An optical system is called symmetric if it possesses an axis of symmetry A and a plane of symmetry containing A. A system K will be called semi-symmetric if it is merely axially symmetric, i.e., if it possesses a screw sense pointing along A. Previous work concerning the consequences of reversibility of symmetric systems is extended to the semi-symmetric case, a “reversal” of K being understood to be its rotation through 180° about a line through A and normal to it, together with a reversal of its screw-sense. It is shown that among the n(n+2) aberration coefficients of order 2n-1 there exist altogether ½ (n-1) (n+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.
H. A. BUCHDAHL, "Optical Aberration Coefficients, XIII. Theory of Reversible Semi-symmetric Systems," J. Opt. Soc. Am. 57, 517-520 (1967)