498 MR. J. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
5th. Small terms arising from C n containing cos n6. These are 
u=0 , v=0 
w — 2 ^} —P cos (@~ x P))( <x h cos “1 a « cos cos m (^~ x ! J ) 
pCtGCyi I 
'dA n+1 
(LL-A 
2 l 
v da 
1 da / 
cos nxjj 
^dA n , 
<4 _ 1 
fdA 
=i™“«r‘ da-^ a \~Ta 
n+ -\ | dA n _-^ 
da 
■ cos mjj. 
Collecting the terms we find 
u 
— i 
i ma ] £Aj cos xjj -f in—l )A M+2 cos (n+l)A^_ 1 cos (n— l)xp) 
+ 
^n/3 n A n -^(cos (n— l)i/»— cos (n-}-l)t//)J . (4) 
^ 2 ma y=^i sin t//+ \fi n ({n 1)A W+ ^ sin sin (n —1)^) 
— \nfi n k n fL{sm (w+1 )i/r+ sin (n— l)t//)j . (5) 
(Ji 
ima\ 2aA 0 —pA 1 + ( 2 a / ,A«+ia„ ; /J ((? 2 ——(n+l)A M+1 )^) cos mp 
w= 
-\-a, 
dA r , 
a- 
da 
/ dA n+l 
dA w _A 
\ da 
h da / 
cos 
#} • ( 6 ) 
Fig. 1. 
Let the figure represent a section of the vortex ring by a plane through its straight 
axis. Let </> be the angle which the radius vector drawn from C the centre of the 
section of the core to any point P on the surface of the ring makes with the straight 
axis of the ring. Let CP=e. 
Then the equations to the surface of the core are 
p=a+2oG cos mp-\-e sin </> 
z=l -\-t/3 u cos cos <j> 
