Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/81

 at any point can be simply expressed as the derivates of the function which is obtained by adding together the masses of all the particles of an attracting system, each divided by its distance from the point; and Laplace had shown that this function V satisfies the equation
 * $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0$$

in space free from attracting matter. Poisson himself showed later, in 1813, that when the point (x, y, z) is within the substance of the attracting body, this equation of Laplace must be replaced by
 * $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 4\pi \rho$$

where ρ denotes the density of the attracting matter at the point. In the present memoir Poisson called attention to the utility of this function V in electrical investigations, remarking that its value over the surface of any conductor must be constant.

The known formulae for the attractions of spheroids show that when a charged conductor is spheroidal, the repellent force acting on a small charged body immediately outside it will be directed at right angles to the surface of the spheroid, and will be proportional to the thickness of the surface-layer of electricity at this place. Poisson suspected that this theorem might be true for conductors not having the spheroidal form—a result which, as we have seen, had been already virtually given by Coulomb; and Laplace suggested to Poisson the following proof, applicable to the general case. The force at a point immediately outside the conductor can be divided into a part S due to the part of the charged surface immediately adjacent to the point, and a part due to the rest of the surface. At a point close to this, but just inside the conductor, the force S will still act; but the forces will evidently