A thin spherical homogeneous shell produces the same attraction at an external point as if its entire mass were concentrated at the centre of the sphere.
This is another famous theorem due to Newton. He gives a beautiful geometrical proof in Section XII. of the first book of the “Principia.” We shall here take it for granted, and we shall consequently assume that—
The attraction by the law of gravitation of a homogeneous sphere on an external point is the same as if the entire mass of the sphere were concentrated at its centre.
§ 2. On the Energy between Two Attracting Masses.
Let m and m´ be two attracting bodies supposed to be small in comparison with their distance x. Let the force between them be ε m m´ ÷ x2 when ε is the force between two unit masses at unit distance. It is required to find the energy necessary to separate them to infinity, it being supposed that they start from an initial distance a. The energy required is obtained by integrating between the limits infinity and a, and is consequently ε m m´ ÷ a.
§ 3. On the Energy Given Out in the Contraction of the Nebula.
We assume that the nebula is contracting symmetrically, so that at any moment it is a homogeneous sphere. We shall consider the shell which lies between the two spheres of radii, r + dr and r respectively.
Let M´ be the mass of the nebula contained within the sphere of radius r, and let dM´ be the mass of the shell just defined. Then it follows from [§ 1] that the condensation of the shell will have been effected by the attraction of the mass M´ solely. The exterior parts of the nebula can have had no effect, for the outer part has always been in symmetrical spherical shells exterior to dM´, and the attraction of these is zero. We see from [§ 2] that the contraction of dM´ from infinity, until it forms a shell with radius r, represents a quantity of energy,
(ε M´dM´)/r ;
for it is obvious that the energy involved in the contraction of the whole shell is the sum of the energies corresponding to its several parts.