78 BELL SYSTEM TECHNICAL JOURNAL 



By introducing polar coordinates a = p cos 6, ^ = p sin 6 it can be shown 

 that the region p > ( more than covers the region in question and that 



Q{a,^)^ (1 + Op-^ (5-24) 



Upon integrating with respect to p and setting in the lower limit (, it is 

 seen that the third contribution is 0(^~^/-). 



We now assume K to be large. Since ^ ^(-Vo, u) ^ 1 we have 



^ e~'''' - (1 - P)'' ^ KP-e'"''' < l/K (5-25) 



The last inequality follows from .v- exp (— .v) < 1 for x ^ 0. A proof of the 

 remaining portions will be found in "Modern Analysis" by Whittaker and 

 Watson, Cambridge University Press, Fourth Edition (1927), page 242. 

 When we observe that replacing [1 — P(.Vo, w)] by 1/K in the right hand 

 side of (5-23) gives an integral whose value is less than 1/K, we see that 



Prob. (P,Q, ■•' ,PkQ> PoQ) (5-26) 



J— q J^ q 



We now take up the problem of expressing the cumulative probability 

 density /*(.Vo, u) in terms of a and /3. When .Vo and u lie in the restricted re- 

 gion of integration shown in (5-6) they are near their average values .fo = 

 (2X + l)r and u = (2X + 1)(1 + r). On the other hand the average value 

 X of .V and the mean square value o-; of (.v — x)- as computed from (4-6), or 

 directly, are 2N + 1 + « and 4.Y + 2 + -iu, respectively. Thus we see that 

 X — .Vo is of the same magnitude as 4iV and becomes much larger than ax as 

 A' -^ 2c . The asymptotic development of Appendix I may therefore be used. 

 In Appendix / (equations (Al-27) and (Al-29)) it is shown that when 

 M(= 2m = 2.Y + 1) is a large number and 1 < < (.f — .Vo) a^ 



P(.Vo, n) = (iirmbo)-"' (1 + ()(l/w)) exp [wF(n)] (5-27) 



where we have introduced the number m = N + 1/2 = g -\- \ to save writ- 

 ing X + 1/2 or (/ -j- 1 repeatedly and where 



2b2 = (1 - lAi)-(l + 4siyi' 

 V, = [! + (! + 4siyiy2s . 



F(v,) = (1 + 4siy' - s - I - logn ^^"-'^ 



.v„ = 2ms = (2A' + l)s, u = 2ml = (2 X + 1)/ 



Comparison of tlie last line in (5-28) with (5-11) shows that ms and ml 

 are equal to r(ij -j- a) — r(m -|- a — 1) and 



(l-^r){q + (3) = (l + r)(w + /3- 1), 



