The invention relates to a method for eliminating axial image deformations &agr;n in electron optical systems, where the extra-axial image deformation of the order n+m with the same behavior in &agr;n, which thus has the form &agr;n&ggr;m, is modified by shifting or tilting the beam path towards the optical axis until compensation of the axial image deformation has been achieved, whereby &ggr; describes the extra-axial image coordinate as a complex number in both sections. The invention also relates to an adjustment method for eliminating all first-, second- and third-order axial image deformations during correction of the third-order spherical aberration in electron optical systems with hexapoles.