Theory of Faults in Co.bles. 69 



condensers, shunted or unshunted, at either or both ends. Let 

 the signalling be from P to Q ; then E x is the battery resist- 

 ance, and E 2 the receiving-instrument's resistance. Let the 

 electromagnetic capacity of the latter be L. Further, let 

 there be n faults of resistances Z 1} Z 2 , . . . at distances a\, x 2j . . . 

 from the beginning P, where a?=0. 



Fig. 4. 



At the time t = let the potential of condenser Ci be V 1} 

 and Y 2 the potential of G 2 . Further, let Y=/( t ?;) be the po- 

 tential of the line when t = 0. Since we have taken into account 

 the magnetic capacity of the receiving-instrument, the speci- 

 fication of the initial state of the system is not complete unless 

 we know the current in E 2 when i= 0. Let this be Gr. Then 

 we want to know v } r 1; v 2 , and g at time t, where r, r 1; v 2 , and 

 g are what Y, Y 2 , Y 2 , and G then become. 



11. Between any two faults let the initial potential be ex- 

 panded in a convergent series of the form 



2Asin/^+&\ 



This can be effected in an infinite number of ways. Then 



XAsin(y + Ae"^ (2) 



where T = ckl 2 , satisfies the partial differential equation (1), 

 and will therefore represent the potential at time t between 

 the same limits, provided the sets of constants A, a, and b are 

 so determined as to make (2) satisfy the conditions imposed 

 by the presence of the faults and the terminal connexions. 

 This, of course, can be done only in one way. 



At each of the faults two conditions are imposed. First, 

 the potential must be continuous at the fault ; secondly, the 

 current in the line going to the fault on the left side exceeds 

 the current growing from the fault on the right side by the 

 current in the fault itself from the conductor to earth ; and 

 the latter is, by Ohm's law, equal to the potential of the line 

 at the fault divided by the resistance of the latter. 



Let Y 0Jl be the initial potential between x = and a?=# l3 

 Y^j-., between x=X\ and x=a; 2j and so on. Then the first 



