176 
MR. J. H. JEANS ON THE CONFIGURATIONS 
may be referred to as “Approximation A.” Inserting numerical values, this stands 
as follows :— 
L = M = n = 1*0273 (y-2)-l*0273 
l = m = 0-3488 (y-2)-0'2467 
N = 0-11845 (y-2)-0*0626 
>(Approximation A). 
(80) 
It will be observed that the error in Approximation A is greatest when y = 2, 
when it is of the order of 2 per cent, of the whole value. 
19. Pear-Shaped Point of Bifurcation .—We proceed now to evaluate the exact 
solution for the configuration at which the Jacobian ellipsoid gives place to the 
pear-shaped figure. At this point we have 
a = 1-88583, b = 0'814975, c = 0-650659, 
2 
n = ——= 0'141999, 9 = 0-413607, 
2 t rp„ 
the lengths again being chosen so that abc = 1. The values of the integrals 
necessary for the evaluation of the potentials have been given in previous papers.* 
The values of L', M', N', l', m', n' are now distinct, and are determined by six 
equations of the type (cf equations (78)) 
4c u -e(^) = |J AA , &c. 
4c 23 —9 J - J BC , &c. 
(81) 
in which the coefficients c n , c 22 , c 33 , c 12 , c 23 and c 31 are now distinct, the condition that 
the potential shall be harmonic being expressed by three linear relations connecting 
them. The values of these six potential coefficients liave been calculated in my 
previous paper; inserting these and solving the six equations (81) I find the set of 
values given in the second column of the following table, the values ‘ given by 
“ Approximation A ” being given in the adjacent column for comparison 
Coefficient. 
Exact solution. 
Approximation A. 
Difference per cent. 
L' 
- 15-4353 
- 10-1768 
-34 
M' 
0-15560 
0-15329 
- 1 
N' 
0-04733 
0-04677 
- 1 
V 
- 0-08468 
- 0-08424 
0 
m’ 
- 0-49850 
- 0-62860 
+ 26 
n 
- 0-95103 
- 1-18103 
+ 24 
* ‘Phil. Trans.,’ A, vol. 215, p. 61, and vol. 217, p. 21. 
