1096 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
The preceding work exemplifies an interesting fact about vortex motion. Wheu 
the configuration of the vortices are known, their various velocities are obtained bv 
linear ecpiations ; just as in the case of gravitating matter, linear equations give the 
accelerations of the various parts. 
§ 43. When there are only two rings, the equations of motion are— 
/ 8q 
TTC.Zi = - log - 
TTCiCi 
TTCoZn = 
«i 
■111:, L be, 
-f log — 
2 V ^ a. 
i + 
— m. 
cR 
clc-^ 
cU 
dh \ 
cR 
rrC^C:2 = 
iR 
(108), 
where 
= f 
CjCo cos (j) cl(j) 
W{(% — rT + ~ cos 0 + r/} 
The two integrals of these equations, already found, are 
= constant . 
(109), 
•> 
Cl 
log ~ - 4 ) + 
Co 
m^m^l = constant 
( 110 ). 
Siqjpose the ring (C^) in front of the ring (Cg). 
Then z^ — z.^ is positive, and 
(R _ P_ fy.2 (q — ::o) cos cf) dcf) _ 
dz Jo \/{(q — + ck ” cos 6 + oy} 
The integral is easily seen to be positive. 
Therefore the radius of the front ring increases, and of the back ring diminishes. 
These changes cause decrease and increase of the velocities, so that it may happen 
that the second ring will overtake the first. 
That the motion is of this character was shown by Helmholtz, ‘ Ckelle’s Journal,’ 
vol. 55, pp. 54, 55. 
