146 ME. J. S. TOWNSEND ON THE DIFFUSION OF IONS INTO GASES. 



n = 



where t is the time occupied by the gas in travelling from the section where its 

 conductivity is n 2 to the section where its conductivity is n. 

 From observation we have : 



aa 



n, = 

 so that 



Let cp be the number of positive ions which recombine with negative ions in the 



tubes T,. Then 



dp/dt = an 2 = emle' 2 *", 

 and p = when t = 0. 



Let P be the value of p at the end of the tubes, the value of t being at that 

 point. 



By integration we obtain : 



Substituting for a its value this equation becomes 



p_ A n\ - n\ 

 ' 



The effect of recombination would obviously be over-corrected for if M were taken 

 equal to P, for although cP ions are lost by encounters with others of opposite sign, 

 still it must be remembered that, had no recombination taken place, some of these 

 cP ions would have lost their charge to the sides of the tube. The number cM 

 should only include those ions which encounter others of opposite sign, and would not 

 subsequently come into contact with the sides if their rate of diffusion were unaltered 

 by the collisions. 



It is easy to see that M is less than P and greater than - - P ; a fairly accurate 



n* 



estimation of its value can be arrived at by the following method : 



At a section z of the tubes, where the conductivity is n, the number that recombine 

 in a time dt is cdp, where dp = an z dt. 



From the formula 10, Section I., it can be seen that of these c8p ions, c8p 



ft 



would not come into contact with the sides as they pass from z to the end of the 

 tubes T,, if their rate of diffusion were unaltered by the recombination. Hence we have 



8M = 8p - 



ft 



