Theory of Faults in Cables. 65 



5. The influence of a fault on the amplitude of reversals 

 may be readily calculated. In the first place, without con- 

 densers. Let contacts, alternately + and — , be made with 

 a battery at the beginning of the line, while the distant 

 end is to earth. If the reversals are sufficiently rapid, the 

 resulting received current is nearly a simple harmonic function 

 of the time. Let c be the capacity and k the resistance per 

 unit of length of cable of length 7, having a fault in the middle 

 of resistance zkl. Also let r be the period of a wave, or the 

 time occupied by a pair of contacts. Then the maximum 

 strength T of the received-current waves is 



r=^ l >*^(e» + e-»-2co$n)-ife n + e-" + 2cosn 



1 1 1 ~i 



+ ^ — (e n -e- n + 2sm?i)+ Tr ^(e n + e- n -2co$n) V , 

 znz v y b?rz N I 



where m = a / c " ^ and E is the electromotive force of the 

 battery. Or, approximately, 



\ Znz b/rz / 



E W2__„ A , 



E 



H ere is the current that would be produced in the line if 



perfectlv insulated, and permanent contact made with the bat- 



8«^2 . , , , ,'/- , 1 , 1 \" f 



terv. e~ n is the reversal-factor, and ( 1 + -=: h Q ., 9 ) 



it \ Znz bn~z~J 



the fault-factor. Xow, if V is the greatest current possible 



with the fault, 



r= 5 . 



u / 1 \' 



therefore T , . 



1 o 

 where -.1 



+ Tz 

 P- 



v-i 



8>i'V 2 



r 



and <f>(n) is the reversal factor. =p represents the proportion 



A o 

 of the maximum received current which is arrived at bv the 



Phil. Mag. S. 5. Vol. 8. Xo. 46. Ji/Zy 1879. F 



