228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



giving 



{w^ cos' t?) = 1.768 (96) 



We can understand the results (92), (94) and (96) by giving the frac- 

 tion of the total energy in ordered motion and the fraction of the energy 

 in motion along the ^-direction. We find for the first ratio 



<^ '""if = 0.559 (97) 



and for the second 



{^1^0^ = 0.751 (98) 



The ratio (97) equals 0.5000 for all mean free time models; the ratio 

 (98) is 0.778 for the mean free time model with isotropic scattering. 

 Thus, the deviations from the earlier results are not drastic. However, 

 in certain derived relations the difference is more noticeable. For instance, 

 a good measure of the anistropy of the diffusion process is furnished by 

 the ratio of the random energy along the field to the energy at right 

 angles.^ From (97) and (98) we find for this number 



{vl^ COS^ d) - {W cos df ^ . - . ,QQS 



KM - (w' cos2 1?)) ^ ^ 



For the mean free time case this number equals 2.50. Hershey^ in his 

 work assumes this number to be 1.000. 



A comprehensive list of velocity averages is attached in Table II. As 

 a comment I may add that the obvious mode of constructing such a 

 table, namely by computing the column v = from (90) and then using 

 the recursion system (86) for the others, runs into some difficulty. First 

 of all, a series of cancellations reduces the accuracy as v increases; 

 finally, at the positions marked "impossible" we find the missing third 

 members of the truncated relations. These elements cannot be com- 

 puted by recursion at all, but would require an explicit solution of the 

 equation system (47) for K+i(c). In the table, this more arduous path is 

 not followed. Instead, the recursion method is used for the numbers in 

 italic type and a few numbers are added by extrapolation. The numbers 

 so obtained will be needed in Section IVB. 



The calculations on the hard sphere model are immediately applicable 

 to the experimental data of Figs. 3 to 7, which exhibit the drift velocity of 



" See below, equations (147) and (165) . 



