Since both [beta] and [alpha] are proportional to the pressure, I and
([beta] - [alpha])^2d/[alpha] 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
1
V = -- [int](pd) (2),
ke
where [int](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 ([beta] - [alpha])d is very
large