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



will again behave as a free ion. The criterion for recom- 

 bination then is the condition that the ion should remain 

 permanently within the region in question. 



The condition that the positive ion should return to the 

 boundary is a well-known result in the theory of attractions, 

 and is that -JraV 2 -f 2^ is positive : where V is the relative 

 velocity of the ions, m is the mass of one of them, and ^ is 

 their potential energy reckoned from the surface of the sphere. 



Since the number of cases in which the positive ion can 

 enter the sphere without a finite velocity component along 

 the radius is infinitesimal, it is evident that at the commence- 

 ment of the path J??iV 2 + 2 s I r is necessarily positive ; so that 

 unless something happens which reduces the value of V 

 recombination will not take place. We shall suppose that 

 the necessary decrease in V is caused by collision with un- 

 charged molecules, and try to find out how many collisions 

 are required to produce the observed experimental results. 



We shall first calculate the probability that an ion pro- 

 jected from the circumference of a sphere of radius r makes 

 a collision within the sphere. To make the calculation 

 possible, we shall make the rather approximate assumption 

 that the path of the ion inside the sphere is straight. It 

 is evident that this assumption cannot change the form of the 

 function representing the probability : it can only modify 

 the values of the constants which enter into it. 



Of any number of ions starting from a given point 

 P (see fig. 1) the fraction which get over a path of length I 

 without collision is e~ l l\ where X = the mean free path of an 

 ion in the gas ; so that the fraction which collide in a path 

 of length I is 1— e~ l K Now the length of the chord 

 Z = 2rcos0, and the ions for which this represents the 



