Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/247

 produce special distortions due to these particles. The nature of these distortions is wholly determined by the direction of displacement, and it is hard to conceive of any reason why the energy of these distortions should not vary with the direction of displacement, like the energy of the general distortion of the wave-motion, which is partly determined by the displacement and partly by the wave-normal.

But the difficulties of the elastic theory do not end with the law of double refraction, although they are there more conspicuous on account of the definite and simple law by which they can be judged. It does not easily appear how the equation of energies can be made to give anything like the proper law of the dispersion of colors. Since for given directions of the wave-normal and displacement, or in an isotropic body, $$\text{B}_{\text{ND}}$$ is constant, and also $$\text{A}_{\text{D}}$$ and $$b_{\text{D}},$$ except so far as the type of the vibration varies, the formula requires that the square of the index of refraction (which is inversely as $$\text{V}^2$$) should be equal to a constant diminished by a term proportional to the square of the period, except so far as this law is modified by a variation of the type of vibration. But experiment shows nothing like this law. Now the variation in the type of vibration is sometimes very important,—it plays the leading rôle in the phenomena of selective absorption and abnormal dispersion,—but this is certainly not always the case. It seems hardly possible to suppose that the type of vibration is always so variable as entirely to mask the law which is indicated by the formula when $$\text{A}_{\text{D}}$$ and $$b_{\text{D}}$$ (with $$\text{B}_{\text{ND}}$$) are regarded as constant. This is especially evident when we consider that the effect on the wave-velocity of a small variation in the type of vibration will be a small quantity of the second order.

The phenomena of dispersion, therefore, corroborate the conclusion which seemed to follow inevitably from the law of double refraction alone.