Let us suppose that the velocity of the ion is proportional to the electric intensity, so that if w is the vertical velocity of the ions, which are supposed all to be of one sign, w = RZ.
Substituting this value of Z, the vertical force on the ions between A and B is equal to
| 1 | ∫∫∫ wρ dxdydz. |
| R |
But ∫∫ wρdxdy = ι, where ι is the current streaming from the point. This current, which can be easily measured by putting a galvanometer in series with the discharging point, is independent of z, the vertical distance of a plane between A and B below the charging point. Hence we have
| F = | ι | ∫dz = | ι | · z. |
| R | R |
This force must be counterbalanced by the difference of gaseous pressures over the planes A and B; hence if pB and pA denote respectively the pressures over B and A, we have
| pB − pA = | ι | z. |
| R |
Hence by the measurement of these pressures we can determine R, and hence the velocity with which an ion moves under a given electric intensity.
There are other methods of determining the velocities of the ions, but as these depend on the theory of the conduction of electricity through a gas containing charged ions, we shall consider them in our discussion of that theory.
By the use of these methods it has been shown that the velocities of the ions in a given gas are the same whether the ionization is produced by Röntgen rays, radioactive substances, ultra-violet light, or by the discharge of electricity from points. When the ionization is produced by chemical action the ions are very much less mobile, moving in the same electric field with a velocity less than one-thousandth part of the velocity of the first kind of ions. On the other hand, as we shall see later, the velocity of the negative ions in flames is enormously greater than that of even the first kind of ion under similar electric fields and at the same pressure. But when these negative ions get into the cold part of the flame, they move sluggishly with velocities of the order of those possessed by the second kind. The results of the various determinations of the velocities of the ions are given in the following table. The velocities are in centimetres per second under an electric force of one volt per centimetre, the pressure of the gas being 1 atmosphere. V+ denotes the velocity of the positive ion, V- that of the negative. V is the mean velocity of the positive and negative ions.