The problems on which we were engaged in the preceding Memoir were the following:—1°, The complete determination of the velocity of transmission of a wave in a fluid of any depth, provided the depth is uniform throughout; 2°, an application of the hypothesis of parallel sections to the problem in its more complicated form, in which the depth is variable in the direction of the breadth of the canal; 3°, The investigation of the motion of a solitary wave. We propose in the following pages to continue the same subjects, and to add the more difficult problem of initial motion.