Errors in the machining of the objective lens in an electron microscope may result in aberrations which prevent the realization of the theoretically possible resolving power of the instrument. This paper treats the electron-optics problem of relating the aberrations to their causative machining defects-elliptic distortion of pole-pieces.The analytical method of mathematical physics is used for obtaining the scalar potential field in a magnetic lens with pole-pieces of elliptic distortion （Fig. 1）. The scalar potential expansion in elliptic-cylindrical co-ordinates is given （from 2.16）

which are exact analytical formulas in term of the coefficients of Mathieu function and convenient to calculate. If the ellipticity of pole-pieces is small when compared with unity

the previous potential expansions in the neighbourhood of the axis are reduced to simpler formulas （from 2.24

2.26） which apparently relate the potential perturbation to the perturbation of pole-pieces.By drawing on the previous resultants

the paraxial electron trajectories of the lens are calculated （Fig. 5）. Then according to W. Glaser

the aberration coefficients α1

α2 （from 3.9

3.11） are obtained.In conclusion

the tolerances for ellipticity are given. For Example: if δ = 2 A

the variation in radius must not exceed 0.15

μ （k2 = 1） or 0.30 μ （k2 = 2）

these tolerances are very difficult to attain.