Recombination of Ions in Gases. 245 



maximum possible path inside the sphere are those contained 

 in the solid angle between two circular cones whose generators 

 are inclined at 6 and 6 + d6 to one another. Hence if P ions 

 start from P in directions uniformly distributed throughout 

 the hemispherical angle 2-7T, the number which lie between 

 the infmitesimally near cones is 



P sin OdO ; 



hence the number which collide within the sphere is 



P (l-e~~~^~Umdd6 



-*W(«-M}- 



On the hypothesis that one collision is sufficient to stop 

 the ion ever getting out o£ the field of force o£ the other ion, 

 we have now calculated a value of € ; for e is the fraction of 

 the number of positive ions, projected into the small sphere 

 surrounding the negative ion, which never succeed in 

 escaping. It is therefore equal to 



2r 



(.-?-!} 



A brief examination of this function is sufficient to show 

 that it is incapable of representing the values of e found 

 experimentally *. 



If we put — =x, then #is proportional to the pressure, and 



our function may be written 



It approaches unity asymptotically as x becomes infinite, and 

 takes the value zero when x = Q. This, however, is not 

 sufficient to satisfy the experimental curves which, as Langevinf 

 has shown, vary as the square of the pressure at low pressures. 

 Expanding / in powers of x in the neighbourhood of the 

 origin, we find the first term =x, which obviously does not 

 satisfy the required conditions. 



We shall now examine what results are obtained on the 

 hypothesis that two is the minimum number of collisions 

 within the prescribed region which are necessary for recom- 

 bination. We have to calculate the probability that an ion 



* Laugevin, These, p. 150. 



f Comptes Rendus, vol cxxxvii. p. 177 (1903). 



