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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 by
r-R=Σ akl cos[k(αw+[open phi])+l(γw+ψ)] z=Σ ckl cos[k(αw+[open phi])+l(γw+ψ)].