Prof. Helmholtz on Integrals expressing Vortex-motion. 507 



and F and E are the complete elliptic integrals of the first and 

 second orders for the modulus k. 

 If we put, for sake of brevity, 



U=-(F-E)-*F, 



where, therefore, U is a function of k, we have 



m, , — dV z—c 



T X=~ * X 



it ' * d/c ' {cf + X) 2 +t z — c T 



If there be a second vortex-filament m at the point %, z 3 and we 



denote by r 1 the velocity in the direction of g which it gives to 



m v we shall find this, if we put in the expression for r instead of 



r x 9 % c m l3 



By this process U and tc are not changed, and we have 



mrx + m 1 T 1 g = (8) 



Let us next determine the value of the velocity parallel to the 

 axis of z } which m v whose coordinates are g and c, produces, and 

 we find 



If we call 2c\ the velocity parallel to z which the vortex-ring m, 

 whose coordinates are z and x> produces at the position of m v 

 we require only to make again the same interchange of letters 

 as before. Hence we find 



2mivx 2 + 2??2 1 w li 9' 2 — mrxz — nn^r^c = V ' gnv . U. . (8a) 



Similar sums can be made for any assumed number of vortex- 

 rings. Denote adg tic in the nth ring by m n and the com- 

 ponents of the velocity it receives from the others by r w and 

 w n , omitting, however, for the time that which each ring- 

 can impart to itself. Call also its radius p n , and X its distance 

 from a plane parallel to xy, which magnitudes, no doubt, 

 correspond in direction to what we have called x anc ^ z > but as 

 belonging to a particular ring they are functions of the time, 

 and not independent variables like x anc ^ z - Finally, let the 

 value of ty, as far as it arises from the other vortex-rings, be 

 ty n . Vie find from (8) and (8 a), by writing out and adding 

 these equations for each pair of rings, 



%(2m n w n p n 2 — m n Tnpn\ n ) =l,(m n p n ylr n ) . 



