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In a proposed experiment, a measurement is to be made of the angular precession of a rotating superconducting gyroscope for the purpose of testing different general relativity theories. For various reasons having to do with the design of the experiment, the superconducting shield surrounding the gyroscope is not spherically symmetric and so produces a torque. There are two distinct features of the shield which lead to a torque on the gyroscope. First, its shape is a sphere intersected by a plane; if the angular momentum of the gyroscope is not parallel to the rotational symmetry axis of the shield, there is a torque which we calculate. Second, there are small holes in the spherical portion of the shield; the earth’s field can penetrate through these holes and give an additional torque. This is also calculated here. In the actual experiment these torques must be accurately known or made very small in order to obtain meaningful results. Our calculation is sufficiently general for application over a wide range of experimental design parameters.