296 BELL SYSTEM TECHNICAL JOURNAL 



The expressions for /»(1), pi2), ■ • • p(K) may be derived in a somewhat 

 similar fashion. 



1.2 Statement of Campbell's Theorem 

 Suppose that the arrival of an electron at the anode at time / = produces 

 an effect F(t) at some point in the output circuit. If the output circuit 

 is such that the effects of the various electrons add linearly, the total effect 

 at time / due to all the electrons is 



/(/) = t. Fit- /,) (1.2-1) 



where the k electron arrives at //.- and the series is assumed to converge. 

 Campbell's theorem states that the average value of 7(0 is 



Uj) = V i F{t)dt (1.2-2) 



J—tc 



and the mean square value of the fiuctuation about this average is 



(/(O - mf = V [ F\i) dt (1.2-3) 



J— 00 



where v is the average number of electrons arriving per second. 



The statement of the theorem is not precise until we define what we mean 

 by "average". From the form of the equations the reader might be tempted 

 to think of a time average; e.g. the value 



Lim 1 f /(/) dt (1.2-4) 



7— oo 1 JQ 



However, in the proof of the theorem the average is generally taken over 

 a great many intervals of length T with t held constant. The process is 

 somewhat similar to that employed in (1.1) and in order to make it clear 

 we take the case of /(/) for illustration. We observe /(/) fcr many, say M, 

 intervals each of length T where T is large in comparison with the interval 

 over which the effect F{t) of the arrival of a single electron is appreciable. 

 Let nl{t') be the value of /(/), t' seconds after the beginning of the n^ in- 

 terval, t' is equal to / plus a constant depending upon the beginning time 

 of the interval. We put the subscript in front because we wish to reserve 

 the usual place for another subscript later on. The value of /(/') is then 

 defined as 



7(0 = Lirrit i [i/(/') + J{t') + • • • + Ar/(/')] (1-2-5) 



M-*oo M 



and this limit is assumed to exist. The mean square value of the fluctua- 

 tion of I{t') is defined in much the same way. 



2 Proc. Camb. Phil. Soc. 15 (1909), 117-136. 310-328. Our proof is similar to one given 

 by J. M. Whittaker, Proc. Camb. Phil. Soc. 2,2> (1937), 451-458. 



