627 Prof. T. H. Havelock on the Stability of 
This gives 
(3n—1- =a , oe GH 
pea Se 
Oo dara 
We shall merely state now the results for the general 
equations of disturbed motion. The equations for *, and 6, 
are the same as in (21), with the following alterations :— 
(i.) write g for p in the coefficients, (ii.) change the sign of 
the last term in each equation, the coefficients of 7,4, and 
6,41 respectively, from — to +, (ili.) change the coefficient 
of 7, in the second equation to 
1 
4nq” 2n2q” 2q 
ee 2 — = — —— 
F (n2—1)—2(n—1) + arnt 
(gy? G9)?’ 
Taking a simple solution of type (22), and proceeding as 
in (24), (25), we obtain, instead of (26), the result 
ase (PON oe 6 5 6 CD 
where 
tie a ek Gi diee) fy n2gr—*(1—g*)? 
1 =Hk(n k) 1—q (1—g")? y 
4nc¢ n2qr—"(1—q*)? 
'=k(n—k)—2(n—1) + —4 == 
Q (n—k)—2(n—1) = Ge 
np GIG) 
1—q" i 
hb (yk a—k 2>n—k(] — 72k 
Ra Mee ated) ) _ ng (LF) (33) 
1—q (l=q")? 
As before, it appears that stability depends upon there 
being values of g less than unity for which Q’ is negative for 
all the values of k. Jt is easily seen that there is no such 
value of g when n>7, and therefore the steady state is 
unstable when there are seven or more vortices in the ring. 
Examining the expressions numerically for smaller values 
of n, we find that the steady state is stable under the 
following conditions :—n=2, b/a<0°386 ; n=3, b/a<0°522 ; 
n=A4, b/a<0°556 ; n=5, b/a<0°579 ; n=6, b/a<0°544. 
These values are slightly less than the corresponding 
limits when the ring is within the circular boundary, but 
there is little difference in the general conclusions. 
340 
