Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

SUPPOSED HOMOGENEOUS. 141
Let the values of x and y, at the point E, be f and g;
and where the canal meets the surface, let the values of
x and y be v and w: then, observing that the pressure at
the surface is = o, we find the pressure at E
PW2 47r'2 v2 Pg 47r2 f2
+- -y -Q --
2 + T3* 2 2 T2 2
But, by the equation to the generating ellipse,
b2 Pw' P b2 P b2 v2
w2 = - (a " -:s) '
a 2 2 a" 2
and, by the demonstration of Prop. 14,
pb2 47r2
Hence, the pressure at E
Pb2 Pg2 4q r f2
2 2 T2 2
This expression, it may be remarked, is independent of the
form of the canal, and of the place at which it terminates
in the surface; and therefore we should have found the same
for the pressure produced by the fluid in any other canal,
as GE. For all canals therefore leading to the same point,
the pressure on that point is the same.
28. From Prop. 15, we find, that if a fluid mass have
the form of an oblate spheroid, there will be no tendency
to disturb the particles at the surface; and from Prop. 16.
it appears that, as the pressure on every particle is equal
in all directions, none of the interior particles will have any
tendency to motion. Every part therefore will be at rest:
and therefore the oblate spheroid is the form of equilibrium,
if the force at the pole: whole force at the equator:: equatoreal axis: polar axis.
29. PROP. 17. To find the proportion of the axes of
the spheroid which is in equilibrium.