Further, if q be distributed over an element dS of a surface, the inverse charge aq⧸f will be distributed over the corresponding element dS' of the inverse surface. But dS'⧸dS = a⁴⧸f⁴ = f'⁴⧸a⁴ where f, f' are the distances of O from dS and dS'. Thus if s be the density on dS and s' the inverse density on dS' we have s'⧸s = a³⧸f'³ = f³⧸a³. When V is constant over the direct surface, while r has different values for different directions of OQ, the different points of the inverse surface may be brought to zero potential by placing at O a charge －aV. For this will produce at Q' a potential －aV⧸r' which with V' will give at Q' a potential zero. This shows that V' is the potential of the induced distribution on S' due to a charge －aV at O, or that －V' is the potential due to the induced charge on S' produced by the charge aV at O. Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. The charges on the conductor (or conductors) after inversion are always obtainable at once from the fact that they are the inverses of the charges on the conductor (or conductors) in the direct case, and the surface-densities or volume-densities can be found from the relations stated above. [Illustration: FIG. 7.] Now take the case of two concentric spheres insulated and influenced by a point-charge q placed at a point P between them as shown in Fig. 7. We have seen at p. 49 how the induced distribution, and the amount of the charge, on each sphere is obtained from the two convergent series of images, one outside the outer sphere, the other inside the inner sphere. We do not here calculate the density of distribution at any point, as our object is only to explain the method; but the quantities on the spheres S₁ and S₂, are respectively －q.OA.PB⧸(OP.AB), －q.OB.AP⧸(OP.AB). It may be noticed that the sum of the induced charges is －q, and that as the radii of the spheres are both made indefinitely great, while the distance AB is kept finite, the ratios OA⧸OP, OB⧸OP approximate to unity, and the charges to －q.PB⧸AB, －q.AP⧸AB, that is, the charges are inversely as the distances of P from the nearest points of the two surfaces. But when the radii are made indefinitely great we have the case of two infinite plane conducting surfaces with a point-charge between them, which we have described above.