Rectilinear Vortices in Ring Formation. 626 
and this is positive if 
(n* + 8n +8) 23 + (3n? —8n — 5) a? + (bn? + 8n—8)a 
Pe —Epsr8 Ss > (29) 
where a =p" = (a/b)”, 
The left-hand side of (29) is always positive for nS 8. 
From similar expressions when n is odd we find that there 
is always a positive value of Qforn=7. Hence we conclude 
that the motion of the ring is unstable when the number of 
vortices is equal to or greater than seven, whatever the radius 
of the outer boundary. For n <7 we shall see that the 
motion is stable provided the ratio of the radius of the ring 
to that of the boundary is less than a certain value in each 
case. We shall examine the cases briefly, noting that in 
each case the mode £=0 means simply aneutral displacement 
of the ring. 
Hor 2=2)/k—1, we find from the previous expressions 
that Q is negative if p < 02137; and as P is positive, it 
follows that the circular motion of the two vortices is stable 
if a/b < 0-462. 
For n=3, k=1 or 2, Q is negative for p < 0:322, and the 
motion is stable for a/b < 0°567. 
Similarly for n=4 we find the critical value of a/b to be 
about 0°575 ; for n=5 it is 0:588, and for n=6 it is 0-547. 
When n=7, which is the critical neutral case when there is 
no boundary, the effect of an outer boundary of any radius 
is to cause instability. 
Single Ring with Inner Boundary. 
5. Suppose now that the fluid is bounded internally by a 
circular barrier (r=6), and that a ring of n vortices is 
rotating in circular motion in a ring of radius a(>6). The 
image of a vortex «x at r=a is a vortex —« at r=67/a, 
together with a vortex « at r=O : this combination makes 
the circulation zero for a circuit enclosing the boundary 
without including any of the actual vortices 
We find the equations of motion of a given vortex, s=0 
in the previons notation, just as in $4. The only differences 
arise (i.) from the additional image vortex nx at the origin, 
and (ii.) in evaluating the various summations, as 6/a is now 
less than unity instead of a/b. For the steady state we have 
; _(n—1)e nk (io PY Lag® 
Ohare Ama un On Tec 1—29C+ q?’ © 
where g=6?/a2 and C= cos (2sm/n). 
(30) 
339 
