Prof. A. Gray: Notes on Hydrodynamics. 17 



If now We write fc = 2co cdcdO, and integrate from # = to 



B = 2ir (remembering that cos0 — .v/c), and £romc=0 to c = r, 



so as to find the effect of all the filaments in the ring, we obtain 



without difficulty 



2ft)7rr 2 , 8a ,^ .. 



v== —. log — (14) 



47ra e r K ' 



Now, for certain reasons which we do not discuss here, 

 the circular axis of the vortex ring is a coaxial circle of 

 radius R given by 



Ro=~~ (15) 



and of distance jjf , from any chosen point on the axis of the 

 system given by 



&= ^E' (16) 



The distance f i >s thus equal to the axial distance of the 

 circular axis of the anchor ring from the same chosen origin, 

 that is the two circular axes specified lie in the same plane. 

 The value of R is equal to the radius of gyration of a 

 uniform circular lamina of radius r about an axis in its plane 

 and at distance a from its centre. We have therefore 



R = v /a 2 +j-r* 



=* + g£ ( 16 ) 



approximately. 



In consequence of the spin co the rate of advance of the 

 circular axis of the vortex ring is loss than the value of v 

 above obtained by ^wr 2 ja» Thus we obtain finally, writing /c 

 for the strength 2©7rr* of the whole ring, 



d£o k /. Sa 1\ „_. 



l = 4^( l0 «T-4> • • • • < 17 > 



the value given by Lord Kelvin. 



5. Any other point in the phme of the circular axis of the 

 vortex and near that axis might have been taken instead of 

 a point on the circular axis of the anchor-ring surface for 

 the specification of the circle at which the axial velocity is 

 calculated. In this case, however, it will be found more 

 convenient for the sake of the integrations to calculate first 

 the whole flow through the circle chosen due to the complete 

 vortex ring, and then determine the velocity sought by 

 differentiation. 



Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. G 



