By applying his construction to the reflection and refraction of light, Huygens accounted for these phenomena in isotropic bodies as well as in Iceland spar. It was afterwards shown by Augustin Fresnel that the double refraction in biaxal crystals can be explained in the same way, provided the proper form be assigned to the wave-surface.
In any point of a bounding surface the normals to the reflected and refracted waves, whatever be their number, always lie in the plane passing through the normal to the incident waves and that to the surface itself. Moreover, if α1 is the angle between these two latter normals, and α2 the angle between the normal to the boundary and that to any one of the reflected and refracted waves, and v1, v2 the corresponding wave-velocities, the relation
sin α1/sin α2 = v1/v2
(7)
is found to hold in all cases. These important theorems may be proved independently of Huygens’s construction by simply observing that, at each point of the surface of separation, there must be a certain connexion between the disturbances existing in the incident, the reflected, and the refracted waves, and that, therefore, the lines of intersection of the surface with the positions of an incident wave-front, succeeding each other at equal intervals of time dt, must coincide with the lines in which the surface is intersected by a similar series of reflected or refracted wave-fronts.
In the case of isotropic media, the ratio (7) is constant, so that we are led to the law of Snellius, the index of refraction being given by
µ = v1/v2 (8)
(8)
(cf. equation 1).
9. General Theorems on Rays, deduced from Huygens’s Construction.—(a) Let A and B be two points arbitrarily chosen in a system of transparent bodies, ds an element of a line drawn from A to B, u the velocity of a ray of light coinciding with ds. Then the integral ∫u−1 ds, which represents the time required for a motion along the line with the velocity u, is a minimum for the course actually taken by a ray of light (unless A and B be too far apart). This is the “principle of least time” first formulated by Pierre de Fermat for the case of two isotropic substances. It shows that the course of a ray of light can always be inverted.