he positive ion is to that of the negative ion. Thus
the force at the negative plate is greater than that at the positive.
The falls of potential V1, V2 at the two layers when 1/[epsilon] is
large can be shown to be given by the equations

/[epsilon]\^3/2 /k1\^1/2
V1 = 8[pi]^2( --------- ) k1 ( -- ) i^2,
\q [alpha]/ \k2/

/[epsilon]\^3/2 /k2\^1/2
V2 = 8[pi]^2( --------- ) k2 ( -- ) i^2,
\q [alpha]/ \k1/

hence V1/V2 = k1^2/k2^2,

so that the potential falls at the electrodes are proportional to the
squares of the velocities of the ions. The change in potential across
the layers is proportional to the square of the current, while the
potential change between the layers is proportional to the current,
the total potential difference between the plates is the sum of these
changes, hence the relation between V and i will be of the form

V = Ai + Bi^2.

Mie (_Ann. der. Phys._, 1904, 13, P. 857) has by the method of
successive approximations obtained solutions of equation (8) (i.) when
the current is only a small fraction of the saturation current, (ii.)
when the current is nearly saturated. The results of his
investigations are represented in fig. 12, which represents the
distribution of electric force along the path of the current for
various values of the current expressed as fractions of the saturation
current. It will be seen that until the current amounts to about
one-fifth of the maximum current, the type of solution is the one just
indicated, i.e. the electric force is constant except in the
neighbourhood of the electrodes when it increases rapidly.

Though we are unable to obtain a general solution of the equation (8),
there are some very important special cases in which that equation can
be solved without difficulty. We shall consider two of these, the
first being that when the current is saturated. In this case there is
no loss of ions by recombination, so that using the same notation as
before we have

d
--(n1k1X) = q,
dx

d
--(n2k2X) = -q.
dx

The solutions of which if q is constant are

n1k1X = qx,

n2k2X = I/e - qx = q(l - x),

if l is the distance between the plates, and x = 0 at the positive
electrode. Since

dX/dx = 4[pi](n1 - n2)e,

we get

1 dX^2 / 1 1 \ l