oduced by coating the cathodes with these
oxides is exceedingly large; in some cases investigated by Tufts and
Stark (_Physik. Zeits._, 1908, 5, p. 248) the current was increased
many thousand times by coating the cathode with lime. No appreciable
effect is produced by putting lime on the anode.

_Conduction when all the Ions are of one Sign._--There are many
important cases in which the ions producing the current come from one
electrode or from a thin layer of gas close to the electrode, no
ionization occurring in the body of the gas or at the other electrode.
Among such cases may be mentioned those where one of the electrodes is
raised to incandescence while the other is cold, or when the negative
electrode is exposed to ultra-violet light. In such cases if the
electrode at which the ionization occurs is the positive electrode,
all the ions will be positively charged, while if it is the negative
electrode the ions will all be charged negatively. The theory of this
case is exceedingly simple. Suppose the electrodes are parallel planes
at right angles to the axis of x; let X be the electric force at a
distance x from the electrode where the ionization occurs, n the
number of ions (all of which are of one sign) at this place per cubic
centimetre, k the velocity of the ion under unit electric force, e the
charge on an ion, and i the current per unit area of the electrode.
Then we have dX/dx = 4[pi]ne, and if u is the velocity of the ion neu
= i. But u = kX, hence we have kX/4[pi] . dX/dx = i, and since the
right hand side of this equation does not depend upon x, we get
kX^2/8[pi] = ix + C, where C is a constant to be determined. If l is
the distance between the plates, and V the potential difference
between them,
_ _____ _ _
/ l 1 /8[pi] | |
V = | Xdx = --- / ----- | ( il + C )^3/2 - C^3/2 |.
_/0 i \/ k |_ _|

We shall show that when the current is far below the saturation value,
C is very small compared with il, so that the preceding equation
becomes

V^2 = 8[pi]l^3i/k (1).

To show that for small currents C is small compared with il, consider
the case when the ionization is confined to a thin layer, thickness d
close to the electrode, in that layer let n0 be the value of n, then