A computing scheme involving only 19 entries per surface is developed for the calculation ab initio of the coefficient of tertiary spherical aberration (and incidentally also the primary and secondary coefficients), and the contributions thereto by the individual (spherical) refracting surfaces of a centered system. A detailed example is provided.
H. A. BUCHDAHL, "Optical Aberration Coefficients. I. The Coefficient of Tertiary Spherical Aberration," J. Opt. Soc. Am. 46, 941-941 (1956)