MacCullagh and Neumann felt that the great objection to Fresnel's theory of reflexion was its failure to provide for the continuity of the normal component of displacement at the interface between two media; it is obvious that a discontinuity in this component could not exist in any true elastic-solid theory, since it would imply that the two media do not remain in contact. Accordingly, they made it a fundamental condition that all three components of the displacement must be continuous at the interface, and found that the sine-law and tangent-law can be reconciled with this condition only by supposing that the aether-vibrations are parallel to the plane of polarization: which supposition they accordingly adopted. In place of the remaining three true boundary-conditions, however, they used only a single equation, derived by assuming that transverse incident waves give rise only to transverse reflected and refracted waves, and that the conservation of energy holds for these—i.e. that the masses of aether put in motion, multiplied by the squares of the amplitudes of vibration, are the same before and after incidence. This is, of course, the same device as had been used previously by Fresnel; it must, however, be remarked that the principle is unsound as applied to an ordinary elastic solid; for in such a body the refracted and reflected energy would in part be carried away by longitudinal waves.
In order to obtain the sine and tangent laws, MacCullagh and Neumann found it necessary to assume that the inertia of the luminiferous medium is everywhere the same, and that the differences in behaviour of this medium in different substances are due to differences in its elasticity. The two laws may then be deduced in much the same way as in the previous investigations of Fresnel and Cauchy.
Although to insist on continuity of displacement at the interface was a decided advance, the theory of MacCullagh and Neumann scarcely showed as yet much superiority over the quasi-mechanical theories of their predecessors. Indeed, MacCullagh himself expressly disavowed any claim to regard
