338 BELL SYSTEM TECHNICAL JOURNAL 



Now by (244) we can expand \/Zx{p); it is 



Z,{p) K ' "^' K 



g-7(2x+.i:) Q-y(As-x) 



+XmiM2 ^ h XmiM2" 



+XAti^M2' 



K ' ^'^' K 



p-y(,As+x) 



K ^ 



(246) 



Now we observe that e~'^^ jK is simply the operational formula for 

 the indicial admittance at point x of an infinitely long line with unit 

 e.m.f. impressed directly on the line at x = 0. This will be denoted 

 by axiC). Similarly e~'^^^^~'^^ jK is the operational formula for the 

 indicial admittance at point i^s — x) with unit e.m.f. impressed directly 

 on the line at x = 0. This will be denoted by ats-x^^, etc. 



Recognition of this fact allows us to derive a formal solution in 

 terms of a series of reflected waves. For let a set of functions 

 i^o,'0\,v-i,vz, .... satisfy and be defined by the operational equations 



yo = X(^)=X 



Z'1=Xa12 

 Z'2=XAti)U2 



vz = \yLxA^ etc. 

 It then follows from the preceding and theorem II that 



(247) 



'-«=.4X'' 



Vo{l - r)ax{j) -\-Vx{^t — T)ais-x{r) 



+ Z'2(^-r)a25+*(T)+ . . . 



(248) 



If, therefore, we know the indicial admittance of the infinitely long 

 line with unit e.m.f. directly applied and if we can solve the operational 

 equations (247), then AxiS) is given by (248) by integration. This 

 solution may well present formidable difficulty in the way of com- 

 putation. It is, however, formally straightforward and the numerical 

 computation is entirely possible, the only question being as to whether 

 the importance of the problem justifies the necessary expenditure of 

 time and effort. Without any computations, however, the solution 

 (248) admits of considerable instructive interpretation by inspection. 

 The first term represents the current at point x of an infinitely long 

 Hne in response to a unit e.m.f. impressed at x = through an im- 

 pedance Zx\ Vo = Vo{t) is the corresponding voltage across the line termi- 

 nals proper. The second term is a reflected wave from the other 



