The computation of the tertiary aberration coefficients of symmetrical optical systems composed of spherical surfaces has been fully dealt with previously. To show that it is quite feasible to calculate in the same way coefficients of still higher order, the present paper deals with the coefficient of quaternary (i.e., ninth order) spherical aberration. The intrinsic contributions are first brought into a remarkably simple form. Thereafter the required iteration formula is developed, and, by way of example, the actual computations are carried out for a certain triplet, the computing scheme in this instance being treated as an appendix to that for the set of tertiary coefficients.
H. A. BUCHDAHL, "Optical Aberration Coefficients. IV. The Coefficient of Quaternary Spherical Aberration," J. Opt. Soc. Am. 48, 757-757 (1958)