An introduction to the mathematical theory of attraction ...

Ellipsoid of Small Ellipticity. 141
Substituting for V from (1), neglecting terms which are
omitted in (2), and remembering that
/ d dy)
-y -x- r = 0,
we get,
3L 3L
N3 = y 2 J (2 - y2) dm= 3L (B - A) xy.
We have then for the three moments required
N (C - B) y
3L
2- 3- (A - C) z, ) * (3)
osra)
N3= (B - 4) j
80. Ellipsoid of Small Ellipticity.-In the case of
a homogeneous ellipsoid of small ellipticity, equations (2)
and (3) are approximately true, no matter how near the point
P is to the surface of the ellipsoid.
To prove this, let a, b, c be the semi-axes of the ellipsoid
whose mass is M, and a', b', c' those of a confocal ellipsoid
inside M whose mass is M', and moments of inertia A', B', C';
then if V' be the potential of the latter ellipsoid at P, by
Mli
Ex. 5, Art. 75, we have V =M' V'
The largest value of r' for the mass M' is a', and if P be
outside M the smallest possible value of r is c. Hence,
r/2 a2
taking M' as the acting mass, we have, < -. Now, if the
ellipticity -- of the ellipsoid M be denoted by E, on the
hypothesis that E2 is negligible a2- c2= 2c2, whence a'=c' + 2Ec2,
a, 2,/2
and = -2 + 2; but c' may be made as small as we please,
r2, ad r
and therefore -^ < 2e, and - is negligible, so that
rz rr