Motion of a Liquid Ellipsoid under its own Attraction. 41 



where the nine quantities x l} yi,... z% are functions of the 

 time, and initially x 1 = a , y 2 = b , z z — c Q , and the rest are zero. 

 Then Dedekind's theorem states that another motion of the 

 ellipsoid will be expressed by the equations 



a b c 



a o c 



(2) 



l/o 



z=zi- +z 2 t + z 3 - 



' J 



The simplest proof of this is that given by Brioschi, and I 

 here give a version of it. 



The kinetic energy of the liquid is T, where 



2T=.p$$$(P+y 2 + z 2 )dV; .... (3) 



the integration extending through the volume, and p being 

 the density. In virtue of the equation of continuity, which is 



2T becomes 



4:7ra Q b Q c 



x x y x z 1 



x 2 y 2 z 2 



&* yz z s 



= a b c , 



(4) 



(x 2 + hi + x? + y x 2 +y 2 +y 3 2 + z? + z 2 2 + i 3 2 ). (5) 

 Again, the potential energy due to the attraction of the ellip- 



15 



soid on itself is 



. 4:ira b c 



15 



where 



X = 27rypa 



Jo v (a 2 + ' 



dyjr 



. (6) 



f){b 2 + ir){c 2 + ^) 



and 7 is the constant of gravitation. 



We have to express % in terms of the x 1} yi, . . . z 3 . 

 Let the free surface at any instant be 



(a',b',c>,f,g>,h>Xxyzy = l, 



the equation giving a 2 , b 2 , c 2 is the discriminating cubic 



( i+ S( l+ p)( l+ ?) =1+ ^ +&/+c ^ 



+ (A' + B' + C'>P + A^ 3 = 0, 



where A', B', C are the minors of a f , b', d in the discriminant 

 A' of {a'Vc'f'g'h'Jxyz) 2 . 



