CIRCUIT inr.oKV .ixn oi'r.h'.iiKKWir. c.ii.cui.vs (m 



For completeness wo write down ilu- fnllowiiii^ »(|iiiv;ileiils of (20) 

 ami (20-a) 



l(t)=A{o)E{t)-\- f A'(I-t)E{t),It, (20-b) 



Jo 



= A{o)E{t)+ f'A'ir)f-:(l-T)dT, (2(r-c) 



»'o 



=E(o)A(t)+ f'E'(t-r)A{T)dr, (20-d) 



= Eio]A(l)+ rE\T)A(t-T),lT. (20-e) 



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where the primes denote differentiation with respect tt) the artjii- 

 ment. Thus A'(t) =d dt A{t). 



These ec|iiations are the fundamental formulas which mathematic- 

 alK- relate the current to the type of applied electromotive force and 

 the constants and connections of the system, and constitute the first 

 part of the solution of our problem. The most important immediate 

 deductions from these formulas are expressed in the following theorems. 



1. The indicial admittance of an electrical network completely 

 determines, within a single integration, the behavior of the network 

 to all ty(x>s of applied electromotive forces. As a corollary, a knowledge 

 of the indicial admittance is the sole information necessary to com- 

 pletely predict the performance and characteristics of the system, 

 including the steady state. 



2. The applied e.m.f. and the inidical admittance are similarly 

 and coequally related to the resultant current in the network. As a 

 corollary the form of the current may be modified either by changing 

 the constants and connections of the network or by modifying the 

 form of the applied e.m.f. 



3. Since the applied e.m.f. may be discontinuous these formulas 

 determine not only the building up of the current in response to an 

 applied e.m.f. but also its subsidence to equilibrium when the e.m.f. 

 is removed and the network left to itself. In brief, forinulas (20) 

 reduce the whole problem to a determination of the indicial admittance 

 of the network. In addition, as we shall see, they lead directly to 

 an integral equation which determines this function. 



It is of interest to show the relation between formulas (20) and the 

 usual steady state equations. To do this let the e.m.f., applied at 



