8 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
where 
P ~ 3 2 
q = &' = W2 
fip 3 ) _ p J-L {ip 2 ) 
3 “ (iP) - p AA GP) 
s = 
1 
1 6 
0 6 
P = - 4 
a 4 
(IP 8 ) - p 
0 f 0 >/ 
0 6 
0 f 0 , 2 0 ^ 
0 4 
(IP 3 ) 
&c., 
^, a ,(PQ) -PdfeiQ 
Lcfdr 
0 fS, 
, &c., 
a = Rff, &c. 
Relation (37) can accordingly be put in the form 
v 2 (g 3 + nsO + 22 T P f || 2 + W 2 V 2 P = | [»VD» (£P 2 ) - -J-D 2 (PQ) + DR + 8P«'J. 
Equation (36), after transformation to £ >?, £ co-ordinates and integration with 
respect to A, gives 
w, = - i [*D» (i-p) - iD 2 (PQ) + DR] -«/„ P - i f/«4 V 2 P <i\.. . (38) 
Jo 
Let us now put 
fw' 2 V 2 PdA = £A,.. (39) 
Jo 
so that A will be a function of A only. We then have 
f/»' 2 V s P dx = =/fA-f(f 2 + , 2 +C) f AdX, . 
Jo Jo 0A Jo 
• (40) 
whence, on operating with V 2 , 
V 2 |/ W 'V 2 P d A=^|^ + | + 5 jj o A^|. . 
Equation (29), transformed to £ >/, f co-ordinates, now becomes 
= AffMA' s 
• (41) 
dw f , 
0A 
3 
- IV 2 w 9 
_ i 
— 32 
fe ® D» (JP) -i ^ D 2 (PQ) + ^ DR 
ax 
ax 
+ i J; (?N + *f |; {(| +1 +1 )1 A dx }- 
