In this part, the third order radial ion trajectory and the second order axial ion trajectory for a crossed toroidal electric and inhomogeneous magnetic field are studied. The formulae for the third order trajectory and slope are derived, which cover most cases of sector fields and in special cases agree with those already presented by previous authors. Therefore, these results are general and may be of use in designing mass spectrometers. The effects of oblique entrance and exit, curved field boundary and fringing field have not been studied yet.
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