CHAPTER XI.

FRESNEL'S THEORY OF DOUBLE REFRACTION.

(178) IT has been stated (60, 66), that soon after the discovery of double refraction in Iceland crystal, Huygens succeeded in embracing its laws in the theory of waves, by a bold and happy assumption. He had already shown that the form of the wave which gives rise to the ordinary refracted ray, in glass and other uncrystallized substances, was the sphere; or, in other words, that the velocity of undulatory propagation was the same in all directions. One of the rays in Iceland crystal, too, was found to obey the same law; and, judging that the law which governed the other, though not so simple, was yet next in simplicity, he assumed the form of its wave to be the spheroid; that is, he supposed the velocity of propagation to be different in different directions, in accordance with the following construction :-"Let an ellipsoid of revolution be described round the optic axis, having its centre at the point of incidence; and let the greater axis of the generating ellipse be to the less in the ratio of the greatest to the least index of refraction: then the velocity of any ray will be represented by the radius vector of the ellipsoid which coincides with it in direction." We have already seen that the construction for the direction of the rays, derived from this assumption, was verified by experience; and we have here another instance, to which the history of science affords many parallels, of the value of analogical principles in directing scientific research.

(179) The law of Huygens was found to hold in many crystals besides that to which it was originally applied ; and in

all of these there was one optic axis, or one line along which a ray of light passed without division. But when the researches of Brewster made known a class of crystalline bodies, having two optic axes, or two lines of no separation, Huygens's law was found not to be general; and it was ascertained that one of the rays, at least, in biaxal crystals, followed some new and unknown law.

In this state of the question, the problem of double refraction was taken up by Fresnel; and by the aid of a natural and simple hypothesis, combined with the principle of transversal vibrations, he has been conducted to its complete solution,—a solution which not only embraces all the known phenomena, but has even outstripped observation, and predicted consequences which were afterwards verified by experiment.

(180) Fresnel sets out from the supposition, that the elastic force of the vibrating medium, in every crystal, is different in different directions. This is, in fact, the most general supposition that can be made; and whether we suppose that the vibrating medium is the ether within the crystal, or that the molecules of the body itself partake of the vibratory movement, there will be obviously such a connexion and mutual dependence of the parts of the solid and those of the medium in question, that we cannot hesitate to admit for the one, what has been already established on the clearest evidence for the other.

It is easy to see, generally, that the phenomenon of double refraction is a necessary consequence of this hypothesis, and of the principle of transversal vibrations.

Let us take, for example, the simple case of a ray of light proceeding from an infinitely distant point, and falling perpendicularly on the surface of a uniaxal crystal, cut parallel to the axis. The incident wave being plane, and parallel to the surface of the crystal, the vibrations are also parallel to

the same surface; and we may conceive them to be composed of vibrations parallel and perpendicular to the axis of the crystal. Now, the elasticity brought into play by these two sets of vibrations being different, they will be propagated with different velocities; and there will be two waves within the crystal, in which the vibrations are parallel to two fixed directions at right angles to one another,—or two rays oppositely polarized. If the second face of the crystal be parallel to the first, the two rays will emerge perpendicularly; and the only effect produced will be, that one will be retarded more than the other, in its progress through the crystal. But if the second face be oblique to the direction of the rays, they will be both refracted at emergence, and differently; and they will therefore diverge from one another.

(181) To return to the general theory. Let us suppose a disturbance to be produced in a medium such as we have been considering, and any particle of the medium to be displaced from its position of rest. The resultant of all the elastic forces which resist the displacement will not, in general, act in the direction of the displacement (as would be the case in a medium uniformly elastic), and therefore will not drive the displaced particle directly back to its position of equilibrium. Fresnel has shown, however, that there are three directions at right angles to each other, in every elastic medium, in each of which the elastic forces do act in the direction of the displacement, whatever be the nature or laws of the molecular action. He assumes that these three directions are parallel throughout the crystal. In fact, the first principles of crystallization oblige us to admit, that the arrangement of the molecules of the crystalline body is similar in all parallel lines throughout the crystal; and the same property must belong to the ether within it, if (as we have reason to presume) its elasticity be dependent on that of the crystal itself. These three directions Fresnel denominates axes of elasticity; and

he concludes that they are also axes of symmetry, with respect to the crystalline form.

If, on each of these axes, and on every line diverging from the same origin, portions be taken, which are as the square roots of the elastic forces in their directions, the locus of the extremities of these portions will be a surface, which Fresnel denominates the surface of elasticity. Its equation is

r2 = a2 cos2 a + b2 cos2 ß + c2 cos2 y:

a2, b2, c2, being the elasticities in the directions of the three axes; r the radius vector of the surface; and a, ß, the angles which it makes with the axes.

This surface determines the velocity of propagation of the wave, when the direction of its vibrations is given. For, the ethereal molecule vibrating in the direction of any radius vector, r, of this surface, the elastic force which governs its vibration will be proportional to r2; and, since the velocity of wavepropagation is as the square root of the elastic force, it must, in this case, be represented by the radius vector of the surface of elasticity in the direction of the vibrations. Hence, if we conceive the vibration in the incident wave to be resolved into two within the crystal, performed in two determinate directions, these will be propagated with different velocities; and, as a difference of velocity gives rise to a difference of refraction, it follows that the incident ray will be divided into two within the crystal, which will in general pursue different paths. Thus, the bifurcation of a ray, on entering a crystal, presents no difficulty, provided we can explain in what manner the vibration comes to be resolved.

(182) To understand in what manner this takes place, let us conceive a plane wave advancing within the crystal. By the principle of transversal vibrations, the movements of the ethereal molecules are all parallel to the wave. But the motion of each molecule, when thus removed from its position of

equilibrium, is resisted by the elastic force of the medium; and that force is, in general, oblique to the direction of the displacement. If the plane containing the direction of the force and that of the displacement were normal to the plane of the wave, the force would be resolvable into two,- one perpendicular to the plane of the wave, which (by the principle of transversal vibrations) can produce no effect; and the other in the direction of the displacement itself, which will thus be communicated from particle to particle without change. But this, in general, is not the case. Fresnel has shown, however, that the displacement may be resolved in two directions in the plane of the wave, at right angles to one another, such that the elastic force called into action by each component will be in the plane passing through the component, and normal to the wave; and thus each component will give rise to a wave, in which the direction of the vibrations is preserved, and which therefore will be propagated with a constant velocity.

The two directions, above alluded to, are those of the greatest and least diameters of the section of the surface of elasticity made by the plane of the wave; so that if the original displacement be resolved into two, parallel to these directions, each component will give rise to a plane wave, in which the vibrations preserve constantly the same direction. The velocity of propagation being represented by the radius vector of the surface of elasticity in the direction of the displacement, the velocities of the two parts of the wave will be proportional to the greatest and least diameters of the section of the surface of elasticity, to which the vibrations are parallel. Thus it appears that an incident plane wave, in which the vibrations. are in any direction, will be resolved into two within the crystal; and these will be propagated with different velocities, and consequently assume different directions.

(183) The vibrations in these waves being parallel to two