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A digital computer program written in the IBM 7090 FORTRAN programming system is described and then employed in the analysis of several axially symmetric electron guns. The digital program employs relaxation techniques while alternately computing electric fields and trajectories. One cycle through the program consists of three parts. 1) Calculation of the voltages within the electron gun by solving Poisson's equation, in difference form, on a matrix containing a maximum of 10 000 points on which electrode potentials are laid out. 2) Calculations of new current densities along the cathode and the setting up of new trajectory starting points along the cathode to simulate these current densities. 3) Calculations of trajectories, including magnetic fields and relativistic effects if desired, and comparison of the new beam with the previous one. Thermal effects and direct particle-particle interaction effects are ignored although space-charge effects, introduced through the field calculations, tend to prevent excessive or usual trajectory crossings. Execution time for the entire program is between 4 and 12 minutes, although for most guns 5 to 7 minutes is usually sufficient. Results produced by the program are presented and compared with experimental and analytical results.