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. 153
Substituting these, we find, at length, the attraction of A
in the direction of x
4C 1 + 2e - e C(12h2 _- f2 - 3g2)}
3 P(f2 g2 5 (f2 + g2 + h2)
that in the direction of y
47r- c3 c2(12h2 _ 3f2 _ - g)
3 p(f+ 2 + ) +2e 5(f2+ g+ h2)2 g
that in the direction of z
4r cc3. c2(6h2 - f2 gg2) h.
- 3p (fl2+ g + h2)5 (f2 +e g2 - h2)2
S 5(f-U if+~+y gh'
48. PROP. 22. To find the attraction of an oblate
spheroid on a point without it; the spheroid being heterogeneous; and all the surfaces passing through points at
which the density is the same, being spheroidal, of variable ellipticity.
Let EF, (fig. 15) be the spheroid; let E'F' be a
spheroidal surface, at every one of whose points the density is the same, and E"F" a spheroidal surface very near
the former, of different ellipticity, at all of whose points
the density is the same, but differing from that at the
surface E'F'. Let CF'= c, CF"= c + oc. Since the ellipticity varies when the semi-axis of the spheroid is varied,
e must be a function of c. Now the density of all the
matter included between E'F' and E"F" is not uniform;
but by diminishing 3c, it may be made to approximate as
nearly as we please to uniformity. Conceive, now, for the
moment, the interior matter of the spheroid E'F' to be of
the same density as that at its surface, or to be equal to
p; let its attraction in the direction of = p. A. Then
the attraction of the spheroid E"F" in the same direction,
will be the value which A receives when c + 3c is put for
c, and when, instead of e we put the value of the ellipticity in the spheroid E"F". But if we consider e as a
function of c, this is included in considering the variation
which it receives in consequence of the variation of c.