11] IONIZATION THEORY OF GASES 41 
Let uw, and uw be the velocities of the positive and negative 
ions for a potential gradient of 1 volt per cm. If the velocity is 
proportional to the electric force at any point, the distance dr 
traversed by the negative ion in the time dt is given by 
dr = X updt, 
or log, u rar 
hp ast at 
Vu, 
Let 7, be the greatest distance measured from the axis of the 
tube from which the negative ion can just reach the electrode A 
in the time ¢ taken for the air to pass along the electrode. 
) tog 
Then t= Teva Ofer - 
If p, be the ratio of the number of the negative ions that reach 
the electrode A to the total number passing by, then 
Ts — oe 
P2 b2 — a2 

Therefore 
ps (bt —a) log, ? 
a 
FINO Pepe te woo eect eee eee ee rc eec st qd ). 
Similarly the ratio p, of the number of positive ions that give 
up their charge to the external cylinder to the total number of 
positive ions is given by 
i 
Q b 
p (B? — a?) loge = 
2V it 
In the above equations it is assumed that the current of air is 
uniform over the cross-section of the tube, and that the ions are 
uniformly distributed over the cross-section ; also, that the move- 
ment of the ions does not appreciably disturb the electric field. 
Since the value of t can be calculated from the velocity of the 
current of air and the length of the electrode, the values of the 
velocities of the ions under unit potential gradient can at once be 
determined. 
The equation (1) shows that p, is proportional to V,—z.e. ie 
U,= 
