6 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
There are an infinite number of such sets of equations, of which we shall need only 
the set for w 3 , w' 3 , ... in addition to the two sets above. To simplify the equations as 
much as possible, let us limit ourselves to the type of distortion which leads to the 
pear-shaped series of figures of equilibrium. For this, as we saw in the previous 
paper, u x is of degree 3 in £ >?, f, so that w 1 must by equation (-24) be of degree unity, 
and from equation (25) w\ must vanish. Similarly u 2 is of degree 4, so that w 2 is of 
degree 2, w' 2 is of degree zero, and w" 2 = 0. Again u 3 will be of degree 5, ^v 3 o^ degree 
3, w\ of degree unity, w" 3 = 0 ; and so on. 
The set of equations for w 3 , v/ 3 ... now reduces to 
4 = — V 2 w 3 —4A ^(^- 2 
1 
4 2 
A dx + ax/ 
x dw'i dv/. 
A dx + 3x ' ~ 2 Ws 
w" = w"' = ... = 0 . 
(28) 
(29) 
7. Let us now introduce the operator D, already used in the previous paper (§ 14), 
defined by 
32 32 32 
(30) 
. r .51 
D “ + B a,f + c sf 
By differentiation with respect to A, we have 
0A A 2 0f + B 2 0, ? 2 C 2 0f 
so that 0D/0X is the same "as V 2 transformed into £ ??, £ co-ordinates. We can now 
solve equations (24) to (29) at once by transforming into £ co-ordinates. 
Equation (24) becomes in £ >?, £ co-ordinates 
. dw j 0D 0D jy 
4 = —yr u \— w- B, 
oX 0X 0X 
and since P is a function of £ f only, this has the integral 
Wl =-|DP .(31) 
No constant of integration must be added, for D vanishes when A = 0 and iv 1 must 
also be made to vanish when X = 0 (cf. equations (23)). Thus equation (31) gives 
the true value of w x and we have also seen above that w\ — w" 2 = ... = 0. 
The value of v x is accordingly 
v x = iv x = — ^DP = — t (APf|+BP^ + CP K ) 
(32) 
