whence 



Equilibrium of Rotating Fluid, 

 k— tan -1 Is 



K = 



A 3 



675 



or 



^^3 + F 3_ 



which is the familiar result for the spheroids*. 



Incidentally we see that we may re-write equations (A) 

 in the, for subsequent purposes, rather more convenient 

 fonn 



■l 



rP 1 ^r = Ka 1 , 

 -l 



£ 



\ T P 2 log[r 2 (l + ^ 2 )]^ = K «2+(l + ^ 2 )(K,-K), 

 and, n > 2, 



%j —1 



where K is the value of K for the spheroid. 



3. To determine points of bifurcation on the series of 

 spheroids we need retain only the first powers of a lf a 2 , &c. 

 in equations (B), which we shall suppose, for the moment, 

 to be denoted by 



Ka n =f n (a L , a 2 , a z ) [n=l, 2, 



Points of bifurcation will then be determined by the equation 



(B) 





-K 



cM 2 



C^2 

 B«2 



K ^ 



~da 3 



C^3 



d«3 



= 0. (5) 



Putting a =l> as we ma 7 w ^ tn 110 l° ss °f generality, re- 

 taining only the first powers of a u a 2 ... , and denoting 



P n /(1 + *V)*» 



* If k= tana, the eccentricity of the spheroid is sin a. 



