4[pi]X /

where Q0 = n0le is the charge received by the plate when the electric
force is large enough to prevent recombination, and [epsilon] =
[alpha]4[pi]e(R1 + R2). We can from this result deduce the value of
[epsilon] and hence the value of [alpha] when R1+R2 is known.

_Distribution of Electric Force when a Current is passing through an
Ionized Gas._--Let the two plates be at right angles to the axis of x;
then we may suppose that between the plates the electric intensity X
is everywhere parallel to the axis of x. The velocities of both the
positive and negative ions are assumed to be proportional to X. Let
k1X, k2X represent these velocities respectively; let n1, n2 be
respectively the number of positive and negative ions per unit volume
at a point fixed by the co-ordinate x; let q be the number of positive
or negative ions produced in unit time per unit volume at this point;
and let the number of ions which recombine in unit volume in unit time
be [alpha]n1n2; then if e is the charge on the ion, the volume density
of the electrification is (n1 - n2)e, hence

dX
-- = 4[pi](n1 - n2)e (1).
dx

If I is the current through unit area of the gas and if we neglect any
diffusion except that caused by the electric field,

n1ek1X + n2ek2X = I (2).

From equations (1) and (2) we have

1 / I k2 dX \
n1e = ------- ( - + ----- -- ) (3),
k1 + k2 \ X 4[pi] dx /

1 / I k1 dX \
n2e = ------- ( - - ----- -- ) (4),
k1 + k2 \ X 4[pi] dx /

and from these equations we can, if we know the distribution of
electric intensity between the plates, calculate the number of
positive and negative ions.

In a steady state the number of positive and negative ions in unit
volume at a given place remains constant, hence neglecting the loss by
diffusion, we have

d
--(k1n1X) = q - [alpha]n1n2 (5).
dx

d
- --(k2n2X) = q - [alpha]n1n2 (6).
dx

If k1 and K2 are constant, we have from (1), (5) and (6)

d^2X^2 / 1 1 \
------ = 8[pi]e(q - [alpha]n1n2)( --- + --- ) (7),
dx^2 \ k1 k2 /

an equation which is very useful, because it enables us, if we know
the distribution of X^2, to find whether at any point in the gas the