376 BELL SYSTEM TECHNICAL JOURNAL 



the mth elementary disturbance being supposed zero until / = /,„. 

 If we now write 



4>„,{t)e^"'dt, (5) 







it is easy to show by the methods employed in my previous paper 

 that 



1 



A'- 1 .V 

 + 2 X^ Z aradniCmCn + S^n^r) COS a)(/„ — /,„) (6) 



TO=1 n=m+l 



N-\ N 

 + 2 1] X! aynOiniCmSn — 5,„C„) sin Cj(/„ — /,„) . 

 m=\ n=m-\-\ 



This is more compactly expressible as 

 1 



+ 2E' i {0,nan■frn{i0:)■fn(-ic^)e"^'''^-''^'}ne.^V.ri■ (6a) 



Now, obviously, if the amplitudes ai, • • • , a,v and the wave form of 

 the elementary functions 0i, • • • , 0,v are specified, G(w) is uniquely 

 defined and determined by the preceding formula. This, however, is 

 not the case in the problem under consideration, where at best the 

 functions are specified only statistically by probability considerations. 

 Under such circumstances, when the problem is correctly set and 

 sufficient statistical information is furnished for its solution, we 

 introduce the idea of the statistical energy-frequency spectrum i?(co) 

 defined as follows: 



The statistical energy-frequency spectrum Rico) is equal to the weighted- 

 average of G(w) for all possible values of G(w), the weighting being in 

 accordance with the probability of the occurrence of each particular 

 possible value. 



For example, the statistical value of a function f(xi, X2, •••, Xn), 

 where the variables Xi, • ■ • , .r„ are defined only by probability con- 

 siderations, is, in accordance with the foregoing definition, 



/1 00 nx /'CO 



I dXipi(Xi) • 1 dX2p2{X2) ■ ■ ■ I dXnpniXn)-f{Xi, X2, • • -, Xn), 

 *J — aa U — ji *J — jn 



where pm{.Xm)dxm is the probability that x^, lies between .v,„ and 



Xm ~r O'Xm' 



