The simple formula suggested below contains, as a special case, the so-called Petzval formula, which gives the field curvature of an optical system in a form which contains a general invariant of the optical system and which is expressed by the powers alone, independent of the center distances.
Let x, y, z, be the coordinates of the object point, and the coordinate of a diapoint where the origins are assumed at the centers of the first and last surfaces, and where the z axis has the direction of the axis of symmetry. If ξν, ην, ζν are the direction cosines of the ray in the νth medium multiplied by nν, equations can be derived of the form [Equation] whereas 1/z′ζ′ is given as a continued fraction containing ϕν/ζνζν′ and the center distances cν multiplied by ζν.
Equation (1) permits the computation of the contributions of the single surfaces to the diapoint errors. The values ϕν in (1) are the powers of the different surfaces for the ray. The quantity ϕν is practically equal to the Gaussian power of the surface (nν′-nν)/rν, an approximate equality permitting the prediction of the effect of a surface change on the quality of the image.
M. HERZBERGER, "The Contributions of the Single Surfaces to the Diapoint Coordinates," J. Opt. Soc. Am. 42, 544-544 (1952)