646 



BELL SYSTEM TECHNICAL JOURNAL 



Following usual conventions, the real part of the complex expression repre- 

 sents the physical situation, and the exponential time function will usually 

 be omitted in what follows. 



Similarly to Appendix III, the induced current in the gap is 



/2 - - ^(^2)/ 



The gap voltage is still the same, viz. 



T- • , T- .;(w<2-(t/2)) 1 2'oX y(a)<.,- 



1 sni wto ^^V e = H — e - 



^ e 



(t/2)) 



(el4) 



Accordingly, the electronic admittance to the fundamental will be 



V _ -^2 _ /o ,2 d j((irl2)-e) 



V Vo X 



(^' -f ■'"- Kf-'"' + ?■"')) ^^^-^^ 



The argument of /i and its derivatives is understood to be .Y. The 

 derivatives can be evaluated in terms of Jo and /i by repeated use of the 

 Bessel function recurrence relations (see Jahnke-Emde, Funktionentafeln, 

 p. 144). The result is 



Vo A 



(■'.-i[(-f)^--^»] 



-^[ux' + x')A-ix'jo'^+ •••) 



(el6) 



The real part of this will be of interest; it is 

 ^ I ^0 ^2 . ^/, , cot^r/. X 



I lux' + x')j, - ixVo]) 



(el7) 



The power generated by the electron stream is given by 



P = -hGeV 



= —lol'o ~r I sin 



K- 



A - ^, ((X' + X\r, - 2A'\Ao)] 



+ 



cos 6 



[0-T)"-f4 



(el8) 



Jo + 



