MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 193 



(26b) is required. The linear range of this plot is not as informative as 

 the high field one. The slope unity is common to all formulas (27), and 

 the temperature dependence of the mobility is needed to give the correct 

 interpretation with the methods developed here. There is a certain 

 likelihood that the parameter a of equation (25) drifts from to 1 as 

 the speed of the ions is reduced ; this was pointed out for the special case 

 of He"*" in He in section lA. A qualitatively similar situation appears to 

 prevail for the other noble gases. 



Part II — The Motion of Uniform Ion Streams in the High 



Field Case 



II A. formulations of the boltzmann equation 



The dimensional analysis of the last section shows that there is an 

 intrinsic simplicity to the high field case which is comparable to the low 

 field case, while the intermediate case is more difficult. With one excep- 

 tion,^ however, theoretical analysis has occupied itself with the low 

 field case only. We shall try to remedy this in the following. To begin 

 with, a tractable but accurate formulation of the problem has to be 

 found. Such a formulation cannot treat the field term of equation (13) as 

 a perturbation term, but must try instead to make use of the basically 

 simple features of the problem, notably those exhibited by the dimen- 

 sional analysis of Section ID. 



The equation governing the high field properties of the ions is ob- 

 tained simply by substituting 5-functions for the Maxwellian velocity 

 distributions in equation (13). This gives 



a • + 



dcz 



_L /i(c) = i- ff 5(C0«c0 -^ n(x) d%> dO (31) 

 t{c) 47r J J t{c ) 



A reduction of the number of integrations from five to two must be 

 possible in the integral term of (31), owing to the presence of the 6-f unc- 

 tion. To achieve this we must transform the variables of integration so 

 as to make three of the differentials equal to dC. We do this in the 

 following way. First observe that 



r = c - c 



and that c is a constant vector. Hence we may replace dC by dy. The 

 five-fold integration reads then 



dQy' dC = 1 dy dOy dQy (32) 



that is, it goes over the magnitude y which the two vectors have in 



