Analytic solutions for aplanatic four-spherical-mirror systems are obtained, and the optical aberrations of the systems are examined numerically. These systems consist of Cassegrainian front parts and inverse Cassegrainian rear parts. The front part is afocal and has high light-collecting capacity. Aberra-tions of the inverse Cassegrainian system are counterbalanced with those of the front part. A germanium corrector lens may be used in these systems for improved performance in the thermal infrared region.