1883.] Mr Hicks, On the motion of a mass of liquid, etc. 309 



(4) On the motion of a mass of liquid under its own attraction, 

 when the initial form is an ellipsoid. By W. M. Hicks, M.A. 



In three communications to this Society Mr Greenhill has fully 

 discussed the problem of the motion of a mass of liquid which 

 moves in space so as to keep its external form unaltered. His 

 method differs from that of Dirichlet, who was the first to consider 

 the question, in that he uses the Eulerian method in preference to 

 the Lagrangian employed by Dirichlet, Dedekind and Riemann, 

 and thus makes the theory simpler and clearer. I propose in the 

 present communication to shew that the same method is applicable, 

 with even greater simplicity, to the case where the mass of fluid 

 has no rotation, and is initially in the form of an ellipsoid. It is 

 known from Dirichlet's investigation that it will always keep the 

 ellipsoidal form, the axes continually altering their lengths, yet so 

 as to keep the volume constant. 



The velocity potential for the motion of the fluid inside an 

 ellipsoidal shell which changes its axes, without altering its volume, 

 has been given by Bjerknes. If a, b, c are the rates of change of 

 the axes a, b, c then 



*=i.( 



a o b „ 6 



- x 2 + T y 2 + - 

 a b J c 



• t d b c i . 



with - + f + - = (1). 



a b c v ' 



This is easily verified. The gravitation potential for points within 

 the ellipsoid may be written 



v= -k f n + - — £ - n - —I 



( a 8a b 8b c 8c y 

 where H 



Jo */{{a* + \)(b a + \)(c 2 +\)} t 



and k is constant when the volume is constant, as here. If we now 

 suppose the fluid enclosed in an ellipsoidal shell which changes its 

 form, the pressure at any point will be given by 



a of 



p p [\a a ) J - \ a 



p V a a 8a J \" b b 8b ) J \ 2 c c 8c J 



