10 Mr Greenhill, On the motion of a liquid ellipsoid [Oct. 25, 



Therefore, integrating, 



-£ + h (A'x 2 + B'y 2 + C'z 2 ) = constant, 



P 

 and therefore the surfaces of equal pressure are the similar and co- 

 axial quadrics 



A'x 2 + B'y 2 + C'z 2 = constant. 



(Putting | = 0, r)=0, f=0, we get the particular case con- 

 sidered, p. 240, Proceedings of the Cambridge Philosophical Society, 

 Vol. in.) 



If we can make A', B', C constant, and 

 A'a 2 =B'b 2 =C'c 2 , 



the surfaces of equal pressure are similar to the external surface. 



A free surface would therefore be possible, and the external 

 case might be removed. 



We should then have a liquid ellipsoid moving under the 

 gravitation of its parts, but not about a principal axis. 



Apparently however this is not possible, except for rotation 

 about a principal axis ; and supposing this the axis of z, we must 

 put £l 1 = 0, Xl 2 = 0, £ = 0, t] = ; and then we come to the cases 

 considered on p. 245, Proc. Cam. Phil. Soc. Vol. in., supposing 

 i2 3 = a> and £ = &/. 



Hyper-elliptic functions are required for the general solution of 

 equations (D) and (6r); but if the ellipsoid be of revolution, the 

 solution can be given by elliptic functions. 



For if a = b, the third equations of (D) and (G) both lead to 



§=^(fl 1? -n,„). 



We may put O s = 0, and then from (G) and (D) 

 d% 2a 2 



dt a 1 + c 

 drj_ 2a 2 

 dt~ a 2 + c s 



AS 





dn t a 2 + c 2 



W= n ^-a^?^' 



dt ~~ n ^~ a 2 -c 2 ^ 



