The value of p, when X is given, for which [alpha] is a maximum, is proportional to X; this follows at once from the fact that [alpha] is of the form X·F(X/p). The value of X/p for which F(X/p) is a maximum is seen from the preceding table to be about 420, when X is expressed in volts per centimetre and p in millimetres of mercury. The maximum value of F(X/p) is about 1/60. Since the current passing between two planes at a distance l apart is i0[epsilon]^{[alpha]l} or i0[epsilon]^{XlF(X/p)}, and since the force between the plates is supposed to be uniform, Xl is equal to V, the potential between the plates; hence the current between the plates is i0[epsilon]^{VlF(X/p)}, and the greatest value it can have is i0[epsilon]^{V/60}. Thus the ratio between the current between the plates when there is ionization and when there is none cannot be greater than [epsilon]^{V/60}, when V is measured in volts. This result is based on Townsend's experiments with very weak currents; we must remember, however, that when the collisions are so frequent that the effects of collisions can accumulate, [alpha] may have much larger values than when the current is small. In some experiments made by J. J. Thomson with intense currents from cathodes covered with hot lime, the increase in the current when the potential difference was 60 volts, instead of being e times the current when there was no ionization, as the preceding theory indicates, was several hundred times that value, thus indicating a great increase in [alpha] with the strength of the current. Townsend has shown that we can deduce from the values of [alpha] the mean free path of a corpuscle. For if the ionization is due to the collisions with the corpuscles, then unless one collision detaches more than one corpuscle the maximum number of corpuscles produced will be equal to the number of collisions. When each collision results in the production of a corpuscle, [alpha] = 1/[lambda] and is independent of the strength of the electric field. Hence we see that the value of [alpha], when it is independent of the electric field, is equal to the reciprocal of the free path. Thus from the table we infer that at a pressure of 17 mm. the mean free path is 1/325 cm.; hence at 1 mm. the mean free path of a corpuscle is 1/19 cm. Townsend has shown that this value of the mean free path agrees well with the value 1/21 cm. deduced from the kinetic theory of gases for a corpuscle moving through air. By measuring the values of [alpha] for hydrogen and carbonic acid gas Townsend and Kirby (_Phil. Mag._ [6], 1, p. 630) showed that the mean free paths for corpuscles in these gases are respectively 1/11.5 and 1/29 cm. at a pressure of 1 mm. These results again agree well with the values given by the kinetic theory of gases.