102 Proceedings of the Royal Society 
than the volume of the solid ring itself, the momenta of the several 
configurations of motions we have been considering will exceed by 
but insensible quantities the momentum when the ring is fixed. 
The value of this momentum is easily found by a proper application 
of Green’s formulee. Thus the actual momentum of the portion 
of fluid carried forward (being the same as that of a solid of the 
same density moving with the same velocity), together with an 
equivalent for the inertia of the fluid yielding to let it pass, is 
approximately the same in all these cases, and is equal to a 
Green’s integral expressing the whole initial impulse on the film. 
The equality of the effective momentum for different velocities of 
the ring is easily verified without analysis for velocities not so great 
as to cause sensible deviations from spherical figure in the portion 
of fluid carried forward. Thus, in every case, the length of the 
axis of the portion of the fluid carried forward is determined by 
finding the point in the axis of the ring at which the velocity is 
equal to the velocity of the ring. At great distances from the 
plane of the ring this velocity varies, as does the magnetic force 
of an infinitesimal magnet on a point in its axis, inversely as the 
cube of the distance from the centre. Hence the cube of the radius 
of the approximately globular portion carried forward is in simple 
proportion to the velocity of the ring, and therefore its momentum 
is constant for different velocities of the ring. To this must be 
added, as was proved by Poisson, a quantity equal to half its own 
amount, as an equivalent for the inertia of the external fluid; and 
the sum is the whole effective momentum of the motion. Hence 
we see not only that the whole effective momentum is independent 
of the velocity of the ring, but that its amount is the same as the 
magnetic moment in the corresponding ring electro-magnet. The 
same result is, of course, obtained by the Green’s integral referred 
to above. 
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 
