Page:Optics.djvu/198

 where $$v$$ represents the ordinary velocity, $$V$$ the extraordinary, $$\theta$$ the angle between the extraordinary ray and the axis, and $$K$$ is a coefficient which is constant for any one given crystal. Introducing this law of the velocity in the equations of the principle of least action, he obtained immediately Huygens's law. This law had been completely verified only for Iceland spar, but M. Biot has found it true for quartz and beril; only the coefficient $$K$$ is positive in crystals of attractive double refraction, and negative in the others. Its absolute value is different in different substances, and it is even found to vary in specimens of the same mineralogical species; but with these modifications it is probable that the law applies equally to all crystals with one axis.

As to those having two axes, it is clear that the extraordinary velocity $$V $$ must depend on the two angles $$\theta$$ and $$\theta'$$ made by the refracted ray with the two axes. Analogy leads us to try whether the square of the velocity $$V$$ cannot be expressed here also by a function of the second degree, but more general, that is, depending on both the angles; now in such crystals the refractions become equal when the ray coincides with one or the other axis. This proves that the ordinary velocity must then be equal to the ordinary. This condition limits the generality of the function, and reduces it to the following form:

that is, there must remain only the product of the two sines. Introducing this formula into the equations of the principle of least action, the path and motion of the rays is found for all cases, and it remains only to try whether it is conformable to experiment. M. Biot has done this for the white topaz which has two axes of double refraction, and the formula agreed perfectly with observation. One may, besides, judging by other phænomena that will be hereafter indicated, be convinced that the same law applies to other crystals with two axes on which experiments have not yet been made; and it is highly probable that it is universally applicable.

It may be remarked that the general law comprises Huyghens's as a particular case, for crystals with only one axis, considering these as having two axes which coincide, for then $$\theta$$ and $$\theta'$$ become equal, and the equation for $$V$$ contains the square of $$\sin \theta.$$