We shall, however, to begin with, assume that the current is so small
that this cumulative effect may be neglected.

Let us now consider the rate of increase, *dn*/*dt*, in the number of
corpuscles per unit volume. In consequence of the collisions,
*f*(X*e*λ)*nu*/λ corpuscles are produced per second; in consequence
of the motion of the corpuscles, the number which leave unit volume
per second is greater than those which enter it by *d**dx*(*nu*); while in a
certain number of collisions a corpuscle will stick to the molecule and
will thus cease to be a free corpuscle. Let the fraction of the number
of collisions in which this occurs be β. Thus the gain in the number
of corpuscles is *f* (X*e*λ)*nu*/λ, while the loss is *d*/*dx*·(*nu*) + β*nu*λ hence

dn | f(Xeλ) | nu |
− | d |
(nu) − | βnu | . |

dt | λ | dx | λ |

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

d | (nu) = | 1 | (f(Xeλ) − β) nu. |

dx | λ |

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* = Cε^{αx}, where α = {*f* (X*e*λ) − β}/λ. 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 *i*_{0}/*e* if *i*_{0} 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* = *i*_{0}ε^{αx}. 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* = *i*_{0}ε^{α 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 α for different
values of X and for different pressures for air, hydrogen and carbonic
acid gas (*Phil. Mag.* [6], 1, p. 198). Since λ varies inversely as the
pressure, we see that α may be written in the form *p*φ(X/*p*) or
α/X = F(X/*p*). The following are some of the values of α found by
Townsend for air.

X Volts per cm. | Pressure .17 mm. |
Pressure .38 mm. | Pressure 1.10 mm. |
Pressure 2.1 mm. | Pressure 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 | .28 |

200 | 3.4 | 2.8 | 1.6 | .5 | |

240 | 2.45 | 3.8 | 4.0 | 2.35 | .99 |

320 | 2.7 | 4.5 | 5.5 | 4.0 | 2.1 |

400 | 5.0 | 6.8 | 6.0 | 3.6 | |

480 | 3.15 | 5.4 | 8.0 | 7.8 | 5.3 |

560 | 5.8 | 9.3 | 9.4 | 7.1 | |

640 | 3.25 | 6.2 | 10.6 | 10.8 | 8.9 |

We see from this table that for a given value of X, α for small pressures increases as the pressure increases; it attains a maximum at a particular pressure, and then diminishes as the pressure increases. The increase in the pressure increases the number of collisions, but diminishes the energy acquired by the corpuscle in the electric field, and thus diminishes the change of any one collision resulting in ionization. If we suppose the field is so strong that at some particular pressure the energy acquired by the corpuscle is well above the value required to ionize at each collision, then it is evident that increasing the number of collisions will increase the amount of ionization, and therefore α, and α cannot begin to diminish until the pressure has increased to such an extent that the mean free path of a corpuscle is so small that the energy acquired by the corpuscle from the electric field falls below the value when each collision results in ionization.

The value of *p*, when X is given, for which α is a maximum, is
proportional to X; this follows at once from the fact that α is of the
form X·F(X/*p*). The value of X/*p* for which F(X/*p*) is a maximum
is seen from the preceding table to be about 420, when X is expressed
in volts per centimetre and *p* in millimetres of mercury. The
maximum value of F(X/*p*) is about 160. Since the current passing
between two planes at a distance *l* apart is *i*_{0}ε^{αl} or *i*_{0}ε^{XlF(X/p)},
and since the force between the plates is supposed to be uniform,
X*l* is equal to V, the potential between the plates; hence the
current between the plates is *i*_{0}ε^{V·F(X/p)}, and the greatest value
it can have is *i*_{0}ε^{V/60}. Thus the ratio between the current between
the plates when there is ionization and when there is none cannot
be greater than ε^{V/60}, when V is measured in volts. This result is
based on Townsend’s experiments with very weak currents; we
must remember, however, that when the collisions are so frequent
that the effects of collisions can accumulate, α may have much larger
values than when the current is small. In some experiments made
by J. J. Thomson with intense currents from cathodes covered
with hot lime, the increase in the current when the potential difference
was 60 volts, instead of being e times the current when there was no
ionization, as the preceding theory indicates, was several hundred
times that value, thus indicating a great increase in α with the
strength of the current.

Townsend has shown that we can deduce from the values of α the
mean free path of a corpuscle. For if the ionization is due to the
collisions with the corpuscles, then unless one collision detaches
more than one corpuscle the maximum number of corpuscles produced
will be equal to the number of collisions. When each collision
results in the production of a corpuscle, α = 1/λ and is independent
of the strength of the electric field. Hence we see that the value of
α, when it is independent of the electric field, is equal to the reciprocal
of the free path. Thus from the table we infer that at a pressure
of 17 mm. the mean free path is 1325 cm.; hence at 1 mm. the mean
free path of a corpuscle is 119 cm. Townsend has shown that this
value of the mean free path agrees well with the value 121 cm.
deduced from the kinetic theory of gases for a corpuscle moving
through air. By measuring the values of α for hydrogen and carbonic
acid gas Townsend and Kirby (*Phil. Mag.* [6], 1, p. 630) showed
that the mean free paths for corpuscles in these gases are respectively
111.5 and 129 cm. at a pressure of 1 mm. These results again agree
well with the values given by the kinetic theory of gases.

If the number of positive ions per unit volume is *m* and *v* is the
velocity, we have *nue* + *mve* = *i*, where *i* is the current through unit
area of the gas. Since *nue* = *i*_{0}ε^{nx} and *i* = *i*_{0}ε^{nl}, when *l* is the distance
between the plates, we see that

*nu*/

*mv*= ε

^{nx}/ (ε

^{nl}− ε

^{nx}),

n | = | v | · | ε^{nx} | . |

m | u | ε^{ne} − ε^{nx} |

Since *v*/*u* is a very small quantity we see that *n* will be less than *m*
except when ε^{nl} − ε^{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* = *i*_{0}ε^{αl} 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

i = i_{0}ε^{αl} + | qe |
(e^{αl} − 1). |

α |

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 ions by collisions, for then we should have a kind of regenerative process by which the supply of corpuscles could be continually renewed. To maintain the current it is not necessary that the ionization resulting from the positive ions should be anything like as great as that from the negative, as the investigation given below shows a very small amount of ionization by the positive ions will suffice to maintain the current. The existence of ionization by collision with positive ions has been proved by Townsend. Another method by which the current could be and is maintained is by the anode emitting corpuscles under the impact of the positive ions driven against it by the electric field.

J. J. Thomson has shown by direct experiment that positively