103 
of Edinburgh, Session 1866 - 67 . 
vortex core or of the fluid filling all space round it, is perfectly de- 
termined 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 magni- 
tude and direction, with the magnetic moment of the corresponding 
electro-magnet. Hence, still considering, for simplicity, only an in- 
finitely 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 circuits 
thus obtained (to be reckoned, according to He Morgan’s rule, when 
automic, as they will generally be), are the components of momen- 
tum perpendicular to these three planes. The verification of this 
result will be a good exercise on “ multiple continuity.” The 
author is not yet sufficiently acquainted with Riemann’s remark- 
able researches on this branch of analytical geometry, to know 
whether or not all the kinds of “multiple continuity” now sug- 
gested are included in his classification 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 com- 
parison with the radius of curvature of its axis and with the dis- 
tance 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 incompressible fluid agrees in 
direction and magnitude with the magnetic moment of the cor- 
responding electro-magnet in Helmholtz’s theory. The author 
hopes to give the mathematical formulae expressing and proving 
this statement in the more detailed paper, which he hopes soon to 
be able to lay before the Royal Society. 
The question early occurs to any one either observing the phe- 
nomena of smoke-rings or investigating the theory — What condi- 
tions determine the size of the ring in any case? Helmholtz’s 
