Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/245

Rh the different coefficients in the function varying separately, because $$\text{G}_{\text{D}}$$ and $$f_{\text{D}}$$ will not in general be similar functions. If we consider a constant direction of displacement while the period varies, $$\text{G}_{\text{D}}$$ and $$f_{\text{D}}$$ will only vary so far as the type of the motion varies, i.e., so far as the manner in which the flux distributes itself among the ponderable molecules and intermolecular spaces, and the extent to which the molecules take part in the motion are changed. There are cases in which these vary rapidly with the period, viz., cases of selective absorption and abnormal dispersion. But we may fairly expect that there will be many cases in which the character of the motion in these respects will not vary much with the period. $$\frac{\text{G}_{\text{D}}}{\text{F}}$$ and $$\frac{f_{\text{D}}}{\text{F}}$$ will then be sensibly constant and we have an approximate expression for the general law of dispersion, which agrees remarkably well with experiment.

If we now return to the equation of energies obtained from the elastic theory, we see at once that it does not suggest any such relation as experiment has indicated, either between the wave-velocity and the direction of displacement, or between the wave-velocity and the period. It remains to be seen whether it can be brought to agree with experiment by any hypothesis not too violent.

In order that $$\text{V}^2$$ may be a quadratic function of any set of direction-cosines, it is necessary that $$\text{A}_{\text{D}}$$ and $$b_{\text{D}}$$ shall be independent of the direction of the displacement, in other words, in the case of a crystal like Iceland spar, that the direct action of the ponderable molecules upon the ether, shall affect both the kinetic and the potential energy in the same way, whether the displacement take place in the direction of the optic axis or at right angles to it. This is contrary to everything which we should expect. If, nevertheless, we make this supposition, it remains to consider $$\text{B}_{\text{ND}}.$$ This must be a quadratic function of a certain direction, which is almost certainly that of the displacement If the medium is free from external stress (other than hydrostatic), $$\text{B}_{\text{ND}},$$ as we have seen, is symmetrical with respect to the wave-normal and the direction of displacement, and a quadratic function of the direction-cosines of each. The only single direction of which it can be a function is the common perpendicular to these two directions. If the wave-normal and the displacement are perpendicular, the direction-cosines