Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/256

But the mental processes by which we satisfy ourselves of the validity of our results (if we do not work out the whole problem in the general case of no assumption in r^ard to the velocity of the missing wave) certainly involve conceptions of a higher degree of difficulty on account of the circumstances mentioned. Perhaps this ought not to aifect our judgment with respect to the question of the truth of the hypothesis.

Although the two theories give laws of exactly the same form for monochromatic light in the limiting case, their deviations from this limit are in opposite directions, so that if the phenomena of optics differed in any marked degree from what we would have in the limiting case, it would be eiwy to find an experimentum crucis to decide between the two theories. A little consideration will make it evident, that when the principal indices of refraction of a ciystal are given, the intermediate values for oblique wave-planes will be less if the velocity of the missing wave is small but finite, than if it is infinitesimal, and will be greater if the velocity of the missing wave is very great than if it is infinite.^[1] Hence, if the velocity of the missing wave is small but finite, the intermediate values of the indices of refraction will be less than are given by Fresnel's law, but if the velocity of the missing wave is very great but finite, the intermediate values of the indices of refraction will be greater than are given by Fresnel's law. But the recent experiments of Professor Hastings on the law of double refraction in Iceland spar do not encourage us to look in this direction for the decision of the question.^[2]

In a simple train of waves in a transparent medium, the potential energy, on the elastic theory, may be divided into two parts, of which one is due to that general deformation of the ether which is represented by the equations of wave-motion, and the other to those deformations which are caused by the interference of the ponderable particles with the wave-motion, and to such displacements of the ponderable matter as may be caused, in some cases at least, by the motion of the ether. If we write $h$ for the amplitude, $l$ for the wavelength, and $p$ for the period, these two parts of the statical energy (estimated per unit of volume for a space including many wavelengths) may be represented respectively by

${\frac {\pi ^{2}{\text{B}}h^{2}}{l^{2}}}$and${\frac {bh^{2}}{4}}\cdot$

1. This may be more clear if we consider the stationary waves formed by two trains of waves moving in opposite directions. The case then comes under the following theorem:
   "If the system undergo such a change that the potential energy of a given configuration is diminished, while the kinetic energy of a given motion is unaltered, the periods of the free vibrations are all increased, and conversely." See Lord Rayleigh's Theory of Sound, vol. i, p. 85.
2. Am. Jour. Sci., ser 3, voL xxxv, p. 60.