solving problems of discontinuous motion .. 123 132133. Transformation by means of complex variables .... 124 134. Particular cases of transformation........ 126 135. Motion of a jet escaping from a slit....... 127 136. Motion of a jet escaping through a small tube..... 129 137. Coefficient of contraction of a jet can never be less than ½ 130 138. Stream of liquid flowing past a rectangular lamina. Pressure on the lamina............ 131 139. Conditions of stable and unstable equilibrium of the lamina ... 134 140. Intrinsic equation of the surface of discontinuity..... 135 Examples............ 135 ^ ON THE KINEMATICS OF SOLID BODIES MOVING IN A LIQUID. 141. Conditions to which the velocity potential must be subject ... 137 142. Boundary conditions for the case of a single solid..... 137 143144. Velocity potential due to the motion of a sphere .... 138 145. do. due to the motion of the solid formed by the revolution of two spheres cutting orthogonally...... 139 146. do. due to the initial motion of two concentric spheres . 140 147. do. due to the motion of an ellipsoid..... 140 148. do. due to the motion of an ellipsoid of revolution .. 143 149. do. due to the motion of a circular disc .... 144 150. do. due to liquid contained in a rotating ellipsoidal cavity . 145 151. do. due to liquid contained between two confocal ellipsoids. 145 152. Magnetic potential of a spherical bowl....... 146 153. Velocity potential due to the motion of liquid about a spherical bowl . 147 154. do. due to a source situated on the axis of the bowl .. 149 155. do. due to the motion of the bowl in an infinite liquid . 149 156. Electrostatic potential of a bowl placed in a field of force symmetrical with respect to the axis of the bowl...... 150 157. Electrostatic potential when the bowl is placed in a uniform field of force parallel to the axis.152 158. Interpretation of the result......... 153 159. Electrostatic potential when the bowl is placed in a uniform field of force perpendicular to a plane containing the axis ....154 160. Current function due to the motion of a solid of revolution parallel to its axis............ 155 Examples............ 156 ^ ON THE GENERAL EQUATIONS OF MOTION OF A SYSTEM OF SOLID BODIES MOVING IN A LIQUID. 161. The motion can be determined by Lagrange's equations ... 159 162. Acyclic motion........... 159 163. Kinetic energy is a homogeneous quadratic function of the velocities of the solids alone.......... 160 164. Proof that Lagrange's equations can be employed..... 161 165. Impulse of the motion.......... 162 166. Hamiltonian equations.......... 163 167. Kirchhoff's equations.......... 164 1689. Geometrical equations......... 165 170. Cyclic motion........... 167 171. Kinetic energy is the sum of a homogeneous quadratic function of the velocities of the solids and a similar function of the circulations . 169 172. Expressions for the generalised components of momentum due to the cyclic motion........... 170 173. The modified Lagrangian function........ 171 174. Interpretation of the result. Product of the circulation and density is a generalised component of momentum...... 175 175. The flux through an aperture relative to the solid is, the generalised velocity corresponding to the product of the density and the circulation through that aperture....... 176 176. Modified function for a single solid having one aperture ... 177 177178. Modified function for a system of cylinders..... 178 179. Explanation of results.......... 181 ^ ON THE MOTION OF A SINGLE SOLID IN AN INFINITE LIQUID. 180. General expression for kinetic energy....... 182 181. Kinetic energy due to the motion of an ellipsoid and of a solid of revolution........... 183 182. Motion of a sphere under the action of gravity..... 183 183. Motion may become unstable owing to the formation of a hollow .. 185 184. Initial motion of a sphere, when the liquid is enclosed within a con centric spherical envelope........ 185 185. Motion of a circular cylinder when there is circulation ....186 186. Determination of the motion by means of Lagrange's equations .. 188 187. Motion of an elliptic cylinder........ 189 188. Motion of a cylinder whose cross section is a curve such as a cardioid . 193 189. Motion of an ellipsoid.......... 193 190. Stability and instability of steady motion parallel to an axis .. 194 191. Calculation of the coefficients of inertia of an ellipsoid .... 195 192. Motion of an ellipsoid when two of its axes remain in a plane .. 195 193. Motion of a ringshaped solid of revolution through whose aperture there is circulation.......... 196 194. Motion of a ring produced by an impulsive couple about a diameter . 197 195. Angular motion expressible in terms of the time by means of elliptic functions............ 199 196. Explanation of results by means of general principles .... 202 197. Steady motion and stability of a ring moving parallel to its axis .. 202 198. Steady motion and stability when the centre of inertia describes a circle 203 199200. Helicoidal steady motion........ 205 201. Stability of helicoidal steady motion....... 207 202. Expressions for the kinetic energy in the case of an isotropic helicoids and other solids.......... 208 203. Three directions of permanent translation for every solid ... 209 204. Wrenches and screws.......... 210 205. Infinite number of steady motions when the impulse consists of a twist about a screw........... 210 206. Motion of a solid is determinate when the impulse consists of a couple 212 Examples............ 213 ^ ON THE MOTION OF TWO CYLINDERS. 207. Expression for kinetic energy......... 219 208. Motion of a cylinder in a liquid bounded by a fixed plane ... 219 209. Ratio of initial to terminal velocity when the cylinder is in contact with the plane and is projected from it...... 221 210. Conditions that the cylinder may or may not strike the plane .. 221 211212. Motion of one cylinder when the other is fixed .... 222 213. Motion of a cylinder in a liquid bounded by a fixed plane, when there is circulation........... 223 214. Steady motion of a cylinder when the liquid is bounded by a horizontal plane............ 224 215216. Deduction of results from general reasoning..... 226 Examples............ 227 ^ ON THE MOTION OF TWO SPHERES. 217. Kinetic energy consists of nine terms only...... 229 218. Motion along the line of centres........ 230 219. Calculation of the coefficients of the velocities in the expression for the kinetic energy, by the method of images...... 231 220221. Complete values of the coefficients....... 232 222. Approximate values of the coefficients, in terms of series of powers of the reciprocal of the distance between the spheres .... 236 223. Motion perpendicular to the line of centres...... 237 224225. Calculation of the coefficients by means of images .... 237 226. Calculation of the images μ_{1}, v_{1}, μ'_{1}....... 239 227. Transference theorem in spherical harmonics...... 240 228. Kinetic energy depends solely upon harmonics of the first degree .. 242 229. Calculation of the velocity potential, and of the approximate values of the coefficients........... 243 230. Motion of a sphere in a liquid bounded by a fixed plane ... 244 231. Expressions for the components of the pressure upon the sphere .. 245 232. Small oscillations of two spheres........ 246 233. Oscillations of the second order. Thomson's theorem .... 247 234. Pulsations of two spheres......... 248 235. Velocity potential due to pulsations, is equivalent to that due to an infinite system of images.........249 236. Formula for determining the pressure upon the spheres ... 249 237238. Determination of that portion of the pressure which does not depend upon the square of the velocity...... 250 239240. Approximate values of the velocity potential, and of the portion of the pressure which depends on the square of the velocity .. 252 241. The spheres attract one another when their phases differ by less than a quarter of a period.......... 255 Examples............ 256 APPENDIX. I. Proof of the equation p = kp^{γ}......... 259 II. Value of a qseries in terms of elliptic functions..... 260 III. Determination of the azimuthal motion of a solid of revolution by means of Weierstrass's functions....... 261 