304 BELL SYSTEM TECHNICAL JOURNAL 



I\t) = a\F\t') + alF\l' - xi) + • • • + aK^xF\t' - x, Xk) 



+ 2aia2F(l')F{t' - xi) + ■ ■ ■ + 2aiaK+iF{t')F{t' - x, ■ ■ ■ -xk) 



+ 2a2aiF{t' - xi)F{l' - xi - X2) -\ 1 



where i' = t — ti . If we integrate I (t) over the entire interval < t' < T 

 and drop the primes we get approximately 



I l\l)dt = (fl^ + . . . + aU,)<p{!d) 



+ 2aia-ifp{xi) + 2aiaz<p{xx + X2) + • • • + 2aiaK+\ip{x\ + • • • + Xr) 

 + 2a2a3<f(x2) + ••• + ••• + 2aKaK+iif(xK) 

 where 



<P 



(x) = f F{i)F{t - x) 

 J— 00 



dx 



When we divide both sides by T and consider K and T to be very large, 

 K ai -i- ' • • -h Oirix 



K 



<^(0) = t-aVlO) 



1 /^ 



7j:,[a\a2<p{xi) + a2a3<p(^2) + • • • + flKaK+Ki^CA;;!:)] = ^ average Oi-CA+i^fxi) 



= pcf j i^{x)p(x) dx 

 Jo 



* j^ 1 



- [aiai(p{xi + 0:2) + • • •] = — TjT- ave. akOk+^ipixk + -n+i) 



= m^ / dxi / </jC2/'(ari)/)(x2)v:(^i + ^^2) 

 Jo Jo 



Consequently 



/^ = Lim 4, f /'(/) dt 



= vaVCO) + 2m' 



/ p{x)ip{x) dx 

 + j dxi j dx.2p{xi)p(x2)^(xi + xz) + ■■■ \ 



For our special exponential form (1.5-7) for 7^(/), 





