A more difficult case is presented by the ellipsoid[5]. We have first to determine the mode in which electricity distributes itself on a conducting ellipsoid in free space. It must be such a distribution that the potential in the interior will be constant, since the electric force must be zero. It is a well-known theorem in attractions that if a shell is made of gravitative matter whose inner and outer surfaces are similar ellipsoids, it exercises no attraction on a particle of matter in its interior[6]. Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a + da), (b + db), (c + dc), where da/a = db/b = dc/c = [mu]. The potential of such a shell at any internal point is constant, and the equipotential surfaces for external space are ellipsoids confocal with the ellipsoidal shell. Hence if we distribute electricity over an ellipsoid, so that its density is everywhere proportional to the thickness of a shell formed by describing round the ellipsoid a similar and slightly larger one, that distribution will be in equilibrium and will produce a constant potential throughout the interior. Thus if [sigma] is the surface density, [delta] the thickness of the shell at any point, and [rho] the assumed volume density of the matter of the shell, we have [sigma] = A[delta][rho]. Then the quantity of electricity on any element of surface dS is A times the mass of the corresponding element of the shell; and if Q is the whole quantity of electricity on the ellipsoid, Q = A times the whole mass of the shell. This mass is equal to 4[pi]abc[rho][mu]; therefore Q = A4[pi]abc[rho][mu] and [delta] = [mu]p, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane. Hence [sigma] = Qp/4[pi]abc (3). Capacity of an ellipsoid. Accordingly for a given ellipsoid the surface density of free distribution of electricity on it is everywhere proportional to the length of the perpendicular let fall from the centre on the tangent plane at that point. From this we can determine the capacity of the ellipsoid as follows: Let p be the length of the perpendicular from the centre of the ellipsoid, whose equation is x²/a² + y²/b² + z²/c² = 1 to the tangent plane at x, y, z. Then it can be shown that 1/p² = x²/a^4 + y²/b^4 + z²/c^4 (see Frost's _Solid Geometry_, p. 172). Hence the density [sigma] is given by Q 1 [sigma] = -------- --------------------------------, 4[pi]abc [root](x²/a^4 + y²/b^4 + z²/c^4) and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression, _ _ / [sigma]dS Q / dS V = | --------- = -------- | --------------------------------- (4). _/ r 4[pi]abc _/ r[root](x²/a^4 + y²/b^4 + z²/c^4)