Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/198

To explain their method, we shall suppose the incident light to be polarized in the plane of incidence. According to Fresnel's sinc-law, the amplitude of the light (polarized in this way) reflected from a transparent body is to the amplitude of the incident light in the ratio

$J={\frac {\sin(i-r)}{\sin(i+r)}}$,

where i denotes the angle of incidence and r is determined from the equation

$\sin i=\mu \sin r$.

MacCullagh and Cauchy assumed that these equations hold good also for reflexion at a metallic surface, provided the refractive index μ is replaced by a complex quantity

$\mu =\nu (1-\kappa {\sqrt {-1}})$     say,

where ν and κ are to be regarded as two constants characteristic of the metal. We have therefore

$J={\frac {\tan i-\tan r}{\tan i+\tan r}}={\frac {(\mu ^{2}-\sin ^{2}i)^{\frac {1}{2}}-\cos i}{(\mu ^{2}-\sin ^{2}i)^{\frac {1}{2}}+\cos i}}$

If then we write

$\nu ^{2}(1-\kappa {\sqrt {-1}})^{2}-\sin ^{2}i=U^{2}e^{2\nu {\sqrt {-1}}}$,

so that equations defining U and ν are obtained by equating separately the real and the imaginary parts of this equation, we have

$J={\frac {Ue^{\nu {\sqrt {-1}}}-\cos i}{Ue^{\nu {\sqrt {-1}}}+\cos i}}$

and this may be written in the form

${\bar {J}}e^{\delta {\sqrt {-1}}}$

where

${\begin{cases}{\bar {J}}^{2}&=&{\frac {U^{2}+\cos ^{2}i-2U\cos \nu \ \cos i}{U^{2}+\cos ^{2}i+2U\cos \nu \ \cos i}}\\\tan \delta &=&{\frac {2U\cos i\ \sin \nu }{U^{2}-\cos ^{2}i}}.\end{cases}}$