ionization is greater or less than the recombination of the ions. We
see that q - [alpha]n1n2, which is the excess of ionization over
recombination, is proportional to d^2X^2/dx^2. Thus when the ionization
exceeds the recombination, i.e. when q - [alpha]n1n2 is positive, the
curve for X^2 is convex to the axis of x, while when the recombination
exceeds the ionization the curve for X^2 will be concave to the axis of
x. Thus, for example, fig. 11 represents the curve for X^2 observed by
Graham (_Wied. Ann._ 64, p. 49) in a tube through which a steady
current is passing. Interpreting it by equation (7), we infer that
ionization was much in excess of recombination at A and B, slightly so
along C, while along D the recombination exceeded the ionization.
Substituting in equation (7) the values of n1, n2 given in (3), (4),
we get
_ _
d^2X^2 | [alpha] / k^2 dX^2\ / k2 dX^2\ | / 1 1 \
------ = 8[pi]e |q - ----------------- (1 + ----- ---- ) (1 - ----- ---- )| ( --- + --- ) (8).
dx^2 |_ e^2X^2(k1 + k2)^2 \ 8[pi] dx / \ 8[pi] dx /_| \k1 k2 /

[Illustration: Fig. 11.]

This equation can be solved (see Thomson, _Phil. Mag._ xlvii. P. 253),
when q is constant and k1 = k2. From the solution it appears that if
X1 be the value of x close to one of the plates, and X0 the value
midway between them,

1
X1/X0 = -------------------
[beta]^2 - 2/[beta]

where [beta] = 8[pi]ek1/[alpha].

Since e = 4 X 10^-10, [alpha] = 2X10^-6, and k1 for air at atmospheric
pressure = 450, [beta] is about 2.3 for air at atmospheric pressure
and it becomes much greater at lower pressures.

Thus X1/X0 is always greater than unity, and the value of the ratio
increases from unity to infinity as [beta] increases from zero to
infinity. As [beta] does not involve either q or I, the ratio of X1 to
X0 is independent of the strength of the current and of the intensity
of the ionization.

No general solution of equation (8) has been found when k1 is not
equal to k2, but we can get an approximation to the solution when q is
constant. The equations (1), (2), (3), (4) are satisfied by the
values--

n1 = n2 = (q / [alpha])^1/2

k1
k1n1Xe = ------- I,
k1 + k2