428 Mr. O. Heaviside on Telegraphic 



of current through e to be strictly dependent on the rise or fall 

 of potential of the end of A. 



To find an expression for the potential and the current at any 

 point of a cable insulated at one end, at any time after contact 

 is made with a battery at the other end, the only way, as far as 

 I am aware, is to follow the method given by Sir William Thom- 

 son in 1855 (Proc. Roy. Soc), making the necessary alterations 

 to suit the changed conditions of the problem. It is to express 

 the actual potential at any time as the difference of two func- 

 tions, one being the known final distribution of potential, and 

 the other the departure from the final potential, the latter being 

 expressed by an infinite convergent series every term of which 

 is of the form 



sin #.e~'. 

 Let / be the length of the line, 



k the electrical resistance of the conductor per unit of 



length, 

 c its electrostatic capacity per unit of length, 

 k l the resistance of the dielectric per unit of length to 



conduction in a radial direction, 

 V the electromotive force of the battery, the resistance of 



which is neglected, 

 v the potential, and 



C the current at any point x of the conductor, measured 

 from the battery-end, at the time / from the moment 

 of making contact. 



The differential equation of conduction in a telegraphic line is 



cl %=%~ h% > (1 > 



where h= \f _L ; and we must find a solution of this to satisfy 



the following conditions, which are given by the circumstances 

 of the case. 



1. v =V when x = 0. 



2. ~r=0 when x=l. 

 ax 



3. v =0 when £=0, except when #=0. 



4. v =/(#) when t= oo. 



To find the function f(x) expressing the permanent distribu- 



av 

 tion of potential after an infinite time, make -j- =0 in (1), and 



integrate the resulting equation 



