224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



circles and horizontal lines and replacing the integration 



hoiw) = 2t / h{w) d (cos t?) 



by a summation over grid points. The function ht(w) so obtained is 

 compared in Table I with ho^^\w)j ho^^\w) and ho^^\w) as defined by 

 (82) and the numbers following. We observe that the first approximation 



Table I 



Comparison of the Monte Carlo hoiw) for hoiw) with successive approximations 

 obtained by the new method. 



is an improvement over the zeroth one, while the second one makes little 

 difference, considering the accuracy to which h'^(w) is given. In indi- 

 vidual cases the sequence drifts away from htiw) ; this is not surprising 

 because the latter function is very rough ; this is to be expected from its 

 mode of derivation. 



The application of this method to the hard sphere model of ion-atom 

 collisions offers no new feature of principle. The actual working out of 

 results is somewhat more complicated, mainly because the connection 

 diagram for the recursion system (49) is more involved. According to 

 equation (26b) the dimensionless variable to be used in this work is 



w = 



\/a\ 



(85) 



We denote its averages (w'P, (cos t?)) by {s, v) as previously. The equa- 



