128 BELL SYSTEM TECHNICAL JOURNAL 



(5.2) 



and io, ^1, ^2 are given in Appendix II. 



Integrating with respect to R gives the probabiUty density for ^, B' . 

 Expanding exp {^hR cos Q) in powers of R and integrating termwise, 



J,//) /}/\ _ ^ I ^ \ c ain^-bibopl B 



^ n + 1 f b cos e Y 



n=0 -p 



When we integrate ^ from — x to tt to obtain p{$') the terms for which n 

 is odd disappear and we have to deal with the series, writings for &-/o, 



Z ^-^4^ (t cos' ^)'" = (27 cos' e + 1) exp (7 cos' ^) 



Thus, the probabiUty density of 0' is 



in^e-hycoa'i e-bzboplB i^ 





From (5.2) 



>2 - 2bie' 



c = p 



62 - 2^10' + M'2 

 c + 7 ^26op pb2 - IbiB' + 2M'' 



(5.5) 



2 5 2 62 - 2^>l^' + M'2 



It will be noted that for large values of | ^' | the probability density of 6' 

 varies as | d' |~'. Although this makes the mean square value of 6' infinite, 

 the average values 6' and |~^ of 6' and | 6' \ still exist. In order to obtain 

 e* it is convenient to return to (4.5) and write 



^ = j[ (f^l rfi? j[ dR' j de' e'p{R, R\ e, e') (5.6) 



