electrified particles when they strike against a metal plate cause
the metal to emit corpuscles (J. J. Thomson, *Proc. Camb. Phil.*
*Soc.* 13, p. 212; Austin, *Phys. Rev.* 22, p. 312). If we assume
that the number of corpuscles emitted by the plate in one second
is proportional to the energy in the positive ions which strike
the plate in that second, we can readily find an expression for
the difference of potential which will maintain without any
external ionization a current of electricity through the gas.
As this investigation brings into prominence many of the most
important features of the electric discharge, we shall consider it
in some detail.

Let us suppose that the electrodes are parallel plates of metal at
right angles to the axis of *x*, and that at the cathode *x* = 0 and at the
anode *x* = *d*, *d* being thus the distance between the plates. Let us
also suppose that the current of electricity flowing between the plates
is so small that the electrification between the plates due to the
accumulation of ions is not sufficient to disturb appreciably the
electric field, which we regard as uniform between the plates, the
electric force being equal to V/*d*, where V is the potential difference
between the plates. The number of positive ions produced per
second in a layer of gas between the planes *x* and *x* + *dx* is α*nu*·*dx*.
Here *n* is the number of corpuscles per unit volume, α the coefficient
of ionization (for strong electric field α = 1/λ′, where λ′ is the mean
free path of a corpuscle), and *u* the velocity of a corpuscle parallel
to *x*. We have seen that *nu* = *i*_{0}ε^{αx}, where *i*_{0} is the number of
corpuscles emitted per second by unit area of the cathode. Thus
the number of positive ions produced in the layer is α*i*_{0}ε^{αx} *dx*. If
these went straight to the cathode without a collision, each of them
would have received an amount of kinetic energy V*ex*/*d* when
they struck the cathode, and the energy of the group of ions would
be V*ex*/d·α*i*_{0}ε^{dx} *dx*. The positive ions will, however, collide with
the molecules of the gas through which they are passing, and this
will diminish the energy they possess when they reach the cathode.

The diminution in the energy will increase in geometrical proportion
with the length of path travelled by the ion and will thus
be proportional to ε^{−βx}, β will be proportional to the number of
collisions and will thus be proportional to the pressure of the gas.
Thus the kinetic energy possessed by the ions when they reach the
cathode will be

^{−βx}·V(

*ex*/

*d*) · α

*i*

_{0}ε

^{αx}

*dx*,

and E, the total amount of energy in the positive ions which reach the cathode in unit time, will be given by the equation

E = ∫d0
ε^{−βx} · V(ex/d) · α i_{0}ε^{αx} dx = |
Veα i_{0} |
∫d0
ε^{−(β−α)x} x dx |

d |

= | Veα i_{0} |
{ | 1 | − ε^{−(β−α)d}{ | 1 | + | d |
}} (1). |

d | (β − α)^{2} |
(β − α)^{2} | (β − α) |

If the number of corpuscles emitted by the cathode in unit time is
proportional to this energy we have *i*_{0} = *k*E, where *k* is a constant;
hence by equation (1) we have

V = | (β − α)^{2} |
· | d | , |

ke α | I |

where

^{−(β−α)d}(1 +

*d*(β − α)).

Since both β and α are proportional to the pressure, I and (β − α)^{2}*d*/α
are both functions of pd, the product of the pressure and the spark
length, hence we see that V is expressed by an equation of the form

V = | 1 | ƒ (pd) (2), |

ke |

where ƒ(*pd*) denotes a function of *pd*, and neither *p* nor *d* enter into
the expression for V except in this product. Thus the potential
difference required to produce discharge is constant as long as the
product of the pressure and spark length remains constant; in
other words, the spark potential is constant as long as the mass
of the gas between the electrodes is constant. Thus, for example,
if we halve the pressure the same potential difference will produce
a spark of twice the length. This law, which was discovered by
Paschen for fairly long sparks (*Annalen*, 37, p. 79), and has been
shown by Carr (*Phil. Trans.*, 1903) to hold for short ones, is one of
the most important properties of the electric discharge.

We see from the expression for V that when (β − α)*d* is very large

^{2}

*d*/

*ke*α.

Thus V becomes infinite when *d* is infinite. Again when (β − α)*d*
is very small we find

*ke*α

*d*;

thus V is again infinite when *d* is nothing. There must therefore
be some value of *d* intermediate between zero and infinity for which
V is a minimum. This value is got by finding in the usual way the
value of *d*, which makes the expression for V given in equation (1)
a minimum. We find that *d* must satisfy the equation

^{−(β−α)d}{1 + (β − α)

*d*+ (β − α·

*d*)

^{2}}.

