Reetilinear Vortices in Ring Formation. 622 
round the origin, the angular velocity being Q(r), and the 
transverse fluid velocity at a distance r being 7. 
Then, referring to equations (4) for the motion of a typical 
vortex in the ring, the only difference is that we have to add 
aQ(a) +r {O(a) +aO'(a)} 
on the right-hand side of the second equation. The angular 
velocity of the ring in the steady state is now 
(n—1)«/4ma? + O(a). 
Following the same procedure, we obtain, instead of (8), 
the equations 
(Arra/«)a=k(n—h)@, 
(41ra*/) B= {k(n —k)—2(n— 1) + (4ma*/x)O'(a)}, (13) 
and hence we have 
M=k(n—k) {k(n—k) —2(n—1)+ (A7ra*/x)Q'(a)}., (14) 
with k=0, 1, ... n~1. 
It follows that the steady state can be stabilized for any 
value of n, provided Q'(a) is negative and sufficiently large. 
Two spécial cases may be noted. First, if the fluid is 
rotating like a rigid body—that is, if O(r) is constant—the 
conditions for stability are unaffected. In the second place, 
suppose there is in assignei vortex fixed at the origin, so 
that Q.(r)=«'/2rr? ; then. if x! is of the same sign as x, we 
can make the steady state stable for any value of n by taking 
«' large enough. 
Single Ring with Outer Boundary. 
4. Suppose the liquid is contained within a circular 
boundary of radius 6, the vertices being in the steady state 
on a concentric circle of radius a (<b). The motion in the 
liquid is due to the given vortices and their images in the 
circular boundary. 
Taking the steady state first, the radius of the image ring 
is 6?/a, the strength of each image vortex being —x, Writing 
down the velocity at any vortex in the given ring, the angular 
velocity in the steady state is given by 
n—1 2 
(m=Ne 5 (0?/a)C—a - (15) 
am Ata 2m .=o b4/a? + a? — 22°C’ 
where C=cos(2sz/n). 
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