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A simple geometrical analysis has been developed which describes the lowest-order transverse mode of any large-Fresnel-number optical resonator located in the unstable or high-loss region of the resonator mode chart. Such resonators include, for example, resonators in which one or both of the mirrors are divergent spherical surfaces. The lowest mode in such a resonator is assumed to consist of two oppositely traveling divergent spherical waves which uniformly illuminate the end mirrors. The centers of curvature of these spherical waves do not, in general, coincide with the mirror centers of curvature, but are found by requiring that each center be the image of the other upon reflection from the appropriate mirror. The resonator losses are found from purely geometrical considerations, and are given by simple analytical expressions. These losses turn out to be independent of the mirror sizes, so that hyperbolic universal equiloss contours can be drawn on the resonator mode chart. The losses agree well with more exact results obtained by Fox and Li for a few specific cases. Experimental results in good agreement with the analysis have been obtained using a ruby laser rod having a divergent spherical surface ground directly onto one end of the laser rod. Unstable resonators, particularly the "Cassegrainian" unstable configuration used in the experiments, appear potentially useful for diffraction output coupling applications, and possibly also for transverse mode control, in ruby and other high-gain lasers.