STATISTICAL ENERGY-FREQUENCY SPECTRUM 377 



To apply the foregoing concept and definition of the statistical 

 value of a function to the problem at hand it is necessary to suppose 

 that the typical impulse /m(*w) is a function of co and certain parameters 

 Xi, X2, • • ', Xn, and that these parameters are statistically specified by 

 probability considerations. Thus we suppose that pm0^m)d\m is the 

 probability that X„, lies between X,„ and X,„ + dK,. G(co) will then be 

 a function of co and Xi, X2, • • •, X„, the amplitudes ai, • • •, ay being 

 regarded as parameters, when defined by probability functions. We 

 then have, in accordance with the foregoing, 



d\ipi{\) ■ d\2p2{^0 



x f" — 00 



Xoo 

 d\npni\n)C{cO, Xj, Xo, •••, X..^ (7) 

 ■00 



X 



(9) 



II 



To apply the foregoing to the simplest possible case let us suppose 

 that the elementary impulses are all identical; Qi = 02 = • • • o.v = 1. 

 and that their distribution in time is purely random. With these 

 assumptions it follows at once from (6) that 



V . , . . , ^ V- t ^, . . ,„l — cos uT „ .Qs 



R{co) ^- f{io^) -^ + 2 •- |/Oco)P :^ , r-^ CO. (8) 



If f{iO) 9^ 0, this has a singularity at co = ; however 



Lim -=. ^''dt = R{(^)do: 



= V I 4>-dt -^ u-\ I 4>dt 



Here v -^ N/T = mean frequency of occurrence of the elementary 

 impulses. This formula is in entire agreement with Fry's results for 

 the Schroteffekt (I.e.). 



To consider a somewhat more involved problem, we shall suppose 

 that the durations of the individual impulses and their amplitudes are 

 distributed at random. We further denote the probability that the 

 duration of any impulse, selected at random, lies between X and 

 X + dX by p{\)d'K. Correspondingly, q{a)da denotes the probability 

 that its amplitude lies between a and a + da. The durations and the 

 amplitudes are then the statistically specified parameters. 



W^e now postulate that $(/) is an alternating series of impulses of 



