. ELECTRIC CIRCUIT THEORY 51 



The solution of (1G4) for V was considered in some detail in the 

 preceding chapter; it is, by (129) 



F=-^/ '—-dr (166) 



where T = 4//Q: = 4//.x-i?C Series expansions of this solution were 

 also given. Another equivalent form is, by (131) 



F=l 





(167) 



This last form, recognizable also from inspection of the series expan- 

 sion (132), is useful because the integral term is what is called the error 

 function and has been completely computed and tabulated. 



Before discussing these formulas and the light they throw on propa- 

 gation phenomena in the non-inductive cable, we shall derive the 

 solution for the current. A very simple way of doing this is to make 

 use of the differential equation (57) 



Rdx 

 Now from (166) and the relation 



we get 



9 _dT d 



dx dx dr 



a^ 1 g-i''^ d 4/ 



ax- 



whence 



^e-^i-^. (168) 



wRt 



It is worthwhile verifying the formula by direct solution from the 

 operational equation (165), From formula (g) of the table of in- 

 tegrals, we have 



' C 

 R 





