THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 49 



while the additional term which has to be inserted on the right of equation (58) is 



Hence the additional equation which has to be satisfied, in addition to equations 

 (59) to (61), is 



A AT>p = ^575~2 ( 62 ) 



Jo ii -rt-Dv-' Ct C 



Eliminating from this equation and (61) we obtain, as an equation which must 

 be satisfied if the distorted ellipsoid is to be a figure of equilibrium, 



= (63) 



This obviously cannot be satisfied, for the integrand is positive for all values 

 of A. We conclude that the distortion now under consideration cannot possibly give 

 rise to a figure of equilibrium. 



21. There remain nine terms for consideration in the general cubic function. 

 Inspection will show, or it will soon become apparent as we proceed with the analysis, 

 that these fall into three groxips, as in 16, and that the three terms of any one 

 group just suffice to give a possible figure of equilibrium when combined with a 

 term to restore the centre of gravity to its position on the axis of rotation. We 

 shall accordingly consider a distortion in which the cubic terms are those already 

 written down in equation (56). These terms are seen (18) to move the centre 

 of gravity parallel to the axis of x, and to correct this we shall add a term (</. 16), 



KX , 



-T- tO tt 1 . 



Thus, for the distorted ellipsoid now under consideration, the boundary will be 



999 / 't O 9 



x 2 if , z 2 .. / x ] O xir xz" x \ ,.. N 



~+ih,H s 1 +e a +p , +y . +K -} (64) 



a b c \ a ao arc, a?/ 



As far as terms in , the value of the potential at infinity is (cf. 18) 



2 a_ 

 5 a 2 



so that for the centre of gravity to remain at the origin we must have 



(65) 



\rOL. CCXV. A. H 



