(194-a) 



62 BELL SYSTEM TECHNICAL JOURNAL 



We start with the fundamental formula 



= £Kt-r)h'{r)dr 



pro\ided h{o) =0, which is the case in the cable. 

 \i f{i) =sin o)t, we write 



Xsit) =sin wt I cos wt.h'{i)dt 

 Jo 



— cos ut j sin wt.h'{t)dt. 



Similarly, if the impressed e.m.f. is cos wt, 



Xc{t) =COS ut j COS wt.li'{t)dt 



+ sin wt \ sin <xit.h'{t)dt. 

 Jo 



The investigation of the building-up of alternating currents and 

 \-oltages, therefore, depends on the progressive integrals 



r 



C= f cos ict.h\t)dt, 



^, (194-c) 



S = j sin wt.h'{t)dt. 



For the case of the voltage waves on the non-inductive, non-leaky 

 cable these integrals, by aid of equations (169), become, if we write 

 co' = aco/4, 



1 f^ g-i/^ cos CO V 

 ^ ~ ~7^ i 7= dr, 



V TT t/o r\/ T 



(194-d) 

 „_ 1 f^ e-^/^sin coV 

 'J — — ^— / 7=- cfr, 



•V/ TT »>'0 TVT 



(194-b) 



V 



where, as before, t = 4//q:. 



For the current wave we have, by (170), 



2 /'Yi 1 \ g~^/'^cos cjV 



2 r^l 1 \ g-^/^sin cjV 



(194-e) 



