Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/233

depend upon the period of vibration, that is, upon the color of the light.^[1]

We may therefore write in vector notation

$[{\mathsf {E}}]_{\text{Ave}}=\Phi [{\mathsf {U}}]_{\text{Ave}}+\Psi [{\dot {\mathsf {U}}}]_{\text{Ave}}$

(7)

where $\Phi$ and $\Psi$ denote linear functions.^[2]

The optical properties of media are determined by the form of these functions. But all forms of linear functions would not be consistent with the principle of the conservation of energy.

In media which are more or less opaque, and which therefore absorb energy, $\Psi$ must be of such a form that the function always makes an acute angle (or none) with the independent variable. In perfectly transparent media, $\Psi$ must vanish, unless the function is at right angles to the independent variable. So far as is known, the last occurs only when the medium is subject to magnetic influence. In perfectly transparent media, the principle of the conservation of energy requires that $\Phi$ should be self-conjugate, i.e., that for three directions at right angles to one another, the function and independent variable should coincide in direction.

In all isotropic media not subject to magnetic influence, it is probable that $\Phi$ and $\Psi$ reduce to numerical coefficients, as is certainly the case with $\Phi$ for transparent isotropic media.

9. Comparing the two values of $[{\mathsf {E}}]_{\text{Ave}}$ have

${\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}[{\mathsf {U}}]_{\text{Ave}}-\nabla [q]_{\text{Ave}}=\Phi [{\mathsf {U}}]_{\text{Ave}}+\Psi [{\dot {\mathsf {U}}}]_{\text{Ave}}.$

(8)

This equation, in connection with that by which we express the solenoidal character of the displacements, if we regard them as necessarily solenoidal, or in connection with that which expresses the relation between the electrostatic potential and the displacements, if we reject the solenoidal hypothesis, may be regarded as the general equation of the vibrations of monochromatic light, considered as oscillating electrical fluxes. For the symbol Pot, however, we must substitute the symbol representing the operation by which electromotive force is calculated from acceleration of flux, with the negative sign, if we are not satisfied with the law provisionally adopted.

1. The relations between the displacements in one of the small spaces considered and the average electromotive force is mathematically analogous to the relation between the displacements in a system of a high degree of complexity and certain forces exerted from without, which are harmonic functions of the time and under the influence of which the system vibrates. The ratio of the displacements to the forces will in general vary with the period, and may vary very rapidly.
   An example in which these functions vary very rapidly with the period is afforded by the phenomena of selective absorption and abnormal dispersion.
2. A vector is said to be a linear function of another, when the three components of the first are homogeneous functions of the first degree of the three components of the second.