Negative Ions at Low Pressures. 323 



one-collision electron traversing x without becoming a 

 negative ion is 



c^- 1 ) 



■ X 



The expectation that the electron makes two collisions and 

 no more is 



j: 



a£/X«£ a <^T 



■a(ar-£)/X. 



X X c 



for e~ a ^ A (af/\) is the expectation that it should have made 

 one, and only one, collision before reaching f ; adf/\, the 

 expectation that it should make a collision between f and 

 g+dg; and e- a (*""©A, the expectation that it should not make 

 a collision in the rest of its journey. The value of the 

 integral is 



*•"*(?)• 



The expectation that neither of the collisions should result 

 in union is ((n — l)/n) 2 . Hence the expectation for a two- 

 collision electron without recombination is 



t -ax/\ 



1 ln—\ «./ Y 

 2 V ~^ xj 



Similarly the expectation for a three-collision electron 

 without union is 



r/X 



1 /n — 1 a.i'Y 

 273 \n~ X ) ' 



and so on. Hence, since any electron must have made 

 collisions varying in number from nought to infinity, the 

 expectation of the electron traversing the distance x in a 

 free state is 



■axf\ 



f, . n— lax l(n-lux\ 2 1 /n-la/iA 3 , 1 



t 1+ 5 I + i (T l ) + 2 . 3 (7j + -j 



, - ox/X (n — 1 )ax/>i\ __ - ax/n\ m 



Thus if N electrons start from #=0 the number which 

 traverse .? without recombination is Ne~ aa '/"'\ 



In experiments on the mobility of ions the quantity 

 measured is, in many cases, the charge of electricity received 

 by one plate during an interval T when a layer of gas close 

 to the other plate has been ionized at the beginning of the 



Y2 



