Abstract
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.
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