This can be found also by calculating the energy of the system, which will be found to consist of three terms, one representing the energy of the sphere with charge Q uninfluenced by an external charge, one representing the energy on a small conductor (not a point) at P existing alone, and a third representing the mutual energy of the electrification on the sphere and the charge q at P existing in presence of one another. By a known theorem the energy of a system of conductors is one half of the sum obtained by multiplying the potential of each conductor by its charge and adding the products together. It is only necessary then to find the variation of the last term caused by increasing f by a small amount df. This will be the product F.df of the force F required and the displacement. Either method may be applied to find the forces of attraction and repulsion for the systems of electrified spheres described below. The problem of two mutually influencing non-intersecting spheres, S₁, S₂ (Fig. 5), insulated with given charges, q₁, q₂, may now be dealt with in the following manner. Let each be supposed at first charged uniformly. By the known theorem referred to above, the external field of each is the same as if its whole charge were situated at the centre. Now if the distribution on S₂, say, be kept unaltered, while that on S₁ is allowed to change, the action of S₂ on S₁ is the same as if the charge q₂ were at the centre C₂ of S₂. Thus if f be the distance between the centres C₁, C₂, and a₁ be the radius of S₁, the distribution will be that corresponding to q₁ + a₁q₂⧸f uniformly distributed on S₁ together with the induced charge －a₁q₂⧸f, which corresponds to the image-charge at the point I₁ (within S₁), the inverse of C₂ with respect to S₁. Now let the charge on S₁ be fixed in the state just supposed while that on S₂ is freed. The charge on S₂ will rearrange itself under the influence of q₁ + a₁q₂⧸f ( = q') and －a₁q₂⧸f, considered as at C₁ and I₁ respectively. The former of these will give a distribution equivalent to q₂ + a₂q'⧸f uniformly distributed over S₂, and an induced distribution of amount a₂q'⧸f at J₁, the inverse point of C₁ with regard to S₂. The image-charge －a₁q₂⧸f at I₁ in S₁ will react on S₂ and give an induced distribution －a₂(－a₁q₂⧸f)f', (I₁C₂ = f') corresponding to an image-charge a₂a₁q₂⧸ff' at the inverse point J₂ of P₁ with respect to C₂S₂. Thus the distribution on S₂ is equivalent to q₂ + a₂q'⧸f － a₂a₁q₂⧸ff' distributed uniformly over it, together with the two induced distributions just described. [Illustration: FIG. 5.] In the same way these two induced distributions on S₂ may now be regarded as reacting on the distribution on S₁ as would point-charges －a₂q₁⧸f and a₂a₁q₂⧸ff', situated at J₁ and J₂ respectively, and would give two induced distributions on S₁ corresponding to their images in S₁.