500 MR. J. J. THOMSON ON THE VIBRATIONS OP A VORTEX RING, 
If we equate tlie value of w from equation (6) to that given by equation (7) 
we get 
\ma ^2aA 0 —pA x + a„[(n —1)A w _ x — (n +1)A W+1 ] cos nxjj 
“b i 22ja« 
cos —e sin<£.<£> 
adA n j 
i — 1 
+ 
4 
V 
dA„_ 1 \ 
da 
v da 
da j 
cos mfj 
Since tbe term 2aA 0 — pA l is not multiplied by any small quantity we cannot 
suppose the A’s to have the same value as for the undisturbed vortex, we must 
substitute for 2aA 0 —pA, 
cl 
2a^t 0 — a^—a n cos nxjj^ + a tt cos w//—(2a^ 0 —ag^) —e sin 
Since the other terms are multiplied by small quantities, we may substitute for the 
A’s their undisturbed values. Making these substitutions we get 
r . r ci 
ma j 2ct ( @L Q —aWi 1 —e sin + ta a cos nxp —(2 
adMn a[ dg{ n+ i . d&n-l 
da , 
d~2H w + \{{ n l)^«-i (^d-l)^«+i)4 2 ^ da 
=%—e sin </><£ -\-'Z/3 u cos mp 
Equating constant terms and the coefficients of sin <f> and cos nxp, we get 
ima 2 (2^ 0 -a i ) = j.(11) 
d 
i maa ^[j p ( 2a ^o— c ^i)+ 2 ^k—(n+l)8L n+l 
d%X n 
fi— ct' 
da 
f d^i nJr Y 
■ dM n A 
V da 
h da j 
= Pn 
( 12 ) 
The first of these equations gives the velocity of translation of an undisturbed 
circular vortex ring, the second is the same as the one we previously obtained for <E>. 
We must now proceed to find the values of the ift's supposing the transverse 
section of the vortex core to be small compared with its aperture. 
Since 
cos (6—\fj)+ . . . ^t»cos n(6—\p) 
{cd -\- p + %~ — 2po cos (0 — ^)} 
7rj 
cos nx-dx 
(cd + p° +£ 3 —2pa Q,o% xf 
