510 
ME. J. J. THOMSON ON THE VIBRATIONS ON A VORTEX RING, 
This expression vanishes when t— — oo ; it begins by being positive but soon changes 
sign; it is negative when t =0 and remains negative for all greater values of t. 
The last terms are the only ones that do not vanish when t— oo. Putting t=co we 
find 
A= 
m'a' z a 
4 c : V sm j>e 
r (sin Je+ sin fe) 
m'a' z a 
chi 
cos 2 Je 
Now (3Ja is the angle through which the plane of the vortex is turned, and since 
the wort ex moves at right angles to its plane this will be the angle through which the 
direction of motion of the vortex ring is turned. 
Since (3 / is negative, the part of the vortex ring C D where cos 0 is positive is 
tilted backwards; now as we have taken it, cos 6 is positive for the upper part of the 
vortex, hence this part of the vortex is tilted backwards, and the normal to its plane, 
which is the direction in which the vortex ring moves, is bent towards r pp, the 
direction of motion of the vortex ring A B through an angle whose circular measure 
m'cfi 
vc' 
A* 
Thus the deflection, other things being the same, varies inversely as the cube of 
the least distance between the vortices. 
Let us now consider the effect of the vortex ring C D on the vortex A B. Let us 
take the perpendicular to the plane of C D as the new axis of z, the perpendicular to 
this drawn upwards in the plane of the paper as the new axis of x. The work we 
went through before consisted in finding expressions for the velocities along the axes 
of coordinates due to one vortex at a point on the other in terms of f, h, l, m, n, and 
then finding the velocities perpendicular to the plane of the vortex and along its 
radius vector in terms of the time by substituting from the equations 
/= 
h= 
■c sin \e—v sin e.t 
c cos -Je— v(l — cos e)t 
} 
1= cos e cos 6 
m= — sin 6 
n— sin e sin 6 
velocity perpendicular to the plane of the vortex —y cos e—a sin e, velocity along the 
radius vector —a. cos € cos 9—fi sin 9-\-y sin e cos 6. 
Now in finding the effect of the vortex ring C D on the vortex A B, the general 
expressions giving the velocities will be the same as before, as we have taken corre- 
