Theory of Faults in Cables. 173 



is generally greater than v in (47) at any time. In the ex- 

 treme, when s is large, the potential of the line according to 

 (46) becomes nearly E everywhere, and afterwards settles 



down to E( 1— yj, thus, 



kit 



V = E-^(1- 6 "1T) + 



whereas according to (47) it rises, thus, 

 z v \ kit 



v= En-^J(i- e -L) + ... 



In spite, however, of this great difference in the phenomena 

 of the charge, the current at x = I rises in precisely the same 

 manner in both instances, as will be seen on differ entiating (46) 

 and (47), and making sc=l. 



23. In the following example we have to deal with a single 

 imaginary root. Suppose the line is initially charged to po- 



tential -j- y that the end x = is to earth, and that the current 



entering the cable at x = l after £ = 0is simply proportional to 

 the potential there at any moment. That is, v = at x = 0, 



and v=ml-j- at , x = l, where m is a + constant. At time t 



the solution is 



_, 2E(??i — l)cosa . ax - a 21 /ao . 



a(l— m cos a) I 7 x J 



where 



tan a = 7iia. 

 There is one particular case where the potential remains un- 

 changed, viz. when m= 1 . All terms in the expression for v in 

 (48) vanish except the first, for which a=0. The limiting 

 value of m ax 



_„ 2E(sina — a cos «)sin-y- 



A. UX x . / 



sin 



l a(a — J sin 2a) 



IT V 



when a = is -j-; so that (48) is simply 



when m =1. If m is greater than 1, v ultimately vanishes ; but 

 if m is less than 1, an imaginary root a — n V— 1, where n is the 

 + root of 



6 n — € -n 



■—— =mn, 



