242 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



possibility; for we know of no mechanical force which realizes this ar- 

 rangement. This model was already taken as the basis of the Monte Carlo 

 calculation in Section IID. It will be seen now that it has a wider sig- 

 nificance than one might anticipate. The necessary angular averages are 



(1 - cos x> = 1 (134) 



(sin' x) = M ■ (135) 



This yields for (117) 



, , M + m , . 



\Cz} = — -^ — ar (136) 



As usual, the formula for the energy does not involve the law of scattering 

 if written in the form (122). If we choose the form (118) instead we get 

 in agreement with (79) 



W) = 2kT + ^^^^ aV (137) 



The partition formula (58) becomes 



e^:ey:e^ = M:M:(M + 6m) (138) 



the partition formula (60) which counts random energy only becomes 



e.ieyiet = (M + w):(ilf + m):(ilf + 4m) (139) 



Comparison of these expressions with the ones for the polarization 

 force shows that the difference between it and the isotropic model is 

 remarkably small from a kinetic standpoint. We may see this by com- 

 paring (132) and (138) or (133) and (139). For the other formulas, we 

 may compare more specifically the polarization results with an isotropic 

 case having its mean free time r given by 



T = 0.9048 Ts (140) 



Equation (136) becomes then identical with (131) and because of (122) 

 the same identity persists for the energy formula (137). In the light of 

 this we may say that it is very nearly correct to state that scattering is 

 isotropic for the polarization force. This qualitatively correct fact was 

 repeatedly made use of in the preceding sections of the paper. The 

 reason for it is chiefly the predominant effect of spiralling collisions. 

 Indeed, equation (140) shows that a modification of Ts by only 10 per 

 cent takes into account the main influence of hyperbolic collisions. 

 From the discussion in Section lA it may be seen that the results 



