Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/267

Rh (see p. 26) denotes the wave-length in vacuo of light of the period considered, which we doubt not is the intention of the author, $$n$$ must be the wave-length in vacuo divided by the wave-length in the crystal, i.e., the velocity of light in vacuo divided by the wave-velocity in the crystal With these definitions of $$u, v, w,$$ and $$n,$$ equation (13) expresses a law which is different from Fresnel's. Applied to the simple case of a uniaxial crystal, it makes the relation between the wave-velocity of the extraordinary ray and the angle of the wave-normal with the principal axis the same as that of the radius vector and the angle in an ellipse. The law of Huyghens and Fresnel makes the reciprocal of the wave-velocity stand in this relation.

The law which our author has deduced has come up again and again in the history of theoretical optics. Professor Stokes (Report of the British Assoc., 1862, part i, p. 269) and Lord Rayleigh (Phil. Mag., (4), vol. xli, p. 525) have both raised the question whether Huyghens and Fresnel might not have been wrong, and it might not be the wave-velocity and not its reciprocal which is represented by the radius vector in an ellipse. The difference is not very great, for if we lay off on the radii vectores of an ellipse distances inversely proportional to their lengths, the resultant figure will have an oval form approaching that of an ellipse when the eccentricity of the original ellipse is small. Rankine appears to have thought that the difference might be neglected (see Phil. Mag., (4), vol. i, pp. 444, 445) at least he claims that his theory leads to Fresnel's law, while really it would give the same law which our author has found. (Concerning Rankine's "splendid failure," and the whole history of the subject, see Sir Wm. Thomson's Lectures on Molecular Dynamics at the Johns Hopkins University, chap, xx.) Professor Stokes undertook experiments to decide the question. His result, corroborated by Glazebrook (Pro. Roy. Soc., voL xx, p. 443; Phil. Trans., voL clxxi, p. 421), was that Huyghens and Fresnel were right and that the other law was wrong.

To return to our author, we have no doubt from the context that he regards $$u, v, w,$$ and $$n$$ as relating to the ray and not to the wave-normal. We suppose that that is the meaning of his remark that the expression for the vibrations (quoted above) is to be referred to the direction of the ray. It seems rather hard not to allow a writer the privilege of defining his own terms. Tet the reader will admit that when the vibrations have been expressed in the above form an inexorable necessity fixes the significance of the direction determined by $$u, v, w,$$ and leaves nothing in that respect to the choice of the author.

The historical sketches of the development of ideas in the theory of optics, enriched by very numerous references, will be useful to