RECENT PROGRESS IN HYDRODYNAMICS. 55 



ellipsoid. Clebsch was unacquainted with Green's work, and rediscovered 

 his values of the velocity-potential for a motion of translation. In addition 

 to the potentials for translation and rotation he gives the equations of the 

 lines of flow for translation, in terms of eUipsoidal co-ordinates, in the 



form log y = fV(^)<^^> ^liere X.fi.v are the ellipsoidal co-ordinates of a 



point (xAj.z), with a similar expression for log z, which is so related to 

 the former that yz may be expressed in terms of an elliptic integral of the 

 second kind. The case of rotation is a more complicated one, and was 

 only completed in a note^ to this paper. Here the co-ordinates fi.i' are 

 expressed as integrals of functions of X, and thus the solution is reduced 

 to a question of quadratures. All these simplify very much when tho 

 ellipsoids are spheroids. Another period of nearly twenty years followed, 

 until Bjerknes ^ completed the general solution by investigating the poten- 

 tials when the boundary itself changes its form, yet so as to remain ellip- 

 soidal. He considers the generalised problem of motion in space of ?* 

 dimensions, and extends the former results to this case. The second paper 

 is divided into two parts, the first devoted to the motion in the infinite 

 fluid outside the ellipsoid, the second to that inside. His results are 

 given below, for space of three dimensions. 



As the results are important, and extremely interesting, I have thought 

 it would be well to give a short notice of the results of the three foregoing 

 writers, expressed in a consistent notation. 



n O 2 



Let E = 1 - ~ — > so that Eq = o is the equation to the 



a^ + X b'^ + X c^ + X 



boundary at any time, the axes being a.h.c. Further, let D denote the 

 product ./[l + 4) (^ + p) (■'■■'■ ^) • '^^^^ ^^^ the velocity-potentials 

 can be expressed in terms of fl where 



B 



o = .|" 



dX 

 D 



viz., the constants A.B.C. being properly chosen, we have for translation 

 parallel to the axis (a) 



fl) =^ A '— (Green) ; 



dm 



for rotation about the axis (a) 



^^Bfy^-z^) (Clebsch); 



V"" dz dyj 



for variation of the axis (a) 



<6 = Ca— (Bjerknes). 



'^ da 



In the last case, if the axes vary so as to keep the volume constant, 

 then the sum of the C must vanish, whereas if they vary so that the 



' Crelle, liii. p. 287. 



2 ' Verallgemeinerung des Problems von dem ruhenden Ellipsoid, in emer 

 bewegten unendlichen Fliissigkeit,' G6tt. Nach. (1873) p. 448. 'Verallgemeinerung 

 des Problems von den Bewegungen welche in einer ruhenden, nnelastischen Fliissig- 

 keit die Bewegving eines Ellipsoids hervorbringt,' IMd, p. 829. 



