CONTENTS. 



Xlll 



CHAPTER VII. 

 VORTEX 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. Kirchhoff s 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. Recapitulation 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 



