120 BELL SYSTEM TECHNICAL JOURNAL 



After some rearrangements this can be put into the form 

 E 



where 



R 



\ 



(32) 



e-'" = 



R+ 2Ro 

 Expanding equation (32) in the form of a series, there results 



E 



R-\-3Ro Ro/2R-\-2Ro 



iwo - i [ ir^ ; (1+.— +e— + 



R+Ro 

 Summing up 7i terms of this series, we have 



E 



z = 



R + 3Ro Ro / 2R -\- 2Ro\ / I - e-2"(F+p) 



R + RolR+2Ro R\R-{-2Ro J\l - e-^c^+i-) J 



(SS) 



Since in the above expression Ro -> 0, we can obtain the value of F 

 by writing the first terms of the expansion for the exponential 



l-2F+<^ + 



R _ . _ 2Ro . 

 R+2Ro~ R 



Hence 



F-> 



Ro 

 R 



If now in equation (33) we proceed to the limit, letting 



Ro 



Ro-)0; P->0; 



there results the equation 



1 ^ J_ 



jcoC' ^ ~ 2D 



'=R 



1 - 



1 _ g-t{llRC+j<^) 



(34) 



1 + jcoCR 

 This equation is the symbolic solution of the integral equation 



Ri + ^ fidt = Eoe^^^^+'K 



If we wish the solution corresponding to the impressed voltage, 

 Eo cos (w/ + 6), we take the real part of (34), obtaining 



i = £i 



where 



cos (o^t + d + 8) - [sin (d - 8) tan dje-"^^^ 

 1 



tan 5 = 



^RC 



