Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/154

Eliminating h between these two equations, and using the formulae

${\frac {\sin ^{2}r}{\sin ^{2}i}}={\frac {c_{2}^{2}}{c_{1}^{2}}}={\frac {\rho _{1}}{\rho _{2}}}$,

we obtain the equation

${\frac {f}{g}}=-{\frac {\sin(i-r)}{\sin(i+r)}}$.

Thus when the light is polarized in the plane of reflexion, the amplitude of the reflected wave is

${\frac {\sin(i-r)}{\sin(i+r)}}\times$ the amplitude of the incident vibration.

Fresnel shows in a similar way that when the light is polarized at right angles to the plane of reflexion, the ratio of the amplitudes of the reflected and incident waves is

${\frac {\tan(i-r)}{\tan(i+r)}}$

These formulæ are generally known as Fresnel's sine-law and Fresnel's tangent-law respectively. They had, however, been discovered experimentally by Brewster some years previously. When the incidence is perpendicular, so that i and r are very small, the ratio of the amplitudes becomes

$\mathrm {Limit} {\frac {f}{g}}=-{\frac {i-r}{i+r}}$,

or ${\frac {\mu _{2}-\mu _{1}}{\mu _{2}+\mu _{1}}}$, where μ₂ and μ₁ denote the refractive indices of the media. 'This formula had been given previously by Young^[1] and Poisson,^[2] on the supposition that the elasticity of the aether is of the same kind as that of air in sound.

When i + r = 90°, tan (i + r) becomes infinite: and thus a theoretical explanation is obtained for Brewster's law, that if the incidence is such as to make the reflected and refracted rays

1. Article Chromatics, Encycl. Britt. Suppl.
2. Mem, Inst. ii. (1817).