MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 237 



namely that all averages of products of integer powers of the Cartesian 

 velocity components, which were shown to be computable in the high 

 field case, can be computed for the intermediate and low field range as 

 well. The calculation proceeds as follows. Suppose we wish to compute 

 the velocity average 



(crc^cf) = j cJ^CyVfic) dc (115) 



for m, n, p integer or zero. We apply the convolution theorem (107) to 

 /(c), decompose the three factors into 



Cx" = {u.+ (Cx-i/.)r 

 Cy" = luy+ (cy - Uy)}"* 



Cz^ = {Wz + (Cz — Uz)Y 



and expand each of them by the binomial theorem. We find 



<'--'^-«-'> = 5Si.t)t)C) 



(116) 



The second integral is a thermal average, the first a high field average 

 computable by the method of Section IIB. Thus the average (116) is a 

 finite sum of products of computable averages and is itself computable. 

 When formula (116) is applied to the averages (52), (53), (54), (57) 

 and (59) very simple results are found because of the symmetry of the 

 function m(v). For the drift velocity (cz) we get from (52) 



This is the same formula as (52) which is thus proved to hold inde- 

 pendently of the gas temperature. In the energy formulas we find simple 

 addition of the thermal and high field values because the middle term in 

 (116) drops out by symmetry. Inserting (53), (54), (57) and (59) we find 



{mcl) = kT+ j-^ ^^-T— r (119) 



2 / 3M sin' X + 4ot(1 - cos x) \/l^^c^x\ 



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ar 



