182 WAVES IN LIQUIDS. [CHAP. VII. 



as a representative of (3) at any particular point of the canal is 

 improved by making 



where k is the elevation in the neighbourhood of that point. If 



we neglect the square of y , this will agree with (14). 



ii/ 



153. We proceed to investigate, on the same assumptions as 

 before, the equations of motion of long waves in a uniform canal 

 when, in addition to gravity, small disturbing forces whose hori 

 zontal and vertical components are X and Y act on the fluid. We 

 have in this case, 



[ h+r &amp;gt; 

 =g(h+r)-y) I Ydy + const., 



J y 



p 



- 



P 



so that the equation of horizontal motion of a particle in the 

 position (x, y), viz. 



df dp 

 P dt^-d 

 becomes 



If X and Y be of the same order, and their rates of variation 

 small, the second and third terms on the right-hand side of this 

 equation may be neglected in comparison with the other two. We 

 then have 



where, for the same reason, X may be supposed a function of x 

 only. This equation being independent of y, the particles which 

 at any instant lie in a plane perpendicular to x lie always in such 

 a plane. The equation of continuity (4) then applies; and (15) 

 becomes 



where c 2 = gh, as before. 



