394 NEWTON S PRINCIPIA. 



following each other. However, we may suppose the 

 grand features of the motion to be similar in the two cases. 

 The above expressions show that the sign of F will be the 



same as or different from that of -~ according as -, 



the velocity of the wave, is greater or less than */g~h* 

 Taking the former supposition, F must be positive, or act 

 in the direction of propagation where the water is rising, 

 that is, along the anterior surface of the elevation of the 

 wave, and must act in the opposite direction where the 

 water is falling, that is, along the posterior surface of the 

 elevation. If, on the contrary, the velocity of the wave 



is less than V g h, then F must act in the direction of 

 propagation along the posterior slope, and in the opposite 

 direction along the anterior slope. A boat can only push 

 the water forwards ; hence it must be on that side of the 

 slope on which a forward force is required ; that is, it will 

 travel behind or before the wave, according as its velocity 

 is less or greater than Vg h. There will be no force 

 acting on the other slope of the wave ; but the wave will 

 undergo a trifling change of form, and thus similar effects 

 will be produced whether this force does or does not act. 



If the velocity of the boat is V g h, no force is neces 

 sary. If the boat ride on the crest of the wave, it acts 

 on both slopes. If these actions were equal, the particles 

 would have their changes of motion accelerated in front 

 and retarded in rear, and thus the whole change would be 

 the same as if the forces did not act. But the head of 

 the boat, Airy observes, presses the water more than the 

 stern ; hence, on the whole, we have a small accelerating 

 force acting on the anterior slope. Hence, that the boat 

 may ride evenly on the crest of the wave, the velocity 



must be a little greater than ^ g h. This is true in prac 

 tice. 



Many mathematicians have investigated the effects of 

 gradual changes in the depth and breadth of the canal on a 



