THE FIGUKES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 55 



It will be noticed that with the value we have assumed for <a, the value of u 2 

 becomes a function of (-, q, , of degree 4 involving only even powers of , ;, f ; its 

 value is 



rf) + s. . . (95) 



The values of fw and of f*w' are easily seen to be similar in form, so that all the 

 second-order terms in $ are of this form (cf. equation (50)). Multiplying by i/r (\) dX 

 and integrating from to o>, we obtain as the terms of the second order in the 

 potential an expression of the form 



- Trpctbce 2 (c u o; 4 + c 22 ?/ 4 + o^z 4 + c vf x V + c^/Y + o 31 2 V + c^.K 2 + c^/y 2 + dj? + d 4 ). 



If this figure can be a figure of equilibrium at all, it will be for a rotation differing 

 only by a second-order quantity from that of the original ellipsoid. Let us suppose 



that for it - 5 = >i + e 2 it," ; then at the boundary, as far as e 2 , 

 2-irpabc 



+ e 2 (c n x l + c 2 . 2 y* + c.^ + c v jL ? if + f^/V + c m z 2 x 2 + <l } .i- 2 + d.jf + c7 ;( 2 2 + d t 

 + third-degree terms in c, the same as before} ...... (96) 



At the boundary, 



so that for the figure to be one of equilibrium, the right-hand member of equation (96) 

 must be identical with 



f a Ji __a / 2 _2 2 



I />* ?/ & I 'Y 1 1 V 



1 /\ \ **' i / i 1 / "^ i O / ft 



1 Cf'^ \)^ C \ CJ** Cl'^^ (t (*^ ft 



a 



. . (98) 



