623 Prof. T. H. Havelock on the Stability of 
We shall have occasion to use the following summations, 
which can easily be proved : 
a —p? 2n 
a oS e =p" iT i Gay) 
fs bee BON ed ele 
s=1 L—2pC +p? 1—p” 1—p’ 
Gagne wet ag 
can (Bate Cay spr 
with 0 <p< 1. 
Writing p=a’/b?, we find from (15) 
a= Fe (-2-1). se a GD 
For small displacements from the steady state we have 
for each vortex « at a point 
a+7s+1, wt + 2sm/n+ Os41, 
an image vortex —« at the point 
62 
a 
(1 — a) » @t+2s7/n+ O41. 
Considering the motion of the vortex given by s=0, we 
have for the radial velocity the expression (2), together with 
FE ee . WS 
20 s=0 H 
and for the transverse velocity we have (3), together with 
K nal (b?/a)(1—1541/a) cos 6— (a+7)) 
Qe so ig 5 0 (gy 
where 
p= 2s7/n+ Os41—9}, 
and 
bt Ts+1 2 b? Ts+t 
Bas =o /f TL a Se Oo) Vesa oy Sete 7 
E =a(1 = ) + (a+m)?-2- (i : )(a+n) eos. 
The steps in the reduction of the equations of motion need 
not be reproduced here; making use of the summations 
given in (16), and writing 
p=a?/b?; S=sin (2sz/n) ; C=cos(2sa/n), . (20) 
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