method of analysis is formulated for determining the radiant interchange among specularly reflecting surfaces, some or all of which may be nonplanar. The emitted radiant energy is diffusely distributed. The formulation makes use of the exchange factor, which has many of the properties of the diffuse-angle factor. Indeed, the exchange factor is constructed as a series, each term of which is a diffuse-angle factor multiplied by a poiuer of the surface reflectance. A general formalism is developed for deriving the exchange factor. Application of the method of analysis is made to the cylindrical cavity of finite depth and to the conical cavity. Numerical results are then obtained both for the local and overall heat transfer for a wide range of the geometrical and radiative parameters. In general, it is found that specularly reflecting cavities are more efficient emitters and absorbers of radiant energy than are diffusely reflecting cavities.