In this part, the ion optical theory for the third order trajectory in a crossed toroidal electric and inhomogeneous magnetic field is treated in detail. The transfer matrix are derived up to the third order. Consequently, for an ion optical system possessing the second and third order aberrations, the determinant of the transfer matrix is shown to be equal to 1. In order to transform the transfer matrix between different deviations the transform matrixes are derived. Some related equalities and corresponding relations for the elements of S and H matrixes are given. These results may be of use in designing mass spector meters.
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