354 BELL SYSTEM TECHNICAL JOURNAL 



If o.v(/) denote the indicial admittance of the Hne at point x, then 

 the current at point x is simply axif), which is given by formula 

 (210-a). But if V{t) is the voltage at point x, the current is also 

 given by 



'Lfv{r)aS-r)dr. 

 at Jo 



Equating these two expansions, we get the integral equation of the 

 problem 



'^- rV{T)ao{t-T)dT = a,{t). 

 tJo 



dK 



xR IT; 

 Now if we write 2^ = pt — A where p = R/2L and A ='— ^_^ then 



and in terms of the relative time T, the integral equation is reducible to 



4^ r V{T-T)e-^Io{r)dr = e-iT^-'Uo{VnT^2A) 

 dl Jo 



while the exact formula for V is by (211-a) 



V(T)=e ^-^Ae -^ / , — dr. 



^ ^ Jo VT(r+2^) 



From the integral equation it is easy to establish superior and inferior 

 limits for V{T)\ it is 



V{T) <e-^ ^°^^^^|^+^^^ = Va(T), 



re~'^ Io{T)Va{T)dT 



>4r =Vi,{T). 



/ e~T h{T)dT 

 Jo 



Both formulas give the correct initial and final values of V; namely 

 e~'^ and unity. Since V lies between Va and Vb, the mean value 

 {Va~\-Vb)/2 also has correct initial and final values and should be a 

 better approximation than either. The table given below shows the 

 orders of approximation obtainable from the case where A =3. It 

 is evident from this table that the foregoing approximate formulas 

 exhibit the form of V(T) qualitatively in a quite satisfactory manner. 



