22 THE THEORY OF IONIZATION OP GASES 



As the force increases a acquires a small value and 

 subsequently increases still further, the ions during these 

 stages acquiring sufficient velocity along some of their 

 free paths to produce others by collisions. With the 

 smaller forces such velocity is acquired only along the 

 longer free paths, but with large forces the number of 

 paths along which an ion gains this velocity becomes 

 more numerous. When a becomes large its rate 

 of increase with the force diminishes, and finally 

 a approaches a maximum value which should be 

 attained when a new pair of ions is produced at each 

 collision. 



11. Maximum value of a ; mean free path of negative ions. 



The maximum value cannot be obtained experi- 

 mentally by these methods, as sparking would ensue 

 when large forces are used, unless the amount of gas 

 between the electrodes is very small, and in this case 

 the calculations would be inaccurate, as the distance 

 between the* plates would not be large compared with 

 the free paths. The maximum values of a for a pressure 

 of 1 millimetre may nevertheless be deduced from the 

 curves by the following method. 



Let it be assumed that one pair of ions is formed when 

 the velocity u at collision exceeds a certain value. Also 

 let it be supposed that the velocity of a negative ion is so 

 much reduced by collision with a molecule that it prac- 

 tically starts from rest along its new path. An ion will 

 then acquire the requisite velocity under the force X 

 when it travels freely along a distance y such that 



