214 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



could account for the entire discrepancy, particularly in the mean 

 squares, although it must be emphasized that the runs with high r make 

 a more than proportional contribution to the total average. The angles 

 of scattering have not been subjected to a similar analysis so that we 

 cannot make a statement whether the aimed at isotropy in the law of 

 scattering was realized or not. We conclude therefore by saying that 

 while the Monte Carlo calculation gives results in general agreement 

 with the deductive theory there are small but noticeable systematic 

 errors in it whose origin is only partly explained. Similar errors must 

 exist in the new results which cannot be compared with theoretical 

 predictions. 



(b) In this part we will discuss the velocity distribution function which 

 may be constructed from the Monte Carlo results. In constructing such 

 a function we make use of the fact that, between collisions, the velocity 

 is accelerated at a uniform rate. Thus, in each period between two 

 collisions, the velocity vector traces out a straight line parallel to the Wz 

 axis covering equal distances in equal times. The Monte Carlo calculation 

 furnishes us with a number of such straight lines as shown in Fig. 11. The 

 density of these straight line tracks in velocity space is the velocity 

 distribution function. The actual procedure used to obtain it was to lay 

 a grid with a mesh of 0.23 in a half-plane with coordinates Wz and 

 Wp = ■\/wl + wl and to count the number of lines crossing each hori- 

 zontal square edge. When the resultant count is converted to density 



w. 



Fig. 11 — Straight line pattern in the w, — w, half plane from which the veloc- 

 ity distribution is constructed; the Monte Carlo calculation furnishes the initial 

 and final velocities (dots and rings). 



