ELECTRIC CIRCLTI TllliORV 381 



Then 



whereby the time integral is transformed into an integral with 

 respect to frequency. 



A corollary of this theorem is as follows : 



If two functions 4>i(t), 02(0 supposed to exist only in the epoch 

 0</<7", are formulated by the Fourier integrals 



1 r°° 



<^i(0 =— / |/i(w) I • cos [o)t-di (w)] r/co, 



TT ,/0 



1 r°° 



<^2(/)=— / |/2(co)|-COS [a)/-9o(cj)]rfco, 



TT Jo 



f 4>i{t)<i>2{t)dt=^ f |/i(co)| .|/2(C0)| .COS(01-02)^W. 



then 



The applications of these theorems to circuit theory proceeds as 

 follows : — 



If an electromotive force E{t), supposed to exist only in the epoch 

 0<t<T, is applied to a network of complex impedance Z(ico) = 

 I Z{iu}) I e^ ^^^ we know from the preceding discussion of the Fourier 

 integral, that the electromotive force E{t) and current I{t) are 

 expressible as the Fourier integrals 



1 r'" 



;(/) = — j |/(co) j . cos (co/-e(w))rfa;, 



TT Jo 



(326) 



iCOs(co/ — ^(w) — /3(co))(fco. 



Z{i(jo) 

 It follows at once from Rayleigh's theorem that 



rr.,t^irimiL,^, 027) 



Jo ttJo !Z(?w) I 2 



Now let /„ be the current absorbed in branch w; let 2(^0;) = |c(i'w)|e'"'"' 

 be the impedance of that branch and let E„(t) be the potential drop 

 across that branch. It follows at once from the corollary to Ray- 

 leigh's theorem that 



W^ I En{t)I„{t)dt = ^ / °° I ^i^!^^\ ' 1 g(^co) I cos a(co)^a;. (328) 



