90 Pear-shaped Configurations of Equilibrium [CH. v 



series of Jacobian ellipsoids. Equating terms in e we obtain again equations 

 (216) which determine the position of the point of bifurcation and the ratio 

 of the coefficients a, /3, 7 and K. On finally equating terms in e 2 we obtain 

 the system of equations : 



, OL , BM < 6N 



(241), 



, C 12 = J Y~A 



C 4 a 4 a 4 o 4 J 



v_ 



be 2 a 4 



! /) 



(242). 



92. In starting computations, we may first determine the ratios a:/3:y:ic 

 from equations (218) (221), which are equivalent to equations (216). Assign- 

 ing to a the arbitrary value a = a 2 , the values of a, ft, 7, K are found to be 



a = - 3-556343, /3 = 0*204689, 7 = 0'0679189, K = 0-506278 . . .(243). 



The potential coefficients c u , c 12 , ... may now be evaluated in equation 

 (238). ' These coefficients cannot be completely determined, but they reduce 

 to linear functions of the still unknown coefficients L, M, N, I, m, n, so that 

 equations (241) become a series of six simultaneous linear equations in the 

 six variables L, M, N, I, m, n. 



Solving these equations, the values of these six coefficients are found 

 to be* 



L = - 11-71505, M = - 0-00583504, N = - 0'000808592 } 



I 



Z =-0-00214448, m = 0-232659, n =0'653198 J 



The values of d l} d z and d 3 may now be evaluated from equation (238), 

 and expressed as linear functions of p, q and r. Equations (242) now become 

 three linear equations connecting the three variables p, q, r, and on solving 

 these, we find 



(245), 



-0-044997 - 7-83600 - ............. (246), 



r = - 0-0140132 - -1647351 ..... (247). 



2-7373 



* Details of this computation and of the checks on its accuracy will be found in a paper 

 already referred to. Phil Trans. 215 A (1915), p. 27. 



