466 BELL SYSTEM TECHNICAL JOURNAL 



When the series in (2.18) of type corresponding to the left-hand members of 

 (2.20) and (2.21) are replaced by the equivalent righthand members, positive 

 exponents containing the squared variables of integration are introduced 

 which cancel the negative exponents already present in the integrand. The 

 resulting integral may be written: 



G{a) =^1\ du\ iu'- /)[/i(l + a, »)/.(! - a, v) 



4 Jo *'o 



+ M1 + a,u)Ml - a,v)]dv, (2.22) 

 where 



/i(a, o;) = 1 + 2 2^ exp — — — cos — - (2.23) 



m=l Zk I 



00 2 2 



/2(a, x) = 1 + 2 2^ i-r exp — — — cos -— (2.24) 



m=l Zk I 



The integrations may now be performed without difficulty. The complete 

 result, which as we shall immediately show is hardly ever necessary to use 

 in full is: 



Cr{a) = -1 2 -„ exp ( - ^^ ) sin/2 



A 2 2 



4:n IT a 



-\r -^1^ l^yrn 7^ n) — exp 



A / 2 2\ 2 joooo 1 



. , 4(w — nW a «v^v^/ ^n 1 



s,n/, - ^^ 2 E {m ^ n) ^^ _ ,^, _ ^^ _ .^, 



exp -^'^-' - ^)' + ^^ - ^^'^' sin/^ -^t^- - i^' - ^- - ^'^'- " (2.25) 



An alternative derivation of (2.25), subsequently suggested by Mr. S. O. 

 Rice, is based on the fact that €(/) as defined by (2.4) or Fig. 3 is a periodic 

 function of Ei which can be expanded in a Fourier series with period Eq. 

 Substituting the series in (2.5) leads to an expression for €(/) e {t -\- t) as the 

 product of two Fourier series. After proof that it is permissible to write 

 this product as a double series and to calculate the average sum as the sum 

 of the averages of the individual terms the problem is reduced to a double 

 series in which the typical term is proportional to the average value of exp 

 i(uVi + VV2) where u and v are constants depending on the position of the 

 term in the series. Rice has shown^^ ^^j^^t the average value of such a term 

 is exp [— (w^ + :^)^o/2 — uv\1/t\. Summation of these terms leads again 

 to (2.25). 



