CIRCUIT THF.ORY AND OPf.RATIOSAL CALCULUS 747 



prM-ided the function k=k(t), defined by the operational equation k = F{p), 

 and the infinite integrals 



I t'k{l)dl (u= I, 'J ) 



exist. , " 



We shall now apply the foreKoinyj tlieory to a physical pr()l)leni 

 discussed in llie last section : narneU', the current entering an infinitely 

 lonj; line of inductance L, resistance R and capacity C [wr unit length. 

 It will he recalled (see equation (100) ) that the integral equation of 

 this prohleni is 



J^ , ^ == r e-P>I(t)dt 

 \ L Vt*'+2\p Jo 



1 the soluti<in 



C L 



where X = R 21., and thai the soluti<in is 

 / = J-f«. ^'/„(X/). 



We can derive the snluiinn in anoilu-r form ,i|)pr<)priate for our pur- 

 poses 1>\- writing 



Now since 



y/p Jo y/wt 

 and 



^^_ = / e-^'-.r dl 



VP+2\ Jo y/i^t 



it I'dHows from Corel's theorem tliat 



\c 1 n e"^^^ 



' L, wJo -y/r y/t — T 



Now subject this definite integral (omitting the factor \/C/L) to 

 the same process applied to (117) : we get 



^Vl 'Jo VT 2tJo 1! ^ *" 



1.3 



(2/) 



^ (2/)Vo 2! '^ at-t ... ^ 



