93 
of Edinburgh, Session 1878-79. 
tides in the same vertical. This implies that the greatest depth 
must he small in comparison with the distance that has to be 
travelled to find the deviation from levelness of the water-surface 
altered by a sensible fraction of its maximum amount. In the 
present short communication I adopt this restriction ; and farther, 
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 greatest, 
small in comparison with the least diameter. Let this basin and 
the water in it rotate round a vertical axis with angular velocity to 
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 a / jjr , where g denotes gravity, 
and A the greatest diameter of the basin. The equations of 
motion are 
du 
a - 2av 
dv 
Tt + 2 ““ 
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 0 y revolving with the basin ; p the pressure at 
any point x, y , 2 , of this column ; and q the uniform density of the 
liquid. The terms u 2 x, w 2 y, which appear in ordinary dynamical 
equations referred to rotating axes represent components of centri- 
fugal 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 
dp 
dx 
dp 
Xy 
( 1 ) 
d(Du) d( J)v) d,h 
dx + dy ~ dt ' ' 
Lastly, by the condition that the pressure at the free surface is 
VOL. x. 
N 
