4 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
5. We now return to equation (8). By a transformation given in the previous paper 
(p. 37), it is found that this is satisfied by taking 
and 
v = 0 when X = 0, 
V 2 u+fV 2 v + 4 
v X dv dv 
a dx ax. 
3 
a oar 
= A av- 
(14) 
(15) 
where <x may be any function of x, y, z and X, which vanishes when X = 0 and when 
X = Xf For equation (15) we may try provisionally a solution 
v = w -4- fw' + f 2 w" + ...-+- f n w n + ... 
where w, iv\ w" ... are quantities satisfying 
2 A ^ 
A ax' • 
/ 
[ 2 
X 
A 
dw 
dx 
dw\ 
dx/ ~ “ 
1 
X 
a w' 
dw'\ 
A 
dx 
ax/ 
/z 
X 
Sw. 4 
9w s \ 
\ 
A 
dx 
ax / 
n+ I 
&c. 
(16) 
(17) 
(18) 
(19) 
After a good deal of simplification, the left-hand member of equation (15) is found to 
reduce to 
V 2 u+fV 2 v + 4(z£~+|^)= -A# 
J \ A dx ax/ ax 
0 + j^(fw'+ 2f 2 w" + ... + nf n w n + ...) 
( 20 ) 
The quantity 6 is so far undetermined. Let it be given by 
0 = 
4 / u 
(w'+2fw"+...+nf n hv n +...), .(21) 
A \1 +v; 
so that, the quantity in the square bracket on the right of equation (20) becomes 
.( 22 ) 
i (/+ fid T"’ + 2 f w "+-"+ n f'~ x + ■ • •) 
When X = X', this vanishes through the factor f -\——. It will vanish when X = 0 
1 ~h V 
through the last factor if we make 
w' — w" = w'" — ... = 0, when X = 0. 
(23) 
If this last condition is satisfied, expression (22) satisfies completely the conditions 
which have to be satisfied by a in equation (15). Hence the value of v given by 
