The present memoir is a continuation of that which the author presented to the Society in April 1839. It is divided into four sections. In the first (which is marked Section 4, in continuation of the preceding memoir) the general problem of wave motion is treated of, and the equations to the surface are obtained by two different processes, giving results which agree with those obtained in the former memoir. In the second, the problem of wave motion in a canal of constant width and constant depth in the direction of the width, but of variable depth in the direction of motion, is treated by the method of the variation of parameters. The following are the approximate results:—1. The length of the wave diminishes directly as the depth diminishes. 2. That the velocity of transmission at any point is directly proportional to the square-root of the depth at that point. 3. That, in a channel uniformly and gradually shelving, the whole time of transmission of the wave from end to end, is exactly double what it would be if the depth were uniform; and, 4. That the elevation of the crest of the wave is inversely as the depth of the fluid.