250 Dr. 0. W. Richardson on the Rate of 



A different view of the nature of e, which explains its 

 variation at low pressures, has been given by Langevin*. 

 On that view the criterion for recombination is that the two 

 ions should undergo a direct collision, considered quite apart 

 from their previous history. If we imagine a sphere described 

 round each negative ion, the radius 8 being some quantity of 

 the order of, and proportional to, the mean free path ; then 

 the probability of any positive ion projected into the sphere 

 striking the opposite ion at the centre is the chance that its 

 originally straight trajectory would pass through a sphere 

 about the negative ion of radius 



s =s 



'i+:£: U 



C. mv 2 s j 



the mass of the two ions being supposed equal. In this 

 formula e is the charge on an ion, m its mass, v the original 

 relative velocity, and s the radius of the sphere of action of 

 the two ions. It can easily be shown that the probability 

 of an ion projected from the circumference of a sphere of 

 radius 8 striking the opposite ion is 



-(HOT. 



so that Langevin's theory leads to the formula 



In this formula 8 varies inversely as the pressure and s' is 

 independent of it, so that e is of the form 1 — (1 — ap 2 )*. It 

 evidently varies as the square of the pressure for small values 

 of the pressure. 



The function 1 — (1— /3 2 x 2 )% is the dotted curve E in fig. 2, 

 where the abscissas have been chosen so as to make the curve 

 fit the three lowest experimental values. The agreement is 

 good enough at the three lowest pressures, but at higher 

 pressures, as is readily deduced from an examination of the 

 function, its rate of increase with x goes up continually as 

 x increases, instead of falling off as the experiments require. 

 For values of x > 1//3 the function is imaginary. For these 

 reasons this view does not appear to afford a satisfactory 

 explanation of the phenomena. 



Returning to the exponential expression for e, we shall 

 now deduce a value of r, the radius of the spheres within 

 which collisions have to occur, to determine recombination. 



Since the charge on an ion appears to be the same for all 

 gases, and r is determined by the field of force, presumably 



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



f Richardson, Cainb. Phil. Proc. vol. xii. p. 144 (1902). 



