This problem Fresnel endeavoured to solve by regarding the ether within a crystal as made up of distinct particles acting on one another with forces which are functions of the distances, and considering in the first instance the motion of a single particle supposed to be alone disturbed. The result is, however, meant to be applied to a whole plane of particles constituting a wave, and this application is kept in, view throughout the investigation. The force of restitution called into play by the displacement is accordingly resolved in a direction parallel and perpendicular to the front of the wave, and it is assumed that the latter component produces no effect, because although a single particle would be as free to move iu that as in any other direction if impelled, a plane of particles could not so move without compression, whereas vibrations which are strictly transversal take place without compression, to which Fresnel supposes the ether would oppose an immense resistance. Accordingly account is taken only of that com ponent of the force of restitution which lies in the plane of the wave, and which therefore the particle, considered as one of a plane of particles, would be free to obey. It is shown that for either of two rectangular displacements parallel to the front of the wave, the com ponent of the force of restitution which is parallel to the front is also in the direction of displacement, but for a given displacement the force of restitution is different in these two directions. If now the initial displacement be parallel to the front of the wave, but otherwise arbitrary, and if it be resolved in these two directions, the components will be propagated independently of each other, but with different velocities. This accounts both for the double velocity of propagation and for the polarisation, in rectangular planes, of the disturbance pro pa,gated with the two velocities respectively.
These results were mainly deduced from a consideration of the force of restitution called into play by an absolute displacement, whereas it belongs to the fundamental conception of the mechanism of an undu lation that it is propagated by forces called into play by relative dis placements. This difficulty by no means escaped Freenel, who endeavoured to show, by probable reasoning, that the general results would still be the same.
The actual results which follow from Fresnel's theory may be enunciated in the following laws : (1.) In every crystal there exists a system of three rectangular axes (axes of elasticity), with respect to which the optical phenomena are symmetrical. (2.) Let a, b, c, be three parameters belonging to these axes respectively, and representing certain velocities of propagation ; construct the ellipsoid a'x'-'+ and cut it by a diametral plane parallel to the front of a wave, the reciprocals of the semi-axes of the elliptic section will represent the two normal velocities of the waves which can travel independently in the given direction, and planes perpendicular to the wave front and to the respective semi-axes will be the corresponding planes of polarisation.
These laws are of a nature to admit of comparison with experiment, either directly or by the consequences which mathematically flow from them. The direction of vibration in pelarised light is not Itself cognisable by the senses. In the theory from which Fresnel deduced the above laws, it is supposed that the direction of vibration is perpen dicular to the plane of polarisation.
The deduction of the form of the wave-surface becomes now a mere geometrical problem of envelopes. Fresnel did not succeed in solving the problem directly on account of the difficulty of the elimination, but ho gave a very elegant construction by points of a surface which he afterwards proved to be the wave-surface required, by showing that it satisfied the requisite condition as to tangent planes, and ho thus obtained its equation. The construction is as follows. Construct the ellipsoid + fr = cut it by a diametral plane, and from the centre perpendicular to this plane draw lines equal respectively to the semi-axes of the elliptic section : the locus of their extremities will be the wave-surface. In the `Cambridge Philosophical Trans actions,' vol. vi., p. 85, Mr. A. Smith has very simply obtained the equation of the surface, regarded as an envelope, by direct elimination. The equation is— where 0. It is readily seen that when two of the para meters, a, b, c, become equal, the wave-surface of Fresno! becomes the sphere and spheroid of Huygens. For an admirable dynamical investigation of the problem of double refraction, the reader is referred to a paper by Green, in the 7th vol. of the Cambridge Philosophical Transactions.' The length to which this article has already run, compels us to omit the subjects of conical refraction, the application of the undulatory theory to the determination of the Intensities of reflected and refracted polarised light, and of the change of phase which accompanies total internal reflection, the properties of metals in relation to the reflection of light, the optical properties of syrup of sugar and other active liquids, and those of transparent unerystallised media subject to the action of a powerful magnet. For these, reference must be made either to the original memoirs of those who have investigated these subjects, or to some of the extensive treatises which have been written on the undulatory theory.