Space charge free potential functions for a particular type of thermionic vacuum tube are derived and represented graphically for four sets of boundary conditions. The geometry under consideration comprises two spaced parallel equipotential metallic planes between which are three symmetrically interposed planes of spaced metallic strands. The strands are mutually parallel in their respective planes, though the strands in planes adjacent to the equipotential surfaces are both orthogonal to those in the central plane. This structure is divided principally into two planes, due to the orthogonal relationship between the strands in the central and adjacent planes. These two planes are each conformally transformed such that the analysis thereof yields the desired potential functions for the original structure, neglecting edge effects. Finally, the theoretical results are applied to a typical commercial thermionic vacuum tube.