the potential difference between the
terms in the gas will be E0 - Ri. Let ABC (fig. 22) be the
"characteristic curve," the ordinates being the potential difference
between the terminals in the gas, and the abscissae the current. Draw
the line LM whose equation is E = E0 - Ri, then the points where this
line cuts the characteristic curves will give possible values of i and
E, the current through the discharge tube and the potential difference
between the terminals. Some of these points may, however, correspond
to an unstable position and be impossible to realize. The following
method gives us a criterion by which we can distinguish the stable
from the unstable positions. If the current is increased by [delta]i,
the electromotive force which has to be overcome by the battery is
R[delta]i + dE/di . [delta]i. If R + dE/di is positive there will be
an unbalanced electromotive force round the circuit tending to stop
the current. Thus the increase in the current will be stopped and the
condition will be a stable one. If, however, R + dE/di is negative
there will be an unbalanced electromotive force tending to increase
the current still further; thus the current will go on increasing and
the condition will be unstable. Thus for stability R + dE/di must be
positive, a condition first given by Kaufmann (_Ann. der Phys._ 11, p.
158). The geometrical interpretation of this condition is that the
straight line LM must, at the point where it cuts the characteristic
curve, be steeper than the tangent to characteristic curve. Thus of
the points ABC where the line cuts the curve in fig. 22, A and C
correspond to stable states and B to an unstable one. The state of
things represented by a point P on the characteristic curve when the
slope is downward cannot be stable unless there is in the external
circuit a resistance greater than that represented by the tangent of
the inclination of the tangent to the curve at P to the horizontal
axis.

[Illustration: FIG. 22.]

If we keep the external electromotive force the same and gradually
increase the resistance in the leads, the line LM will become steeper
and steeper. C will move to the left so that the current will
diminish; when the line gets so steep that it touches the curve at C',
any further increase in the resistance will produce an abrupt change
in the current; for now the state of things represented by a point