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 HETEROGENEOUS. 163
couche de niveau, is found therefore by making v constant,
dv
or - = 0
df
dg dh
or F+Gf +H d= o.
df df
The equation to the external surface, found above, is only
a particular case of this.
d(V) dv dv
It follows also that, as -- = p - = function of v x df
df df df
V or the pressure will also be a function of v. But p is
a function of v; consequently V is a function of p. That
is, the surfaces of equal density will also be surfaces of
equal pressure. And the equations of equilibrium will
therefore be the sane whether the density depend on the
pressure or not.
57. PiOP. 26. To find the ellipticities of the spheroids
of equal density which will satisfy the equation of equilibrium.
For this purpose we shall find the differential equation
to a spheroid, and shall make it coincide with the equation
found above from the condition of equilibrium.
The polar and equatoreal semi-axes being y and y(1 + e),
the equation to the spheroid is
f2 -+ g h2
_=.l ~: ' = _L
72 (1 + 0)2 'y2
and consequently its differential equation is
dg dht
f g + (i + C)2.h - = 0.
That this may coincide with the equation
dg dh
F+G dg + H -h = 0,
df df
we must have
f: g: ( +e)2.h:: F:G:H;
or f: g: (1+2e).h:: F:: H.
11 -2