Let us suppose that the electrodes are parallel plates of metal at right angles to the axis of x, and that at the cathode x = 0 and at the anode x = d, d being thus the distance between the plates. Let us also suppose that the current of electricity flowing between the plates is so small that the electrification between the plates due to the accumulation of ions is not sufficient to disturb appreciably the electric field, which we regard as uniform between the plates, the electric force being equal to V/d, where V is the potential difference between the plates. The number of positive ions produced per second in a layer of gas between the planes x and x+dx is [alpha]nu·dx. Here n is the number of corpuscles per unit volume, [alpha] the coefficient of ionization (for strong electric field [alpha] = 1/[lambda]', where [lambda]' is the mean free path of a corpuscle), and u the velocity of a corpuscle parallel to x. We have seen that nu = i0[epsilon]^[alpha]x, where i0 is the number of corpuscles emitted per second by unit area of the cathode. Thus the number of positive ions produced in the layer is [alpha]i0[epsilon]^[alpha]x dx. If these went straight to the cathode without a collision, each of them would have received an amount of kinetic energy Vex/d when they struck the cathode, and the energy of the group of ions would be Vex/d·[alpha]i0[epsilon]^dx dx. The positive ions will, however, collide with the molecules of the gas through which they are passing, and this will diminish the energy they possess when they reach the cathode. The diminution in the energy will increase in geometrical proportion with the length of path travelled by the ion and will thus be proportional to [epsilon]^-[beta]x, [beta] will be proportional to the number of collisions and will thus be proportional to the pressure of the gas. Thus the kinetic energy possessed by the ions when they reach the cathode will be [epsilon]^{-[beta]x} · V(ex/d) · [alpha]i0[epsilon]^{[alpha]x} dx, and E, the total amount of energy in the positive ions which reach the cathode in unit time, will be given by the equation _ /d E = | [epsilon]^{-[beta]x} · V(ex/d) · [alpha]i0[epsilon]^{[alpha]x} dx _/0 _ Ve[alpha]i0 /d = ----------- | [epsilon]^{-([beta]-[alpha])x}·x·dx d _/0 Ve[alpha]i0 / 1 / 1 d \ \ = ----------- { ---------------- - [epsilon]^{-([beta]-[alpha])d} { ----------------- + ---------------- } } (1). d \([beta]-[alpha])² \([beta]-[alpha])² ([beta]-[alpha])/ / If the number of corpuscles emitted by the cathode in unit time is proportional to this energy we have i0 = kE, where k is a constant; hence by equation (1) we have