ULTRA-HIGH-FREQUENCY VACUUM TUBES 



643 



same result may be obtained from (41), (42) and (43) above by 

 writing e~^ in series form. The result is : 



n=0 



J ^ KTKn + 2) ^ a^TKn + 2) ^ u^T' 



(w + 4) ! 



(w + 3)! ' (« + 2)! 



, , aaT'^in + 1) , UaT 



Mi 



fli = L 



(-^)" 



n=0 W! 



Ml = L 



(-^)" 



re=0 «1 



/r 



jr- 



{n + 2)\ '(« + !)! 



xr''(w + 1) y 



(w + 2)! ;J' 



1 KT' 1 



w + 1 2mo (w + 3) 



+ ai 1 



2Mr 



Ml 



1 



aoT 1 



w + 2 2uo (w + 3) 

 ooT 

 2uo 



+ a,T 1 



+ M1 1 





«0 /J 



(41a) 



(42a) 



(43a) 



These latter forms of expression clearly show what happens at low 

 frequencies and they prove that the equations do not give infinite 

 values for any of the components so long as aoT/uo and KT^/uo remain 

 finite. Now in general 



aoT ^ {KT-^aa)T 



Wo 



and 



Kin 



Wo 



J>2 

 i^ y + aaT + Ua 



-K: Y + aaT + Ua 



and these remain finite for all finite positive values of T, K, aa and Ua- 

 Thus the difficulty at the origin which was discussed * in the Proceedings 

 of the Institute of Radio Engineers, February, 1935 is overcome by the 

 generalized definition of T in (10). It is true that 5i still tends toward 

 infinity when T approaches zero if aa and Ua are zero and m is different 

 from zero. This means that the ratio of the variation in transit time, 

 5i, to the transit time T tends toward infinity when T approaches zero, 

 and when variations in initial velocity are still present with no constant 

 initial acceleration. This is a logical and expected result, but it leads 

 also to the conclusion that the electrons actually halt their forward 



