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The general theory is developed and the exact solutions obtained for the design of reflecting microscope objectives having two or three concentric spherical surfaces. It is shown that the only serious optical defect of reflecting objectives, low contrast due to the high obstruction ratio, can be satisfactorily controlled in three-mirror systems. Non-concentric objectives are discussed, but in practice prove to be difficult to adjust.

A geometrical method is developed for calculating the distribution of intensity with angle of the light leaving an optical system, when the angle at which a ray from a point on the optical axis leaves the system is known as a function of the aperture. The case of a point source on the axis of the system is treated exactly, and an approximation is given for that of a small finite source ; the method is applicable to systems with spherical aberration. The distribution of illumination across any plane normal to the axis is treated by analogous methods. The results are compared with measurements of the light distribution from an optical system possessing considerable spherical aberration.