(24) 



THEORY OF MULTI-ELECTRODE VACUUM TUBE CIRCUITS 673 

 and (22), respectively, and 



In (24) the introduction of the angles i/' is convenient when it is desired 

 to evaluate the real parts of the expressions (23). 



The expressions (21) to (24) can be used to obtain any second-order 

 current of frequency (coa — cob) by replacing coi with oia and co2 with oib- 

 The remaining second-order currents are found by a process similar 

 to that above. For instance, ip2{2oi^ and ig2{2wi) are given by the 

 right-hand expressions in (23) provided the e.m.f.'s [e] are those of 

 frequency (2coi) and the impedances Z are evaluated at (2wi). In 

 passing it may be remarked that equations similar to those in (23) 

 and (24) also occur when third and higher-order effects are calculated. 



Special Cases 



If the impedances Zi, Z2, and Z3 are infinite the case treated above 

 reduces to that considered by Llewellyn,^ and after proper simplifica- 

 tions the previous equations give results identical with those obtained 

 by him. For instance, if we take the limiting values of Cpi in (12) as 

 Zi, Z2, and Z3 tend to infinity, and if we then divide the quantity 

 inside the summation sign by — Zp{coh), we get an expression for ipi 

 which may be shown to be identical with equation (33) in Llewellyn's 

 paper, except for differences in notations. Similarly, the plate current 

 ipiioii — C02) in (23) reduces to a value which may be shown to be 

 equal to the sum of his equations (35) and (36), evaluated for this 

 type of second-order current. 



Another special case is that when the impedances Zi, Z2, and Z3 are 

 all finite but conductive grid current is absent. We then have Hg 

 equal to zero, and Rg equal to infinity, and the previous general equa- 

 tions are simplified correspondingly. 



We arrive at the case treated by Peterson-Evans ^ by maintaining 

 the assumption of no conductive grid current but by assuming Zi, Z2, 

 and Z3 to be infinite. For instance, if then ipi and ^^2(^1 — C02) are 



