Coma of modified Gregorian and Cassegrainian mirror systems

The equivalence of the classical Newtonian, Cassegrainian, and Gregorian mirror systems with respect to the first two Seidel aberrations is rederived by means of a simple congruence. The effects of arbitrary small modifications of the two mirror systems are then studied and general formulas are derived for the effects of such modifications on the spherical aberration and coma. Spherical aberration is corrected to the third order if the amount of glass removed from one surface is replaced at the corresponding zone of the other surface. Modifications in which one surface is made spherical while the other is adjusted to eliminate spherical aberration result in large increases of coma for systems having the usual amplifying ratios.