  A transverse multipole expansion is derived, including the longitudinal
components necessarily present in regions of varying magnetic field profile. It
can be used for exact numerical orbit following through the fringe field
regions of magnets whose end designs introduce no extraneous components, {\it
i.e.} fields not required to be present by Maxwell's equations. Analytic
evaluations of the deflections are obtained in various approximations. Mainly
emphasized is a ``straight-line approximation'', in which particle orbits are
treated as straight lines through the fringe field regions. This approximation
leads to a readily-evaluated figure of merit, the ratio of r.m.s. end
deflection to nominal body deflection, that can be used to determine whether or
not a fringe field can be neglected. Deflections in ``critical'' cases (e.g.
near intersection regions) are analysed in the same approximation.
