Rectilinear Vortices in Ring Formation. 628 
Double Alternate Rings. 
6. In the previous sections we have been considering in 
effect a double symmetrical ring, in which the motions of 
one ring—the image ring—are constrained in accordance with 
those of the actual ring. We shall leave on one side the 
general case of a free double symmetrical ring, and prozeed 
to two alternate rings in an unlimited liquid. 
Let there be n positive vortices, each of strength x, equally 
spaced round a circle of radius a, and n negative vortices of 
strength x! equally spaced round a concentrie circle of radius 
6 (>a), the arrangement of the vortices being alternate. 
Thus, if the vortices in the inner ring are given by polar 
coordinates a, 2sm/n, those of the outer ring are given by 8, 
2(s+2)a/n, with s=0, 1, ..., n—1. 
Examine first the possibility of a steady state with the 
two rings rotating with equal angular velocity, the relative 
configuration remaining unchanged. The radial velocity of 
any vortex is zero. The transverse velocity of a vortex in 
the inner ring is given by 
(n—l)e «21 bC’—a 
Ara 27 = 6? +a2—2abC”’ ) 
and in the outer ring by 
@=1)e' ik" a5 (35) 
eae SS a? + 62?—2abC'’ 
where C’=cos{2(s+2)ar/n}. 
We shall require the following summations, with 
p=al/b<1: 
n—l1 1—pC! n 
o=0 L~2pC' +p? ~ 14 py? 
rep lap __ n(l—p”) 
) 
0 L—2pC’+p2 1+,” 
Dail (1+ p2)C'— 2p Re 
0 (L—2p0'+p?)2 (1+ p)2" 
The condition for equal angular velocity of the two rings 
then becomes 
(36) 
6 n 2 
1) IE al FO (n= Lp - & 
It can be seen that for a given ratio of «' to « we obtain 
from this equation a corresponding value of p less than unity, 
