252 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



is made between a small density gradient and a small change in the 

 applied field. Thus we would interpret (148) to mean 



Du^^-^l{c.')-{c,f] (165a) 



da 



which, with (85), (92) and (96), becomes 



Du ^ 0,26a"V^ (165b) 



The error of formula (165) is thus 18 per cent, when compared to (164). 



Part V — Concluding Observations 



The present article is supposed to contain the essentials of a kinetic 

 theory of charged particles moving through a gas in the presence of an 

 intermediate or high electric field. An effort was made to make the theory 

 general, yet many irksome restrictions will become apparent to those 

 who will try to apply it to their particular problem. Especially those who 

 have in mind application to electrons will find the article unsatisfactory. 

 It is true that many sections leave the masses variable; however, the 

 assumption of elastic collisions, which is made throughout, is almost 

 fatal to all but the most elementary applications. Thus most of the 

 material is slanted for ions. Within this domain, numerous awkward 

 restrictions are still found here. The most important ones are pre- 

 sumably the restriction to D.C. conditions, the assumption of ''low" 

 ion density, and the omission of all magnetic effects. It is my general 

 impression, which I gained from the convolution theorem Section IIIA 

 and which is confirmed by a recent publication^^ that much can be done 

 to remove these three restrictions provided the mean free time assump- 

 tion is made for collisions. To many the adoption of the mean free time 

 condition will in itself appear an awkward restriction. In a rigorous sense 

 this is true, and calculations are made in this article for the more ap- 

 propriate hard sphere model when quantitative comparison with experi- 

 ment is contemplated (equations (100) and (164)). Indications are even 

 given for a treatment which dispenses altogether with the use of models 

 (equation (103)). However, for rapid advance and easy handling, the 

 mean free time assumption does appear essential. It is therefore im- 

 portant to point out that in a wider semiquantitative sense, the use of 

 this model is no barrier to application. In other words, there is in the 

 mean free time formulas information which suggests a wider validity. 

 This is particularly true for equations which do not contain model 

 parameters, such as (55), (56), (122) and (148). Even formulas which 



