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The conception of random positive ion velocities corresponding to ion temperatures in a plasma has serious theoretical difficulties and is lacking in direct experimental verification. It is more reasonable to assume that each ion starts from rest and subsequently possesses only the velocity which it acquires by falling through a static electric field which is itself maintained by the balance of electron and ion charges. This new viewpoint thus ascribes motions to the positive ions which, for long free paths, are ordered rather than chaotic, each negative body in contact with the discharge collecting ions from a definite region of the plasma and from it only. The resulting integral ? the plasma-sheath potential distribution have been set up for plane, cylindrical, and spherical plasmas, for long, short and intermediate length ion free paths, and for both constant rate of ionization throughout the plasma and rate proportional to electron density, and these equations have been solved for the potential distribution in the plasma in all important cases. The case of short ion free paths in a cylinder with ion generation proportional to electron density gives the same potential distribution as found for the positive column by Schottky using his ambipolar diffusion theory, with the advantages that ambipolarity and quasineutrality need not appear as postulates. The calculated potential distribution agrees with that found experimentally. The potential difference between center and edge of plasma approximates $\frac{{T}_{e}}{11,600}$ volts in all long ion free path cases. The theory yields two equations. One, the ion current equation, simply equates the total number of ions reaching the discharge tube wall to the total number of ions generated in the plasma, but it affords a new method of calculating the density of ionization. The second, the plasma balance equation, relates rate of ion generation, discharge tube diameter (in the cylindrical case), and electron temperature. It can be used to calculate the rate of ion generation, the resulting values checking (to order of magnitude) those calculated from one-stage ionization probabilities. The potential difference between the center of the plasma and a non-conducting bounding wall as calculated from the ion current equation agrees with that found experimentally.The solution of the general plasma-sheath equation has been extended into the sheath surrounding the plasma to determine the first order correction which is to be subtracted from the discharge tube radius to obtain the plasma radius. The wall sheath in the positive column is several times the thickness given by the simple space charge equation.Actually the ions do not start from rest when formed but have small random velocities corresponding to the gas temperature, ${T}_{g}$. In the long ion free path cases this leads to an error of the order of only $\frac{{T}_{g}}{{T}_{e}}$ in the calculated potential distributions.In the plasma surrounding a fine negatively charged probe wire the potential difference between plasma potential maximum and sheath edge may be so small that the ions generated within the plasma potential maximum are not trapped but can traverse the maximum by virtue of their finite initial velocities. This justifies the use of a sufficiently fine negatively charged wire in the usual way to measure positive ion concentrations, although certain difficulties appear which are thought to be connected with the collector theory rather than the present plasma theory.