176 Compressible and Non-Homogeneous Masses [CH. vu 



figures. There are points of bifurcation on the sequence of compressible 

 figures; at the first of these the pseudo-spheroidal shape gives place to a 

 pseudo-elliptical shape, and at x the second the pseudo-elliptical shape gives 

 place to a pear shape. The configurations at these points of bifurcation are 

 arrived at by distortion from the corresponding configurations for the in- 

 compressible mass, and it is these configurations for which I have calculated 

 the exact values of L, M, N, I, m, n. 



At the first point of bifurcation 



a = 6= 11972, c = "69766, ;^-= '18712, = -47125, 



the scale of length, which is at our disposal, being chosen so as to make 

 r = (abc)^ = 1. The exact solution is found, by direct solution of equations 

 (468) to be 



L = M = n = 1-0273 (7-2)-T0466 



l = m = 0-3488 (7- 2) -0'23784 

 ^=0-11845 (7 - 2) - 0-06328, 



while the approximate solution (470) is found to be 

 L = M=n = 1-0273 (7 -2)- 1-0273^1 



(exact) ......... (471), 



( 7 - 2)- 0*2467 Happrox. A) ...(472). 

 ^=011845 (7 - 2) - 0626 j 



It will be seen that the error is of the order of two per cent, in the 

 terms which do not involve 7. 



175. At the second point of bifurcation, at which the pseudo-ellipsoid 

 gives place to the pear-shaped figure, I find for the exact solution 



L = 6-3238 (7-2)- 15-4353 

 M= 0-22057 (7 -2)- 0-15560 



^ = 0-08962(7-2)- 0-04733 | 



I =0-14059(7-2)- 0-08468 

 m = 0-75280 (7 - 2) - 0*49850 

 n =1-18103(7-2)- 0*95103; 



while the approximate solution is found to be 



L =6-3238 (7 -2) -10-1768 

 M =0-22057 (7-2)- 0-15329 



^ = 0-08962(7-2)- 0*04677 ^ (474) 



L =0-14059(7-2)- 0*08424 

 m = 0*75280 (7 - 2) - 0*62860 

 n =1-18103(7-2)- 1*18103; 



