44 Motion of a Liquid Ellipsoid under its oion Attraction. 

 — k* ( x cos Jet —y - sin kt ) + ( Ax cos kt — By- sin kt ) ~ = 0, 



— k*( x j sin kt +y cos AM + ( Ax j sin kt + By cos fa J -j— = 0, 



c^-^=o. 



c 

 Thus" 



p. const. +10^(1-5-^-^.) 



and substituting for cr, we obtain two equations, 



%Aa 2 -Cc 2 -£ 2 a 2 ) cos 2 kt+ (B£ 2 -Cc 2 -W) sin 2 fe] 



• + ^sin^cos^[Aa 2 -B^ 2 -A; 2 (a 2 -^ 2 )]=0, 

 and 



f| [(Aa 2 -Cc 2 -A 2 a 2 ) sin 2 kt+ (Bi 2 -Cc 2 - W) cos 2 &] 

 ^o 



+ *| sin fo cos kt [ Aa 2 - BZ> 2 - k\a* - i 2 ) ] = 0. 



which are both satisfied by 



„ = A^C f = B^-C f (1Q) 



These are the same conditions as those required for Jacobi's 

 ellipsoid. 



Another very simple proof can be derived by Greenhill's 

 method. Suppose the liquid occupying an ellipsoidal case to 

 be rotating, as if rigid, about the least principal axis with 

 angular velocity <w, and let an equal and opposite angular 

 velocity be imparted to the case. "We shall show that Dede- 

 kind's condition is the condition that the surface may be free. 



The velocity-potential of the motion set up by rotating the 

 case is a 2 — Z> 2 



Thus the velocity-components of the liquid are 

 a 2 -A 2 



a 2 -A 2 

 w= 0. 



