Prof. T. H. Havelock 618 
which the number of vorticesis finite. The obvious analogous 
problems arise when the vortices are equally spaced round 
the circumference of one or more concentric rings; the 
problems are not perhaps of special importance, but they 
are of some interest, and, further, one may obtain the 
infinite straight rows as limiting cases of ring tormation. 
We examine first the motion of a single ring of vortices, 
a problem which attracted attention many years ago in 
connexion with the vortex theory of atoms. Kelvin* worked 
out the case of three vortices, but failed to obtain a solution 
for a larger number; it was in this connexion that he drew 
attention to the now well-known experiments of Mayer with 
floating magnets. Shortly afterwards the problem was 
attacked by J. J. Thomson +, and it is usually stated that 
he proved the configuration to be stable if, and only if, the 
number of vortices does not exceed six. He,in fact, worked 
out the small oscillations for the particular cases of three, 
fonr, five, six, and seven vortices, obtaining an instability in 
the last case. It appears that the equations for the general 
case are capable of asimple explicit solution, and thisis given 
in § 2; a ring of seven vortices is neutral for small displace- 
ments, with less than seven it is completely stable, and for 
more than seven unstable. In §3 the effect of an assigned 
velocity field in addition to that of the vortices is examined 
briefly. 
In the next two sections we work out the effect of a 
concentric circular boundary upon the stability of a single 
ring, the boundary being either interna] or external to the 
ring. In both cases the stability is diminished, seven or 
more vortices being unstable whatever the radius of the 
bonndary. Fora smaller number there is a limiting ratio 
of the radius of the ring to the radius of the boundary for 
stability in each case. For an external boundary the motion 
is unstable in any case if the radius of the boundary is less 
than about twice the radius of the ring, and there is a similar 
result for aninternal boundary. The effect of the boundary, 
estimated in.this way, seems larger than might have been 
anticipated. 
In the remaining sections we examine the motion of two 
concentric rings of vortices, of opposite rotations, the vortices 
being spaced alternately. A steady stateis possible in which 
the rings rotate and retain their relative positions unaltered, 
but there are always modes of disturbance which give rise 
* Kelvin; Math. and Phys. Papers, iv. p. 185 (1878). 
+ J. J. Thomson, ‘ Treatise on Vortex Rings,’ p. 94 (1883). 
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