The problem of space‐charge limited currents in a diode with plane‐parallel electrodes is considered for the case where space‐charge limitations are counteracted by the introduction of ions at the cathode. The ``no‐collision'' approximation is used, in which inelastic and short‐range elastic collisions are neglected, and each particle interacts with the Coulomb field of all the other particles. The electrostatic potential is determined, in a self‐consistent manner, from Poisson's equation. The equations are simplified by transforming to reduced variables in a manner described by Langmuir and others. The first integrations of Poisson's equation are given for the six most important cases. The first of these has been reported by P. L. Auer, but rigorous analysis of the remaining five cases has not been previously reported. Analytical solution of the first‐order differential equations derived is impossible. Numerical solutions have been obtained for illustrative purposes on the IBM‐704 computer. More complete numerical solutions are being compiled.