ions by collisions, for then we should have a kind of
regenerative process by which the supply of corpuscles could be
continually renewed. To maintain the current it is not necessary that
the ionization resulting from the positive ions should be anything like
as great as that from the negative, as the investigation given below
shows a very small amount of ionization by the positive ions will
suffice to maintain the current. The existence of ionization by
collision with positive ions has been proved by Townsend. Another method
by which the current could be and is maintained is by the anode emitting
corpuscles under the impact of the positive ions driven against it by
the electric field. J. J. Thomson has shown by direct experiment that
positively electrified particles when they strike against a metal plate
cause the metal to emit corpuscles (J. J. Thomson, _Proc. Camb. Phil.
Soc._ 13, p. 212; Austin, _Phys. Rev._ 22, p. 312). If we assume that
the number of corpuscles emitted by the plate in one second is
proportional to the energy in the positive ions which strike the plate
in that second, we can readily find an expression for the difference of
potential which will maintain without any external ionization a current
of electricity through the gas. As this investigation brings into
prominence many of the most important features of the electric
discharge, we shall consider it in some detail.

Let us suppose that the electrodes are parallel plates of metal at
right angles to the axis of x, and that at the cathode x = 0 and at the
anode x = d, d being thus the distance between the plates. Let us also
suppose that the current of electricity flowing between the plates is
so small that the electrification between the plates due to the
accumulation of ions is not sufficient to disturb appreciably the
electric field, which we regard as uniform between the plates, the
electric force being equal to V/d, where V is the potential difference
between the plates. The number of positive ions produced per second in
a layer of gas between the planes x and x+dx is [alpha]nu.dx. Here n is
the number of corpuscles per unit volume, [alpha] the coefficient of
ionization (for strong electric field [alpha] = 1/[lambda]', where
[lambda]' is the mean free path of a corpuscle), and u the velocity of
a corpuscle parallel to x. We have seen that nu = i0[epsilon]^[alpha]x,
where i0 is the number of corpuscles emitted per s