﻿402 Dr. Norman Campbell on 



X=V/Z. Let N = N p be the number of collisions made by 

 an ion in travelling 1 cm. in the direction of x ; p is the 

 pressure of the gas. Then, if q is the number of ions 

 starting from the origin and q the number arriving without 

 previous collision at x, we have, since Ndx is the number of 

 collisions made in a layer dx, 



-£=**• w 



or 9 = ?o« _S * (2) 



Let 6 be the smallest distance through which the ions must 

 fall freely in order to be able to ionize at the next collision, 

 so that 



V7 V 

 ? '=T=X < 3 > 



Then e~ is the chance that an ion starting from a given 

 layer will travel a distance b parallel to x without making a 

 collision, and N e~^ b dx is the chance that, after travelling 

 freely for that distance, it will make a collision in a layer of 

 thickness dx. Following Townsend, we shall write 



a =a p='Ne- m (4) 



5. Townsend now proceeds to argue as follows : — Let n l 

 be the number of negative ions liberated per second at the 

 surface of the positive electrode, n p dx the number liberated 

 per second by the external ionizing agent (rays of various 

 descriptions) in any layer of the gas of thickness dx at a 

 pressure^. Let / (x) be the number of ions entering from 

 the positive side a layer of thickness dx at a distance x from 

 the positive electrode per second. Then f (V) dx is the 

 number of ions liberated per second in that layer. These 

 ions are due to two agencies : first, the external ionizing 

 agent, giving a number n G pdx' } second, the collisions of 

 ions liberated in previous layers which have travelled a dis- 

 tance not less than b before arriving at the layer. The 

 number of these collisions Townsend makes equal to the 

 number of ions passing through the layer per second multiplied 

 by the chance that an ion travels a distance not less than b 

 and then makes a collision. He thus obtains 



/'(■'-') = »o >>+«/(*) (5) 



Or, since f(x) =n l} when # = 0, 



«*)=(»!+ -°W- n °. • • • (0) 



