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A thorough investigation of axially symmetric two-electrode electrostatic lenses is given. The study is based on the analysis of axial potential distributions, according to the principle of synthesis. The aberration coefficients of a large class of axial potential distributions are examined and plotted. General conclusions are drawn for the axial potential functions and electrode shapes that have small aberration coefficients. The electrode shapes of an optimized lens with small spherical aberration are given. General criteria are established for the design of two-electrode lenses.