432 Mr. A. Russell on the Magnetic 



the surface o£ the vortex ring we have r 2 =r, and we may 

 write rj = 2«, cos0=j, E = l, and F=log(8a/r). Hence 

 points on the surface of the ring have a velocity m/nr, that 

 is, cor about the circular axis and a linear velocity iv upwards, 

 where 



W= £a{ l °S 8 r - 1 ) W 



We see, therefore, that the vortex ring moves bodily 

 upwards with the velocity w. We see also at once that the 

 velocity of the fluid at the centre of the vortex ring is 

 ^irija x ?>?/27n, that is, m/a. 



Again, if the components of the velocity of an element of 

 the volume dv of the fluid be u and w, and if p be the density 

 of the fluid, we have 



07T I 

 777 2 V87T/ 



where T x is the kinetic energy of the moving fluid and 

 11= the resultant magnetic force at dv in the electrical 

 problem. But by Kelvin's formula 



2(R 2 /87r)dr = Ii 2 /2, 



and thus 



T ^ 



2tt 



T, = &, 



where LP/2 is the energy due to the linkages of the lines of 

 force with the whole of the electric current. 

 Hence, by the preceding section 



T 1 = ^(lo 8 ^- 2 ), 

 = 2pm 2 a(logy-2). 



Let us now suppose that the core of the vortex-ring is 

 solid so that the angular velocity of every point in it round 

 the circular axis is constant. Ihis corresponds to assuming 

 that the current density over the cross section of the ring in 

 the electrical problem is constant. Let us also suppose that 

 the density of the core is the same as that of the fluid. The 

 kinetic energy T 2 of the core is given by 



T 2 = \ 9 irrHira{ (r 2 /2) co' 2 + w 2 } , 



= ^pm 2 a + small terms. 



