392 NEWTON S PKINCIPIA. 



and the mass moved is dx ; hence the accelerating force is 



dy 

 9 Tx 



or am 2 hg sm (ntmx) 



or m 2 ^^X; 



hence the force varies as the displacement. This is well 

 known to lead to the very law assumed for X, provided 



= n ; 



hence the velocity of the wave, which we know is . must 



m 



The general equation for the motion of any long wave 

 may be obtained by a generalisation similar to that em 

 ployed in the proportion corresponding to this in sound. 

 The resulting equations are even the same : we arrive at 



&amp;gt;J = ~ 

 d x 



and whatever be the form of the wave, provided only it be 

 very long, and the height of the wave be small compared 

 with the depth of the water, the velocity of transmission 

 will be always the square root of the product of the depth 

 and gravity. 



If each particle of the fluid be under the action of forces, 

 the motion may still retain the characteristics of a wave. 

 Such a wave, however, is called a &quot; forced wave,&quot; and there 

 is no necessary relation between the velocity of such a 

 wave and the depth of the water. As an example, let 

 each particle be acted on by a horizontal force 



F=/. sin. (it mx). 



