Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/231

 inductive capacity may be seen by what follows, which is substantially a translation into electrostatical language of Poisson's theory of induced magnetism.

Let ρ denote volume-density of clectric charge. For each of Faraday's "small shot" the integral

integrated throughout the shot, will vanish, since the total charge of the shot is zero: but if r denote the vector (x, y, z), the integral

will not be zero, since it represents the electric polarization of the shot: if there are N shot per unit volume, the quantity

will represent the total polarization per unit volume. If d denote the electric force, and E the average value of d, P will be proportional to E, say

By integration by parts, assuming all the quantities concerned to vary continuously and to vanish at infinity, we have

where φ denotes an arbitrary function, and the volume-integrals are taken throughout infinite space. This equation shows that the polar-distribution of electric charge on the shot is equivalent to a volume-distribution throughout space, of density

Now the fundamental equation of electrostatics may in suitable units be written,