Mr. 0. Heaviside on the Extra Current. 139 



expresses the current at time t. If the wire is originally every- 

 where at potential zero and without current, the potential xt and 



clQ' 

 current — p- at time t after the end P is raised to potential Y, 



the end Q being to earth, are 



dQ[_ Y_dQ 

 dt "Id dt' 



where v and -p have the values given in (13) and (14). 



Suppose 47T 2 -q > 1, then the first series in (13) and (14) dis- 

 appear, and we have 



2 V -Isl - iwxf , 1 . \tm'i /1K . 



V= . € 2a 2 tSIII-t— ( COS -\ rSink-, . (lO) 



7r i % I \ m f i / 2a 7 



dQ V -i 4V -If 1 form . tm' { nr , 



.t ( lb ) 



dt ~ Id Id ' i m'i I 2a' 



The extra current is exhibited in (16) as consisting of two 



V -i 

 parts. One, a current y-j . e * , uniform at all parts of the wire, 



which dies away without oscillations with a rapidity propor- 

 tional to -. This current is due entirely to the momentum of 

 a 



the original current jj. The other part, 



4V _ -L » 1 iirx . tm'i 

 jj .e 2a 2 • — cos - T - . sin -^— ? 



is due entirely to the original charge of the wire, and consists 

 at any point x of an infinite series of currents alternately posi- 

 tive and negative, which die away with only half the rapidity. 

 These oscillations are of greatest intensity at the end P, and 



least at the end Q. They are insensible both when -5 is very 



small and when it is very large. In the former case only the 

 higher terms in (13) and (14) are periodic with respect to the 

 time ; and in the latter case they become very rapid and weak 

 in the same proportion. But when the time-constants a and /3 

 are not very different, the oscillations are of considerable 

 strength, and may become observable by proper means. Sup- 



