sii.ixiirr. ciKcriis .ixn >/.///( i.\ 1 1 kii ki m i. m 



that * (/) as (li'hiK'tl l)y (1")) is tin- most ni-iicral typo of (lisliirl)ance 

 |)ossil)lf. Thi- only assiim|)tion made as yi't is that thi- instants of 

 inciili-nit' t\ . . . t^ are (listril)iilc(l at random ii\tr ilio epoch o^t^T; 

 an .issumplion vvhicli is dearly in aicorilanc c wlili the facts in the 

 case of static interference. If we write 



(V(,w) = / (/>,(/) COS wt (It, 



Sr{<ji) = / </>,(/) sin wl (It. (16) 



•'O 



it follow^ from (2) and (15), after some easy rearrangements that 



.V .V 



l-\^) ;■-•= N^ N^cosa)(/,-/,)(G(<.))C(a,) + 5,(a,)5.(w)] = 



r-l i-\ 



V^G» + 5,=((o) (17) 



+ ^ ^,nsa>(/,-/,) [G(u,)G(a.)+5r(u)) 5,(0,)], r?t«. 



The first summation is .-impl\- ^ |/r(a,) ]-. The double summa- 

 tion in\ol\es the factor cos a, (t, — li). Now by virtue of the assump- 

 tion of random time distribution of the elementary disturbances, it 

 follows that /, and /,, which are independent, may each lie anywhere 

 in the e()och o^t^T with all values equally likely. The mean value 

 of j F (u>) |- is therefore gotten by a\eraging- with respect to tr and fj 

 o\er all possible values, whence 



1 FM 1== ^l/,(a,) ;.+2/T' ^~^"^ 



X ]^ ^ [CMC,M+Sr{(^)S,M] (18) 



and 



/- ' \^ /•°° |/r(fa)) i' J , 2 \^ \^ /•°° l-cosoir f^, ^ . 



+5»5,(co)li '^'" 



1 Ziico) 



' The averaging process with respect to the parameters /,. and /, employed above 

 logically applies to the average result in a very large number of epochs during which 

 the system is ex()ose<l to the same set of disturbances with different but random 

 time distributions. Otherwise stated, the averaging process gives the mean value 

 corresponding to all possible ecjually likely times of incidence of the elementary 

 disturbances. The assumption is, therefore, that if the epoch is made sufficiently 

 large, the actual ctTect of the unrelated elementary disturbances will in the long 

 run be the same as the average effect of all possible and equally likely distributions 

 of the elementary disturbances. 



