206 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



provisionally that 



we see that we can drop the terms in Kj^i in the system (47) . The integral 

 in (47) is evaluated in zero order with the help of (61) and (62); it 

 becomes then 



K{c) 1 r 



—r'll P. (cos x)n(x) sin X ^X 

 t{c) 2 Jo 



This means that we may solve explicitly for hy{c) in terms of K-i{c). 

 The formula is 



hXc) ^ ^^ ,, p / ,, (66) 



2j/ — 1 / I — Py (cos x) \ 



\ aric) I 



To estimate the order of magnitude of this we may neglect Fy (cos x) as 

 compared to 1 and assume v large. The operation in the numerator will 

 lead to two kinds of terms: some of the form 



c 



and others of the type 



m c 7 / \ 



M {(aT(c)j2 



coming from differentiation of an exponent of the type (65). Now we 

 find from (65) that the overwhelming majority of particles have speeds 

 c which in order of magnitude satisfy 



,1/2 



c ^ {M/my"ar(c) (67) 



because this is the range within which the exponent remains comparable 

 to 1. Applying this to the two types of terms arising from (66) we find 

 for them in order of magnitude 



1 



©■'■'^ 



and 



yaric) - /i._i(c) ^ ,/ ( ^^ ) /i^i(c) 



"^(''^ F K^^ ''-'^'^ ~ (f) '''-'^'^ 



