4 BELL SYSTEM TECHNICAL JOURNAL 



where, denoting d/dt A(t) by A'(t), 



a{co, t) =A(o) + / cos coy A'(y)dy 

 and IV 



b(co, t) = — J sin coy A'{y)dy. 



The ultimate steady-state amplitudes are evidently the limits of 

 the foregoing as t approaches infinity. Thus if we write the steady- 

 state current as 



then 



and 



I = a{co) sin (w/ + e)+/8(w) cos (w/ + 6), 

 a{u>)—A{6)-\- J cos coy A'(y)dy 



j8(o>) = — / sin coy A f (y)dy. 

 Jo 



For the derivation and a fuller discussion of the foregoing formulas 

 the reader is referred to " Theory of the Transient Oscillations 

 of Electrical Networks and Transmission Systems," Proc. A. I. E. E., 

 March, 1919. 



In the majority of the more important selective networks A(o) =0; 

 that is to say the initial value of the current is zero and the current 

 in response to the applied sinusoidal voltage of the frequency co/2ir 

 is built up entirely from the progressive integrals 



and 



a(co, t) = I cos coy A' (y)dy 

 Jo 



b(co, t) = — I sin coy A '{y)dy 



in accordance with formula IV. The derivative A'(t)=d/dt A{t) of 

 the indicial admittance which appears in the integrals will be termed 

 the impulse function of the network to indicate its direct physical 

 significance; it is equal to the current in response to a "pulse" of 

 infinitesimal duration and moment (or time integral) unity, or, stated 

 in the terminology of the radio engineer, it is equal to the response of 

 the network to "shock-excitation." These formulas therefore estab- 

 lish a definite quantitative relation between the selective properties 

 of the network and its response to "shock-excitation" ; a relation which 



