A method of analysis has been devised for determining the radiant interchange among surfaces, each of which may have both specular and diffuse reflectance components. The formulation uses and generalizes the exchange factor concept (which was initially devised for specularly-reflecting surfaces) and the radiosity concept (which was initially devised for diffusely-reflecting surfaces). Various forms of the analytical method are presented that are suitable either for overall engineering-type computations or for more detailed local investigations. Specific analytical and numerical consideration is given to radiant interchange in cylindrical and conical cavities and to radiant transport through a circular tube. Results are presented for various subdivisions of the surface reflectance into specular and diffuse components. In general, it is found that the radiant efflux from a cavity increases as the specular component becomes a larger fraction of the surface reflectance. A similar statement applies for the transmission of radiant energy through a tube.