ELECTRIC CIRCUIT THEORY 81 



with F{t) as a continuous function. Now, in accordance with (207), 

 perform the operation of differentiation upon (209-a) : we get 



dF 



-v^ = ^6{t-x/v)e-P'Io{(r\/t^-xyv') 

 ax at 



-v<l)it-x/v)-$-e-p'Io(<T\/t^-x~/v^). 

 The first expression follows from the fact that 



dx V ot 



pi 

 We observe also that — 4>{t — x/v) =0 except at t = x/v, when it is in- 



finite. We also observe that, for f^x/v, 



i 



, 6{t-x/v)dt = l 

 ot 



and that the whole contribution to the integral occurs at t = x/v. 

 With these points clearly in mind, the expression 



V=-v I ^dt 



•J r — 



Jo dx 

 reduces to (211) without difificulty. 



CHAPTER VIII 



Propagation of Current and Voltage in Artificial 

 Lines and Wave Filters 



The artificial line here considered is a periodic structure, com- 

 posed of a series of sections connected in tandem, each section con- 

 sisting of a lumped impedance 2i in series with the line, and a lumped 



Fig. 19 



impedance Z2 in shunt across the line. In the artificial line which 

 we shall consider it will be assumed that the voltage is applied at 

 the middle of the initial or zeroth section, as shown in Fig. 19. This 

 termination is chosen because of its practical importance, and be- 



