On the Induction of Electric Currents. 141 



removed to the position which it occupied at time r—dr ago, 

 the value of P corresponding to the sudden introduction and 

 removal of the system P' will be 



f{z + RdT, t) -f(z, T) =Bgg./fc r)d*. 



At the instant under consideration (that is, after an interval 

 r—dr), the value of this is 



R±f{z+B(T-dT), r\dr 



= B^Az + Rr,T)dT. 



ultimately. Summing up all similar terms between the limits 

 t and of t, we have finally 



P = e!J , /> + Kt,tMt+/(z,0), • • (!) 



the last term being the potential of the currents generated at 

 the instant at which we are observing them. 



If the electromagnetic system is rotating about the axis of 

 z in the same direction as that in which cf> is measured, and 

 with uniform angular velocity co, the value of f(z -f- B/r, t) , 

 expressed in cylindrical coordinates, will be 



f(z + ~RT,p, <£ + o)t) ; 



and the value of P, after the currents have become steady, 

 will be obtained by integrating between the limits go and ; 

 whence 



P=^Mj^ + ^,p,<l> + ayr}dr+Mp,4>), . . (2) 



which represents a spiral trail of images. 



3. From the foregoing expression for P we can at once 

 obtain Niven's results. For 



dzJ Q dr \ dz d<j>/ T=z0 \d(j>) T= o ^ 



dP +6) ^P 

 dz d<\> 



since /=0 when t = <x> . 



The last result is perfectly general ; but at the surface, 



