f the electric field exceeds a certain
critical value, which depends not merely upon the nature of the gas but
also upon the pressure and the dimensions of the vessel in which it is
contained. The variation of the electric field required to produce
discharge can be completely explained if we suppose that the ionization
of the gas is produced by the impact with its molecules of corpuscles,
and in certain cases of positive ions, which under the influence of the
electric field have acquired considerable kinetic energy. We have direct
evidence that rapidly moving corpuscles are able to ionize molecules
against which they strike, for the cathode rays consist of such
corpuscles, and these when they pass through a gas produce large amounts
of ionization. Suppose then that we have in a gas exposed to an electric
field a few corpuscles. These will be set in motion by the field and
will acquire an amount of energy in proportion to the product of the
electric force, their charge, and the distance travelled in the
direction of the electric field between two collisions with the
molecules of the gas. If this energy is sufficient to give them the
ionizing property possessed by cathode rays, then when a corpuscle
strikes against a molecule it will detach another corpuscle; this under
the action of the electric field will acquire enough energy to produce
corpuscles on its own account, and so as the corpuscles move through the
gas their number will increase in geometrical progression. Thus, though
there were but few corpuscles to begin with, there may be great
ionization after these have been driven some distance through the gas by
the electric field.

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