82 BELL SYSTEM TECHNICAL JOURNAL 



With this assumption (3.8-5) yields 



p{R, 0, R") = R\2Tr"'BT"' f dd' (3.8-6) 



J—oo 



exp —^[B,F: + R{[Bo^ + IB-ARS" - 2B,R") + B,{R" - RO'^] 



The probability that a maximum occurs in the elementary rectangle dR 

 dt is, from (3.8-1), p{t, R) dR dt where 



p{t, R) = -! p{R, 0, R")R" dR" (3.8-7) 



We put (3.8-6) in this expression and make the following change of variables. 



r1/2 „1/2 



X = -^ Re'\ y = -4^ R" 



V2B -^ V2B 



z = ^ R = -4^ i? (3.8-8) 



\/254 B V2Bi 



^ ^ _ (^22 + 2^o) 



25 6; 



2 _ Bo 2Bi _ bobi 



3 Z»o64 



.2 ~ 2^J 



= K3 - a' 



where we have used the expressions for the B's obtained by setting bi and 

 bz to zero in (3.8-3). Thus 



Pit, R) = -1- (PY r y dy f X-'" dx (3.8-9) 



bobl \27r/ ^0 ^0 



exp [ — a" 2^ + 2bzx + 2zy — (x -{- y)^] 



As was to be expected, this expression shows that p(t, R) is independent of /. 

 A series for p(t, R) may be obtained by expanding exp 23(y + bx) and 

 then integrating termwise. We use 



[ dy [ dxx'y'^e-^'^"'" = 

 Jq Jq 



Vtt r(7 + i)r(M + 1) 



2M+7+2 



r 



(._t^3) 



which may be evaluated by setting 



X = p cos* if, y = p~ sin' (p 



