Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/266

undertaken from any narrow point of view. Any faults of detail will be readily forgiven, if the author shall give the theory of optics the πού στώ which it has sought so long in vain. We may add that if this effort shall not be judged successful by the scientific world, the author will at least have the satisfaction of being associated in his failure with many of the most distinguished names in mathematical physics.

We have sought to test the proposed theory with respect to that law of optics which seems most conspicuous in its definite mathematical form, and in the rigor of the experimental verifications to which it has been subjected, as well as in the magnificent developments to which it has given rise : the law of double refraction due to Huyghens and Fresnel, and geometrically illustrated by the wave-surface of the latter. We cannot find that the law of Fresnel is proved at all in this treatise. We find on the contrary, that a law is deduced which is different from Fresnel's, and inconsistent with it. We do not refer to anything relating to the direction of vibration of the rays in a crystal, which is a point not touched by the experimental verifications to which we have alluded. We shall confine our comparison to those equations from which the direction of vibration has been eliminated, and which therefore represent relations subject to experimental control. For this purpose equation (13) on page 299 is suitable. It reads

${\frac {u^{2}}{n_{x}^{2}-n^{2}}}+{\frac {v^{2}}{n_{y}^{2}-n^{2}}}+{\frac {w^{2}}{n_{z}^{2}-n^{2}}}=0,$

$n_{x},n_{y},n_{z}$ being the principal indices of refraction. This the author calls the equation of the wave-surface or surface of ray-velocities. It has the form of the equation of FresneUs wave-surface, expressed in terms of the direction-cosines and reciprocal of the radius vector, and if $u,v,w$ are the direction-cosines of the ray, and $n$ the velocity of light in vacuo divided by the so-called ray-velocity in the crystal the equation will express Fresnel's law. But it is impossible to give these meanings to $u,v,w$ and $n.$ They are introduced into the discussion in the expression for the vibrations (p. 295), viz..

$\rho ={\mathfrak {A}}\cos 2\pi \left({\frac {t}{\text{T}}}-{\frac {n(ux+vy+wz)}{\lambda }}\right)\cdot$

The form of this equation shows that $u,v,w$ are proportional to the direction-cosines of the wave-normal, and as the relation $u^{2}+v^{2}+w^{2}=1$ is afterwards used, they must be the direction-cosines of the wave-normal. They cannot possibly denote the direction-cosines of the ray, except in the particular case in which the ray and wave-normal coincide. Again, from the form of this equation, $\lambda /n$ must be the wave-length in the crystal, and if $\lambda$ here as elsewhere in the book