672 BELL SYSTEM TECHNICAL JOURNAL 



the equations (12) can be written 



ai(co;,) 



eg\ + figepl = Z 





ZM 



Z(wh) 



kh COS [oiht -\- Kh — ^l(w/,)J 



kh COS [cO;>/ -\- Kh — iPi{oih)~\ 

 kh COS [co^/ + K.h — <^3(wa)] 



Z(wft) 



kh COS [cO;,/ + K/t — V'4(wa) J 



(20) 



This concludes our consideration of effects of the first order and we 

 now turn to those of the second order. For this purpose we substitute 

 the values given by (20) in the right-hand side of the expressions (10) 

 for the grid and plate e.m.f.'s, and we then obtain two expressions, 

 each of which is equal to a sum of sinusoidal terms. If we limit our 

 attention to the terms of frequency (001 — 002), it is readily shown that 

 the grid e.m.f. of this frequency is equal to the real part of 



rgT2 



aiioii) / a4(co2) 



Z(cOi) \ Z(C02) 



l/aM£ I dfXg \ azM / 0:2(0^2) 

 ■^ 2 \ a£p "^ ^^ dEg I Z(coi) V Z(co2) 



1 dlXf, [0:1(001) ( Oil{w2) 



+ 



"1(002) \ 0:2(001) 



2 dEg \ Z(ooi) \ Z(C02) / \ Z(o02) / Z(ooi) 



and the plate e.m.f. is equal to the real part of 



^1^2«'(^"»-"2>'+''l-''S^ (21) 



r'P: 



0.3(001) / 03(002) 



Z(ooi) \ Z(0O2) 



1 / dfXp 



diJLp \ ai(wi) / 01(002) 



+ 2 \ dEg "^ ^^ dE„ I Z(ooi) V Z(a;2) 



1 dup r oi(ooi) / 02(002) \ , / oi(oo2) \ 0:2(001) 



2a£plZ(0Oi) \ZMl \Z(0O2),/ Z(oOi) 



kik2e'^^''^-"^'^'+'^-'^\ (22) 



where a bar over a quotient indicates its conjugate complex. 



It follows from Fig. 3 by straightforward calculations that the 

 currents ipi and igi produced by the e.m.f.'s (21) and (22), are 



*'p2(00l — OO2) = R 

 ^02(001 — OO2) = i? 



[ej 



+ 



[^p] 



Z6(ooi — CO2) Zd(ooi — W2) 



(23) 



Za(cOi — 002) Zc(0Oi — OO2) 



where [e^] and [e^,] are abbreviations for the complex quantities (21) 



