Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/172

Fresnel, that the vibrations are executed at right angles to the plane of polarization, and that the optical differences between media are due to the different densities of aether within them.

It now remained for Green to discuss the case in which the incident light is polarized at right angles to the plane of incidence, so that the motion of the acthereal particles is parallel to the intersection of the plane of incidence with the front of the In this case it is impossible to satisfy all the six boundary-conditions without assuming that longitudinal vibrations are generated by the act of reflexion. Taking the plane of incidence to be the plane of yz, and the interface to be the plane of xy, the incident wave may be represented by the equations

$e_{y}=A{\frac {\partial }{\partial z}}f(t+lz+my)$; $e_{z}=-A{\frac {\partial }{\partial y}}f(t+lz+my)$;

where, if i deote the angle of incidence, we have

$l={\sqrt {\frac {\rho _{1}}{n}}}\cos i$, $m=-{\sqrt {\frac {\rho _{1}}{n}}}\sin i$.

There will be a transverse reflected wave,

$e_{y}=B{\frac {\partial }{\partial z}}f(t-lz+my)$; $e_{z}=-B{\frac {\partial }{\partial y}}f(t-lz+my)$;

and a transverse refracted wave,

$e_{y}=C{\frac {\partial }{\partial z}}f(t+l_{1}z+my)$; $e_{z}=-C{\frac {\partial }{\partial y}}f(t+l_{1}z+my)$,

where, since the velocity of transverse waves in the second medium is $\textstyle {\sqrt {n/\rho _{2}}}$, we can determine l₁ from the equation

$l_{1}^{2}+m^{2}={\frac {\rho _{2}}{n}}$;

there will also be a longitudinal reflected wave,

$e_{y}=D{\frac {\partial }{\partial y}}f(t-\lambda z+my)$; $e_{z}=D{\frac {\partial }{\partial z}}f(t-\lambda z+my)$,