NOTE ON THE MOTION OF WAVES IN CANALS. 277 



The first of these conditions is due to the vertical side, and 

 the second to the inclined one, since at these extreme limits the 

 fluid particles must move along the sides. 



Now from what has been shown in our memoir, it is clear 

 that we may satisfy the equation (B) and the two conditions 

 just given, by 



< = &+& Q/ 2 +* 2 ) ..................... ( c ) 



<f> and <^> / being two such functions of x and t only that 



It now only remains to satisfy the condition due to the 

 upper surface. Let therefore 



<> = -&.< 



be the equation of this surface. Then the formula (A) of our 

 paper before cited gives 



or neglecting (disturbance) 



c being'the vertical depth of the fluid in equilibrium. 



Also at";the upper surface p = 0, therefore by continuing to 

 neglect (disturbance) 2 (A) gives 



(when z = c). 



Hence, by eliminating f, we get 



which by (c) becomes, when we neglect terms of the order y* 

 and z* compared with those retained, 



Or eliminating <, by means of ((7), 