We find by a process of trial and error that (β − α)*d* = 1.8 is approximately
a solution of this equation; hence the distance for minimum
potential is 1.8/(β − α). Since β and α are both proportional to the
pressure, we see that the critical spark length varies inversely as
the pressure. If we substitute this value in the expression for V
we find that V, the minimum spark potential, is given by

V = | β − α | · | 2.2 | . |

α | ke |

Since β and α are each proportional to the pressure, the minimum potential is independent of the pressure of the gas. On this view the minimum potential depends upon the metal of which the cathode is made, since k measures the number of corpuscles emitted per unit time by the cathode when struck by positive ions carrying unit energy, and unless β bears the same ratio to α for all gases the minimum potential will also vary with the gas. The measurements which have been made of the “cathode fall of potential,” which as we shall see is equal to the minimum potential required to produce a spark, show that this quantity varies with the material of which the cathode is made and also with the nature of the gas. Since a metal plate, when bombarded by positive ions, emits corpuscles, the effect we have been considering must play a part in the discharge; it is not, however, the only effect which has to be considered, for as Townsend has shown, positive ions when moving above a certain speed ionize the gas, and cause it to emit corpuscles. It is thus necessary to take into account the ionization of the positive ions.

Let *m* be the number of positive ions per unit volume, and *w*
their velocity, the number of collisions which occur in one second
in one cubic centimetre of the gas will be proportional to *mwp*,
where *p* is the pressure of the gas. Let the number of ions which
result from these collisions be γ*mw*; γ will be a function of *p* and
of the strength of the electric field. Let as before *n* be the number
of corpuscles per cubic centimetre, *u* their velocity, and α*nu* the
number of ions which result in one second from the collisions between
the corpuscles and the gas. The number of ions produced per
second per cubic centimetre is equal to α*nu* + γ*mw*; hence when
things are in a steady state

d | (nu) = αnu + γmw, |

dx |

and

*e*(

*nu*+

*mw*) =

*i*,

where *e* is the charge on the ion and *i* the current through the gas.
The solution of these equations when the field is uniform between the
plates, is

*enu*= Cε

^{(α−γ)x}− γ

*i*/ (α − γ),

*emw*= −Cε

^{(α−γ)x}+ α

*i*/ (α − γ),

where C is a constant of integration. If there is no emission of
positive ions from the anode *enu* = *i*, when *x* = *d*. Determining C
from this condition we find

enu = | i |
{αε ^{(α−γ) (x−d)} − γ
}, emw = | αi |
{1 − ε ^{(α−γ) (x−d)}
}. |

α − γ | α − γ |

If the cathode did not emit any corpuscles owing to the bombardment
by positive ions, the condition that the charge should be
maintained is that there should be enough positive ions at the cathode
to carry the current *i.e.* that *emw* = *i*; when *x* = 0, the condition
gives

i | {
αε^{−(α−γ)d} − γ} = 0 |

α − γ |

or

^{αd}/α = ε

^{γd}/γ.

Since α and γ are both of the form *p*ƒ(X/*p*) and X = V/*d*, we see that
V will be a function of *pd*, in agreement with Paschen’s law. If we
take into account both the ionization of the gas and the emission
of corpuscles by the metal we can easily show that

α − γε^{(α−γ)d} | = | kαVe | [ | 1 | − ε^{−(β+γ−α)d}{ |
1 | + | d | }], |

α − γ | d |
(β + γ − α)^{2} | (β + γ − α)^{2} |
β + γ − α |

where *k* and β have the same meaning as in the previous investigation.
When *d* is large, ε^{(α−γ)d} is also large; hence in order that the left-hand
side of this equation should not be negative γ must be less
than α/ε^{(α−γ)d}; as this diminishes as *d* increases we see that when
the sparks are very long discharge will take place, practically as
soon as γ has a finite value, *i.e.* as soon as the positive ions begin to
produce fresh ions by their collisions.

In the preceding investigation we have supposed that the electric field between the plates was uniform; if it were not uniform we could get discharges produced by very much smaller differences of potential than are necessary in a uniform field. For to maintain the discharge it is not necessary that the positive ions should act as ionizers all along their path; it is sufficient that they should do so in the neighbourhood of cathode. Thus if we have a strong field close to the cathode we might still get