Sir William Thomson ,on Vortex Atoms, 23 



The synthetical method just explained is not confined to the 

 case of a single circular ring specially referred to., but is equally 

 applicable to a number of rings of any form, detached from one 

 another, or linked through one another in any way, or to a single 

 line knotted to any degree and quality of " multiple continuity," 

 and joined continuously so as to have no end. In every possible 

 such case the motion of the fluid at every point, whether of the 

 vortex core or of the fluid filling all space round it, is perfectly 

 determined by Helmholtz's formulae when the shape of the core 

 is given. And the synthetic investigation now explained proves 

 that the effective momentum of the whole fluid motion agrees in 

 magnitude and direction with the magnetic moment of the cor- 

 responding electromagnet. Hence, still considering for simpli- 

 city only an infinitely thin line of core, let this line be projected 

 on each of three planes at right angles to one another. The 

 areas of the plane circuit thus obtained (to be reckoned according 

 to De Morgan's rule when autotomic, as they will generally be) 

 are the components of momentum perpendicular to these three 

 planes. The verification of this result will be a good exercise on 

 " multiple continuity." The author is not yet sufficiently ac- 

 quainted with Riemann's remarkable researches on this branch 

 of analytical geometry to know whether or not all the kinds of 

 "multiple continuity" now suggested are included in his clas- 

 sification and nomenclature. 



That part of the synthetical investigation in which a thin solid 

 wire ring is supposed to be moving in any direction through a 

 fluid with the free vortex motion previously excited in it, requires 

 the diameter. of the wire at every point to be infinitely small in 

 comparison with the radius of curvature of its axis and with the 

 distance of the nearest of any other part of the circuit from that 

 point of the wire. But when the effective moment of the whole 

 fluid motion has been found for a vortex with infinitely thin core, 

 we may suppose any number of such vortices, however near one 

 another, to be excited simultaneously; and the whole effective 

 momentum in magnitude and direction will be the resultant of 

 the momenta of the different component vortices each estimated 

 separately. Hence we have the remarkable proposition that the 

 effective momentum of any possible motion in an infinite incom- 

 pressible fluid agrees in direction and magnitude with the mag- 

 netic moment of the corresponding electromagnet in Helmholtz's 

 theory. The author hopes to give the mathematical formulae 

 expressing and proving this statement in the more detailed paper, 

 which he expects soon to be able to lay before the Royal Societ}^. 



The question early occurs to any one either observing the 

 phenomena of smoke-rings or investigating the theory, — What 

 conditions determine the size of the ring in any case ? Helm- 



