621 Prof. T. H. Havelock on the Stability of 
From (9), when k=0 we have A=0. If we examine this 
case we find that the displacement consists of a rotation of 
the ring combined with a small change in its radius ; the 
result is a new steady state with a corresponding small 
change in the angular velocity. The condition for stability 
is that X? must be negative for all the other values of &, 
namely, 1, 2,...n—1. Hence, from (9), the steady state is 
stable if 
KO) AG) 56 556 6 3 CO) 
is negative for all the values of k, and this is the case if it is 
negative for k=4n when n is even, or $(n+1) when nis odd. 
It follows at once that the steady state is completely stable 
when n<7. When n=7 the expression (10) is zero for 
k=3 or 4; while for n >7 there are always some values 
of k for which 2? is positive, and hence the system is 
unstable. 
Whatever the value of n there are always two modes of 
possible small oscillations, namely, those given by k=1 and 
i=) 
When k=2 we have 
v= 45.) (0-2), Leh et Oi cict) 
while for k=1 
pepe euiaeies oie 
We notice that in the latter case the period of the small 
oscillation is the same as the period of rotation of the ring 
in the steady state; this motion was worked out for the 
particular case of three vortices by Kelvin in the paper 
already quoted, and it is illustrated ina characteristic manner 
by the description of a working model to show the motion 
ot the vortices. 
The single infinite straight row of vortices may be obtained 
by making both n and a become infinite, with the ratio n/27a 
finite and becoming in the limit equal to the distance between 
consecutive vortices ; the usual results then follow from (6) 
and (9). 
Single Ring in assigned Field. 
3. We have so far considered the vortices to be moving 
solely under their mutual actions. Suppose now that there 
is an assigned velocity field which is maintained indepen- 
dently ; for simplicity we suppose the flow to be in circles 
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