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When a cylindrical or spherical electrode (collector) immersed in an ionized gas is brought to a suitable potential, it becomes surrounded by a symmetrical space-charge region or "sheath" of positive or of negative ions (or electrons). Assuming that the gas pressure is so low that the proportion of ions which collide with gas molecules in the sheath is negligibly small, the current taken by the collector can be calculated in terms of the radii of the collector or sheath, the distribution of velocities among the ions arriving at the sheath boundary and the total drop of potential in the sheath. The current is independent of the actual distribution of potential in the sheath provided this distribution satisfies certain conditions."Orbital Motion" equations for spherical and cylindrical collectors.---General formulas for the current are derived and the calculations are then carried out for collectors in a group of ions having velocities which are (A) equal and parallel; (B) equal in magnitude but of random direction; (C) Maxwellian; (D) Maxwellian with a drift velocity superimposed. In all cases the collector current becomes practically independent of the sheath radius when this radius is large compared with that of the collector. Thus the volt-ampere characteristics of a collector of sufficiently small radius can be used to distinguish between the different types of velocity distribution. General equations are also given by means of which the velocity distribution can be calculated directly from the volt-ampere characteristics of a sphere or cylinder.Special properties of the Maxwellian distribution.---For a collector of any shape having a convex surface, the logarithm of the current taken from a Maxwellian distribution is a linear function of the voltage difference between the collector and the gas when the collector potential is such as to retard arriving ions, but not when this potential is accelerating. This is a consequence of the following general theorem: Supposing for simplicity of statement that the surface of an electrode of any shape immersed in a Maxwellian distribution is perfectly reflecting, then the ions in the surrounding sheath will have a distribution (called ${D}_{M}$) of velocities and densities given by Maxwell's and Boltzmann's equations, even in the absence of collisions between the ions, provided that there are in the sheath no possible orbits in which an ion can circulate without reaching the boundary; but if such orbits exist, the distribution will be ${D}_{M}$ except for the absence of such ions as would describe the circulating orbits. As another corollary of this theorem there is deduced an equation relating the solution of problems having inverse geometry. Finally it is indicated how the theorem can be applied to calculate the volt-ampere characteristic of A. F. Dittmer's "pierced collector" when placed in a Maxwellian distribution.The effect of reflection of ions at the collector surface in modifying currents calculated by the preceding equations is discussed.