An introduction to the mathematical theory of attraction ...

20 Resultant Force.
where z is the distance of P from the centre of the ellipse, a and b its semiaxes,
a2- b2
e its eccentricity, and K2 =a2- b2
a2 + z2'
a zi
Here n = i sin 0d dp =| (i -cos 0) dp = 2-r- 4 cos 0 dp.
If m be the central radius vector of the ellipse, and if we put
3cos <p = a cos, Z sin =bsin i,
we have a2 cos2~ + b2 sin2/ == zm2 = z2 tan2O,
and the limiting values of p are the same as those of p. Hence by substitution
we have the result above.
18. Ellipsoidal Shell.-A homogeneous shell bounded
by similar, concentric, and coaxal ellipsoids, is called a thick
homcoid. Such a shell exercises no
attraction at a point P inside its
internal surface.
To prove this, suppose a cone of
infinitely small angle, having its
vertex at P, and extending in both
directions to the outer boundary of the shell. The ellipsoids
being similar, the plane CML conjugate to the axis of this cone
is the same for the outer and the inner ellipsoid, and therefore
the intercepts on this axis on opposite sides of P between the
inner and outer boundaries of the shell are equal. Hence,
by Art. 9, the mass inside the cone exercises no attraction at
P, but by supposing an infinite number of such cones the
whole mass of the shell may be exhausted; hence the total
attraction of the shell at P is zero.
It is plain that we can show in a similar manner that,
in the case of a uniplanar distribution of mass attracting
inversely as the distance, a homogeneous band bounded by
similar concentric and coaxal ellipses exercises no attraction
at a point inside its inner boundary.
A spherical shell, whose attraction at an internal point
bas been investigated in Art. 12, is obviously a particular
case of an ellipsoidal shell bounded by similar ellipsoids.
When the external and internal surfaces of a homoeoidal
shell approach infinitely near to each other, the shell is called
simply a homosoid.