nit energy,
and unless [beta] bears the same ratio to [alpha] for all gases the
minimum potential will also vary with the gas. The measurements which
have been made of the "cathode fall of potential," which as we shall
see is equal to the minimum potential required to produce a spark,
show that this quantity varies with the material of which the cathode
is made and also with the nature of the gas. Since a metal plate, when
bombarded by positive ions, emits corpuscles, the effect we have been
considering must play a part in the discharge; it is not, however, the
only effect which has to be considered, for as Townsend has shown,
positive ions when moving above a certain speed ionize the gas, and
cause it to emit corpuscles. It is thus necessary to take into account
the ionization of the positive ions.

Let m be the number of positive ions per unit volume, and w their
velocity, the number of collisions which occur in one second in one
cubic centimetre of the gas will be proportional to mwp, where p is
the pressure of the gas. Let the number of ions which result from
these collisions be [gamma]mw; [gamma] will be a function of p and of
the strength of the electric field. Let as before n be the number of
corpuscles per cubic centimetre, u their velocity, and [alpha]nu the
number of ions which result in one second from the collisions between
the corpuscles and the gas. The number of ions produced per second per
cubic centimetre is equal to [alpha]nu + [gamma]mw; hence when things
are in a steady state

d
--(nu) = [alpha]nu + [gamma]mw ,
dx

and

e(nu + mw) = i,

where e is the charge on the ion and i the current through the gas.
The solution of these equations when the field is uniform between the
plates, is

enu = C[epsilon]^{([alpha]-[gamma])x} - [gamma]i/([alpha] - [gamma]),

emw = -C[epsilon]^{([alpha]-[gamma])x} + [alpha]i/([alpha] - [gamma]),

where C is a constant of integration. If there is no emission of
positive ions from the anode enu = i, when x = d. Determining C from
this condition we find

i / \
enu = ----------------- {[alpha][epsilon]^{([alpha]-[gamma])(x-d)} - [gamma] },
[alpha] - [gamma] \ /

[alpha]i / \
emw = --