758 BEI.I. SYSTEM TECHNICAL JOURNAL 



and denoting x by x. 



If, on the other hand, the impressed e.m.f. is cos a;/, then by formula (i) 



ViP)=j^, (148) 



and 



Now let us consider the operational expansion suggested by the 

 Hea\'iside processes : 



H^ (:-)■+ {^)'-( !■)'+■■■; (7^5 <-> 



and 



=-i(:-r-(-5)'+(^)'-(-5)"+-!'/7iF)- o-^') 



Now let us identify \'II{p) with /;(/) and rcjilace p" by d'/dt': we get 



and 



\ \ d 1 <f3 , 1 rf^ I ,,,, 



^'='^d7--^^573 + ;7rfr^----f''^'^ (152) 



\ d^ I d* ,1 d^ ) ,,,, ,,,„- 



CO' dt' a)-" rf/-* co" rf/'' I 



We have now to inquire into the significance of equations (152) and 

 (153), derived from the operational equations of the response of the 

 system of an e.m.f. sin oit and cos co/ respectively, impressed at time 

 / = (). From the mode of derivation of these expansions from the 

 operational equations it might be inferred that they are the divergent 

 of asymptotic expansions of the operational equations (147) and 

 (149). This would certainly not be an unreasonable inference in the 

 light of the Heaviside expansions we have just been considering. This 

 inference is however, not correct: on the other hand, the series (152) 

 and (153) have a definite physical significance, as we shall now show 

 from the explicit equations of the problem.. 



