The paper discusses the possibility of the propagation of elastic waves, analogous to Rayleigh waves or Love waves, over a (0, 0, 1) surface of a cubic crystal. An examination of symmetrical cases in which the direction of propagation is parallel to the x-axis or makes an angle of 45 degrees with it shows that waves with amplitude falling off exponentially with distance from the free face do not exist for every set of values of the three elastic constants chosen at random. For crystals of aluminium and copper these Rayleigh-type waves do not exist, but for rock salt their existence is demonstrated both for the symmetrical cases and for an asymmetrical case. In the latter the particles describe ellipses in a plane inclined to the direction of propagation. Waves of Love type can exist only in the two symmetrical cases cited; it is shown that then the general Rayleigh-type motion degenerates into a superposition of Rayleigh waves and Love waves. It is easily seen that for all types of crystals the general result will be the same, but that the algebraical and arithmetical work involved will be extremely heavy.