## Abstract

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 *ξ** _{ν}*,

*η*

*,*

_{ν}*ζ*

*are the direction cosines of the ray in the*

_{ν}*ν*th medium multiplied by

*n*

*, equations can be derived of the form*

_{ν}whereas 1/*z*′*ζ*′ is given as a continued fraction containing *ϕ** _{ν}*/

*ζ*

_{ν}*ζ*

*′ and the center distances*

_{ν}*c*

*multiplied by*

_{ν}*ζ*

*.*

_{ν}Equation (1) permits the computation of the contributions of the single surfaces to the diapoint errors. The values *ϕ** _{ν}* in (1) are the powers of the different surfaces for the ray. The quantity

*ϕ*

*is practically equal to the Gaussian power of the surface (*

_{ν}*n*

*′−*

_{ν}*n*

*)/*

_{ν}*r*

*, an approximate equality permitting the prediction of the effect of a surface change on the quality of the image.*

_{ν}© 1952 Optical Society of America

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