Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/168

 remained to supply the boundary-conditions at an interface, which are required in the discussion of reflexion, and the relations between the elastic constants of the solid, which are required in the optics of crystals. Cauchy seems to have considered the question from the purely analytical point of view. Given certain differential equations, what supplementary conditions must be adjoined to them in order to produce a given analytical result? The problem when stated in this form admits of more than one solution, and hence it is not surprising that within the space of ten years the great French mathematician produced two distinct theories of crystal-optics and three distinct theories of reflexion, almost all yielding correct or nearly correct final formulae, and yet mostly irreconcilable with each other, and involving incorrect boundary-conditions and improbable relations between elastic constants.

Cauchy's theories, then, resemble Fresnel's in postulating types of elastic solid which do not exist, and for whose assumed properties no dynamical justification is offered. The same objection applies, though in a less degree, to the original form of a theory of reflexion and refraction which was discovered about this time almost simultaneously by James MacCullagh (b. 1809, d. 1847), of Trinity College, Dublin, and Franz Neumann (b. 1799, d. 1895), of Königsberg. To these authors is due the merit of having extended the laws of reflexion to crystalline media; but the principles of the theory were originally derived in connexion with the simpler case of isotropic media, to which our attention will for the present be confined.