Considerations of geometrical optics indicate that the limiting rays travelling along the interface between two homogeneous elastic media should carry inappreciable amounts of energy. Nevertheless, the ``first arrival'' waves in refraction shooting processes, which give linear time distance curves, are usually interpreted as being due to such limiting rays travelling along the interface with the velocity of the lower medium. To clear up the situation an analysis of the problem has been made both from the points of view of geometrical optics and wave theory. It is pointed out that although the assumed ``refraction'' paths are minimal time paths in the sense of Fermat's principle the applicability of the latter along a surface of discontinuity may be open to question. On the other hand, it is proved that, geometrically, they are necessarily real, since they are the only type of path that can give the observed linear time distance curves. The wave theory treatment of the problem for fluid media modified slightly from the operator method analysis previously given by Jeffreys, is then presented to show that the wave theory actually gives waves whose geometrical interpretations are exactly those of the refraction shooting process. Their amplitudes vary inversely as the square of the distance from the source and they give vertical displacements of the same order as those of the directly reflected waves. The analysis is extended to the case of general elastic media and it is shown thatfourtypes of ``refracted'' waves will be produced upon the incidence at the interface of either a longitudinal or transverse wave pulse. Two of the waves will be recorded as longitudinal waves and the other two as transverse. One of each pair effectively travels in the second medium with the longitudinal velocity of the medium and the other two travel with the transverse velocity of the lower medium. The velocities and accelerations produced by these ``refracted'' waves will vary over the pulse thickness exactly as the displacements and velocities in the incident pulse.