154 
MR. L. N. G. FILO^v ON THE ELASTIC EQUILIBRIUM OF 
Consider now the equation 
+ Wfy = 0. 
Let be any solution of the equation 
Then if y., be a solution of 
it is also a solution of 
(c;. q_ !)■:) y ^ 0. 
(.92 + 1 = 0. 
Now if ijy — IqZi what is the condition that w'e can obtain a second product solution 
y., = ILZo ? 
We have, substituting 
dTL lyz, _ IQ Zj 
R, Z, “Ib'z, 
(llX 
or function of r only + function of z only =: product function in r and 
Tf ( 11 ) is to be identically satisfied, this })roduct function must be a function of r 
only or of i only. 
Case (i.). Zj = aZ^ where a is a constant. 
We find 
or, since IQ is a solution of 
^ IQ 
" ~ a '■ 
(d' — C') IQ = 0 
( 12 ), 
( 13 ), 
which is the same as (*J), IQ is a solution of 
{.P - /f-QUQ = 0, 
which is not at the same time a solution of (13). 
Now the solutions of tliis eipiation are 
dh 
But 
and similarly 
dl; 
K, (kr) = rK„ (kv) - f K, (kr ). 
dk 
The four independent integrals are therefore 
Ki(^''’), ^‘lu(^’^’)= ’’Ko(^’'’)> 
and therefore the required values of IQ are rQ,, rK^,. 
