CONTENTS. Xlll 



CHAPTER VII. 



VOKTEX MOTION. 



ART. PAGE 



142. ' Vortex-lines ' and ' vortex-filaments' ; kinematical proper- 



ties . ... . . . . . .222 



143. Persistence of vortices 224 



144-146. Conditions of determinateness of vortex-motion. Deter- 

 mination of motion in terms of expansion and rotation. 

 Electro-magnetic analogy 227 



147, 148. Case of a single isolated vortex. Velocity-potential due 



to a vortex 231 



149. 'Vortex-sheets' 234 



150-153. 'Impulse' and energy of a vortex system. . . . 236 



154, 155. Rectilinear vortices. Special Problems .... 243 

 156. Vortex- pair; 'impulse' and energy. KirchhofFs form of 



the theory 248 



157,158. Stability of a cylindrical vortex. Kirchhoff's elliptic vortex 250 



159. Vortices in a curved stratum of fluid .... 253 



160,161. Circular vortices; energy and 'impulse.' Stream-function 254 



162. Isolated vortex-ring. Stream-lines. Velocity of transla- 



tion 257 



163. Mutual influence of vortex-rings. Image of a vortex in 



a sphere 260 



164. General conditions for steady motion of a fluid. Examples. 



Hill's spherical vortex 262 



CHAPTER VIII. 

 TIDAL WAVES. 



165. Introduction. Kecapitulatioii of the general theory of 



small oscillations 266 



166-170. Waves in canal of uniform section. Equations of motion. 

 Integration and interpretation. Wave- velocity. Mo- 

 tion in terms of initial circumstances. Physical 

 meaning of the various approximations . . .271 



171. Energy of a wave-system. In progressive waves it is 



half potential and half kinetic ..... 278 



172. Artifice of steady motion 279 



173. Superposition of waves. Reflection 280 



174-176. Effect of disturbing forces. Free and forced oscillations 



in a canal of finite length ...... 281 



177. Canal theory of the tides. Disturbing potential . . 286 



