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.

162 FIGURE 0F THE EARTH
the integral of which with respect to f is a function of
r, = - f.p(r). The same thing may be shewn, if the forces
arise from attractions to any number of particles; and consequently, if they arise from the attractions to every particle of a solid or fluid mass.
The same thing is true if a part of the forces is produced by centrifugal force. If, (as above) the resolved
47r2
part of centrifugal force in the direction of f = -r f; in
47r2
that of g= T~g: and in that of h- 0o: the corresponding
part of
+ Gdg dh
s:7T2, +~fJ'dg
df
iS T2 (f+gd 27
the integral of which with respect to f is - (f2 + g2).
56. It appears then that the only forces which we
dg dh
have to consider will make F+ G d + Hd- a complete
df df
differential without assigning any relation between f, g, and
dv
h. Call it then -. Then our first condition of equilidf
brium requires that p -d be integrable with respect to f.
This can happen only if p be a function of v. Our first
condition then amounts only to this,
p = a function of v,
dv d g' dh
where = F + G d+ H d.
df df df
Frorn this it follows that p is constant while v is constant. The equation to a surface of equal density, or