CONTENTS. XI 



ART. . PAGE 



73-80. Discontinuous motions. Investigations of von Helmholtz 

 and Kirchhoff. Applications to Borda's mouth-piece, 

 the vena contracta, the impact of a stream on a lamina. 



_> ; BobylefTs problem 100 



81. Flow in a curved stratum ... 114 



CHAPTER V. 



IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN 

 THREE DIMENSIONS. 



82,83. Spherical harmonics ; Maxwell's theory ; poles. . . 117 



84. Transformation of the equation v 2 $ = 0to polar coordinates. 120 



85, 86. Zonal harmonics. Hypergeometric series. Legendre's 



functions. Velocity- potential of a double-source. 



Functions of the second kind 121 



87. Tesseral and sectorial harmonics 125 



88. Conjugate property of surface harmonics ; expansions . 127 



89. 90. Hydrodynamical applications. Impulsive pressure over a 



sphere. Prescribed normal velocity over a sphere. 



Energy 127 



91, 92. Motion of a sphere in an infinite liquid. The effect of 

 the fluid is on the inertia of the sphere. Sphere 

 in a liquid with a concentric spherical boundary . 130 

 93-96. Stokes' stream-function. General formulae for stream- 

 function. Stream-lines of a sphere. Image of double- 

 source in a sphere. Rankine's method . . .133 

 97, 98. Motion of two spheres in a liquid ; kinematic formulae . 139 



99. Sphere in cyclic region 143 



100-103. Ellipsoidal harmonics for ovary ellipsoid. Motion of a 

 liquid due to an ovary ellipsoid (translation and 



rotation) 145 



104-106. Ellipsoidal harmonics for planetary ellipsoid. Flow through 

 a circular aperture. Motion of fluid due to a planet- 

 ary ellipsoid (translation and rotation). Stream-lines 

 of a circular disk . . . . . . . .150 



107. Motion of fluid in ellipsoidal case (unequal axes) . . 155 



108. General expression of v 2 $ = in orthogonal coordinates . 156 



109. Confocal quadrics ; ellipsoidal coordinates. . . . 158 

 110-112. Motion of fluid due to a varying ellipsoidal boundary . 159 



Flow through an elliptic aperture. Translation and rota- 

 tion of an ellipsoid. 

 113. References to other researches . . . 166 



