508 
MR. J. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
hence '*= coefficient of cos nO in the expression for the velocity perpendicular to the 
plane of the vortex C I). To find /3 lt we must therefore express the velocity due to 
the vortex A B as a function of the time. In order to make the work as simple as 
possible we shall suppose that the vortices in their undisturbed states had equal 
strengths and radii. In the small terms which express the velocity at the vortex C D 
due to the vortex A B we may as a first approximation calculate the quantities on the 
supposition that the motion is undisturbed. In order to make the expressions as 
simple as possible, let us measure the time from the instant when the distance between 
the centres of the vortices has its least value (it is easy to see that this will be when 
the line joining the centres of the vortices is parallel to the line bisecting the angle 
between their directions of motion), then the square of the distance between their 
centres will be expressible in the form c 2 +/3t 2 when c is the least distance between 
the centres, t the time that has elapsed since the centres were this distance apart; let 
v be the velocity of translation of either vortex when undisturbed, then 
f= — c sin Je —v sin e.t 
h= c cos — cos e)t 
Therefore 
f 2 -\-h 2 =c 2 -\-Av 2 sin 2 t 2 
Making these substitutions we find that the velocity perpendicular to the plane of 
the vortex C D 
i 
m'a'° 
2 /„ 3 
(c~ + 4A sin 2 ^ei 3 )’ 
f {-|c 2 (3+ cos e) + 2v 2 sin 2 ^e(l — 3 cos e)£ 2 } 
rnfa'-a 
+ eos () t( ( .wi^5 A,! +«+ Cf + D i 
-j- cos 20f 
/ /-■> o 
m a ~a~ 
(c 2 + 4r 2 sin 2 t 2 ) 
rfA'^+B'^+OT+m+E'} 
where 
A=8# sin 3 fe(cos fe—5 cos fe) 
B = cv 2 sin 2 fe(l5 sin fe—sin fe) 
C =c 2 v sin fe(15 cos fe-}- cos fe) 
D= — fc :3 (5 sin fe-f- sin fe) 
A'= — 10y 4 sin 4 fe(2 cos 3e+- 8 - cos 2e+7 cos e + 4 8 -) "" 
B'= 10 cv 3 sin 3 fe(3 sin e—f sin 2e) 
Cf= — 10 c 2 v 2 sin 2 f e cos 3e 
D'= —§c s v sin e(!4 sin e+ sin 2e) 
E'= -fc 4 sin 2 ie(l 1 + cos e-fi 8 cos 2e) 
(27) 
