106 Sir William Thomson on Vortex Statics. 



demonstrating 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 dis- 

 tributed in a ring still 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 than when the ring is symmetrical. But now let the 

 figure of the cross section of the ring, instead of being approxi- 

 mately circular, be made considerably oval. This will diminish 

 the energy with the same vorticity and the same impulse. 

 Thus from the figure of steadiness w r e 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-minimum, and not of absolute maxim urn, 

 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 sub- 

 ject 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 respect to stability of vortex co- 

 lumns (to be given in a later communication to the Eoyal 

 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 his- 

 tory 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 pro- 

 positions, definitions, and principles used in the preceding 

 abstract, of which some appeared in my series of papers on 

 vortex motion communicated to the Royal Society of Edin- 

 burgh in 1867, -68 and -69, and published in the Transactions 

 for 1869. The rest will form part of the subject of a con- 

 tinuation 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 por- 

 tion of liquid which is all vortex-core, and has contiguous with 

 it over its whole boundary irrotationally moving liquid, is 

 called a vortex. A vortex thus defined is essentially a ring of 

 matter. That it must be so was first discovered and published 

 by Helmholtz. Sometimes the word vortex is extended to in- 

 clude irrotationally moving liquid circulating round or moving 



