Analytical results are presented for radiation interchange in cavities having nonisothermal walls in order to implement experimental methods for determining the emittance of solids. The cavity may be visualized as a hole that is machined into the exposed face of a solid whose emittance is to be determined, with the solid being heated uniformly from behind. The results were obtained by solving the coupled problems of radiant transport in the cavity and Two-Dimensional heat conduction in the solid that bounds the cavity. From the solutions, results for the radiant flux leaving the base surface of the cavity are presented as a function of the emittance, of the Depth-Radius ratio of the cavity, and of a temperature gradient parameter. This information can be used in conjunction with measurements of the radiances of the base surface and the exposed face of the solid to determine the emittance. A method is also described for simultaneously determining both the emittance of the material and the temperature of the exposed face.