164 Mr. 0. Heaviside on the 



Hence the potential v at time t after the electromotive force 

 E which produced the initial state is removed is, from #=Q 



to x=-j 



2 . ax 



v = 2El t J— 6 ~t; .... (22 j 



a— sin a 



and from #= - to x=l, 



. a . /# 1\ 



sinrsmaly- ~ ) 



(7 '2? 



u = 2E2 4-e" T" + 2E2 — - X . LJ e~ t 



a — sm a za a — sm a 



which may be transformed into 



— cos iir 



^ = 2EX^_"r"siiwe""r- .... (23) 



a— sm a 



(where x r = l—x) by making use of (21). 

 Let T be the current at Q. Then 



r= 2 4s 



kl 1 sin a 

 a 



If T is the initial current, 

 E 



r = 



(-£)• 



^ 



therefore 



1 + 4:Z ^ — cos far - a2t 

 1 



I--^- 2 7-lin-a e T > •"• ' • ^ 



from which the arrival curve of the current may be calculated ; 



r 



for 1 — yr is the proportion of the final current received at Q at 



o 

 time t after contact has been made with the battery at P. 



17. The most easily calculated cases are z=co and 2=0. 

 When £ = go, there is no fault, a i =far, (22) and (23) both 

 become 



2ES1 . farx _*£ 



V= — -2,-r Sin-r- 6 T 

 IT i I I 



and (24) becomes 



^-=2S— COS17T6 • T (25) 



1 n l 



