--------------- {1 - [epsilon]^{([alpha]-[gamma])(x-d)} }.
[alpha] - [gamma] \ /

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 / \
----------------- {[alpha][epsilon]^{-([alpha]-[gamma])d} - [gamma] } = 0,
[alpha] - [gamma] \ /

or

[epsilon]^{[alpha]d}/[alpha] = [epsilon]^{[gamma]d}/[gamma].

Since [alpha] and [gamma] are both of the form pf(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

_
[alpha]-[gamma][epsilon]^{([alpha]-[gamma])d} k[alpha]Ve | 1
--------------------------------------------- = ---------- | -------------------------- -
[alpha] - [gamma] d |_ ([beta]+[gamma]-[alpha])^2
_
/ 1 d \ |
[epsilon]^{-([beta]+[gamma]-[alpha])d} { -------------------------- + ---------------------- } |,
\([beta]+[gamma]-[alpha])^2 [beta]+[gamma]-[alpha]/ _|

where k and [beta] have the same meaning as in the previous
investigation. When d is large, [epsilon]^{([alpha]-[gamma])d} is also
large; hence in order that the left-hand side of this equation should
not be negative [gamma] must be less than [alpha]/[epsilon]^
{([alpha]-[gamma])d}; as this diminishes as d increases we see that when
the sparks are very long discharge will take place, practically as soon
as [gamma] 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