110 Sir William Thomson on Gravitational 



adopt this restriction ; and, further, instead of supposing the 

 water to cover the whole or a large part of the surface of a 

 solid spheroid as does Laplace, I take the simpler problem of 

 an area of water so small that the equilibrium-figure of its 

 surface is not sensibly curved. Imagine a basin of water of 

 any shape, and of depth not necessarily uniform, but, at great- 

 est, small in comparison with the least diameter. Let this 

 basin and the water in it rotate round a vertical axis with an- 

 gular velocity &> so small that the greatest equilibrium-slope 

 due to it may be a small fraction of the radian : in other words, 



the angular velocity must be small in comparison with \/ yr? 



where g denotes gravity, and A the greatest diameter of the 

 basin. The equations of motion are 



du ~ 1 dp 



— —2cov=— - -f, 

 at p ax 



dv a 1 dp 



dt p dy 



(i) 



where u and v are the component velocities of any point of the 

 fluid in the vertical column through the point (xy), relatively 

 to horizontal axes Ox, Oy revolving with the basin ; p the 

 pressure at any point x,y, z of this column; and p the uniform 

 density of the liquid. The terms co 2 x, co 2 y, which appear in 

 ordinary dynamical equations referred to rotating axes, repre- 

 sent components of centrifugal force, and therefore do not 

 appear in these equations. Let now D be the mean depth and 

 D + h the actual depth at any time t in the position (xy). The 

 " equation of continuity " is 



d(Du) d(Dv) = dh (2) 



dx dy dt' ' 



Lastly, by the condition that the pressure at the free surface 



is constant, and that the difference of pressures at any two 



points in the fluid is equal to g x difference of levels, we have 



dp __ dh 



dx dx , o\ 



dp __ dh^ 

 dy~ 9P dy' 

 Hence for the case of gravitational oscillations (1) becomes 



du dh 



dt 



dh ^ 

 2(OV= -2dx> I 





dv , dh i 



