ELECTRIC CIRCUIT THEORY 343 



Now refer back to equation (248) giving the required current in 

 terms of Vo,Vi,V2 . . . and the admittances ax{t),a'2s-x(t), .... It 

 follows at once by substitution of the preceding that 



the functions Vo, vi, Vz being zero for negative values of the argument. 

 This result may possibly require a little explanation. 

 Consider the expression 



£f(t-T)-l{r)dT 



where 1(/) denotes a function which is zero for t<to and unity for t>to. 

 It is evidently identical with the admittance a.x(0 provided the 

 proper value is assigned to to- 



Now since l(/)=0 for t<to and unity for t>io, the preceding may 

 be written as zero for t<to, and 



JJtf'f^^-^y^ for/>/o 



which is equal to f{t — to) . 



If we set x = 0, we get the current entering the line; thus 





This has been computed for the case where \/5LCo= 10 and is shown 

 in Fig. 26. Referring to this figure we see that the current jumps 

 at / = to the value \/C/L = l/k, and keeps this constant value for a 

 time interval 2s/v. At this instant the first reflected wave arrives and 

 the current takes another jump, of 2/k. Thereafter it begins to 

 decrease very slowly until time t = 4:s/v at which time it takes another 



