THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 39 



(32 32 ^2 \ 



5 a + 5-5 + 5-5 ) , which in the 

 dx* 3?^ 3z J A = co,, s . 



new variables becomes 



_LJH JL^., JLJ!! 



A a af B 2 av cw ' 



and this will be denoted by V 2 f)|f . 



Equations (32) and (35) in the new co-ordinates, reduce to 



/ -\ 

 ' ..... (37) 



(38) 



We are assuming that f+<f> = when X has the value appropriate to the values of 

 >i, so that in the first equation /"may be replaced by -- </>, but in the second 

 equation this may not be done. 



13. It is convenient, for the purposes of the present paper, to suppose the distortion 

 to start from the undistorted ellipsoid, and to proceed in powers of a. parameter e. 

 Thus we assume 



u = 



and in the equation (37) since f+ (u+fn) = 0, it is clear that, when e is small, f will bo 



a small quantity of the smallness of e. As far as e, equation (37) reduces to ~ = 0, 



CA 



giving ,. Equation (38) then gives r, ; equation (37) taken as far as e~ will then 

 give u 2 , and (38) will give r a ; (37) taken as far as e :i will give 3 , (38) will give r. ( 

 and so on. 



2 



As far as e only, equation (37) reduces to ^- L = 0, of which the solution is 



(j\ 



u i = X (f> '/> f) where x is the most general function of f, >i, and f. At the boundary 

 reduces to (wJ A = or k>x ( , K, J, so that the generality of the function x enables 



\\Jv D G I 



us to deal with the most general small displacement possible. 



At present we shall consider only solutions for which x is algebraic and of degree 

 not greater than 3 in f, >;, f. For these solutions equation (38) shows that i\ will 

 be of degree not greater than 1 in ^, f, sa that V J Vj = 0, and the equation reduces 

 to 



A^I _ v ( l ^ i a 2 i o 2 



4 ' f " tWl= " ' 8 



