

MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 219 



averages for an arbitrary law of force between the ions and the gas 

 molecules, and an arbitrary mass ratio. The application will be limited 

 to the case of the mass ratio 1 whose study was begun in the preceding 

 section. 



The basis of the method is an observation on the equation system (46) 

 or (47), which is the form taken by the Boltzmann equation after in- 

 serting the Legendre decomposition (43). It would appear at first sight 

 that these recursion relations are of such a structure that an arbitrary 

 function }U)(c) could be substituted into the "zeroth" equation and that 

 the relations would then successively determine /ii , /i2 , /^a • • • . Upon 

 closer inpsection this is found not to be the case. Suppose we have 

 obtained somehow functions ho , hi , h2 - - • hn and we are trying to use 

 the nth equation to determine hn+i . This equation is of the form 



— ^ + hn+i = known material - (77) 



dc c 



We solve for hn+i by multiplying with c"""*"^ and integrating. This gives 



c'^^^n^iic) = I (known material) dc 



The left-hand side is of such a structure that it must vanish both for 

 c = and c = oo . It follows that the right-hand integral when taken 

 between the limits and oo must equal zero. This condition is indeed 

 obeyed for any ho{c) when n = 0. The integral condition reads in this 

 case 



iro^.crn(x)sinx.x'^(^Y-r^c^<io=o 



2 Jo Jo t{c) \c J h t{c) 



If we invert the order of integration in the double integral, then intro- 

 duce c' as variable of integration by equation (41) and finally invert 

 again this becomes 



f^^cUc W f n(x) Anxdx - l] = 

 Jo t{c) [_l Jo J 



This equation is trivally obeyed because the square bracket vanishes in 

 virtue of the definition of n(x). For values of n higher than 0, the in- 

 tegrability condition deduced from (77) is not generally obeyed for any 

 function ho(c). Such a statement may be proved by examples; these 

 examples will arise in the course of the calculations to follow. Thus 

 we find that except in the passage from ho{c) to hi{c), the recursion system 

 is such that at each stage it imposes a condition upon the K's already 

 determined if the new hn+i(c) is to exist at all. With such an infinity of 



