702 REI.I. SYSTEM TECHSICAL JOURNAL 



Equating the two expressions (2G) and (27) for /(/) and dividini; 

 through by e*" we get 



■^+y{t)e-^'=pfj A{r)e'f'dT-pj\4{T)e-f'dT+A{t)e-P'. (28) 



This equation is \alid for all \alues of /. Consequently if we set 

 /= 30, and if the real part of p is positive, only the first term on the 

 right and the left hand side of the equation remain, the rest vanishes, 

 and we get 



pm=r^^'^''""- ^''^ 



This is an itilei^ral equation ^ valid for all positive real values of p, 

 which completely determines the indicial admittance A{t). It is on this 

 e(|uation that we shall base our discussion of operational methods and 

 from which we shall derive the rules of the Operational Calculus. 

 Equations (20) and (29) constitute a complete mathematical formula- 

 tion of our problem, and from them the complete solution is obtainable 

 without further recourse to the differential equations, or further con- 

 sideration of boundary conditions. 



To summarize the preceding: we have reduced the determination 

 of the current in a network in response to an electromotive force 

 E{t), impressed on the network at reference time t = o, to the mathe- 

 matical solution of two e(|iiations: first the integral equation 



Wipr r ^^^^^-"'^ (29) 



and second, the deiinite integral 



l{t)=j^£A(t~r)E{T)dT. (20) 



It will be observed that in deducing these equations we have merely 

 postulated (1) the linear and invariable character of the network and 

 (2) the existence of an exponential solution of the type e'" / Z{p) for 

 positi\e values of p. Consequently, while we ha\e so far discussed 

 these formulas in terms of the determination of the current in a (inite 

 network, they are not limited in their application to this specific 

 problem. In this connection it may be well to call attention explicitly 

 to the following points. 



'An integral equation is one in wliiili the iMikiiowii liiiution appears iiiuler (lie 

 sign of integration. (29) is an integral ccpiation of tlie I.aplace type. If Z(p) is 

 specilied, A(i) is unicpiely determined. Melliods for solving the integral ecjualions 

 arc considered in detail later, in connection with the exposition of the Operational 

 Calculus. The phrase "all positive values of p" will be understood as meaning all 

 values of p in the right hand half of the complex plane. 



