The theory of operation of the three-electrode thermionic vacuum tube given in this paper is based on the fundamental equation of the family of characteristic curves for various plate and grid voltages. This equation, which may be written I= a( R_ EB + 2; E,+ C) where E b and E c , are the plate and grid voltages respectively, and α, μ 0 . and e are the structural parameters of the device, is obtained empirically with the help of the relation previously discovered by the author, which states that the voltage between filament and plate bears a linear relation to the effective voltage produced by it between the filament and a plane coincident with that of the grid. From this it follows that an electromotive force e impressed upon the grid circuit produces an electromotive force μ 0 e in the plate circuit. With the help of these fundamental relations the amplification equations of the tube are derived in terms of the structural parameters of the tube and the constants of the circuit. Methods are given for experimentally determining the constants of the tube, the two most important of which are the amplification constant μ 0 and the internal output impedance. Experiments are described which were performed to test these equations, and they indicate that the first order approximation made give results sufficiently accurate for amplification purposes.