Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/280

magnetic effects of the current produced by discharging the jar. The resulting value was nearly

c = 3·1 × 10¹⁰ cm./sec.;

which was the same, within the limits of the errors of measurement, as the speed with which light travels in interplanetary space. The coincidence was noticed by Kirchhoff, who was thus the first to discover the important fact that the velocity with which an electric disturbance is propagated along a perfectlyconducting aerial wire is equal to the velocity of light.

In a second memoir published in the same year, Kirchhoff^[1] extended the equations of propagation of electric disturbance to the case of three-dimensional conductors.

As in his earlier investigation, he divided the electromotive force at any point into two parts, of which one is the gradient of the electrostatic potential φ, and the other is the derivate with respect to the time (with sign reversed) of a vectorpotential a; so that if i denote the current and k the specific conductivity, Ohm's law is expressed by the equation

$i=k(c^{2}\mathrm {grad} \ \phi -{\dot {\mathbf {a} }})$.

Kirchhoff calculated the value of a by aid of Weber's formula for the inductive action of one current element on another; the result is

$\mathbf {a} =\iiint {\frac {dx^{\prime }\ dy^{\prime }\ dz^{\prime }}{r^{3}}}(\mathbf {r.i^{\prime }} )\mathbf {r}$,

where r denotes the vector from the point (x, y, z), at which a is measured, to any other point (x′, y′, z′) of the conductor, at which the current is i′; and the integration is extended over the whole volume of the conductor. The remaining general equations are the ordinary equation of the electrostatic potential

$\nabla ^{2}\phi +4\pi \rho =0$

(where ρ denotes the density of electric charge), and the equation of conservation of electricity

${\frac {\partial \rho }{\partial t}}+\mathrm {div} \ \mathbf {i} =0$.

1. Ann. d. Phys, cii (1857), p. 629: Ges. Abhandl., p. 154.