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MR. J. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
Problem II. To find the action upon each other of two vortex rings which move so 
as never to approach nearer than a large multiple of the diameter of either. 
For the sake of simplicity we shall suppose that the normals to the planes of the 
vortices intersect. 
Fig. 2. 
Let the plane of the paper contain p p and q q the normals to the two vortices, 
let A B be the vortex moving along p p, C D the vortex moving along q q . 
Let the figure of the circular axis of the vortex A B be given by 
p'=a %cL a ' cos n6' 
^—1 d -tfr,' cos n6' 
where z' is measured along and p perpendicular to p p'. Since the vortices never 
approach near to one another a,/ and f3 u ' will be small compared with a; they will be 
functions of the time which we shall have to find. 
Let the figure of the circular axis of C D be given by 
p—a- j-2a„ cos n9 
z = % -|-2/L cos nd 
where z and p are measured respectively along and perpendicular to q q. For the 
same reason as before a n and /F will be small compared with a. To find how the vortex 
C D is affected by the vortex A B we shall have to find the velocities of the fluid along 
z and p due to the vortex A B; in doing this we may as a first approximation assume 
that the axis of A B is circular and in one plane, i.e., we may calculate the velocities 
as if a n ' and /3» were both zero. 
Let e denote the angle between p p' and q q\ Let p p be taken as the axis of z, 
the perpendicular to p p drawn upwards through the centre of the vortex A B, being 
the axis of x'. 
Let l, m, n be the direction cosines referred to these axes of a radius vector in the 
plane of the vortex ring C D, drawn from the centre of the vortex ring and making 
an angle 9 with C D the intersection of the plane of the vortex ring with the plane of 
the paper. 
To find l, m, n through the centre of a sphere draw planes parallel to the two vortex 
rings and let these be H K H', L K IF, the former being parallel to the ring A B and 
the latter to C D. Let A G be the poles of these great circles. 
