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Proceedings of the Poyal Society 
comparison with the core’s diameter. Consider, for example, the 
simplest case (§ 12, fig. 5) of two simple rings linked together. 
18 Gro back now to the simple circular Helmholtz ring. It is 
clear that there must be a shape of absolute maximum energy for 
given vorticity and given impulse, if we introduce the restriction 
that the figure is to be a figure of revolution, that is to say, 
symmetrical round a straight axis. If the given vorticity be given 
in this determinate shape the motion will be steady, and there is 
no other figure of revolution for which it would be steady (it being 
understood that the impulse has a single force resultant without 
couple) . If the given impulse, divided by the cyclic constant, be 
very great in comparison with the two-thirds power of the volume 
of liquid in which the vorticity is given, the figure of steadiness is an 
exceedingly thin circular ring of large aperture and of approximately 
circular cross section. This is the case to which chiefly attention is 
directed by Helmholtz. If, on the other hand, the impulse divided 
by the cyclic constant be very small compared with the two-thirds 
power of the volume, the figure becomes like a long oval, bored 
through along its axis of revolution and with the ends of the bore 
' 
rounded off for trumpeted) symmetrically, so as to give a figure 
something like the handle of a child’s skipping-rope, but sym- 
metrical on the two sides of the plane through its middle 
perpendicular to its length. It is certain that, however small 
the impulse, with given vorticity the figure of steadiness thus 
indicated is possible, however long in the direction of the axis 
and small in diameter perpendicular to the axis and in aperture 
it may be. I cannot, however, say at present that it is certain 
that this possible steady motion is stable, for there are figures 
not of revolution, deviating infinitely little from it, in which, 
with the same vorticity, there is the same impulse and the same 
energy, and consideration of the general character of the motion 
is not reassuring on the point of stability when rigorous demon- 
stration is wanting. 
19. Hitherto I have not indeed succeeded in rigorously demon- 
strating the stability of the Helmholtz ring in any case. With 
given vorticity, imagine the ring to be thicker in one place than in 
another. Imagine the given vorticity, instead of being distributed 
in a symmetrical circular ring, to be distributed in a ring still, 
