/" 



730 BEI.I. SySlEM TECHNICAL JOURXAL 



y.o\\ f(l) may l)e of such form that the infinite integral 

 }{t)e-'"dt 

 can be evaluated and has tlie \alue F{p) p: thus 



r mc-"dt = ]-F{p). (55) 



^0 P 



This is possible, of course, for many important types of applied forces, 

 including the sinusoidal. 



It follows at once from Theorem I\' that x{t) satisfies and is de- 

 termined by the integral equation 



We have thus succeeded, by virtue of Theorem IV in expressing the 

 response of a network to an arbitrary e.m.f. impressed at time l = o, 

 by an integral equation of the same form as that expressing the 

 response to a "unit e.m.f." That is to say we have, at least formally, 

 extended the operational calculus explicitly to the case of arbitrary 

 impressed forces. 



We now translate the foregoing into the rorrespntuiiiit; Operational 

 Theorem. 



Theorem VIII 



If the operational equation 



h = \iHit>) 



expresses the response of a network to a "unit e.m.f." and if an arbitrary 

 e.m.f. E impressed at time t = o, is expressible by the operational equation 



E=V{p) 

 or the infinite integral 



f 



\it)e-^'dt=^M 

 P 



theti the response x of the network to the arbitrary force is given by the 

 operational equation 



x-Y^ 

 Hipy 



and xU) is determined by the integral equation 



pmp) Jo ^^'^' "'■ 



