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. 139
When the pressures are equal (which, from the nature of
fluids, is necessary for equilibrium),
Pb2= (Q -T2) a2,
or P.b: (Q-4)a:: a: b.
But Pb = force at the pole; (Q - - ) a = the whole force
at the equator; therefore the force at the pole: force at
the equator:: equatoreal radius: polar radius.
26. PROP. 15. When this proportion holds, the whole
force at any point on the surface is perpendicular to the
surface.
Let E (fig. 9) be the point on the surface: take EC to
represent the force in the direction EC, and CN to represent
that in direction EX: then, EN will represent the magnitude
and direction of the whole force at E. Now,
EC: CN:: P. EC: (Q - CW, by (24),
or:: a2.EC: b'.CW,
by the demonstration of Prop. 14;
b2
or EC: CN:: EC: - CW.
Hence, CN = CW = subnormal (by Conic Sections);
therefore, EN is the normal; that is, the whole force is
perpendicular to the surface. It appears also that the whole
force is represented in magnitude by the normal.
27. Pitor. 16. When the same proportion holds, if to
any point within the spheroid canals of any form be drawn,
terminated any where in the surface; the pressure on that
point, found by adding the pressures of successive portions
of any of the canals, will be the same for every canal.