The transport of electrons and ions in a thermionic converter is analyzed through a pair of Boltzmann equations and an electric field equation. Ionization of cesium neutrals in the gap is accounted for by electron impact with a speed-dependent ionization cross section. Transport of ions is approximated by diffusion theory, neglecting the electric field term that is inversely proportional to the ion mass. After linearizing the electric field term in the electron transport equation, a complete set of solutions to the system of homogeneous equations is determined; these homogeneous solutions form the basis for an expansion of the actual electron, ion, and electric field distributions existing in the converter gap. Finally, very general boundary conditions appropriate to thermionic converter operation are applied to an expansion of the solutions.