Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/165

To the latter problem Cauchy soon addressed himself, his investigations being in fact published^[1] in the same year (1830) as the first of his theories of crystal-optics.

At the outset of any work on refraction, it is necessary to assign a cause for the existence of refractive indices, i.e. for the variation in the velocity of light from one body to another. Huygens, as we have seen, suggested that transparent bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity. Cauchy in his earlier papers^[2] followed this lead more or less closely, assuming that the density ρ of the aether is the same in all media, but that its rigidity n varies from one medium to another.

Let the axis of x be taken at right angles to the surface of separation of the media, and the axis of z parallel to the intersection of this interface with the incident wave-front; and suppose, first, that the incident vibration is executed at right angles to the plane of incidence, so that it may be represented by

$e_{z}=f\left(-x\cos \ i-y\sin \ i+{\sqrt {\frac {n}{\rho }}}t\right)$,

where i denotes the angle of incidence; the reflected wave may be represented by

$e_{z}=F\left(x\cos \ i-y\sin \ i+{\sqrt {\frac {n}{\rho }}}t\right)$,

and the refracted wave by

$e_{z}=f_{1}\left(-x\cos \ r-y\sin \ r+{\sqrt {\frac {n^{\prime }}{\rho }}}t\right)$,

where r denotes the angle of refraction, and n′ the rigidity of the second medium.

To obtain the conditions satisfied at the reflecting surface, Cauchy assumed (without assigning reasons) that the x- and y-components of the stress across the xy-plane are equal in

1. Bull. des Sciences Math, xiv. (1830), p. 6.
2. As will appear, his views on this subject subsequently changed.

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