Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/166

the media on either side the interface. This implies in the present case that the quantities

$n{\frac {\partial e_{z}}{\partial x}}$ and $n{\frac {\partial e_{z}}{\partial y}}$

are to be continuous across the interface: so we have

$\textstyle n\cos i^{\prime }.(f^{\prime }-F^{\prime })=n^{\prime }\cos r.f_{1}^{\prime }$;     $n\sin i.(f^{\prime }+F^{\prime })=n^{\prime }\sin r.f_{1}^{\prime }$.

Eliminating f′₁, we have

${\frac {F^{\prime }}{f^{\prime }}}={\frac {\sin(r-i)}{\sin(r+i)}}$

Now this is Fresnel's sine-law for the ratio of the intensity of the reflected ray to that of the incident ray; and it is known that the light to which it applies is that which is polarized parallel to the plane of incidence. Thus Cauchy was driven to the conclusion that, in order to satisfy the known facts of reflexion and refraction, the vibrations of the aether must be supposed executed at right angles the plane of polarization of the light.

The case of a vibration performed in the plane of incidence he discussed in the same way. It was found that Fresnel's tangent-law could be obtained by assuming that e_x and the normal pressure across the interface have equal values in the two contiguous media.

The theory thus advanced was encumbered with many difficulties. In the first place, the identification of the plane of polarization with the plane at right angles to the direction of vibration was contrary to the only theory of crystal-optics which Cauchy had as yet published. In the second place, no reasons were given for the choice of the conditions at the interface. Cauchy's motive in selecting these particular conditions was evidently to secure the fulfilment of Fresnel's sine-law and tangent-law; but the results are inconsistent with the true boundary-conditions, which were given later by Green.

It is probable that the results of the theory of reflexion had much to do with the decision, which Cauchy now made,^[1] to

1. Comptes Rendus, ii. (1836), p. 341.