rying with the inclination of the rays to the axis of the crystal. Lastly, Professor MacCullagh has shown that both the circular po- larization of the rays in the axis, and the elliptical polarization of the rays inclined to it, may be explained by a certain assumed form of the differential equations of vibratory movement, which not only links together the two classes of phenomena, but also affords a ma- thematical expression for their laws. The general theory, to be al- luded to presently, has enabled him to explain the origin of these as- sumed forms of the differential equations.
The theory of reflexion at the surfaces of un crystallized media had been given by Fresnel, although apparently on erroneous prin- ciples. The more complex case of reflexion at the surfaces of cry- stals was left by him to his successors ; and the discovery was made independently, and nearly at the same time, by Professor MacCul- lagh* and M. Newmann of Konigsberg. The discovery is not only important in itself, as bringing within the domain of the wave- theory a large class of hitherto unexplained phenomena, but per- haps still more on account of the physical principles upon which it is based, and the constitution of the luminiferous aether which it ren- ders probable. Thus, it is assumed in this theory, in opposition to the hypothesis of Fresnel, that the vibrations are parallel to the plane of polarization^ and that the density of the cether is the same in all media. These, together with the law of the vis viva, and the beauti- ful principle of the equivalence of vibrations (but half perceived by Fresnel), form the foundation of the theory of crystalHne reflexion, and derive the highest probability from its accordance with pheno- mena. The results of the theory are embodied in geometrical con- structions of great elegance, which determine generally the plane of polarization of the reflected ray, and the amplitudes of the reflected and refracted vibrations.
Hitherto the laws of reflexion at the separating surface of two me- dia were apparently unconnected wdth those which govern the pro- pagation of light in the same medium. It remained to connect these laws as parts of one and the same system, and to trace the hypothe- tical principles upon which each theory was based, up to some higher mechanical principle. This crowning point of the theory has been attained ])y Professor MacCullagh f. Employing the general pro- cesses of analytical mechanics, as laid down by Lagrange |, and li- miting the general theorems solely by the conditions that the density of the aether is constant, and that the vibrations are transversal, he
- " On the laws of crystalline reflexion and refraction/^ Transactions
of the Royal Irish Academy, vol. xviii. This memoir has been honoured by the Medal of the Royal Irish Academy.
f Proceedings of the Royal Irish Academy for December 1839. The complete paper has not yet been published.
X Mr. Green appears to have been the first to apply these methods to the dynamics of light, in a paper on the laws of reflexion and refraction at the surfaces of uncrystallized media, published in the Cambridge Transactions. He has failed, however, in assigning the form of the principal function, and has consequently been led to erroneous results.
