A noniterative numerical method has been found for solving a partial differential equation when its boundary conditions are not known on the entirety of a closed boundary. Under such conditions, the iterative numerical methods which are generally used, such as the relaxation technique, are not applicable. As an example, this noniterative numerical method has been used to calculate the shapes of a pair of beam forming electrodes for use in an electron gun in which rectilinear flow is desired. The calculated shapes are in almost complete agreement with Pierce's experimental results. Equipotential surfaces calculated by this method have been checked by means of an electrolytic trough which employs a semiautomatic detecting and plotting system.