496 MR. j. J. THOMSON ON THE VIBRATIONS OF A VORTEX RINCf 
We can determine the components of the velocity in terms of the quantities we 
have denoted by A 0 . . . A*, we shall for the sake of clearness divide the determination 
up into several steps. 
To determine the parts of u, v, w independent of cl u and /3 n , in which we shall not 
suppose £ small, 
m 
'2 *r 
cos ^{Aq+Aj cos . . . A w cos n(9—\p)}adO 
cos 3 9 cos \JjdO=^mat,A 1 cos \}j 
m 
2 tt 
v= — j sin ^(A^-j-A, cos (6 —'/')+ • • • sin </i 
( 2 ) 
171/ - 
•=r— {cos 9(a cos 9—p cos i//)+ sin 9 (a sin 9—p sin i//)}(A 0 +A 1 cos 9 —.. )ad9 
Ztt j q 
=— (a —p cos (9— i//)}(A 0 -J-A 1 cos (9—1 //)+ • • • )dd 
Air Jn 
= 2^( A o a2v '- A -iP n ) 
=^ma{2A 0 a—A 1 p} ... (3) 
These are the velocities due to the undisturbed vortex, and in using them in the 
second half of the paper we require A 0 , A : determined without supposing £ to be 
small. 
2 nd. The values of u, v, w arising from small terms in els'. 
As far as now concerned, 
ds=d9a n cos n9 
H= 0, v=0 because they involve £a ;<4 
w— — \ (a+p cos (0— i//))[A 0 +A 1 cos (6— . * * ]u n cos nOdO 
'Zirj o 
=-^-\ \aA n cos n(9—\jj)—\pA n+1 cos n(9—\p)—^pA n ^ 1 cos n(9—\}j)~\ a* cos n9d9 
JiTTj q 
— -^17l(X } ^ClA n 2P(^' nJ rl~^~ l)U COS Tlxjj 
