Theory of Faults in Cables. 169 



and find i 



fch 



when Ti = r 2 = 20 , makes 



which follows from (41) ; and find the limit when a = 0. The 



result is 



This, added to -~, what the constant term in (42) becomes 



E 



(>-?) 



which is the constant term in (39), The remainder of (39) is 

 immediately deducible from (42) by making r 1 =r a ==-oo . 



21. The solution (40) for the potential in condenser work- 

 ing could be deduced from that for the current in working with- 

 out condensers. For, in the latter case, the final result of the 

 introduction of an electromotive force at ^=0 is a current in 

 the line of the same strength everywhere, and v = at ^ = 

 and x = I ; and in the former the final result is that the po- 



dv 

 tential of the line is the same everywhere, and — = at x = 



and x = l. Both the current and the potential must satisfy 

 the same partial differential equation. Hence the current in 

 the latter case at x at time t must rise in the same manner as 

 the potential in the former. Now 



y=s{ 1+22cos ^ e ~J- • • • (43) 



is the solution for the current in working without condensers, 

 where 77 is the final uniform current. In the condenser-pro- 



blem the final uniform potential is — j— =Er 1 , substituting 



E 



which for 77 in (43), and changing y into v, equation (40) 



results without a separate investigation. It is also very re- 

 markable that (40) and (43) are capable of expression in an 

 entirely different form, leading to the identity 



€ -z 2 _|_ e -(*-a)2 _|_ € -(*+a)2 _|_ e -(*-2a)2 _j_ € ~(*+2a)2 + ^ 



-Hi 2ttx -llL 2 kirx -^ 6ttx , N 



+ 6 «' COS 1- € « 2 COS ■ H € « 2 COS f- . A 



a a a J 



well known to mathematicians. 



When t — 0, the current as given by (43) is zero every- 

 where, except at x = 0, where it is infinite; and in (40) the 

 potential is zero every where when £ = 0, except at x = 0, where 

 it is infinite. These impossible infinite values arise from the 



~~a~\2 



