_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´_ ÷ _x_^2 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 _d_M´ 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 _d_M´, and the attraction of these is zero. We see from § 2 that the contraction of _d_M´ from infinity, until it forms a shell with radius _r_, represents a quantity of energy, (ε M´_d_M´)/_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. If M be the total mass and _a_ the radius of the nebula always supposed homogeneous M´ = M (_r_^3/_a_^3), and therefore _d_M´ = 3 M (_r_^2/_a_^3) _dr_. Hence the work done in the contraction is (ε/_r_) M (_r_^3/_a_^3) · 3 M (_r_^2/_a_^3) _dr_ = (3 ε/_a_^6) M^2 _r_^4 _dr_. Integrating, therefore, the total work of contraction is ⅗ (ε M^2/_a_) At the present moment a mass of 1 lb. at the surface of the sun would weigh 27 lbs. if tested by a spring balance. Hence ε M/_a_^2 = 27. With this substitution we find the expression for the foot-pounds of work corresponding to the contraction of the nebula from infinity to a sphere of radius _a_ to be, ⅗ · 27 _a_ M = 16 _a_ M very nearly.