NEWTON S PRINCIPIA. 393 



Let us assume the displacement 



X = a sin. (i t m x) 



But as we have already seen, the water seeking to recover 

 its level causes a horizontal force equal to 



w?h g X ; 

 hence the whole force is 



-( ** -I 



By the same reasoning as before, this coefficient must be 

 equal to z 2 , or 



hence a = - - , 



i 2 m? hg 



or the motion in the horizontal direction will be given by 



X = &quot;&quot; ^2 - 2 A * ^ (** 



z 2 in* h g 



and the elevation of the water by 



f m h / . . 



This proposition will be required when we come to con 

 sider the theory of tides. It will also serve to explain in 

 some degree the motion of the waves generated by a boat 

 moving along the surface of a canal. It is not a complete 

 explanation, for the waves observed by Scott Russell were 

 solitary waves ; whereas we have supposed in the above 

 reasoning that there is an infinite succession of waves 



