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.^[1]

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

$\iiint \rho dx\ dy\ dz$,

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

$\iiint \rho \mathbf {r} dx\ dy\ dz$

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

$\mathbf {P} =N\iiint \rho \mathbf {r} dx\ dy\ dz$

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

$\mathbf {P} =(\epsilon -1)\mathbf {E}$.

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

$\iiint \left(P_{x}{\frac {\partial }{\partial x}}+P_{y}{\frac {\partial }{\partial y}}+P_{z}{\frac {\partial }{\partial z}}\right)\phi (x,y,z)dx\ dy\ dz=-\iiint \phi \mathrm {div} \ \mathbf {P} \ dx\ dy\ dz$,

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

${\overline {\rho }}=-\mathrm {div} \ \mathbf {P}$.

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

$\mathrm {div} \ \mathbf {d} =\rho$;

1. W. Thomson (Kelvin), Camb. and Dub. Math. Journal, November, 1845; W. Thomson's Papers on Electrostatics and Magnetism, § 43 sqq.; F. O. Mossotti, Arch. des sc. phys. (Geneva) vi (1847), p. 193: Mem. della Soc. Ital. Modena, (2) xiv (1850), p. 49.

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