Abstract: This paper gives the expression of the vector potential produced by a dipole magnet with a symmetry plane. The second order trajectory and second order aberration are obtained by using the variation theorem. It is verified that Liouville's theorem still holds under such circumstances. The beam ellipsoid deformed by second order aberration becomes a new ellipsoid after appropriate modifications, and still works according to first order transport theory. The seuond order beam optics is therefore established.

Key words: Dipole magnet, second order aberration, second order beam optics, transport theory, second order trajectory