As in his previous investigations, he assumes that the vibrations which constitute light are executed at right angles to the plane of polarization. He adopts Young's principle, that reflexion and refraction are due to differences in the inertia of the aether in different material bodies, and supposes (as in his memoir on Aberration) that the inertia is proportional to the inverse square of the velocity of propagation of light in the medium. The conditions which he proposes to satisfy at the interface between two media are that the displacements of the adjacent molecules, resolved parallel to this interface, shall be equal in the two media, and that the energy of the reflected and refracted waves together shall be equal to that of the incident wave.
On these assumptions the intensity of the reflected and refracted light may be obtained in the following way:—
Consider first the case in which the incident light is polarized in the plane of incidence, so that the displacement is at right angles to the plane of incidence; let the amplitude of the displacement at a given point of the interface be f for the incident ray, g for the reflected ray, and h for the refracted ray.
The quantities of energy propagated per second across unit cross-section of the incident, reflected, and refracted beams are proportional respectively to
,
where c1, c2, denote the velocities of light, and ρ1,ρ2 the densities of aether, in the two media, and the cross-sections of the beams which meet the interface in unit area are
cos i, cos i, cor r
respectively. The principle of conservation of energy therefore gives
.
The equation of continuity of displacement at the interface is
f + g = h.
