218 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



For equal masses the integral on the right runs over a plane in velocity 

 space. Its integrand is always positive; hence the integral can never 

 vanish and is always positive; it is conceivable that it could be infinite 

 for a special position provided the infinity is integrable. Such an infinity 

 would only make matters worse. At any rate the equation shows that 

 h*(c) increases monotonely with increasing Cz . When we approach the 

 origin from negative Cz we get a logarithmic divergence (or worse). As 

 the function h*{c) can nowhere decrease with increasing Cz the infinity 

 along the positive Cz axis is confirmed and its logarithmic nature is made 

 very likely. 



Information obtainable from equation (46) confirms this conclusion. 

 Lowest powers in the entire recursion system can be made to cancel by 

 assuming that for small c 



ho '^ —Alnw 



hi ^ Bi i > 



with suitable relationships existing between these quantities. 



A defect of all three approaches is that they give no information 

 concerning the nature of the infinity for Cz > 0. One is tempted to 

 conclude from Fig. 13 that it cannot be very strong. Something like a 

 singularity is discernible at the origin, particularly if the contour 0.1 is 

 drawn back to cut the Wz-Sixis at a negative value ; this is perfectly com- 

 patible with the available information. For large positive Wz , on the other 

 hand, the picture almost contradicts the theorem just proved. One con- 

 cludes from this that the singularity, for large Cz , becomes a weak and 

 narrow ridge rising more or less abruptly in an otherwise well behaved 

 function. 



nE. THE CASE OF EQUAL MASSES ; A NEW COMPUTATIONAL PROCEDURE 



The foregoing sections have accumulated substantial evidence that 

 there are many analytical details involved when one discusses the 

 structure of a velocity distribution function. These details are of little 

 interest to the experimenter who may want nothing but a formula for 

 the drift velocity or the average energy. In view of this situation it 

 appears very desirable to find a method whereby such quantities can be 

 derived directly and accurately from the Boltzmann equation without a 

 full knowledge of the entire distribution. 



Maxwell's original work shows us how to achieve this for molecules 

 obeying the mean free time condition of Section IIB. In the following, 

 a general method is described which will permit determination of such 



