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



makes two consecutive collisions within the region considered. 

 The general problem, in fact, is to find the chance that an 

 ion, after any one of the above primary collisions, makes a 

 second collision before leaving the spherical enclosure, and to 

 integrate the value of this over all the primary collisions 

 which occur. I have not been able to obtain the exact 

 solution of the problem as thus stated ; but it is very easy to 

 obtain an upper limit to the required number, which can be 

 shown to be very near the true value. 



This may be done by supposing the ions which have 

 undergone a primary collision at any point within the sphere 

 to start on their next path from the centre of the sphere. 

 The error introduced by this supposition is greatest in the 

 case of an ion which has collided at the surface of the sphere 

 and when X is small. In the most unfavourable case, the 

 probability would be increased in the ratio of two to one by 

 supposing the ion displaced to the centre. But the number 

 of cases near the boundary is necessarily small compared 

 with the total number, so that the average error will be small 

 compared with the maximum error. Thus we see that the 

 upper limit obtained in this will not be far from the true 

 value. 



The probability that an ion, starting from the centre of a 

 sphere of radius r, will collide before it reaches the circum- 

 ference is readily seen to be 1 — e~ rlK ; so that, out of P ions 

 projected into the prescribed sphere, we obtain as an upper 

 limit to the number which make two collisions before reaching 

 the circumference the value 



( 1+ -(.-f _,))(,_.-!). 



If two collisions within the prescribed sphere are necessary 

 for recombination to take place, we find that 



{ 1+ |(,-f. 1 )}( 1 _,-; ) , 



= < 1 



= </>(cf) say, where x — r/\. 



Evidently </> approaches unity asymptotically as x becomes 

 infinite ; also on expanding it in powers of x in the neigh- 

 bourhood of x = we see that the first term in the expansion 

 is x 2 , so that <£ = when x = and cf> varies as x 2 for small 

 values of w. Thus the function we have found gives a 

 qualitative explanation of the values of e found by experiment. 



By carrying the argument a step further we see that an 

 upper limit to the number of ions which make three collisions 



