760 DELI- SYSTEM TECHNICAL JOURNAL 



This process can he repeated iiidefinilely, and we get 



' \o>dt o}^dt'^ i^' dt' "^ 0)="-' dC--')^' 



'^h^J, sin '-(--') |2^/'(r)</r. (159) 



The series expansion (159), except for tlie remainder term, is identical 

 with the series expansion (152) derived directly from the operational 

 equation. This series may be either convergent or divergent, de- 

 pending on the frequency 03/2w and the character of the indicial ad- 

 mittance function fi{t). In the important problems of the building-up 

 of alternating currents in cables and lines we shall see that, even when 

 divergent, the series is of an asymptotic character and can be employed 

 for computation. 



We thus arrive at the following theorem: 



If an e.m.f. sin wt is impressed at time / = on a network or system 

 of generalized indicial admittance /?(/), and if the transient distortion, 

 Ts, is defined as the instantaneous difference between the actual re- 

 sponse of the system and the steady-state response, then Ts can be 

 expressed as the series 



\i>} at oi^dt' oi'dt' u)^" ' dt^" V 



with a remainder term 



(-1)" 



J sm uj(r-/)^-^2M^/;(i 



)(/t 



If the impressed e.m.f. is cos co/, the corresponding series for the trans- 

 ient distortion, T^, is 



with a remainder term 



(-1)"/'" . ,N<^^"+', 



(Hil) 



-j cos Oi{T-t)^^;^Jl{T)dT. 



The second part of this theorem, relating to the transient distortion, 

 Tc, in response to an e.m.f. cos ut, is derived from formula (31) by 

 processes precisely analagous to those employed above in deriving 

 the series expansion for T,. The derivation will be left to the reader. 



To summarize the preceding discussion of the divergent solution of 

 operational equations, it may be said that the theory is as yet rather 



