732 BELL SYSTEM TECHNICAL JOURNAL 



The preceding theorems, together with the power series and ex- 

 pansion theorem solutions formulate the most important rules of the 

 operational calculus, and are constantly employed in the solution of 

 the elect rotechnical problems. On the other hand, the table of 

 infinite integrals furnishes the solution of a set of operational equa- 

 tions, which are of the greatest usefulness in the systematic study of 

 propagation phenomena in transmission systems which will engage 

 our attention. Before taking up this study, how^ever, we shall first 

 s(jlve a few specific problems which will serve as an introduction to 

 asymptotic and divergent solutions involving Heaviside's so-called 

 "fractional differentiation." 



Problem A : Current Entering the Non-inductive Cable 



The non-inductive cable is a smooth line with distributed resistance 

 R and capacity C per unit length; for the present we neglect induct- 

 ance and leakage. A consideration of cable problems leads to some 

 of the most interesting questions relating to operational methods, 

 particularly to questions regarding divergent expansions. It would 

 seem best to allow specific problems to ser\c as an introduction to 

 these general questions. 



The dilTcrenlial etjuations of iho calile are 



(57) 



C^V=-i-I 

 dl dx 



where x is the distance, measured along the cable from any fi.xed 

 point, / is the current at point .v, and V the corresponding potential. 

 Replacing d/dl by the operator p, we have 



RI=-^V 



dx 



(58) 



i'Jimiiuiting, successiveh', I'and / from iliese etiuations, we tl>-'l 



and 



pRCI = ^\l 



/>^CF=gr. 



