[Illustration: FIG. 6.] The same reasoning can be applied to determine the electrical image of a point-charge of positive electricity in a spherical surface, and therefore the distribution of induced electricity over a metal sphere connected to earth produced by a point-charge near it. Let +q be any positive point-charge placed at a point A outside a sphere (fig. 6) of radius r, and centre at C, and let P be any point on it. Let CA = d. Take a point B in CA such that CB·CA = r², or CB = r²/d. It is easy then to show that PA : PB = d : r. If then we put a negative point-charge -qr/d at B, it follows that the spherical surface will be a zero potential surface, for q/PA - rq/d · 1/PB = 0 (24). Another equipotential surface is evidently a very small sphere described round A. The resultant force due to these two point-charges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges. It is not difficult to show that E = -(d² - r²)q/rAP³ (25), in other words, the force at P is inversely as the cube of the distance from A. Suppose then we remove the negative point-charge, and let the sphere be supposed to become conductive and be connected to earth. If we make a distribution of negative electricity over it, which has a density [sigma] varying according to the law [sigma] = -(d² - r²)q/4[pi]rAP³ (26), that distribution, together with the point-charge +q at A, will make a distribution of electric force at all points outside the sphere exactly similar to that which would exist if the sphere were removed and a negative point charge -qr/d were placed at B. Hence this charge is the electrical image of the charge +q at A in the spherical surface.