Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 91 



{A) o = g * + ? + *±, 



« % ■ p ' dt 



we have, also, the conditions 



(«) • = jj£ = (when y = 0), 



,, w da % * 



(») — = -rr = - (when - = cot a), 



' v d<f> y y ' 



dy 



a being the angle which the inclined side of the channel makes with 

 the vertical. 



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 (2?) and the two conditions just given, by 



(c) . <p + £ (tf + *), 



<p and <p t 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 



= * - it., 



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

 before cited gives 



o= S~§-§^ (when * =c + £> 



M 2 



