619 Prof. T. H. Havelock on the Stability of 
to instability. By suitable choice of the relative strengths 
of the vortices in the two rings it is possible to limit the 
instability to only one special mode of disturbance ; itis this 
particular configuration which becomes in the limit the 
stable Karman vortex street, when we make the radius of a 
ring and the number of vortices both infinite, keeping their 
ratio finite. 
Single Ring of Vortices. 
2. Let there be n equal vortices, each of strength «, 
equally spaced round the circumference of a ring of radius a. 
In steady motion the ring rotates with a certain angular 
velocity w. Let the vortices be slightly displaced, and 
suppose the disturbed positions to be given in polar co- 
ordinates by 
G+7sii, 2s7/n+ott+Oe41,- - - - (1) 
where s=0,1,...n—1, and r,@ are small radial and angular 
displacements from the steady state. Consider the motion 
of one of the vortices, say that at the point (a+7, @t +0) ; 
its velocity is due to the other vortices, and the radial 
component is 
kK 1 (a+7r541) sin (2sm7/n+ 0541 —9)) 
eo 7 eae) 
while the transverse component is 
RS (a+ 7.41) cos (2s7/n+ 6,41 — 61) —(a +77) (3) 
5 =~ 
27 «=| D? ‘ 
where 
D?=(@+ Ts41)° + (a+ 4)? 
—2(a+7rs41)(a+7) cos (287r/n+O541—4). 
We expand these expressions to the first order terms in 
rand @, and so get the equations of motion of the vortex under 
consideration. After some reduction we obtain 
vie Ana y 1—C; ’ 
K 4 C, 14 iL a 
_ pees LS Az 
(a+r)o+a=qo7 a a 1—C,a 1—C, a J? (4) 
where C,=cos(2s7/7). 
The steady state is given by o= (n—1)k/42ra?, and since 
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