arks not much longer than the critical spark length that we
could hope to detect this difference. When the current through the gas
exceeds a certain critical value depending among other things on the
size of the cathode, the cathode fall of potential increases rapidly and
at the same time the thickness of the dark spaces diminishes. We may
regard the part of the discharge between the cathode and the negative
glow as a discharge taking place under minimum potential difference
through a distance equal to the critical spark length. An inspection of
fig. 16 will show that we cannot regard the electric field as constant
even for this small distance; it thus becomes a matter of interest to
know what would be the effect on the minimum potential difference
required to produce a spark if there were sufficient ions present to
produce variations in the electric field analogous to those represented
in fig. 16. If the electric force at a distance x from the cathode were
proportional to [epsilon]^-px we should have a state of things much
resembling the distribution of electric force near the cathode. If we
apply to this distribution the methods used above for the case when the
force was uniform, we shall find that the minimum potential is less and
the critical spark length greater than when the electric force is
uniform.

_Potential Difference required to produce a Spark of given Length._--We
may regard the region between the cathode and the negative glow as a
place for the production of corpuscles, these corpuscles finding their
way from this region through the negative glow. The parts of this glow
towards the anode we may regard as a cathode, from which, as from a hot
lime cathode, corpuscles are emitted. Let us now consider what will
happen to these corpuscles shot out from the negative glow with a
velocity depending on the cathode fall of potential and independent of
the pressure. These corpuscles will collide with the molecules of the
gas, and unless there is an external electric field to maintain their
velocity they will soon come to rest and accumulate in front of the
negative glow. The electric force exerted by this cloud of corpuscles
will diminish the strength of the electric field in the region between
the cathode and the negative glow, and thus tend to stop the discharge.
To keep up the discharge we must have a sufficiently strong electric
field between the negative glow and the anode to remove the corpuscles
from this re