Achromaticity vs. Isochronicity

An achromatic system is one where the transfer matrix elements for the transverse coordinates do not depend on momentum. An isochronous system is one where the transit time of a trajectory through the system does not depend on the initial coordinates. It is well known that a first-order achromatic system is also isochronous, except for pure momentum dependence. The converse is also true. This result is entended to higher orders. Conditions are found so that for a system whose chromatic terms all vanish up to a certain order the transit time will be independent of the transverse coordinates up the same order. Under the same conditions, the converse will also be true.