n]^(nx)

Since v/u is a very small quantity we see that n will be less than m
except when [epsilon]^nl - [epsilon]^nx is small, i.e. except close to
the anode. Thus there will be an excess of positive electricity from
the cathode almost up to the anode, while close to the anode there
will be an excess of negative. This distribution of electricity will
make the electric force diminish from the cathode to the place where
there is as much positive as negative electricity, where it will have
its minimum value, and then increase up to the anode.

The expression i = i0[epsilon]^[alpha]l applies to the case when there
is no source of ionization in the gas other than the collisions; if in
addition to this there is a source of uniform ionization producing q
ions per cubic centimetre, we can easily show that

qe
i = i0[epsilon]^{[alpha]l} + -------(e^{[alpha]l} - 1).
[alpha]

With regard to the minimum energy which must be possessed by a
corpuscle to enable it to produce ions by collision, Townsend (loc.
cit.) came to the conclusion that to ionize air the corpuscle must
possess an amount of energy equal to that acquired by the fall of its
charge through a potential difference of about 2 volts. This is also
the value arrived at by H. A. Wilson by entirely different
considerations. Stark, however, gives 17 volts as the minimum for
ionization. The energy depends upon the nature of the gas; recent
experiments by Dawes and Gill and Pedduck (_Phil. Mag._, Aug. 1908)
have shown that it is smaller for helium than for air, hydrogen, or
carbonic acid gas.

If there is no external source of ionization and no emission of
corpuscles from the cathode, then it is evident that even if some
corpuscles happened to be present in the gas when the electric field
were applied, we could not get a permanent current by the aid of
collisions made by these corpuscles. For under the electric field, the
corpuscles would be driven from the cathode to the anode, and in a short
time all the corpuscles originally present in the gas and those produced
by them would be driven from the gas against the anode, and if there was
no source from which fresh corpuscles could be introduced into the gas
the current would cease. The current, however, could be maintained
indefinitely if the positive ions in their journey back to the cathode
also produced