Theoretical deduction of the formula for thermionic emission.---The constant $A$ of the thermionic emission formula: $I=A{T}^{2}{e}^{\ensuremath{-}\frac{{b}_{0}}{T}}$ has been shown by Richardson and Dushman by different lines of argument to be a universal constant, and a value for the constant has been deduced by Dushman. The experimental data of Dushman verify this value of $A$ for pure metals, but if the metal is coated there are very wide divergences. In this paper an attempt is made to give a more rigorous deduction of the emission formula, so that it may be plain under what conditions it may be expected that $A$ be universal. Assuming that the entropy of the electron gas is the same as that of an ordinary monatomic gas in the Sackur Tetrode theory, and also assuming that the entropy of a surface charge is zero at 0\ifmmode^\circ\else\textdegree\fi{} Abs. it is shown that an emission formula of the type above necessarily demands that $A$ have the universal value assigned by Dushman, and the necessary and sufficient condition for this is that the difference between the specific heat of the neutral metal and of a charge on its surface vanish (${C}_{\mathrm{pm}}\ensuremath{-}{C}_{p\ensuremath{\rho}}=0$). This condition may also be formulated in terms of the surface heat and is equivalent to $\ensuremath{\sigma}+\frac{d{P}_{s}}{\mathrm{dT}}\ensuremath{-}\frac{{P}_{s}}{T}=0$, where ${P}_{s}$ is the surface heat and $\ensuremath{\sigma}$ the Thomson heat. If $\ensuremath{\sigma}$ can be neglected, the emission formula holds with the form given when ${P}_{s}$ is proportional to absolute temperature.Physically it seems probable that (1) if the metal is uncoated the forces on the surface ions are nearly the same as on the neutral atoms, so that the condition ${C}_{\mathrm{pm}}\ensuremath{-}{C}_{p\ensuremath{\rho}}=0$ is satisfied, but (2) if the surface charge has the properties of a gas, as in an oxygen coated filament, (a) the specific heat may be different from that of the metal, or more probably (b) the entropy of the surface charge may not vanish at 0\ifmmode^\circ\else\textdegree\fi{} Abs.; and (3) if the surface is coated with another more easily ionizable metal there may be electrical forces on the surface ions which modify their specific heat.