The paraxial equation for rotationally symmetric magnetic lenses is integrated numerically by using the Numerov method. The normalized third‐order aperture and second‐order chromatic aberration coefficients (C3NandCcN) are calculated. A Fourier representation of the lens field is used, and a search for minimumC3Nis carried out by varying the Fourier coefficients. Comparisons with comparable quadrupole doublets, and calculations ofC3NandCcNfor actual coil systems are presented. The main conclusion is that magnetic round lenses are about a factor of 2 better than comparable magnetic quadrupole doublets as far as the third‐order aperture aberration is concerned, but the chromatic aberrations in both systems are comparable.