200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



listed below 



s = 1, p = 1 



/I — cos X o\ M + m ,^_ . 



\ r\ — c cos ^) = — — — (50a) 



s = 2, V = ^ 



/I - cos X 2\ {M -\r mY , . . . 



\ -r\ — c / = TJ \^ cos d) (50b) 



\ aric) I Mm 



s = 2, V = 2 



/m sin X + 4m(l - cos x) ^.p^ (^^^ ^A ^ 



\ otCc) / 



(50c) 



= —j^ (c cos tf> 



While the averages entering into (49) are not always the desired 

 ones, it remains true nevertheless that all solution methods evolved in 

 the following use this equation system as a starting point rather than 

 other forms of the Boltzmann equation. 



IIB. THE MEAN FREE TIME MODEL AT HIGH FIELD 



If the angular distribution in the center of mass system is independent 

 of speed and the collision cross section varies inversely as the speed 

 then the developments of the previous section permit actually a solution 

 of the Boltzmann equation. It is a solution in the sense that all signifi- 

 cant velocity averages can be obtained directly without the knowledge 

 of the velocity distribution function. 



Before developing these facts from the equations of the last section, I 

 should point out that the derivation to follow is in a sense artificial. It 

 has been shown already by Maxwell^ ^ for related problems that if the 

 mean free time between collisions is assumed constant specially simple 

 techniques may be employed to get constants of experimental im- 

 portance. These techniques can be employed here ; they consist essentially 

 in multiplying (13) by a suitable multiplier, followed by integration 

 over c. However, if we were to follow this procedure we would have to 

 duplicate for a special model in an unsystematic way the work done 

 systematically for all laws of interactions in the preceding section. A 

 further advantage of using systematic procedure is that we can see at a 



" Maxwell, J. C, Collected Papers, Vol. II, p. 40. 



