ELl.CTRIC CIRCUIT TllliORV 61 



The first two terms of these formulas are clearly the ultimate steady 

 state values of the \oltage and current waves, and can be deter- 

 mined by e\"aluating- the infinite integrals. A far simpler and more 

 direct wa>', however, is to make use of the fact that the ultimate 

 steady values of V and / are gotten from the operational formulas 

 by setting p = 0. That this statement is true is easily seen if we 

 reflect that the steady d.c. voltage and current are gotten from the 

 original differential equations of the problem by assuming a steady 

 state and setting d/dt = 0. 



From the operational formulas we get, therefore, 



(l + x/ dt]v''e~^' = e-^'^ =e~'<'^, (189) 



(l + X r"rf/)/''e-^'=-^|^ e-^'^ = -yj^e-'=^'^. (190) 



Introducing these expressions into (187) and (188) respectively, we get 



F=e-*^'^-X / " Ve-^'dt, (191) 



I^ ^^e~-'-^^'-\ n Pe-^'dL (192) 



The definite integrals can be expanded by partial integration; thus 

 -X / V''e-^'dt= / Vde-^' 



Jt dt 



Continuing this process we get 



^=--^'''-^-Kl + 4 + xW^+--)^- (193) 



?d 



Using the values of V° and /", as given by (169) and (170), it is ex- 

 tremely easy to compute Y and I, for large values of /, from (193) 

 and (194). 



So far we have considered the current and voltage waves in re- 

 sponse to a "unit e.m.f.," impressed on the cable at .t = 0. It is of 

 interest and importance to examine the waves due to sinusoidal 

 e.m.fs., suddenly impressed on the cable, particularly in view of 

 proposals to employ alternating currents in cable telegraphy. 



