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IX. On the Vibrations of a Vortex Ring, and the Action upon each other of Two Vortices 
in a Perfect Fluid. 
By J. J. Thomson, B.A., Fellow of Trinity College, Cambridge. 
Communicated by Lord Rayleigh, F.R.S. 
Received November 16,—Read December 8, 1881. # 
The following paper contains (1) a discussion of the vibrations which take place in the 
axis of the core of a vortex ring whose section is very small in comparison with its 
aperture when the axis is made to deviate slightly from the circular form; and (2) a 
discussion of the action upon each other of two vortex rings which move in such a 
way that they never approach nearer than a large multiple of the diameter of either. 
The fluid in which these vortices exist is supposed to be frictionless and incom¬ 
pressible. 
The method which I have employed is the same in both cases, and is purely kine- 
matical. It is merely the application of the fact that if V(x, y, z, t) = 0 be any 
equation to a surface which always consists of the same particles then 
dF 
dt 
-j —u 
dV dV , d¥ n 
dx +V d V + W d^-° 
where u, v, w are the velocities of the particle at (x, y, z ) along the axis of x, y, z 
respectively, and where the differential coefficients are partial. 
The surface of a vortex ring is evidently a surface of this kind, and the equation 
just written is the condition that F (x, y, z, t) = 0 should be the equation to the surface 
of a vortex ring. I have found that this condition, joined to the ordinary expressions 
for the velocity due to a vortex element, is sufficient to solve the problems discussed 
in this paper. This is an instance of the large number of problems in vortex motion 
which are capable of purely kinematical solution; indeed, a vortex theory of gases 
would be entirely kinematical so long as we only considered the molecules of gas 
themselves and not their effects upon the containing vessel, &c. For example, in this 
theory when two atoms clash, the problem of finding their subsequent motion must be 
capable of solution by purely kinematical considerations, but in the ordinary theory of 
* Since the paper was sent into the Society it has been copied by the author with changes in the 
notation, introduced chiefly to facilitate the printing, but no change of any importance has been intro¬ 
duced into the substance of the paper. 
