Recombination of Ions in Gases. 247 



inside the sphere is given by 



{. + i(.-f-i)}(i-.-£)r. 



and generally, 



is an upper limit for the number which make n collisions. 

 For high values of n, however, the above product will cease 

 to represent, even approximately, the true value of the number 

 required, since the error is proportional to the number of 

 terms in the product. 



We should, expect from the nature of the preceding 

 argument, that out of those molecules which make a sufficiently 

 large number, say n, of collisions within the sphere practically 

 the whole would result in recombination. Of those which 

 made n — 1, but not n, collisions a certain definite fraction 

 would become fixed, and of those which made n — 2, but not 

 n — 1, a smaller fraction, and so on. In fact, if c n be 

 written for 



('+r,( : -M)('--r' 



the number of ions which make n collisions within the sphere, 

 n being sufficiently large, we should expect e to be of the 

 form 



€ == C n + a n-l( C n.-l- C J+ a n-2(. C n-2~ C n-l) + —<*\(<h.-C % ),- 



where a w ^_ 1 ...a 1 are proper fractions which gradually become 

 smaller as the suffix decreases. 



We are now in a position to test the theory by the 

 experimental results. This may be done by referring to 

 fig. 2. The ordinates of the various curves plotted in this 

 diagram represent the following functions : — 



e~ 2x — } 

 A=l + — g^-" =oi, 



c =( 1+ ^)( l ^T=^ 

 D -( 1+ ^)( 1 -W-^ 



E = l-(l-0*)*. 



