ce multiplied by the distance between the plates, the
potential difference will vary as the square of this distance.

The potential difference required to produce saturation will, other
circumstances being the same, increase with the amount of ionization,
for when the number of ions is large and they are crowded together, the
time which will elapse before a positive one combines with a negative
will be smaller than when the number of ions is small. The ions have
therefore to be removed more quickly from the gas when the ionization is
great than when it is small; thus they must move at a higher speed and
must therefore be acted upon by a larger force.

When the ions are not removed from the gas, they will increase until the
number of ions of one sign which combine with ions of the opposite sign
in any time is equal to the number produced by the ionizing agent in
that time. We can easily calculate the number of free ions at any time
after the ionizing agent has commenced to act.

Let q be the number of ions (positive or negative) produced in one
cubic centimetre of the gas per second by the ionizing agent, n1, n2,
the number of free positive and negative ions respectively per cubic
centimetre of the gas. The number of collisions between positive and
negative ions per second in one cubic centimetre of the gas is
proportional to n1n2. If a certain fraction of the collisions between
the positive and negative ions result in the formation of an
electrically neutral system, the number of ions which disappear per
second on a cubic centimetre will be equal to [alpha]n1 n2, where
[alpha] is a quantity which is independent of n1, n2; hence if t is
the time since the ionizing agent was applied to the gas, we have

dn1/dt = q - [alpha]n1 n2, dn2/dt = q - [alpha]n1 n2.

Thus n1 - n2 is constant, so if the gas is uncharged to begin with, n1
will always equal n2. Putting n1 = n2 = n we have

dn/dt = q - [alpha]n^2 (1),

the solution of which is, since n = 0 when t = 0,

k([epsilon]^{2k[alpha]t} - 1)
n = ---------------------------- (2)
[epsilon]^{2k[alpha]t} + 1

if k^2 = q/[alpha]. Now the number of ions when the gas has reached a
steady state is got by putting t equal to infinity in the preceding
equation, and is therefore given by the equation

n0 = k = [root](q/[alpha]).

We see from equation (1) that the gas will not approximate to its