In the formation of finished slide surfaces for slide rails on a track for an electromagnetically levitated vehicle, the track is constructed as a girder-like beam with the slide rails located on cantilevered parts of the upper surface of the beam. The beam can be formed of prestressed concrete. Initially the slide rails are provided with a height oversize. The height oversize part of the slide rails is removed and the slide surface of the rails are finished to the desired degree. The height oversize part is removed by cutting and grinding devices mounted on a machine frame movable over the upper surface of the track. The machine frame is adjustable in the vertical direction from a datum plane formed by the lower surfaces of support stators secured to the undersides of the cantilevered parts of the track. The machine frame rides on the upper surface of the beam and is in contact with the lower surfaces of the support stators.