STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
o 
If, theu, 
we must have 
(4>) = of + V sin 2 <j) + c' sin 4 </>+... , 
a' = \a, V — \b ~ , c’ — , &c. 
K 
Thus when f 0 , f 2 , jQ, &c., are found, it is easy to compute a, b, c, &c., and 
a', b', c', &c. 
Our formulae tend to involve the differences between large numbers, and this defect 
becomes more pronounced as the order of harmonics increases. The fault is mitigated 
by using the forms 
=f 0 sin 1 0—f 2 sin 1-2 0 cos 2 6+... , 
Ct (<£) = a' + b' sin 2 (j) + c r sin 4 <f>+ — 
In the case of a lower harmonic, however, such as the fourth, we may just as well 
use the form for ^ involving a, b, c, &c., and powers of cos 2 6. 
We must now show how to complete the evaluation of the f’s for the zonal 
harmonics. 
It appears, from p. 486 of “ Harmonics,” that, when % is even, we have to solve the 
equation 
q ij/3y{bj 3} {i, 4 } 
P 4.1 2 + /3o-~ 4 .2 2 +/So-—... 
ending with 
i 2 + /3o- 
where { i, j } = (i +j) (i-j +1). 
We are to take that root which vanishes when /3 vanishes. 
Although the equation for /3cr is of order i— 1, yet at least for such an ellipsoid as 
we have to deal with, it is very easy to solve it by successive rapid approximations. 
It is clear that we may write the equation in the form 
(M 2 + 
j/3 2 {i, 3} {i, 4} j(3 2 {i, 5} {i, 6} ~ 
4.2 2 + /3cr— 4.3 2 + /3cr— ... _ 
/3<x = iP 2 {i, 
An analytical approximation is found by neglecting the continued fraction in the 
second term on the left, and we then obtain 
/3cr = -2 + 24l+i i 8 2 (/-l)i(4l)(42)]. 
If this value of /3cr is used in computing the first term of the continued fraction, 
and if the quadratic is solved again, we obtain a closer approximation. We then use 
the second approximation and include one more term in the continued fraction, and 
proceed until ficr no longer changes. 
