60 BELI. SYS I EM TECHNICAL J OURS. IE 



It follows at once by comparison with (183) and the rule that \/p is 

 to be replaced by | dt, that 



V = 



(l-\-\ j'df\V"e-^K (185) 



By a precisely similar procedure with the operational f(jrmula (176) 

 for the current, we get 



/=(l + X r'rf/)/"e-^' (186) 



where /" is the current in the non-leaky cable. Now by formulas 

 (169) and (170) 



V°=—,^ / —^dl, (169) 



\/ -K *JQ t\/i 



/"= .l_^e-°/^ (170) 



which completes the formal solution of the problem. 



Formulas (185) and (186) are extremely interesting, first as showing 

 the superiority of the definite integral to the series expansion — compare 

 (185) with the series expansions (181) — and secondly as exhibiting 

 clearly the effect of leakage on the propagated waves of current and 

 voltage. We see that in both the current and voltage the elifect of 

 leakage is two-fold: first it attenuates the wave by the factor e~ ', 

 {\=^G/C), and secondly it adds a component consisting of the pro- 

 gressive integral of the attenuated wave. This, it may be remarked, 

 is the general effect of leakage in all types of transmission systems. 

 Its effect is, therefore, easily computed and interpreted. 



Formulas (185) and (186) are very easy to compute with the aid 

 of a planimeter or integraph; or, failing these devices, by numerical 

 integration. However, for large values of /, the character of the waves 

 is more clearly exhibited if we make use of the identity 



l'dt= I dt- j dt 

 Jo Jo Jl 



F-(l + X j'^dt\ Ve-^'-xj " Ve-^'dt (187) 



J^^l + X /'"r//)/"c-'^'-X^'"/V-'^'^/. (188) 



whence 



and 



