332 TIDAL WAVES. [CHAP. VIII 



where 



D d i d , d d 



fr.= -ji + U -y- + V-J- + W-J- ............... (1), 



Dt dt dx dy dz 



as usual. These are in directions making with the original axis 

 of x angles whose cosines are 1, nSt, 0, respectively, so that the 

 velocity parallel to this axis at time t + &t is 



u + -r- St n + vSt v + nx 



Hence, and by similar reasoning, we obtain, for the component 

 accelerations in space, the expressions 



......... (2)*. 



In the present application, the relative motion is assumed 

 to be infinitely small, so that we may replace D/Dt by d/dt. 



200. Now let Z Q be the ordinate of the free surface when 

 there is relative equilibrium under gravity alone, so that 



(3), 

 y 



as in Art. 27. For simplicity we will suppose that the slope of 

 this surface is everywhere very small, in other words, if r be the 

 greatest distance of any part of the sheet from the axis of rotation, 

 n?r/g is assumed to be small. 



If ZQ + denote the ordinate of the free surface when disturbed, 

 then on the usual assumption that the vertical acceleration of the 

 water is small compared with g, the pressure at any point (x, y, z) 

 will be given by 



, 1 dp d% 1 dp d 



whence -- = n & 7 -r- , /- = ri*y g -^ . 



/o a 9 das pay dy 



The equations of horizontal motion are therefore 



cfa^ cZ cZO 



-37 %nv = q^~ 5 , 

 dt * dx dx 



dv -" ^^ ^ (5)> 



-,- 

 dt y dy dy 



where U denotes the potential of the disturbing forces. 



* These are obviously equivalent to the expressions for the component accelera- 

 tions which appear on the left-hand sides of Art. 195 (1). 



