HECENT rUOGRESS IN lIYDrvODYNAMlCS. 43 



1*08016 by taking tlie series for the velocity and stream functions given 

 in Thomson and Tait, and expressing the series for the velocities derived 

 from them in the above way. The upper limits of the integrals are the 

 co-ordinates of the point. Definite integral expressions, with constant 

 limits for the velocity and stream functions, and the velocities, have 

 been determined by myself,' by taking the values of these functions for 

 a source inside a rectangle, and distributing sources and sinks over the 

 sides of the rectangle, proportional to the normal motion of the boundary 

 at the point. 



The two functions for a source inside a rectangle were first (1865) 

 determined by Jochmann^ by summation of the corresponding functions 

 for the whole set of images of the source. They depend in general on the 

 Theta functions, and for particular positions take simple forms. The same 

 problem for vortices has been solved by Greenhill,^ who has also found 

 the equation to the path of a single vortex inside the boundary. The 

 form is so simple that I venture to reproduce it here. If the origin be at 

 the centre, and if K : K' be the ratio of the sides, the equation to the path 

 of a vortex is ctn2(Ka;/a., Zc) + ctji^(K'y jb, h') = const, whilst, if the vortex 

 is at the centre, the stream-lines are cn(K,i;/a, h) cn(K'7//6, Jc') = tanh>^/m. 



The Circle. — This naturally was amongst one of the first boundaries 

 for which the velocity potential was fonnd. It was given by Stokes* in 

 1843 as a particular case of the general motion of the surfaces of two 

 concentric circles, and he showed that the mass of the equivalent solid 

 was equal to that of the fluid displaced. A very full discussion of the 

 motion of the particles of the fluid, by Clerk-Maxwell,^ will be found in 

 the ' Proceedings ' of the Mathematical Society, a discussion which is ren- 

 dered all the more insti'uctive from the figures which accompany it. Both 

 these papers treat of non-cyclic motion ; but the space in two dimensions 

 about a circle being cyclic, admits of a many-valued velocity potential. 

 Rayleigh ^ and Greenhill ^ have shown that when there is a cyclic motion 

 about the circle it will itself move in a circle in the same direction as that 

 motion, whilst the latter has shown that if it also moves under the action 

 of gravity it will describe a trochoid. The image of a source in a circle 

 has long been known, and that of a vortex is a natural corollary. 



Two Circles. — This is another boundary for which we owe the first 

 discussion to Stokes^ (1843). He considered them concentric, with any 

 general motion of the points of the surface, and in particular for the 



' ' On velocity and electric potentials between parallel planes,' Quart. Jour, xv, 

 p. 274. 



^ ' Ueber einige Aufgabeu, welcbe die Tbeoric des logarithmisclien Potentials und 

 den Durchgang eines constanten elektrisclien Stroms durcb eine Ebene betreflfen.' 

 Schlbm Z. X. jDp. 48 and 89. Tbey are also given in my paper referred to above, and 

 a particular case when the source and sink bisect opposite sides of the rectangle by 

 Betti. Sopra la diHtrihuziune (Idle corrcnti elettriche in mio lustra rettangolare. 

 N. Cim. (2) iii. (1870) ; also Heine (1874), ' Ueber die constante electrische Stromung 

 in ebenen Flatten,' Bwch. Ixxix. p. 1, and Berl. Mcniats. (1874) p. 186. 



* ' On plane vortex motion,' Quart. Jour. xv. p. 10 ; also ' Solution by means of 

 elliptic functions of some problems in the conduction of electricity and heat in plane 

 figures. Ibid. xvii. p. 284. 



■* ' On some cases of fluid motion,' Camb. Phil. Trans, viii. p. 105 (184.3). 



° ' On the displacement in a case of fluid motion,' Proc. Math. Soc. iii. p. 82 

 (1870). 



" ' On the irregular flight of a tennis ball,' Mess. Math. vii. p. 14 (1877). By 

 some oversight the circle is made to move in the wrong direction. 



' ' Notes on hydrodynamics,' Mess, Math. vs.. p. 11.3, 



8 ' On some cases,' &c. 



