698 BF.LL SYSTEM TECHNICAL JOURNAL 



or a single Iiranrh (for the case of equal subscripts) may be under 

 consideration. 



From the linear character of the network, it is evident that if a 

 steady e.m.f. E = Et is inserted at time t=T, the network being in 

 equilibriimi, the resultant current is 



£,-J(/-t). 



Generalizing still further, suppose that steady e.m.fs. E,,, E\, Ei, . . . En 

 are impressed in the same branch at the respective times tu, t\, t<> 

 . . . Tn\ the resultant current is evidently 



EoA{l)+E,A{t-T,)+ . . +E„A{t-Tn) =^EjA(t-Tj). (18) 



To apply the foregoing to our [)robleni we suppose tiiat there is 

 applied to the network, initially in a state of c(|uilil)rium, an e.m.f. 

 E{t) which has the following properties. 



1. It is identically zero for t<o. 



2. It has the value E{o) {or o<.t<M. 



3. It has the value E{o)+AxE for M<l<2Sl. 



4. It has the value E{o)+AiE+A2E for 2A^</<:U/. 



In oilu-r words it has the increment Aj£ at time / = /A/. 

 Kvidentl>- then the resultant current /(/) is 



EnA(l)+A,EAit-M)+ . . +A„E.A{l~i!M]. 



Now evidentK' if the inter\-al A/ is made shorter and shorter, then 

 in the limit Sl->-<ll and j\1 = t and 



\jE='-^E{T)dT. 

 At 



Passing to the limit in the usual m.ininr this summation becomes a 

 delinite integral and we get 



m=E{o)A{l)^ f'A(l-T)''~E{T)dT. (19) 



Jo (IT 



Finally by obvious transf(jrmalions of tiie e.\|)ression we arri\c at ilie 

 fundamental formula of circuit theory 



I{t) = jjU(t-T)E{T)dr, (20) 



= jJ^'E{t-r)A{r)dT. (20-a) 



