Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/243

amounts to the same thing, but may present a more distinct picture to the imagination, the wave-length may be regarded as enormously great in comparison with the distances between neighboring molecules. Whatever view we take of the motions which constitute light, we can hardly suppose them (disturbed as they are by the presence of the ponderable molecules) to be in strictness represented by the equations of wave-motion. Yet in a certain sense a wave-motion may and does exist. If, namely, instead of the actual displacement at any point, we consider the average displacement in a space large enough to contain an immense number of molecules, and yet small as measured by a wave-length, such average displacements may be represented by the equations of wave-motion; and it is only in this sense that any theory of wave-motion can apply to the phenomena of light in transparent bodies. When we speak of displacements, amplitudes, velocities (of displacement), etc., it must therefore be understood in this way.

The actual kinetic energy, on either theory, will evidently be greater than that due to the motion thus averaged or smoothed, and to a degree presumably depending on the direction of the displacement. But since displacement in any direction may be regarded as compounded of displacements in three fixed directions, the additional energy will be a quadratic function of the components of velocity of displacement, or, in other words, a quadratic function of the direction-cosines of the displacement multiplied by the square of the amplitude and divided by the square of the period.^[1] This additional energy may be understood as including any part of the kinetic energy of the wave-motion which may belong to the ponderable particles. The term to be added to the kinetic energy on the electric theory may therefore be written $f_{\text{D}}{\frac {h^{2}}{p^{2}}},$ where $f_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. The elastic theory requires a term of precisely the same character, but since the term to which it is to be added is of the same general form, the two may be incorporated in a smgle term of the form ${\text{A}}_{\text{D}}{\frac {h^{2}}{p^{2}}},$ where ${\text{A}}_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. We must, however, notice that both ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ are not entirely independent of the period. For the manner in which the flux of the luminiferous medium is distributed among the ponderable molecules will naturally depend somewhat upon the period. The same is true of the degree to which the molecules may be thrown into vibration. But ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ will be independent of the wave-length (except so far as this is

1. For proof in extenso of this proposition, when the motions are supposed electrical, the reader is referred to page 187 of this volume.