The motion of single ions through a perfect quadrupole is governed by linear differential equations in both static (d.c. only) and radio-frequency cases. However, motion in distorted quadrupole fields and higher multipoles (e.g. hexapoles, octapoles etc.) is determined by non-linear differential equations with coupled ( x,y) terms in Cartesian coordinates. In imperfect quadrupoles with electrodes of circular cross-section this non-linearity is “weak”. Motion in multipoles, in contrast, is entirely non-linear and it is to these multipoles that we turn our attention. We are aware that this work may also have significance for quadrupoles. Trajectory plotting can be successfully achieved by numerical methods, e.g. Runge—Kutta and the objective of any such studies is to discover the functional properties of the device apparent in the behaviour of charged particles during transit. In multipole and distorted systems the application of numerical methods is complicated by the substantially increased number of parameters involved. This mitigates against achieving the objective stated above. We have found that a simplification may be introduced which allows a semi-analytical treatment of the static cases and furthermore reduces the number of parameters required for the numerical analysis of the radio-frequency cases. Despite the simplification, the r.f. special case trajectories retain many of the features that occur in general cases. The simplification involves removing the ( x,y) product terms from the differential equations. This is not unreasonable as the equations then describe motion confined to the symmetry planes of the multipole. Results are shown that demonstrate, semi-analytically, the types of function encountered in the approximate solutions for the r.f. case over small increments of time. Trajectory plots are shown for the r.f. case, for which the simplification described above allows the production of phase-space diagrams.