380 BELL SYSTEM TECHNICAL JOURNAL 



The application of the precediniy analysis to this case gives 



Sir 

 plus the real part of 



X (aifi(- ^'co) + a2f2(- i^) + Os/.^C- ^'o;)) (17) 



X Lim {. e' E [I (e'"'' + <?'"'= + ^'"'0 ]"-'"- ' • 



It is to be understood that the real part of the second term is alone to 

 be retained. 

 If we write 



1 1 / V _ 1 V 1 — r"^'"! \ 



j^l.l^lAe -re -re )j \ - x\ N N 1-x J 



and 



Lim |. L E = 7-^: •^- < 



A— »•« ^* i A. 



= T * = '■ 



There is therefore an infinity at co = 0, as we should expect. Its 

 measure, however, is finite. 



The preceding is merely an example which admits of extension to 

 more complicated types of signals, as will be obvious to the reader. 

 For example, the probabilities of the elementary signals need not be 

 the same and their number need not be restricted to three. 



IV 



In all the cases discussed above it will be observed that the dis- 

 turbance is "quasi-systematic" in the sense that the elementary 

 disturbances are all of the same wave-form differing only in duration 

 and amplitude. Indeed, some such assumptions as these are essential 

 to the application of the mathematical theory. In the case of atmos- 

 pheric disturbances we have no reason to suppose any such quasi- 

 systematic character exists. Furthermore, even if for the sake of 

 argument, we suppose that the elementary disturbances, which make 

 up static, have a common wave form at the point at which they 



