ELECTRIC ClRCUn THEORY 365 



is in eciuilibrium prior to the reference time / = 0. The mathematical 

 theory of this network depends on the following proposition: — 



The network described above can be treated as an invariable network 

 by eliminating the variable resistance element r{t) and inserting an electro- 

 motive force —r{t)I„(t) : that is, an electromotive force equal and oppo- 

 site to the potential drop across the variable resistance element. Conse- 

 quently the current in the variable resistance branch is determined 

 analytically by the integral equation 



In{t) = ^ f^E{r)AUt-r)dT - ^£r{T)In{r)Ann{t-T)dr. (288) 



The first component is simply the current Io{t) which would exist in 

 the variable branch if the variable element were absent; hence, drop- 

 ping the subscript n for convenience, the current in the variable 

 branch is given by the integral equation 



I{t)=Ut) - ^£r{r)I{r)A{t-T)dT (289) 



and the voltage across the variable element by 



v{t)=r{t)I{t). (290) 



Having determined I{t) and v{t) from this integral equation, the dis- 

 tribution of currents in the network is calculable as that due to a 

 source E{t) in branch 1 and a source v{t) in branch n of the invariable 

 network : that is, the network with the variable resistance element 

 eliminated. 



A very simple example will serve to illustrate the foregoing : — 

 Into a circuit of unit resistance, and inductance L = \/a, in which 

 a steady current lo is flowing, a resistance r is suddenly inserted at 

 time / = 0: required the resultant current /(/). In this case we have: 



A{t) =indicial admittance of unvaried circuit 



= 1-6-"', 



r{t)=r, 

 and the integral equation of the problem is: 



m=Io-r~£ {l-e-<^y)I{t-y)dy 

 = Io-ra I I{t-y)e-''ydy. 



