210 Prof. Townsend on the Conductivity produced in 



agitation of the negative ions is 80 times the velocity of 

 agitation of the molecules of air; and the velocity acquired by 

 an ion in travelling between two points differing in potential 

 by 4 volts would be 800 times the velocity of agitation of the 

 molecules of air. The latter velocity is therefore so small in 

 comparison with the velocities of the negative ions, that it 

 may be neglected, and the molecules of the air may be con- 

 sidered to be at rest. 



Let us now consider what happens when an ion is con- 

 strained to move with a fixed velocity (greater than its 

 velocity of agitation) through a gas at pressure p. The 

 number of collisions that the ion makes with molecules of the 

 gas in going through a centimetre is independent of the 

 velocity and is proportional to the pressure. Let ftp denote 



the number of collisions ; then -?r- will be the length of the 



mean free path, expressed in centimetres. According to our 

 theory, the new ions are produced by collision, and if the 

 velocity with which the ion is constrained to move is suffi- 

 ciently great, it will produce j3p new negative ions and an 

 equal number of positive ions, in going through one centi- 

 metre of the gas. 



Since the collisions are not all of the same kind, an 

 arbitrary velocity of the ion might be sufficient to produce 

 ions on some occasions without producing the maximum 

 number (/3p). We would expect, therefore, that there is not 

 a fixed minimum velocity of impact necessary to produce ions, 

 but the greater the velocity, the nearer will the number of 

 new ions produced approach the value ftp. 



Using these principles as a basis for our theory, we may 

 proceed in the following manner to find an expression for a. 

 in terms of X and p, 



6. The free paths described by an ion as it moves through a 

 gas wall not be all of equal length. Out of y paths the 

 number which exceed the length x is 



_ X 



ye \ 



where c is the mean free path. 



In going along a centimetre an ion has j3p free paths, so 

 that the number of paths w r hich exceed the length x is 



$pe-&\ . 

 The number of paths intermediate between x i and x 2 is 

 (3p(€-PP x i — e-^ X2 ). 

 Let I P be the velocity acquired by an ion in travelling 



