Let u_{1} and u_{2} 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 = Xu_{2} dt,
or
dt = (log_{e}(b/a) r dr)/(Vu_{2}).
Let r_{2} 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 t taken for the air to pass along the electrode.
Then t = ((r_{2}^2 - a^2)/(2Vu_{2})) log_{e}(b/a).
If ρ_{2} be the ratio of the number of the negative ions that reach the electrode A to the total number passing by, then
ρ_{2} = (r_{2}^2 - a^2)/(b^2 - a^2).
Therefore
u_{2} = (ρ_{2}(b^2 - a^2) log_{e}(b/a))/(2Vt) (1).
Similarly the ratio ρ_{1} 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
u_{1} = (ρ_{1}(b^2 - a^2) log_{e}(b/a))/(2Vt).
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 movement 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 ρ_{2} is proportional to V,—i.e. that
