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which it emanates individually from each particle of the crystal; for what we have said about accounting for the phænomena by the supposition of attractive and repulsive forces emanating from the axes is only the indication of a complicated result, and not the expression of a molecular action. What is known then, in this case, or at least what may be supposed, when the idea of the materiality of light is adopted, is that the forces, whatever they may be, which act on the rays of light, in these as in other circumstances, are attractive or repulsive, or both, and emanate from the axes of the crystal. Now in all cases when a material particle is subjected to the action of such forces, its motion is subjected to a general mechanical condition called the principle of least action. Applying this principle here, and joining the particular condition that the forces are sensible only at insensible distances, M. Laplace has deduced two equations which determine completely and generally the direction of the refracted ray for each given direction of incidence, when you know the law of the final velocity of the luminous molecules in the interior of the medium, at a sensible distance from the refracting surface.
In the case of ordinary refraction the final velocity is constant; for the deviation of the ordinary ray is the same in a given substance in whatever direction the experiment be made, provided the angle of incidence and the nature of the ambient medium be unchanged. Accordingly if the interior velocity is supposed to be constant, the equations deduced from the principle of the least action, show that the refraction takes place in the same plain as the incidence, and that the ratio of the sines is invariable, as it appears to be from all observations hitherto made.
Reasoning by analogy, it appeared natural to suppose that the extraordinary refraction was produced by a velocity varying according to the inclination of a ray to the axes of the crystal. Now taking at first crystals with one axis, we have seen that the extraordinary refraction takes place symmetrically all round the axis, that it disappears when a ray lies along the axis, and is at its maximum when they are at right angles. We must then, in the case of these crystals, limit ourselves to the laws of velocity that satisfy these conditions. M. Laplace has tried the following:
V2=v2+Ksinθ2,
