A theoretical analysis of this little-studied resonator is performed, including a study of its sensitivity as a gyroscope and of transmission spectrum properties. When all its free parameters are optimized (the coupling ratio of the 3x3 coupler, the probe laser wavelength, and the length mismatch between the rings), this sensor is predicted to have the exact same maximum possible rotation sensitivity as a resonant fiber optic gyroscope (RFOG) with the same ring radius and fiber loss, for either a triangular or a planar a 3x3 fiber coupler. Changing the length mismatch (and re-adjusting the coupling ratio to maximize the sensitivity) has very little impact on the best sensitivity: all it does is redistribute the circulating light between rings, but the total number of recirculations is essentially unchanged, and so is the sensitivity. A second type of double-ring resonator is studied in which the output is collected via a 2x2 coupler placed on one of the rings. This two-coupler double-ring resonator gyroscope is found to have a larger rotation sensitivity than a two-coupler RFOG. For exceedingly small loss, the sensitivity enhancement is vanishingly small. For a ring loss of 1 dB, the enhancement is 1.8-fold. As the loss is increased, the enhancement increases asymptotically to either ~2.11 (triangular coupler) or ~1.78 (planar coupler), but the loss is then too high for either sensor to be practical. For small loss, the sensitivity of a two-coupler RFOG is 4 times lower than that of a conventional (one-coupler) RFOG. A two-coupler double-ring resonator gyroscope therefore never surpasses a single-coupler RFOG in sensitivity, but for applications requiring an additional 2x2 coupler, a two-coupler double-ring resonator has a higher rotation sensitivity than a twocoupler RFOG.