A torsional line load applied along a transverse circle travels in the axial direction along the interior of a cylindrical bore in an infinite elastic medium. The line load moves with a constant velocity V which is either greater or less than the velocity of the equivoluminal waves which are found to propagate in the medium. The resulting superseismic and subseismic responses are derived as steady-state solutions which do not change in a coordinate system which, attached to the applied torsional loads, moves with velocity V. Displacement and stress components are obtained for points throughout the medium. In the superseismic case, a sharp S-wave front exists and the behavior at this front is investigated. Numerical results are presented for the displacements and stresses throughout the medium for both superseismic and subseismic cases.