(1.) The investigations of the Astronomer Royal and of some other mathematicians on straight-crested parallel waves in a liquid, are based on the supposition that the displacements of the particles of the liquid are small compared with the length of a wave. Hence it has been very generally inferred that the results of those investigations are approximate only, when applied to waves in which the displacements, as compared with the length of a wave, are considerable. (2.) In the present paper I propose to prove that one of those results (viz., that in very deep water the particles move with a uniform angular velocity in vertical circles whose radii diminish in geometrical progression with increased depth, and consequently that surfaces of equal pressure, including the upper surface, are trochoidal) is exact for all displacements, how great soever.