Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 89 



and consequently, 



supposing for greater simplicity that the origin of the integral is at 

 a - 0. 



Hence the value of x becomes 



x = a + - f*dal (a- t \/gy). 



Suppose o = length of the wave when t = 0; then £(«) = 0, ex- 

 cept when a is between the limits and «. If therefore we consider 

 a point P before the wave has reached it, 



J a dat(a-t^gy) = f;da^a)=r; 



the whole volume of the fluid which would be required to 'fill the 

 hollow caused by the depression £ below the surface of equilibrium 

 when t = 0. Hence we get 



, V 

 x = a H ; 



7 

 x being the horizontal co-ordinate of P, before the wave reaches P. 



Also, let x" be the value of this co-ordinate after the wave has 

 passed completely over P, then 



£da%(a - t \/gv) = 0, and x" = a. 



If £ were wholly negative, or the wave were elevated above the 

 surface of equilibrium, we should only have to write - V for V, and 

 thus 



V 



x' = a , and x" = a. 



7 

 Vol. VII. Part I. M 



