  Magnetostatic fields in accelerators are conventionally described in terms of
multipoles. We show that in two dimensions, multipole fields do provide
solutions of Maxwell's equations, and we consider the distributions of electric
currents and geometries of ferromagnetic materials required (in idealized
situations) to generate specified multipole fields. Then, we consider how to
determine the multipole components in a given field. Finally, we show how the
two-dimensional multipole description may be extended to three dimensions; this
allows fringe fields, or the main fields in such devices as undulators and
wigglers, to be expressed in terms of a set of modes, where each mode provides
a solution to Maxwell's equations.
