An introduction to the mathematical theory of attraction ...

Ivory's Theorem. 165
hence VP Vp: V:1: M'. It is plain that this result holds
good whatever be the law of force, provided the force due to
an element of mass varies as the product of this mass, and
some function of its distance.
If the thickness of one homceoid H' be altered, its potential
Y' and its mass M' are altered in the same ratio. Hence the
theorem proved above is true for any two confocal homoeoids.
It follows from this theorem that the equi-potential surfaces
in external space of a homogeneous hom~ooid are confocal
ellipsoids; and also, that at a point external to both, the
potentials of confocal homoeoids are as their masses.
In fact, if we suppose H' outside H, since a homoeoid has
the same potential at all internal points, V', is constant, and
so therefore is Vp,, whatever be the position of P' on the
surface of H'.
Again, if we have two confocal homoeoids E and H',
through any point P" outside both we can suppose another
confocal homceoid "H described; then,
Vy,, V'P V, Vp,,
M ]I I " lf '
It is now easy to prove Ma laurin's Theorem by the method
given in Ex. 5, Art 75.
90. Ivory's Theorem.-If there be two confocal homogeneous solid ellipsoids, Eand E', of the same density, whose
semi-axes are a, b, c, and C', b', c'; and if Xp, be the component parallel to a of the attraction of E at a point P' on the
surface of E', and X'p the parallel component of the attraction of E' at the point P on E corresponding to P', then,
xp, X'P
bc b'c'
This theorem is independent of the law of attraction, provided
the force due to an element of mass is proportional to this
mass multiplied by some function of the distance.
To prove Ivory's Theorem, let x, y, z be the coordinates
of any point of E, and r' its distance from P', and let f' (r')
denote the force due to a unit of mass at the distance r', then
X, = P (r') d dxdydz = [/ f (r') dydz,
JJJ adx