846 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



The inconvenient electric intensities are now easily eliminated by substitu- 

 tions from the d-c. equations (39), which give 



g,= l^A+^ e-% + -^ (e-^ - l)i (58) 



Ub Ub ecc^Ub 



The value of A is now introduced from (54), with Ea again ehminated by (39), 

 and a grouping of terms then gives the final equation 



This equation gives tlie following values of starred coefficients: 



D* = -^ (fie-f + «"" - i) (60) 



2eUt/ 



These coefficients may now be rewritten in any desired form; and, to con- 

 form with previous articles on electron streams, we replace co by its equiva- 



lent expression — ^; and we also express / in terms of the space charge 



factor r from (38), that is, 



I-^%{Ua+ Ub) (61) 



These substitutions then give the coefficients in the form 



£* = -1 [Ui, - ma + Ui)]e-' (62) 



ft 



F* = 2j£ ( U. + U. \ . 



This is obviously a longer mode of expression, but it has two advantages: 

 it is convenient for circuit calculations, and it permits a direct comparison 

 with previous articles on electron streams. 



The equations for ut and (vb — Va) are obtained by similar substitutions 

 in (46) and (49) ; and the three equations are then written in the symbolic 



