immersed in ionized gas
and the potential difference between the plates. For let q be the
amount of ionization, i.e. the number of ions produced per second per
unit volume of the gas, A the area of one of the plates, and d the
distance between them; then if the ionization is constant through the
volume, the number of ions of one sign produced per second in the gas
is qAd. Now if i is the current per unit area of the plate, e the
charge on an ion, iA/e ions of each sign are driven out of the gas by
the current per second. In addition to this source of loss of ions
there is the loss due to the recombination; if n is the number of
positive or negative ions per unit volume, then the number which
recombine per second is [alpha]n^2 per cubic centimetre, and if n is
constant through the volume of the gas, as will approximately be the
case if the current through the gas is only a small fraction of the
saturation current, the number of ions which disappear per second
through recombination is [alpha]n^2.Ad. Hence, since when the gas is
in a steady state the number of ions produced must be equal to the
number which disappear, we have

qAd = iA/e + [alpha]n^2.Ad,
q = i/ed + [alpha]^n2.

If u1 and u2 are the velocities with which the positive and negative
ions move, nu1e and nu2e are respectively the quantities of positive
electricity passing in one direction through unit area of the gas per
second, and of negative in the opposite direction, hence

i = nu1e + nu2e.

If X is the electric force acting on the gas, k1 and k2 the velocities
of the positive and negative ions under unit force, u1 = k1X, u2 =
k2X; hence

n = i/(k1 + k2)Xe,

and we have

i [alpha]i^2
q = -- + -----------------.
ed (k1 + k2)^2e^2X^2

But qed is the saturation current per unit area of the plate; calling
this I, we have

d[alpha]i^2
I - i = ---------------
e(k1 + k2)^2X^2

or

i^2.d[alpha]
X^2 = -------------------.
e(I - i)(k1 + k2)^2

Hence if we determine corresponding values of X and i we can deduce
the value of [alpha]/e if we also know (k1 + k2). The value of I is
easily determined, as it is the current when X is very large. The
preceding result only applies when i is small compared with I, as it
is only in this case that the values of n and X are uniform throughout