Closed-form solutions for centered imaging systems, corrected for any combination of third-order aberrations, have been derived. The theory is based on Seidel’s formulas for third-order aberrations, which have been reformulated so that they are expressed by mutually independent variables only. Chief importance is laid upon all-reflective systems. Special formulas for well-corrected two- and three-mirror systems are given.
Dietrich Korsch, "Closed-form solutions for imaging systems, corrected for third-order aberrations," J. Opt. Soc. Am. 63, 667-672 (1973)