Radiation produced by sloioly moving Cathode Rays. 369 



The mean kinetic energy of the corpuscles when there is no 

 electric field is, on the assumption that the corpuscles behave 

 like a gas, the same as the mean kinetic energy of the mole- 

 cules of any gas at the same temperature. In an electric field, 

 however, the corpuscles under the action of forces of the mag- 

 nitude of those occurring in exhausted tubes, acquire velocities 

 which are large compared with the average velocity when there 

 is no electric field. Thus in these tubes the energy of the 

 corpuscle is practically equal to that given to it by the field. 

 Now if X is the electric force, A the mean free path of a 

 corpuscle, the mean kinetic energy acquired by the corpuscle 

 in the electric field will be proportional to X^V, the work 

 done on it by the electric field during its free path ; e is the 

 charge of electricity on the corpuscle. A certain fraction of 

 the whole number of collisions will lead to ionization, the 

 value of the fraction depending upon the kinetic energy of 

 the corpuscles ; we shall denote the value of this fraction by 

 /(XeX), where at present all we know about/ is that it cannot 

 exceed unity however great XeA may be, and that it vanishes 

 when XeA falls below a certain limiting value. 



If u is the average velocity of a corpuscle, \/u is the interval 

 between two collisions ; hence each corpuscle makes v/\ col- 

 lisions in unit time, so that if n is the number of corpuscles in 

 unit volume, nu/\ is the total number of collisions in unit 

 volume per unit time, the number of ions produced in unit 



nu 

 volume per second is therefore equal to — f(X.e\). 



A 



The corpuscles ultimately get attached to molecules, thereby 

 losing their mobility and, to a great extent at any rate, their 

 power of ionizing other molecules. We shall suppose that in 

 a certain fraction of the number of collisions the effect of the 

 collision is to leave the corpuscle sticking to the molecule 

 with which it collided, and thus to end its career as an ionizing 

 agent. 



Let /3 be the fraction of the whole number of collisions 

 where this occurs, then the rate at which the corpuscles dis- 

 appear is per unit volume equal to j3nu/\. The rate at 

 \> hich the ions are increasing per unit volume is therefore 

 equal to 



^{/(X,A)-/3K 



If we consider the case of a tube in which the ions are 

 moving parallel to the axis of x, then we have by the 

 equation of continuity, 



