## Abstract

An optical system is called symmetric if it possesses an axis of symmetry <i>A and</i> a plane of symmetry containing <i>A</i>. A system <i>K</i> will be called semi-symmetric if it is merely axially symmetric, i.e., if it possesses a screw sense pointing along <i>A</i>. Previous work concerning the consequences of reversibility of symmetric systems is extended to the semi-symmetric case, a “reversal” of <i>K</i> being understood to be its rotation through 180° about a line through <i>A</i> and normal to it, together with a reversal of its screw-sense. It is shown that among the <i>n</i>(<i>n</i>+2) aberration coefficients of order 2<i>n</i>-1 there exist altogether ½ (<i>n</i>-1) (<i>n</i>+2) relations. These divide themselves into a set of relations, previously obtained in the symmetric case, between the “proper coefficients” alone, and a new set of homogeneous relations between the “skew coefficients” alone. The third- and fifth-order relations are exhibited explicitly, and some special points relating to all orders are considered. As a contribution towards a proper appreciation of the meaning of the results obtained, a fairly detailed discussion is included of the geometrical significance of the various types of aberrations possessed by semi-symmetric systems. This part of the work has been shorn of all irrelevancies and it is essentially an extension of Steward’s elegant presentation.

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