734 BELL SYSTLM TF.CHNICAL JOURNAL 



we ha\t- the integral eciiialinii 



The solution of this is known (see fornnila (c) of the preceding table 

 of integrals) : it is 



Heavisicie arri\e(l at tliis sdliilion from considering the known 

 solution of tlie same ])rol>leni in the tiieor\- of heat flow. He liiere- 

 fore inferred that the operational e(]nation 



has the explicit solution 



This is correct; we, however, ha\e (leri\ed it directly from the integral 

 equation of the problem and the known integral 



We then see from the foregoing that, if a "unit e.ni.f." is impressed 

 on the cable terminals, the current entering the cable is initialK' 

 infinite arid dies away in accordance with the formula y/C/-wRt. 

 The case is, of course, idealized and the infinite initial value of the 

 current results from our ignoring the distributed inductance of the 

 cable, which, no matter how small, keeps the initial current finite, 

 as we shall see later. 



Now let us go a step farther; suppose that in addition to (iistril)iitcd 

 resistance R and capacity C, the cable also has distributed leakage 

 G per unit length. The difTerential equations are now 



RI= -4-V 

 ■dx 



^ (70) 



{Cp + G)V=--°L 



d-v 



Conse(Hienli\- it follows that in the oi)eraiional c(iiiation for the cm-rent 

 entering the cable we need only replace Cp by Cp+G. Tiierefore, 

 when leakage is included, equation ((Hi) is to be replaced by 



where X = G/C. 



