of Edinburgh, Sessio7i 1875 - 76 . 
69 
with a circular axis, but thinner in one part than in the rest. It 
is clear that with the same vorticity, and the same impulse, the 
energy with such a distribution is greater that when the ring is 
symmetrical. But, now let the figure of the cross section of the 
ring, instead of being approximately circular, be made considerably 
oval. This will diminish the energy with the same vorticity and 
the same impulse. Thus, from the figure of steadiness we may 
pass continuously to others with same vorticity, same impulse, and 
same energy. Thus, we see that the figure of steadiness is, as 
stated above, a figure of maximum-minimnm, and not of abso- 
lute maximum, nor of absolute minimum energy. Hence, from 
the maximum-minimum problem we cannot derive proof of 
stability. 
20. The known phenomena of steam rings and smoke rings 
show us enough of, as it were, the natural history of the subject 
to convince us beforehand that the steady configuration, with 
ordinary proportions of diameters of core to diameter of aperture, 
is stable, and considerations connected with what is rigorously 
demonstrable in repect to stability of vortex columns (to be given 
in a later communication to the Koyal Society) may lead to a 
rigorous demonstration of stability for a simple Helmholtz ring 
if of thin enough core in proportion to diameter of aperture. But 
at present neither natural history nor mathematics gives us perfect 
assurance of stability when the cross section is considerable in 
proportion to the area of aperture. 
21. I conclude with a brief statement of general propositions, 
definitions, and principles used in the preceding abstract, of which 
some appeared in my series of papers on vortex motion com- 
municated to the Royal Society of Edinburgh in 1867-68 and 69, 
and published in the Transactions for 1869. The rest will form 
part of the subject of a continuation of that paper, which I hope 
to communicate to the Royal Society before the end of the 
present session. 
Any portion of a liquid having vortex motion is called vortex 
core, or, for brevity, simply “ core.” Any finite portion of liquid 
which is all vortex -core, and has contiguous with it over its 
whole boundary ir rotation ally moving liquid, is called a vortex . A 
vortex thus defined is essentially a ring of matter. That it must 
