unction 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 ([beta] - [alpha])d is very
large

V = ([beta] - [alpha])^2d/ke[alpha].

Thus V becomes infinite when d is infinite. Again when ([beta] -
[alpha])d is very small we find

V = 1/ke[alpha]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

/ \
1 = [epsilon]^{-([beta]-[alpha])d} {1 + ([beta] - [alpha])d + ([beta] - [alpha].d)^2}.
\ /

We find by a process of trial and error that ([beta]-[alpha])d = 1.8
is approximately a solution of this equation; hence the distance for
minimum potential is 1.8/([beta] - [alpha]). Since [beta] and [alpha]
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

_ [beta] - [alpha] 2.2
V = ---------------- . ---.
[alpha] ke

Since [beta] and [alpha] 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 u