Radiation produced by slowly moving Cathode Rays, 371 



before it disappears. The condition XA — a constant implies 

 that whenever a uniform discharge is taking place, the kinetic 

 energy possessed by the corpuscles has a definite value inde- 

 pendent of the pressure of the gas or the current through it. 

 The differential equation giving the distribution of electric 

 force in the discharge-tube when the discharge does not vary 

 with the time may be found as follows. If i is the current 

 through the tube, m the number of positive ions per unit 

 volume, v the velocity of these ions, then 



nu + m v = i. 



As the kinetic energy possessed by the ions is proportional 

 to XA, we may put 



u = k 1 \/X, v = k 2 Vx, 



where k l and k. 2 are constants depending on the density of 

 the gas. Substituting these values, we get 



i 

 «*l + «*a=— 7^ ( 2 ^ 



vx 



Now dX 



where p is the density of the electrification, and p — (m—n)e; 

 the minus sign occurs in this equation because we have taken 

 as the positive -direction of x that from the cathode to the 

 anode, while X is measured in the opposite direction ; thus 



J_dX 



-±7T<? dx 



(3) 



hence from (2) and (3) 



or 



i k 2 dX 



M= *i i+ **^ j* . . . (4) 

 Hence from equation (1) we have 



Aire 3 dx 2 ' \ *• 



= ™j /( X*A)-^ 



a differential equation to determine X. 



We see from this equation that when Sf(Ke\)—l3\ is 



