Now replace S by a spherical shell of metal connected to the earth by a long fine wire, and imagine all other conductors to be at a great distance from it. If this be under the influence of the charge q at P alone, a charge is induced upon it which, in presence of P, maintains it at zero potential. The internal charge －qa⧸f, and the induced distribution on the shell are thus equivalent as regards the potential produced by either at the spherical surface; for each counteracts then the potential produced by q at P. But it can be proved that if a distribution over an equipotential surface can be made to produce the same potential over that surface as a given internal distribution does, they produce the same potentials at all external points, or, as it is usually put, the external fields are the same. This is part of the statement of what has been called the "theorem of replacement" discovered by Green, Gauss, Thomson, and Chasles as described above. Another part of the statement of the theorem may now be formulated. Coulomb showed long ago that the surface-density of electricity at any point on a conductor is proportional to the resultant field-intensity just outside the surface at that point. Since the surface is throughout at one potential this intensity is normal to the surface. Let it be denoted by N, and s be the surface-density: then according to the system of units usually adopted 4πs = N. Let now the rate of diminution of potential per unit of distance outwards (or downward gradient of potential) from the equipotential surface be determined for every point of the surface, and let electricity be distributed over the surface, so that the amount per unit area at each point (the surface-density) is made numerically equal to the gradient there divided by 4π. This, by Coulomb's law, stated above, gives that field-intensity just outside the surface which exists for the actual distribution, and therefore, as can be proved, gives the same field everywhere else outside the surface. The external fields will therefore be equivalent, and further, the amount of electricity on the surface will be the same as that situated within it in the actual distribution. Thus it is only necessary to find for －qa⧸f at P' and q at P, the falling off gradient N of potential outside the spherical surface at any point E, and to take N⧸4π, to obtain s the surface-density at E. Calculation of this gradient for the sphere gives 4πs = －q(f² － a²)⧸ar³. The surface-density is thus inversely as the cube of the distance PE. If the influencing point P be situated within the spherical shell, and the shell be connected to earth as before, the induced distribution will be on its interior surface. The corresponding point P will now be outside, but given by the same relation. And a will now be greater than f, and the density will be given by 4πs = －q(a² － f²)⧸ar³, where, f and r have the same meanings with regard to E and P as before.