The number of ions produced by collisions can be calculated by the following method. Let the electric force be parallel to the axis of x, and let n be the number of corpuscles per unit volume at a place fixed by the co-ordinate x; then in unit time these corpuscles will make nu/[lambda] collisions with the molecules, if u is the velocity of a corpuscle and [lambda] the mean free path of a corpuscle. When the corpuscles are moving fast enough to produce ions by collision their velocities are very much greater than those they would possess at the same temperature if they were not acted on by electrical force, and so we may regard the velocities as being parallel to the axis of x and determined by the electric force and the mean free path of the corpuscles. We have to consider how many of the nu/[lambda] collisions which take place per second will produce ions. We should expect that the ionization of a molecule would require a certain amount of energy, so that if the energy of the corpuscle fell below this amount no ionization would take place, while if the energy of the corpuscle were exceedingly large, every collision would result in ionization. We shall suppose that a certain fraction of the number of collisions result in ionization and that this fraction is a function of the energy possessed by the corpuscle when it collides against the molecules. This energy is proportional to Xe[lambda] when X is the electric force, e the charge on the corpuscle, and [lambda] the mean free path. If the fraction of collisions which produce ionization is [int](Xe[lambda]), then the number of ions produced per cubic centimetre per second is [int](Xe[lambda])nu/[lambda]. If the collisions follow each other with great rapidity so that a molecule has not had time to recover from one collision before it is struck again, the effect of collisions might be cumulative, so that a succession of collisions might give rise to ionization, though none of the collisions would produce an ion by itself. In this case [int] would involve the frequency of the collisions as well as the energy of the corpuscle; in other words, it might depend on the current through the gas as well as upon the intensity of the electric field. We shall, however, to begin with, assume that the current is so small that this cumulative effect may be neglected.