272 TIDAL WAVES. [CHAP. VIII 



omission of the term udu/dx, which is of the second order, so 

 that 



If we put 



then f measures the integral displacement of liquid past the 

 point x, up to the time t ; in the case of small motions it will, to 

 the first order of small quantities, be equal to the displacement 

 of the particle which originally occupied that position, or again 

 to that of the particle which actually occupies it at time t. The 

 equation (3) may now be written 



The equation of continuity may be found by calculating the 

 volume of fluid which has, up to time t, entered the space bounded 

 by the planes x and x + $x\ thus, if h be the depth and b the 

 breadth of the canal, 



The same result comes from the ordinary form of the equation of 

 continuity, viz. 



[vdu , du 



1 hus v = I -j- ay = V -j- . 



*da;* * dx 



if the origin be (for the moment) taken in the bottom of the canal. This 

 formula is of interest as shewing that the vertical velocity of any particle is 

 simply proportional to its height above the bottom. At the free surface we 

 have y = h+r), v = dr)/dt, whence (neglecting a product of small quantities) 



dr,__ d^ 



dt~ dxdt 



Prom this (5) follows by integration with respect to t. 



