## Abstract

The author’s previous work on the computation of algebraic higher order aberration coefficients and their derivatives has been illustrated numerically by providing explicit calculations, relating to a certain simple triplet operating over a relatively small field and at low aperture. Doubts have been expressed as to the usefulness of using such aberration coefficients to describe the (geometrical) behavior of systems operating at higher apertures and over wider fields. In this paper, therefore, all the primary, secondary, and tertiary aberration coefficients, and the coefficient of quaternary spherical aberration are given for two further systems, *viz*. (i) a wide angle objective operating at an aperture of *f*/4.5 and over a field of 70°; and (ii) an objective operating at an aperture of *f*/2.3 over a field of 40°. A number of diagrams compare the predicted displacements, relative to the ideal image point, of the intersection points with the ideal image plane of certain selected (a) tangential pencils, (b) sagittal pencils, and (c) general skew pencils of rays, with the actual displacements as obtained from trigonometrical traces. The agreement is very satisfactory. Even in cases, however, in which the agreement—for the largest apertures or fields considered—leaves something to be desired a knowledge of the coefficients and their derivatives provides a powerful tool for the control over the performance of the system. A detailed elementary example illustrating this assertion is included.

© 1959 Optical Society of America

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