220 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



conditions one can improve indefinitely an initial trial function assumed 

 for ho(c). 



The integrability conditions whose general structure is thus indicated 

 have actually already been written down. They are the equations (49) 

 for the special case s = v. Generally speaking, the relations (49) are also 

 of the recursion type, permitting us to start with arbitrary averages 

 (c'), and computing successively (cPi (cos t?)) etc. At every stage, how- 

 ever, there is the exception mentioned: the equation for which s = v 

 has no third member, and therefore it imposes a condition upon averages 

 already known from the previous equations. We shall refer to this type 

 of equation as a "truncated" relation. 



It is reasonable to assume that l/r(c) can be developed into a power 

 series in c because it equals the known constant polarization value for 

 c = 0. If this can be assumed then each truncated relation s = v is 

 equivalent to a unique relation among velocity averages involving 

 ho(c) only. One obtains this relation by applying to each member in the 

 truncated relation its own recursion formula and repeating this process 

 until V is brought down to zero. This process will never lead into another 

 truncated relation s' = / because at each step s' increases by at least 

 two units with respect to v\ 



In order to test the method for a known case, it will be applied first 

 to the case of constant mean free time. This case is adequately described 

 by the theoretical treatment of Section IIB and the Monte Carlo 

 calculation of Section IID. We have seen that the equations (49) reduce 

 in this case to the form (51) which dovetails as shown in Fig. 9; this 

 dovetailing leads to explicit values for certain averages as shown in 

 Fig. 10. A "computational method" is only needed when one tries to get 

 an average outside this selected list. In the present case the reduction of 

 the truncated relations to a condition on ho(w) is particularly simple as 

 is seen from Fig. 9. A singular relation which starts out as between 

 {c~ P,_i (cos t?)) and {cP, (cos t?)) actually yields the numerical value 

 of the latter because the former has been obtained numerically in a previ- 

 ous stage. This numerical value yields in combination with previous 

 information (c"^^P,-i (cos t?)), {c"^^P,-2 (cos t>)) etc and finally {c"). 

 Thus we end up with the set of even moments of ho{c) which may be 

 used in succession to determine ho(c) more and more closely. There is no 

 guarantee that this procedure converges mathematically, since the 

 general theorems usually require the knowledge of all integer moments.^ 



" Shohat, J. A., and J. D. Tamarkin, The Problem of Moments. Am. Math. 

 Soc, 1943. The original three-dimensional formulation appears a little more 

 favorable for a proof because, in this case, we know indeed all integer moments. 



