Superconductors can in theory be used to detect rotation by Josephson interference or by detection of the London field, a magnetic induction that fills the interior of any rotating bulk superconductor. One might hope to use these properties of superconductors to build a practical inertial guidance gyroscope. A problem arises from the necessity of surrounding the device with superconducting magnetic shielding: the London field generated by a co-rotating shield eliminates the response of the superconducting device within the shield. The present article demonstrates this point more rigorously than has been done before, discussing solutions of Ampe`re’s law for rotating and nonrotating superconductors and paying careful attention to boundary conditions. Beginning with a supercurrent density derivable from either the Ginzburg-Landau or the London theory of superconductivity, the article shows: 1) that a superconducting device cannot distinguish between rotation and an applied magnetic field: 2) that a superconducting device surrounded by a co-rotating superconducting shield cannot detect rotation. The term “superconducting gyroscope” in this article refers only to a device whose working principle is the response of the superconductor itself to rotation, not to any device in which superconducting electronic components are used to detect some other effect. ©1997 American Institute of Physics.