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.

Let us now consider the rate of increase, dn/dt, in the number of
corpuscles per unit volume. In consequence of the collisions,
[int](Xe[lambda])nu/[lambda] corpuscles are produced per second; in
consequence of the motion of the corpuscles, the number which leave
unit volume per second is greater than those which enter it by
(d/dx)(nu); while in a certain number of collisions a corpuscle will
stick to the molecule and will thus cease to be a free corpuscle. Let
the fraction of the number of collisions in which this occurs be
[beta]. Thus the gain in the number of corpuscles is
[int](Xe[lambda])nu/[lambda], while the loss is (d/dx)(nu) +
[beta](nu)/[lambda]; hence

dn nu d [beta]nu
-- = [int](Xe[lambda]) -------- - --(nu) - --------.
dt [lambda] dx