MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 231 



The elimination of hi(c) is achieved by forming the average (101) on the 

 function (102). This leaves the required condition on ho{c); it may be 

 given the following form 



(^/\VW<^.)=.2|, (103) 



The equations (79) and (87) are manifestly special cases of this more 

 general relation. Adaptations of this procedure to other cases are clearly 

 possible whenever the need arises. 



The calculations of this section are meant to suggest that it is possible 

 to compute reliably average values from a Boltzmann equation without 

 solving it completely. The method employed here for this purpose 

 resembles a Ritz method in that it works with trial functions which must 

 be guessed at, and like that method it is capable of indefinite improve- 

 ment. The numerical results suggest strongly that we are converging 

 toward a definite answer; however, a mathematical proof of this fact has 

 not been presented. The method will be applied once more in the section 

 on diffusion. 



Part III — Motion of Uniform Ion Streams in Intermediate Fields 



IIIA. A convolution THEOREM 



Wlienever we deal with the motion of a given type of charged particle 

 in a gas of given composition, then there exists a wide range of densities 

 n and N as discussed in Section lA in which the motion of these particles 

 depends only on a/N and kT. For this range the motion is governed by 

 equation (13). Since deriving that equation, all our efforts were dealing 

 with the "high field" equation (34) or (40), in which the gas temperature 

 is taken to be zero and the electric field often scales out, as in (26), (75) 

 and (85). The accomplished solution of this restricted problem, together 

 with the low field solutions available in the literature, brings us back to 

 the more general equation (13) and the question what can be done with 

 it. The topic of Part III so defined is definitely inferior in importance 

 to the one in Part II. For we are studying here an intermediate range 

 of variables which can be handled qualitatively, both in concept and 

 practice, by some sort of interpolation between the high and low field 

 regimes. For precise measurements, conditions can always be chosen so 

 as to satisfy one or the other of the two extremes. For this reason the 

 intermediate field case will only be pushed as far as it will go con- 

 veniently, without appeal to numerical methods. 



In this Section IIIA we shall give a complete solution of the inter- 



