Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/319

The equations by which φ and a have been defined are equivalent to the equations

$\nabla ^{2}\phi ={\overset {..}{\phi }}/c^{2}=-4\pi \rho$,     (2)

$\nabla ^{2}\mathbf {a} ={\overset {..}{\mathbf {a} }}/c^{2}=-4\pi \mathbf {\iota }$,     (3)

while the equation of conservation of electricity,

$\mathrm {div} \ \mathbf {\iota } +{\overset {.}{\rho }}=0$

gives

$\mathrm {div} \ \mathbf {a} +{\overset {.}{\phi }}=0$.     (4)

From equations (1), (2), (4), we may readily derive the equation

$\mathrm {div} \ \mathbf {E} =4\pi c^{2}\rho$;     (I)

and from (1), (3), (4), we have

$\mathrm {curl} \ \mathbf {H} ={\overset {.}{\mathbf {E} }}/c^{2}+4\pi \mathbf {\iota }$,     (II)

where H or curl a denotes the magnetic force: while from (1) we have

$\mathrm {curl} \ \mathbf {E} =-{\overset {.}{\mathbf {H} }}$.     (III)

The equations (I), (II), (III) are, however, the fundamental equations of Maxwell's theory; and therefore the theory of L. Lorenz is practically equivalent to that of Maxwell, so far as concerns the propagation of electromagnetic disturbances through free aether. Lorenz himself, however, does not appear to have clearly perceived this; for in his memoir he postulated the presence of conducting matter throughout space, and was consequently led to equations resembling those which Maxwell had given for the propagation of light in metals. Observing that his equations represented periodic electric currents at right angles to the direction of propagation of the disturbance, he suggested that all luminous vibrations might be constituted by electric currents, and hence that there was "no longer any reason for maintaining the hypothesis of an aether, since we can admit that space contains sufficient ponderable matter to enable the disturbance to be propagated."

Lorenz was unable to derive from his equations any explanation of the existence of refractive indiecs, and his theory lacks