89-91] Stability 89 



so that 



The constant term \s in this value of Q is not necessary to satisfy the 

 conditions of equilibrium, but is introduced in order to keep the total 

 volume of the distorted surface equal to that of the original ellipsoid. 



The potential of the figure determined by equation (234) can be written 

 down by the method already explained. We have for the second order 

 terms 



If we now put 



(fig i 







in which the coefficients c n , ... are determined by comparison with equation 

 (237), then the potential at the boundary of the figure (234) will, as far as e 2 , 

 be given by 



Trpabc 



+ ex 



+ d. 2 f +4^ + 4] ......... (239). 



The value of &> 2 in this configuration is not necessarily the same as in 

 the ellipsoidal figure, although it must obviously differ only by terms in e 2 . 

 Let us assume the new value to be < 2 + e 2 So> 2 , where the first term refers to 

 the value of o> 2 in the ellipsoidal configuration. Then the equation of equi- 

 librium becomes 



V b + i (a) 2 + e 2 So) 2 ) (a* + f) = - -rrpabc 6 ( - z + + -' - 1 + eP + e 2 Q ) (240) 



\(Z C / 



and this must be satisfied for all values of as, y, z and for all values of e*. 



Equating terms independent of e we obtain merely equations (65) (67) 

 of | 36. These are of course simply the equations which determine the 



* We might have obtained an appearance of greater generality by replacing 6 in this equation 

 by an expression of the form 6 + ed' + e 2 0", but it would have been only an appearance. On 

 equating coefficients we should have immediately been forced to put 0' = 0, and the generality 

 introduced by the undetermined 0" adds nothing to that already involved in the presence of the 

 coefficients j>, q, r and s. 



