240 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Here v equals hu and the parameter /? equals h/hura. . It is the parameter 

 /8 introduced by Hass6. Now by the definition of r we have 



M^\ = N I y<pix) da{y,x) 



= N f yv>{x)b db [' (k 

 Jo Jo 



= irNyb'u^ f vix) d{f) 

 Jo 



From (125) and (126) the factor in front of the integral just equals I/ts ; 

 the integral on the other hand is a computable pure number independent 

 of 7 which is obtained by inserting into it the relationship (127) between 

 X and jS. Hence we may write 



/^\ = 1 f " ^(x) di^r (128) 



\ r / Ts Jo 



The three equations (126), (127) and (128) completely define the nature 

 of the averages appearing in previous sections. The integral (128) has 

 to be computed by numerical methods. It is seen in the course of the 

 evaluations that it naturally decomposes into two parts. The part for 

 which ^ varies from to 1 deals with spiralling collisions and exists for 

 any ^(x). For /? between 1 and oo we get the contribution of the hyper- 

 bolic collisions to the average. This part is only finite if ^(x) vanishes for 

 small angle deflections. 



The averages (52), (53), (54), (57) and (59), as well as (117) to (121) 

 contain numerous averages of the form (128) all of which satisfy the 

 predicted condition <^(0) = 0. They are obtained by linear combination 

 of two basic types: ((1 — cos x)/t) and (sin^ x/r). The first average is 

 given in Hass^.^ Separating the parts due to spiralling and hyperbolic 

 collisions we find 



Jo 



cosx)^(/3') = 0.8979 



J (1 - cos x) d(^^) = 0.2073 

 This combines to give 



^l_-^^ = i. 1.1052 (129) 



