602 BELL SYSTEM TECHNICAL JOURNAL 



emission of electrons in this velocity class may then be defined as 

 follows: 



1 ^ 



«o(wc) = p I^ njuc). (33) 



-t TO=1 



During each interval, the deviation from the average rate of emission is 



dm(uc) = fimiuc) — no(uc). (34) 



The mean square of the instantaneous deviation for all P intervals 

 is then 



1 



p 



^^(uc) = p E L^miuc) - no(uc)y. (35) 



The value of the mean square emission deviation may be found 

 from the following considerations. In previous studies of pure shot 

 noise, it was assumed that the electrons are emitted from the cathode 

 independently of one another, that is, the probability of an electron 

 being emitted during a very small interval of time depends only upon 

 the average rate of emission and the length of the time interval. 

 The classical theoretical equation developed for the shot noise from 

 this assumption was found experimentally to be correct to well within 

 experimental errors. Hence, the same assumption may be made in 

 this analysis of tube noise as the presence of the space charge near the 

 cathode can have but a negligible effect upon the rate of emission of 

 electrons from the cathode. Thus, since the electrons in any velocity 

 class are also emitted independently of one another, the application of 

 probability theory shows that the mean square deviation in the total 

 number of electrons emitted with velocities between Uc and Uc + duc 

 in a time, r, is equal to the average number of electrons of this velocity 

 class emitted in the same time, r. 



That is, 



p Z! r-8m-{Uc) = rno(Uc), 

 or 



L ^m~{Uc) ^—^flaiUc). (36) 



Since the period of time, r, is mathematically small, it can be 

 assumed that the instantaneous deviation from the average rate of 

 emission is constant in each of the P intervals and equal to 8m{Uc). 

 If it is assumed that the function representing the instantaneous 

 deviation repeats itself after a very long period of time, L, the Fourier 

 series for the P square-top pulses comprising the instantaneous 



