Subdivision of the fundamental equation of elasticity into two wave equations appears in most text-books on elasticity theory but the two types of vibration are rarely considered independently. Prescott [1] discussed the possibility of the separate existence of plane dilational and distortional waves in semi-infinite material and, failing to satisfy the conditions at a stress-free boundary, concluded that the two types of motion could not exist independently in such circumstances. He therefore derived solutions using combinations of the two types of vibrations. In this paper it is shown that Prescott's solutions are not unique and that special types of purely dilational and purely distortional vibrations are possible in the presence of a free plane boundary. The problem first investigated by Lamb [2] and later by Cooper [3] of transient vibrations of an infinite plate is then considered. In view of the complexity of the equations involved it is worth while attempting to use the subdivision of the fundamental equation to split the problem into simpler problems. In this connection the possibility of dilational or distortional vibrations alone is investigated and a stable form of distortional waves is discovered. It is seen, however, that subdivision of the general problem is not possible.