Let us now consider the rate of increase, dn/dt, in the number of corpuscles per unit volume. In consequence of the collisions, [int](Xe[lambda])nu/[lambda] corpuscles are produced per second; in consequence of the motion of the corpuscles, the number which leave unit volume per second is greater than those which enter it by (d/dx)(nu); while in a certain number of collisions a corpuscle will stick to the molecule and will thus cease to be a free corpuscle. Let the fraction of the number of collisions in which this occurs be [beta]. Thus the gain in the number of corpuscles is [int](Xe[lambda])nu/[lambda], while the loss is (d/dx)(nu) + [beta](nu)/[lambda]; hence dn nu d [beta]nu -- = [int](Xe[lambda]) -------- - --(nu) - --------. dt [lambda] dx [lambda] When things are in a steady state dn/dt = 0, and we have d 1 / \ --(nu) = --------( [int](Xe[lambda]) - [beta] )nu. dx [lambda] \ / If the current is so small that the electrical charges in the gas are not able to produce any appreciable variations in the field, X will be constant and we get nu = C[epsilon]^{[alpha]x}, where [alpha] = {[int](Xe[lambda]) - [beta]}/[lambda]. If we take the origin from which we measure x at the cathode, C is the value of nu at the cathode, i.e. it is the number of corpuscles emitted per unit area of the cathode per unit time; this is equal to i0/e if i0 is the quantity of negative electricity coming from unit area of the cathode per second, and e the electric charge carried by a corpuscle. Hence we have nue = i0[epsilon]^{[alpha]x}. If l is the distance between the anode and the cathode, the value of nue, when x = l, is the current passing through unit area of the gas, if we neglect the electricity carried by negatively electrified carriers other than corpuscles. Hence i = i0[epsilon]^{[alpha]l}. Thus the current between the plates increases in geometrical progression with the distance between the plates. By measuring the variation of the current as the distance between the plates is increased, Townsend, to whom we owe much of our knowledge on this subject, determined the values of [alpha] for different values of X and for different pressures for air, hydrogen and carbonic acid gas (_Phil. Mag._ [6], 1, p. 198). Since [lambda] varies inversely as the pressure, we see that [alpha] may be written in the form p[phi](X/p) or [alpha]/X = F(X/p). The following are some of the values of [alpha] found by Townsend for air.