Rectilinear Vortices in Ring Formation. 620 
equations (4) give 
n—1 
(41a/«)ry= A6é,— > a0s415 
1 
5 n—-1 
(Ara*/x)0,=Bry— & asreyr, . . - ~ (5) 
1 
where 
1 1 : 
A= g(—1); B= gi —-Y(m—-1)); a,=1/(1—C,). 
There are similar equations for each vortex, giving altogether 
a system of 2n equations. 
The simplest method of treating the equations is to examine 
a possible simple solution of the form 
Poe edle 3 Chanel 5 6 5 4 (8) 
where k=0, 1, 2, ... n—1. 
It may be proved that under the conditions stated 
n—1 e2ksniin 
1 
s=1 1—cos (2s7/n) = g 1) —k(n—k) O° (7) 
Hence, from (5), we find that the equations for a, 8 reduce to 
(4rra/x)e=k(n—k) B, 
(Arra/«)B={k(n—k)—2(n—1)}a. . . « (8) 
Finally, taking « and @ to be proportional to e, these 
give 
n= (45) Mn—H){Mn——2n—-D)} a9) 
It follows that in (6), (8), and (9) we have, in general, 
2n independent solutions of the equations of the system, and 
that we can build up the complete solution for any arbitrary 
small initial displacements of the vortices. 
An alternative method of solution may be noticed briefly, 
namely, the method used by previous writers for particular 
cases ; it may be extended to give the general results, though 
not quite so simply as in (6)-(9). In the 2n equations (5) 
we assume each coordinate to be proportional to e¥, and 
form the determinantal equation for }. The determinant 
can be reduced to one of order n in )2, and it can be shown 
that it is of the type known as a circulant, and can be 
factorized in terms of the nth roots of unity ; after some 
reduction we obtain (9) again, and can deduce the corre- 
sponding simple solutions given by (6). 
