134 BELL SYSTEM TECHNICAL JOURNAL 



wave frequency /g so that ^i and ^3 are zero. Then from equations (3.8-3) 

 and (3.8-^) of Reference A the probabiHty density of xi, x^, - " x% is 



-— -— exp ( -—- [b^ixl + ^4) + 2h2{xiXz + x^x^) 



S^hD \ 2D ^^^^ 



+ {D/b2){xl + xl) + ^0(^3 + xl)]\ 



where D = ^0^4 — ^2 and the bjs are given by equations (A2-1). Replac- 

 ing the .x''s by their expressions in terms of R and d, similar to those just 

 above equation (3.8-5) of Reference A, shows that the probability density 

 for R, R\ R", e, d\ e" is 



/>(/?, R\ R'\ 0, e\ e") = g-^ ^^P ("2^ [^^(^' - 2i?(3 COS e + Q 



+ {D/h2){R'^ + R'e'^) + 2h2{RR" - R^d"") (6.9) 



+ ho{R"^ - 2RR"e'^ + ^R'^e'^ + ^RR'e'e'' + R^e" + r^'"") 



) 



- 2hQ{R" cos e - Rd'^ cos d - 2R'e' sin d - RB'' sin d)] 



It must be remembered that (6.9) applies only to the symmetrical case in 

 which 61 and b^ are zero. 



Integrating R' and R'^ in (6.9) from — oc to qo gives the probability 

 density of R, $, 6\ B" . The integration with respect to R" is simplified by 

 changing to the variable R'' - RB'-. The result is 



p{R, B, B\ B") = R\2Tr)-\bQb2D)-^i\\ + u)-^'^ 



exp ^-^^ 1^7?' - 2i?(2cos ^ + Q' + b^R^B'^/b^ ^^^^^ 



(Qbi sin B + boRB 



n 



(1 + u)D 



where u = Ab2boB'''/D. The expected number of times per second the time 

 derivative of increases through the value B' is 



AT,, = 1 dB j dR I dB''B''p{R, B, B\ B") 



-\b2b/b,f" f dB r rdrf xdx ^^^^^ 



J- X Jo Jo 



= IT 



exp [-yr- -f- 2mcos B - a^ - 8{x + asm B^} 



