where ƒ is an arbitrary function.
The question as to the law of the secondary waves is thus answered by Stokes. “Let ξ = 0, η = 0, ζ = ƒ(bt − x) be the displacements corresponding to the incident light; let O1 be any point in the plane P (of the wave-front), dS an element of that plane adjacent to O1, and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let O be any point in the medium situated at a distance from the point O1 which is large in comparison with the length of a wave; let O1O = r, and let this line make an angle θ with the direction of propagation of the incident light, or the axis of x, and φ with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to O1O, and lying in the plane ZO1O; and, if ζ′ be the displacement at O, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,
| ζ′ = | dS | (1 + cos θ) sin φ ƒ′(bt − r). |
| 4πr |
In particular, if
| ƒ(bt − x) = c sin | 2π | (bt − x) (5), |
| λ |
we shall have
| ζ′ = | cdS | (1 + cos θ) sin φcos | 2π | (bt − r) (6).” |
| 2λr | λ |
It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.
The occurrence of sin φ as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction θ depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin φ = 1; but, if the primary vibration be in the plane of diffraction, sin φ = cos θ. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations of polarized light are executed in a direction perpendicular to the plane of polarization.
The factor (1 + cos θ) shows in what manner the secondary disturbance depends upon the direction in which it is propagated with respect to the front of the primary wave.