629 Prof. T. H. Havelock on the Stability of 
and hence a possible steady state. Consider now the general 
equations for the disturbed motion. Let the positions of the 
vortices in the inner ring be given by polar coordinates 
a(1+741), 2sT7/n+ ot + A541, 
and those in the outer ring by 
b(1+ps41), 2(s+3)m/n+ot+ $eyr 
We form the equations of motion as in the previous 
sections. We choose a typical vortex, s=0, in the inner 
ring, and to simplify the notation we take the vortex s=n—1 
in the outer ring. Expanding the components of velocity to 
first order terms, and reducing the coefficients by means of 
(36), we obtain the equations for these two vortices : 
’ 1 2n2p” 
dati = LGD pga 
6, n-19 i 72 Ce=2 
—K 3 =e, 2 PAC =e Die 
4 8’ 
+x! bs FE eas 
. i Anp” Qn2p” kn 
Ym) Pi A—i1)\—Ome 
4a, “4 é (n2—1) —2(n—1)+ T+ p* ~ d+p"? 
n—1 U 
in VS 2n{(1 +p?) C'—2p} 
IU ~ @ D2 ae 
1 
— 2) os 72) 8! 
ee! A Sy 
‘ 1 Qn2p” \ 
2 S33 = 
Anb®p, = “fan 1) (+p)? Dn 
3 N=19) 1+p2 CV = 
+x! 5 ae = PU ie PS O41 
“5! 2p(— pS, 
pp eI 
BZ D2 stl 
oy aus 3 )—Nerc 4n = 
Amb bn = ls noe (n 1) 2(n 1) ip 1+p" +p")? p 
=! i 2p{ (1+p?)C'—2 
n—1 
—K > 
1 
n 
n-19, — me\n! 
—K > ee 65415 ° ° e ° (38) 
0 
342 
