642 BELL SYSTEM TECHNICAL JOURNAL 



With the restriction (c8) imposed we obtain 



a„ = ~ j h cos n Idi + (f i- 8 1 + ~ sin di j ddi (clO) 



If we let V? = —6 all coefficients b„ will be zero and 



a„ = 2(-\)"IJ„(Xn), (/,, = /,, (ell) 



Thus the first order expansion for the current returning throujj;h the gap is 



/, = /„ [(1 - 2Jr(X) cos a>(/2 - Tu) 



(cl2) 



+ IJ.ilX) cos 2co(/2 - To) • • • ] 



Our principal interest is in the fundamental component, which in complex 

 notation is given by 



{hlf = -2I,JiiX)e'''^''~''^ (cl3) 



It is shown in Appendix II that the circuit current induced in the gap will 

 be given, if account is taken of the phase reversal of k resulting from the 

 reversal of direction of the beam, by 



The gap voltage at the time of return will be v — V sin o^t^ or in complex 

 notation 



Hence the electronic admittance to the fundamental will be 



In the foregoing it was assumed that h was a single valued function of 

 /i . We may generalize by writing (c3) as 



Y^I.dl^ = hdto (cl6) 



For sufficiently large signals there may be several intervals <//i which con- 

 tribute charge to a given interval dt2 and hence we write a summation for the 

 left hand side of (cl6). \\ hen the Fourier analysis is made and the change 

 in variable from I2 to /1 is made the single integral breaks up into a sum of 

 integrals. In Fig. 127 we plot time /, on a vertical scale with the sine wave 

 indicating the instantaneous gaj) voltage. Displaced to the right on a ver- 

 tical scale we plot time Aj . The solid lines connect corresponding times in 

 the absence of signal for increments of time dl^ and df-: . \\ hen sufficiently 



