22 Sir William Thomson on Vortex Atoms. 



ring and then setting the fluid in motion by aid of an instanta- 

 neous film, or by applying the two initiative actions simultane- 

 ously. The whole amount of the impulse required, or, as we 

 may call it, the effective momentum of the motion, or simply the 

 momentum of the motion, is the sum of the integral values of the 

 impulses on the ring and on the film required to produce one or 

 other of the two components of the whole motion. Now it is 

 obvious that as the diameter of the ring is very small in compa- 

 rison with the diameter of the circular axis, the impulse on the 

 ring must be very small in comparison with the impulse on the 

 film, unless the velocity given to the ring is much greater than 

 that given to the central parts of the film. Hence, unless the 

 velocity given to the ring is so very great as to reduce the volume 

 of the fluid carried forward with it to something not incompa- 

 rably greater than the volume of the solid ring itself, the mo- 

 menta of the several configurations of motions we have been con- 

 sidering 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 formulae. 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 mo- 

 mentum for different velocities of the ring is easily verified with- 

 out analysis for velocities not so great as to cause sensible devi- 

 ations 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 that 

 velocity varies, as does the magnetic force of an infinitesimal 

 magnet on a point in its axis, inversely as the cube of the dis- 

 tance from the centre. Hence the cube of the radius of the ap- 

 proximately globular portion carried forward is in simple inverse 

 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 

 electromagnet. The same result is of course obtained by the 

 Green's integral referred to above. 



