Part I. First‐order theory of fields with an axis and a plane of symmetry. By using a symmetry property of the ion orbits in magnetic or electric fields with rotational symmetry, the ion optics inside these fields can be simply deduced. The physical meaning of the various terms in the expressions for the ion orbits is clarified. First‐order imaging properties are derived.Part II. First‐order theory for the general case. The calculations on the ion optics of fields with an axis and a plane of symmetry are generalized by omitting the requirement of the symmetry plane. The trajectories in the field are calculated up to the second order. The general equation of the ion path is given byr−R=&Sgr; akl cos[k(&agr;w+[open phi])+l(&ggr;w+&psgr;)]z=&Sgr; ckl cos[k(&agr;w+[open phi])+l(&ggr;w+&psgr;)].The first‐order imaging properties are discussed.Part III. Second‐order theory of configurations of consecutive fields and the principle of object aberrations. The method is extended to calculate the ion paths up to the second order in instruments having one or more fields with a plane and an axis of symmetry. By introducing the concept of optical aberrations in the object, the calculation of higher‐order aberrations can be simplified to a large extent.