THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 51 



The distorted ellipsoid will be a possible figure of equilibrium, as far as the 

 first order of small quantities, provided the right-hand member of equation (68) is 

 identical with 



- + * + r + . . . (69) 



Equating coefficients, we obtain 







72. J c = 1' ...... ( 70 ) 



C 



(*+*>-*-*- ........ (") 



(74) 



and from these equations, together with equation (G5), we must eliminate or obtain 

 the coefficients. 

 If we put 



a / ft of 7 ' (>7-\ 



c? =a ' &"* ? = y ....... (7j) 



then equations (71) to (73) reduce to 



a'(<yH*)-|8V-'/e.= ^?', ......... (76) 



(78) 



Cv C 



whilst on substitution for K from equation (65) and J A from (70), equation (74) 



reduces to 



(79) 



We have now to deal with four equations (76), (77), (78), and (79), and have 

 to examine whether, and how, these can be satisfied by values of a, /3', and y . 

 Since there must, from general physical principles, be an equation of some sort for 

 points of bifurcation (whether capable of being satisfied by real values or not), we 

 are led to suspect that these four equations are not really independent. 



H 2 



