MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
supposed to be the pear-shaped figure) is taken to be the surface X = 0 in the more 
general family, 
2 2 2 
.( 2 ) 
f + <p — — -— + 7 c/ J- 2 + 0 ” 1 — 0. 
a+X 6 2 +X c +X 
Here <p is a function of x, y , 2 , and X, different values for 0 representing different 
distortions of the fundamental ellipsoid. 
If V f , V 0 are the internal and external potentials at x, y, z (or their analytical 
continuations as explained in the previous paper, § 3), we have seen that V is Vy must 
be of the form 
— yp- (x) (f+ <p) dX, 
(3) 
where 
= 
V — [ ^ 0) {./+$) ^.(4) 
J v 
7 rpCtbc _ 7 Tpctbc 
{(« 2 +x)(6 3 + x)(c 2 +x)} 1/2 
A 
• (5) 
In these integrations, x, y , 2 are treated as constants while X' is the value of X at 
the point x, y, z as determined by equation (2). Furthermore, 0 must not be selected 
at random ; equations (3) and (4) will only give the true potentials if <p is chosen so as 
to satisfy 
-(l + l)I> wv! ^ x+ ^ (x) [ 42 ito + 43 | + 2 (fe) 8 ] = a • ■ (6) 
This equation, as before, is most conveniently solved by a solution 
0 = u+fv, . . . 
in which u, v must satisfy the equations 
\p (X) V-(u+f v) d\ +4:\p (\)v = 0, . 
4 (1 +v) 
y x du du \ r(y x_dv dv 
'^Adx ax/ \ a dx ax 
( 7 ) 
( 8 ) 
(9) 
4. We attack the second equation first. As in the previous paper, let us introduce 
new co-ordinates £ »/, £ defined by 
. x x » 
b 2.x A 5 , 
a +X A 
and the equation is found to reduce to {cf. equation (37) of the previous paper) 
