An introduction to the mathematical theory of attraction ...

156 Surfaces and Curves of the Second Degree.
From the equation of the ellipsoid we have
1 cos2 0 sin2 0 cos2 f sin2 0 sin2 <
r2 + - + 2 --
and 2I 2doe = l ' r2 sin dOd(p.
Hence r2 dw
is a homogeneous function of a2, b2, and c2 of the degree -.
5. Prove that
d21 dI dI
2 ( a2- 2) d( a)2)d(b2) d(a2) d(b2)
6. If the components of the attraction of a homogeneous ellipsoid at an
internal point x, y, z, be denoted by - Ax, - By, - Cz, show that
da db dc
A -+ B -+ Ca b c
is a perfect differential.
Here A = - 4rpabc d-) &c.
Hence, if we put ~, Ij, M for a2, b2, c2, we have to show that
dI dI I/ dI
is a perfect differential; but this follows immediately from Ex. 5, by which
d21 dI dl
2 (t - cl dd 7)
7. Find the potential of a homogeneous focaloid,at an internal point.
If V be the potential of a homogeneous ellipsoid, and U that of a focaloid of
equal mass having the surface of the ellipsoid as its boundary, by Art. 83, at an
internal point x, y, z, we have
Y- U_ K 1- a ---)
where
27rpa2b2c2 3 abc
a2b2 + b2c2 + c2a2 2 a2b2 + b2c2 + c22^'
the mass of the focaloid being 21.
Eence U1= Vo -K+ — 2A + K- + K- C 2
where the values of Vo, A, B, and C are given by (5), Art. 86, and (15), Art.
21.