An image error theory is developed for finite aperture and field. In the limit the formulas reduce to those of the well-known Seidel theory. The constants of the Seidel theory are replaced in the general theory by functions. It is shown that for a given object plane these functions can be reduced to two independent functions, the vanishing of one leading to a symmetrical image, the vanishing of both to a sharp image, and it also is proved that these two functions permit one to compute the caustic surfaces. The image error functions for a given plane can be identified as the second-order derivatives of a characteristic function. To ascertain the image quality for a different plane, transformation formulas are developed which permit one to compute the first- and second-order derivatives of this characteristic function for any shift of the object plane. The authors have found that the second-order derivatives of another characteristic function are invariant with respect to a shift of the object and that a complete system of invariants of the image errors can be obtained. The basic formulas demonstrate that the shifted image-errors are linear functions of the image errors of the original plane and their second- and thirdorder determinants. Applications to classical problems are given.

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