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The outline of a method of electron-gun iterative analysis by computer is given, with particular emphasis on improved methods for cathode-current-density calculations. It is shown how the effects of cathode curvature, magnetic fields at the cathode surface, etc. can be taken into account in estimating the emission currents. Numerical results are compared with those of several previously-published computer analyses. It is also shown how 4th-order correction terms can be included in the finite-difference equations to improve the accuracy of results obtained using coarser meshes.