CIKCl'IT rill-.ORY .1X1) Ofl.h'.tltnx.lt. (■.(/.( r/.C.S 72S 



Iiiletjratiii); llii- lirsl of llii-M- I)\- parts wo li,i\i', 



\vhvTch'(l)=d/dthit). 



If li{o) =0, we have at once 



Comparison with the integral equation for g(/) shows at once that 

 g{t)=h'(t), since the integral equation determines the function 

 uniquely. 



Theorems II and 111 establisli the characteristic Heaviside Opera- 

 tions of replacing 1 '/> bv / dt and p hv d/dt. 



' •'0 



Theornn IV 

 If in the operational equation 



It = I /Hip) 

 the generalized impedance function can be factored in the form 



II{p)=H,{p)-Ih{p) 

 and if the auxiliary operational equations 

 hr = l/mp) 



Jh=l/Ih(p) 



define the auxiliary variables hi and hz, then 



h{t)=jJ\Mh,{t-T)dr 

 ='jJju.{r)hi{t-r)dT. 



This theorem is immediately deducible from Borei's theorem and 

 theorems II and III, as follows. 

 The integral equations are 



kr^7mF)-piwrr''^'^'''''' 



^^=fl,U)e-^.dt 

 ^^=f,.it)e-m. 



