494 
ME. J. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
gases the “ clash of atoms” involves dynamical considerations of very considerable 
complexity. This is a consequence of the vortex theory being of a much more funda¬ 
mental character than the ordinary one that atoms consist of small pieces of solid 
matter. 
Problem I. To find the vibrations of the circular axis of a vortex ring. 
Using cylindrical coordinates let the equations to the axis of the core be 
p=a-\-ta' )i cos nQ 
z= cos nd 
where a n and j3 n are small compared with a the radius of the core when undisturbed ; 
the summation over all integer values of n between zero and infinity. The axis of z is 
perpendicular to the plane of the vortex, and 0 is measured from the axis of x as 
initial line. 
The velocity due to a distribution of vortices is proportional to the magnetic force 
produced by a system of currents arranged in exactly the same way as the vortices 
and of the same strength. 
Now the vortex filaments we are considering are distributed uniformly (or very 
approximately so)'* in a ring the radius of whose transverse section is very small in 
comparison with the radius of the aperture. Now if electric currents flow uniformly 
through a conductor of such a shape the magnetic action at a point outside or on the 
surface of the conductor is the same as if all the currents were condensed into one 
flowing along the axis.t Hence when finding the velocities outside the vortex ring 
we may suppose the vortices condensed into one at the axis of the core. If a> be the 
angular velocity of molecular rotation, e the radius of the transverse section of the 
core, then ire-w is the strength of the vortex which we must suppose placed at the 
axis of the core. We shall for brevity denote 7 re 2 w by m. 
The components (u, v, w ) of the velocity at the point (x, y, z) are given by 
where r is the distance of the point (x, y, z) from the point (x, y, z), a point on the 
vortex whose polar coordinates are (p, 9 ); s' is an arc of the vortex ring. 
* See a note by Sir W. Thomson at tbe end of Helmholtz’s paper on “ Vortex Motion,” Phil. Mag., 
1867. . •' 
t Maxwell’s ‘Electricity and Magnetism,’ 2nd edition, §. 683. 
