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The second-order expression for the torque (or acceleration torque) existing on two arrangements of a modified cryogenic gyro design is derived assuming that an axially-symmetric magnetic field is used for the suspension of the rotor. The modified design consists of interchanging the role of the current coils and the superconductor in the conventional design. In the modified design the coils are now situated on the rotor with the superconductor utilized as the stator (reverse of the conventional design). The two types of arrangements resulting are either the coils, or rotor, surrounding the stator, or the stator surrounding the rotor. For the first arrangement, that of the coils surrounding the rotor, the torque calculations are performed by two methods in order to check the manipulations. For the second arrangement, only one derivation is used. The first-order restoring-force expression for the first arrangement is also presented. The main result for the first arrangement is that when all the components of the first-order restoring force are equal, the second-order torque vanishes. It is also theoretically possible to achieve an ideal, or continuous, distribution of current for the support field such that the constraint of equality on the components of the first-order restoring force is satisfied.