20 
MR. J. El. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
On substituting the values for ffi, tL and tl 3 given by equations (62)-(64), and 
equating coefficients of p, CJ, V in equation (89) to zero, we obtain 
33-fci + T 
3/l’j + A T /l ‘ o 
= 0 . 
3 lc\ + /do + Jc' s 
i e 
3k" 1 + k" 2 + k" 3 = *-f- 4 . 
ere 
With the help of these relations equation (88) reduces quite simply to 
In this equation the coefficients of and 1U depend only on a, b, c the semi-axes 
of the Jacobian ellipsoid, and so are fully known. The quantities W, W however 
depend on the second-order coefficients L, M, N, ... p, q, r, s. These were calculated 
in the previous paper, but p, q, r, s could not be fully determined, since they were 
found to depend on a quantity n", which measured the change in angular velocity 
This it was found impossible to evaluate so long as the investigation was confined 
to second-order terms. It now appears that equation (90) is in effect an equation 
determining n". The equation is linear in n", so that it gives only one value for n". 
When n" has this value we are on the true linear series, but if n" has any other 
value our solution, when we try to extend it to third-order terms, degenerates into a 
solution of the type of (85), with which no progress can be made. Our plan, then, 
is to evaluate the terms which occur in equation (90) and so obtain the value of n". 
On inserting this into the values of p, q, r, s which were obtained in the previous 
paper, we complete the solution as far as the second-order terms, and can then proceed 
to the stability criterion. 
Numerical Computations. 
17. It is at once apparent that the evaluation of C u , C 12 , ... ffi, ... , given by 
equations (58)—(67), can be made to depend on integrals of the same type as occurred 
in the previous paper, namely integrals defined by 
T _ f d\ 
' ABC - Jo A ABC...' 
