VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 519 



and a correction series A, which depends on the derivatives of the 



steady-state admittance F(tco) with respect to frequency and the 



derivatives of the variable frequency ij,{t) with respect to time. 



If the parameter X is sufficiently large and the derivatives of 5 are 



small enough so that C„ may be replaced by the two leading terms, 



we get 



_, .(n— l)n , ^ , da 



Then by (16) and (18) 



= Y(i^) +^F(2)(ifi). (16a) 



The preceding formulas are so fundamental to variable frequency 

 theory and the theory of frequency modulation that an alternative 

 derivation seems worth while. We take the applied e.m.f. as 



E exp ( iwe/ + id + i C /xdn, (23) 



the phase angle 6 being included for the sake of generality. 



Now in any finite epoch — / — 2", it is always possible to write 



exp liCfjLdt)^ f F(ic^)e''''dco, (24) 



thus expressing the function on the left as a Fourier integral. For 

 present purposes it is quite unnecessary to evaluate the Fourier 

 function F(iw). 



Substitution of (24) in (23) gives for the current 



/ = £-exp (twc/ -}- iO)- I F(zco) F(iwc + iw)e^''^dw. 



tJ —txi 



(25) 



We suppose as before that, in the interval Q ^ t ^ T, n{t) and its de- 

 rivatives are continuous. We can then expand the admittance func- 



