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We demonstrate through theoretical analysis that unlike predicted by others, an unbiased coupled resonant optical waveguide (CROW) gyroscope made of N ring resonators has a response to a rotation rate Omega that is proportional to (NOmega)2, and hence its sensitivity to small rotation rates is vanishingly small. We further establish that when proper phase bias is applied to the CROW gyro, this response becomes proportional to NOmega and the sensitivity to small rotation rates is then considerably larger. However, even after optimizing the CROW parameters (N and the ring-to-ring coupling coefficient kappa), the CROW gyro has about the same sensitivity as a conventional fiber optic gyroscope (FOG) with the same loop loss, detected power, and footprint. This maximum sensitivity is achieved for N = 1, i.e., when the CROW gyro resembles a resonant FOG. The only benefit of a CROW gyro is therefore that it requires a much shorter length of fiber, by a factor of about 1/ (2 kappa) , but at the expense of a stringent control of the rings' optical path lengths, as in a resonant FOG. Finally, we show that the slower apparent group velocity of light in a CROW gyro compared to a FOG is unrelated to this shorter length requirement.