[lambda]

When things are in a steady state dn/dt = 0, and we have

d 1 / \
--(nu) = --------( [int](Xe[lambda]) - [beta] )nu.
dx [lambda] \ /

If the current is so small that the electrical charges in the gas are
not able to produce any appreciable variations in the field, X will be
constant and we get nu = C[epsilon]^{[alpha]x}, where [alpha] =
{[int](Xe[lambda]) - [beta]}/[lambda]. If we take the origin from
which we measure x at the cathode, C is the value of nu at the
cathode, i.e. it is the number of corpuscles emitted per unit area of
the cathode per unit time; this is equal to i0/e if i0 is the quantity
of negative electricity coming from unit area of the cathode per
second, and e the electric charge carried by a corpuscle. Hence we
have nue = i0[epsilon]^{[alpha]x}. If l is the distance between the
anode and the cathode, the value of nue, when x = l, is the current
passing through unit area of the gas, if we neglect the electricity
carried by negatively electrified carriers other than corpuscles.
Hence i = i0[epsilon]^{[alpha]l}. Thus the current between the plates
increases in geometrical progression with the distance between the
plates.

By measuring the variation of the current as the distance between the
plates is increased, Townsend, to whom we owe much of our knowledge on
this subject, determined the values of [alpha] for different values of
X and for different pressures for air, hydrogen and carbonic acid gas
(_Phil. Mag._ [6], 1, p. 198). Since [lambda] varies inversely as the
pressure, we see that [alpha] may be written in the form p[phi](X/p)
or [alpha]/X = F(X/p). The following are some of the values of [alpha]
found by Townsend for air.

+---------+----------+----------+----------+----------+----------+
| X Volts | Pressure | Pressure | Pressure | Pressure | Pressure |
| per cm. | .17 mm. | .38 mm. | 1.10 mm. | 2.1 mm. | 4.1 mm. |
+---------+----------+----------+----------+----------+----------+
| 20 | .24 | | | | |
| 40 | .65 | .34 | | | |
| 80 | 1.35 | 1.3 | .45 | .13 | |
| 120 | 1.8 | 2.0 | 1.1 | .42 | .13 |
| 160 | 2.1 | 2.8 | 2.0 | .9 | .