This is the most general solution of the first of the equations of condition for (c), and it satisfies all the rest. Hence the only laws of attraction which have the property in question are those of the direct distance, the inverse square, and a law compounded of these.

PROP. To find for what laws the shell attracts an internal point equally in every direction.

whatever c is, A being a constant independent of c;

These conditions are all satisfied if the first is: this gives

and therefore the inverse square is the only law which possesses this property.

11. The form of the Earth and of the other bodies of the Solar System differing from the spherical, and more resembling the spheroidal, it is desirable to find the attraction of a spheroid upon an external and an internal point.

PROP. To find the attraction of a homogeneous oblate spheroid upon a particle within its mass; the law of attraction being that of the inverse square of the distance.

12. Let a, c be the semi-axes; the minor axis 2c coinciding with the axis of z: then the equation to the spheroid from the centre is

Let fgh be the co-ordinates to the attracted particle, which we shall take as the origin of polar co-ordinates,

r = radius vector of any particle of the attracting mass,

= angle which makes with a line parallel to z,

=

angle which the plane in which is measured makes with the plane cz;

.. x=ƒ+r sin @ cos p, y=g+r sin 0 sin o, z = h +r cos 0, and the equation to the spheroid becomes

(f+r sin cos )2+(g+r sin 0 sin )2, (h+r cos 0)2

+

c2

=

= 1,

a2

(fsin

+

sin2 0 cos2 0

=

The volume of the attracting element sin @drde do as in Art. 2: let p be the density of the spheroid. Then the attraction of this element on the attracted particle is

p sine dr de do:

and the resolved parts of this parallel to the axes of xyz are ρ sin2 0 cos drd0dp, p sin2 0 sin pdr de dø,

p sin cos e dr d0 dp.

Let A, B, C be the attractions of the whole spheroid in the directions of the axes, estimated positive towards the centre of the spheroid. Then these equal the integrals of the attractions of the element; the limits of r being -r' and r", of Ꮎ being 0 and π, of being 0 and π. Hence

Now it is easily seen that if R (sin a, cos2 a) be a rational function of sin a and cos2 a, then

* If the spheroid be prolate, c is >a and the denominator of this must be written c2 (ca2) cos2 0, and the integral would involve logarithms instead of circular arcs.

13. We gather from these expressions, that the attraction is independent of the magnitude of the spheroid, and depends solely upon its ellipticity. Hence the attraction of the spheroid similar to the given one, and passing through the attracted particle, is the same as that of any other similar concentric spheroid comprising the attracted particle in its mass. Hence a spheroidal shell, the surfaces of which are similar and concentric, attracts a point within it equally in all directions. This property can be proved geometrically exactly as in Art. 5.

14. If we put the ellipticity of the spheroide, and suppose e so small that we may neglect its square, we have

If we had taken an ellipsoid instead of a spheroid, the expressions would not have been capable of integration..

15. If we had attempted to find the attraction on an external particle according to the process of the last Article, we should have fallen upon expressions which no known methods have yet integrated: and therefore we are unable by any direct means to obtain the attraction of a spheroid on an external particle. Mr Ivory has, however, devised an indirect method of obtaining it, which we shall now proceed to develop. He has discovered a theorem by which the attraction of an ellipsoid upon an external particle is shown to be proportional to that of another ellipsoid, dependent on the first for form and dimensions, upon a particle internal to it, and therefore (in the case of a spheroid, or ellipsoid of revolution) determinable by the last Proposition.

PROP. To enunciate and prove Ivory's Theorem.