Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/261

light. This is strikingly true of the metals. But the analogy does not stop here. To fix our ideas, let us consider the case of an isotropic body and circularly polarized light, which is geometrically the simplest case although its analytical expression is not so simple as that of plane-polarized light. The displacement at any point may be symbolized by the rotation of a point in a circle. The external force necessary to maintain the displacement ${\mathfrak {F}}$ is represented by $n^{-2}{\mathfrak {F}}.$ In transparent bodies, for which $n^{-2}$ is a positive number, the force is radial and in the direction of the displacement, being principally employed in counterbalancing the dielectric elasticity, which tends to diminish the displacement. In a conductor $n^{-2}$ becomes complex, which indicates a component of the force in the direction of ${\dot {\mathfrak {F}}},$ that is, tangential to the circle. This is only the analytical expression of the fact above mentioned. But there is another optical peculiarity of metals, which has caused much remark, viz., that the real part of $n^{2}$ (and therefore of $n^{-2}$) is negative, i.e., the radial component of the force is directed towards the center. This inwardly directed force, which evidently opposes the electrodynamic induction of the irregular part of the motion, is small compared with the outward force which is found in transparent bodies, but increases rapidly as the period diminishes. We may say, therefore, that metals exhibit a second optical peculiarity,—that the dielectric elasticity is not prominent as in transparent bodies. This is like the electrical behavior of the metals, in which we do not observe any elastic resistance to the motion of electricity. We see, therefore, that the complex indices of metals, both in the real and the imaginary part of their inverse squares, exhibit properties corresponding to the electrical behavior of the metals.

The case is quite different in the elastic theory. Here the force from outside necessary to maintain in any element of volume the displacement ${\mathfrak {E}}$ is represented by $n^{2}{\ddot {\mathfrak {E}}}.$ In transparent bodies, therefore, it is directed toward the center. In metals, there is a component in the direction of the motion ${\dot {\mathfrak {E}}},$ while the radial part of the force changes its direction and is often many times greater than the opposite force in transparent bodies. This indicates that in metals the displacement of the ether is resisted by a strong elastic force, quite enormous compared to anything of the kind in transparent bodies, where it indeed exists, but is so small that it has been neglected by most writers except when treating of dispersion. We can make these suppositions, but they do not correspond to anything which we know independently of optical experiment.

It is evident that the electrical theory of light has a serious rival, in a sense in which, perhaps, one did not exist before the publication