([beta]-[alpha])² d V = ----------------- · --, ke[alpha] I where I = 1 - [epsilon]^{-([beta]-[alpha])d} (1 + d([beta] - [alpha])). 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 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