MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 211 



The equation can be solved explicitly in Cartesian coordinates by the 

 method of separation of variables. The result is 



(73) 



This is a Maxwellian distribution with elliptic distortion and shifted 

 origin, that is, the type shown in Fig. 1 (b). 



The result (73) indicates the main features of the solution for heavy 

 ions. Because of the neglect of derivatives higher than the second in 

 /i(cO it is not certain that (73) is correct in all details, even in the limit 

 of very large m/M. 



IID. THE CASE OF EQUAL MASSES; IONS TRAVELLING IN THE PARENT GAS 



The developments of the previous section show that if the ion mass is 

 either large or small in comparison to the molecular mass, analytical 

 methods can be applied successfully to determine the velocity distribu- 

 tion of the ions. No such possibility was found for the mass ratio unity, 

 which one would judge to be of particular interest because it applies to 

 ions travelling in the parent gas. There exist isolated fragments of such 

 an analytical theory; for instance, if we assume isotropic scattering in 

 (46), that is n(x) = 1, then the zeroth equation (46) becomes explicitly 

 integrable and yields 



Uc) = -\ar(c)''J^ (74) 



This is a curious reversal of the differential relationship (63) derived for 

 electrons and implies a rather strong condition on the structure of 

 hi{c). However, I have not been able to consolidate these fragments into 

 something which can be used successfully in computation. The high 

 field distribution function for mass ratio unity appears, however, suffi- 

 ciently interesting to warrant the use of other methods. 



A numerical determination of the velocity distribution function was 

 undertaken in cooperation Avith R. W. Hamming by the so-called 

 Monte Carlo method. The Monte Carlo method is a way of gaining 



