84 Prof. J. S. Townsend on the Potentials required 



For larger currents dY/di, the rate of increase of the 

 potential with the current, diminishes as i increases, and the 

 relation between V and i may be obtained on the same prin- 

 ciples, when the pressure of the gas between the wire and 

 the cylinder is not very low, and ionization by collision takes 

 place only at points whose distances from the wire are small 

 compared with the radius of the cylinder. 



The condition that the forces near the wire should be 

 sufficiently large to maintain a current is represented by the 

 equation 



< dv, 



-'■ 



where a. and /3 have their usual signification, a being the 

 radius of the wire and c the distance from the axis at which 

 the force becomes so small that a and j3 may be considered 

 to vanish. This condition is independent of the intensity of 

 the current, so that the forces corresponding to the starting 

 potential V , when i is infinitely small, will be sufficient to 

 maintain any current. 



The current affects the field of force owing to the electric 

 charge produced by the separation of the ions in the gas, the 

 principal action being due to ions of the same sign as the 

 charge on the wire, which give rise to a volume distribution 

 of electricity p in the space between the outer cylinder and 

 the region near the wire where the ions are generated. The 

 distribution p' due to ions of the same sign as the charge on 

 the wire that move through a short distance of the order 

 c — a has such a small effect compared with the distribution 

 p that it may be neglected. 



When a current is flowing the distribution p tends to 

 diminish the force near the surface of the wire, and in order 

 to maintain the current it is necessary to counteract this 

 effect by increasing the potential difference between the wire 

 and the cylinder. The relation between V and i may be 

 obtained on the hypothesis that the force at the surface of 

 the wire is equal to X x ( = V /alog bja), the force correspond- 

 ing to the starting potential V . For simplicity it may be 

 assumed that the velocities of the ions are proportional to 

 the electric force, and although this is by no means true for 

 the negative ions moving in gases at low pressures, the 

 formula obtained on this hypothesis, when compared with 

 the experimental results, shows very clearly many properties 

 of the negative ions. 



