320 



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 mo- 

 tion in a canal of constant width and constant depth in the di- 

 rection of the width, but of variable depth in the direction of mo- 

 tion, 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. 



Such of these conclusions as admit of testing, the author has 

 compared with Mr Russell's experiments, given in the seventh 

 volume of the Reports of the British Association. The data are, 

 however, insufficient for effecting a satisfactory examination of the 

 formula. There is, however, a very general agreement between 

 theory and experiment. Section 6th is devoted to the determina- 

 tion of the velocity and force of the wave in a canal, the section 

 of which, perpendicular to the direction of transmission, is some 

 given curve ; whilst the depth estimated in the direction of mo- 

 tion is constant. In the former memoir (Section II.) an approxi- 

 mate solution of this problem had been given, on the hypothesis 

 of parallel sections. The conclusions then arrived at agree re- 

 markably well with experiment, but the hypothesis is too limited 

 to be universally applicable. In the present section, the problem 

 of motion is solved in all its generality, and the condition of the 

 bounding section is introduced in determining the arbitrary con- 

 stants. But, general as the solution in this case appears, it is ne- 

 vertheless subject to a particular hypothesis, viz., that at the side 

 of the canal, and near the surface, the equations of motion are 

 continuous. This hypothesis can hardly be objected to, in all cases 

 where the variation of depth is uniform or gradual. But it to- 

 tally fails when the variation is abrupt, and can with difficulty 

 be conceived to apply when it is subject to much fluctuation. If, 

 for instance, the section is triangular, we can have no hesitation 

 in applying it ; if, on the other hand, it is triangular at the bot- 

 tom, and rectangular at the top, it utterly fails. One remarkable 



