Equilibrium of Rotating Fluid. 683 



The coefficient of 6 in the angular momentum is thus 

 proportional to 



But K = (l-«c)c 2 ; 



rP 2 R 1 ^=i aC (l + 3 < j 2 ) 2 -|c 2 a + 3c 2 ); 



( 1 p 2 R 2 ^= 3 1 ^r«<-3 + 39c 2 + 375c 4 + 525c G ) 



- cos- a(3 + 39c 2 +- 125c 4 + 105c 6 ) -180c 4 -420c G ] . 



Therefore the coefficient of is proportional to 

 «c(l + c 2 ) (9 + 235c 2 + 875c 4 + 1225c 6 ) 

 - 119c 2 - 655c 4 - ^c 6 - 1225c 8 



=o, 



by equation (10) ; whence it follows that the angular 

 momentum is stationary when = 0. 



Added September 18th, 1914. 



It is but right to point out that while correcting the 

 proofs of this paper I have met with a difficulty which I am 

 at present unable to solve. 



We know that w, and therefore K, must be stationary 

 when = 0, so that K t in equation (11) must be zero. In 

 fact it is by varying the total energy, while w remains con- 

 stant, that Poincare derives the bifurcation equations. Or, 

 since the angular momentum and therefore the moment of 

 inertia are stationary for 0=0, we might have begun by 

 making the coefficient of 6 in the moment of inertia vanish. 

 This, since the volume is constant, would lead us to the 

 equation 



7a n = 6a 12 sin 2 a, 



and on substituting in (13), remembering that K^O. we 

 should again find the bifurcation equation, in the form given 

 in § 3a. But the vanishing of K x requires a certain relation 

 between the coefficients of a n , a 12 , &c. in equations (13), 

 (14), and (15), and this relation (which 1 had before 

 assumed to hold) I have been unable to obtain. 



2 Y2 



