An approximate analytic technique for solving the transport equations of thermionic energy converters is developed. Using an assumed parametric form for the ionization rate, the problem is reduced, essentially, to one of the solution of three simultaneous transcendental equations. A numerical iterative procedure for the solution of these equations is derived and implemented in a computer code and some illustrative solutions are obtained. A simple criterion for whether the converter is operating under equilibrium or nonequilibrium conditions and other simple relations describing the character of the solutions are derived. For nonequilibrium conditions, the assumed form for the ionization rate is found to agree well with the actual form, and the solutions obtained are in good agreement with previous results. The empirically observed minimum in the arc drop at the optimum value of the pressure-spacing product is predicted. The present theory is the only one known to the authors which shows this result. The solutions may be systematically improved by numerical integration and iteration and the range of applicability of the solutions can be extended to local thermodynamic equilibrium conditions by assuming a more general form for the electron production and to a greater variation in parameters by incorporating a recently derived improved set of boundary conditions into the analysis.