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130

The Luminiferous Medium,

and by the relation previously found—namely,

${\frac {l^{2}}{{\frac {1}{\epsilon _{1}}}-v^{2}}}+{\frac {m^{2}}{{\frac {1}{\epsilon _{2}}}-v^{2}}}+{\frac {n^{2}}{{\frac {1}{\epsilon _{3}}}-v^{2}}}=0$ .

By the usual procedure for determining envelopes, it may be shown that the locus in question is the surface of the fourth degree

${\frac {x^{2}}{\epsilon _{1}r^{2}-1}}+{\frac {y^{2}}{\epsilon _{2}r^{2}-1}}+{\frac {z^{2}}{\epsilon _{3}r^{2}-1}}=0$ ,

which is called Fresnel's wave-surface.^[1] It is a two-sheeted surface, as must evidently be the case from physical considerations. In uniaxal crystals, for which ε₂ and ε₃ are equal, it degenerates into the sphere

$r^{2}=1/\epsilon _{2}$ ,

and the spheroid

$\epsilon _{2}x^{2}+\epsilon (y^{2}+z^{2})=1$ .

It is to these two surfaces that tangent-planes are drawn in the construction given by Huygens for the ordinary and extraordinary refracted rays in Iceland spar. As Fresnel observed, exactly the same construction applies to biaxal crystals, when the two sheets of the wave-surface are substituted for Huygens' sphere and spheroid.

"The theory which I have adopted," says Fresnel at the end of this memorable paper, "and the simple constructions which I have deduced from it, have this remarkable character, that all the unknown quantities are determined together by the solution of the problem. We find at the same time the velocities of the ordinary ray and of the extraordinary ray, and their planes of polarization. Physicists who have studied attentively the laws of nature will feel that such simplicity and

↑ Another construction for the wave-surface is the following, which is due to MacCullagh, Coll. Works, p. 1. Let the ellipsoid

$\epsilon _{1}x^{2}+\epsilon _{2}y^{2}+\epsilon _{3}z^{2}=1$

be intersected by a plane through its centre, and on the perpendicular to that plane take lengths equal to the semi-axes of the section. The locus of these extremities is the wave-surface.

[1] Another construction for the wave-surface is the following, which is due to MacCullagh, Coll. Works, p. 1. Let the ellipsoid

$\epsilon _{1}x^{2}+\epsilon _{2}y^{2}+\epsilon _{3}z^{2}=1$

be intersected by a plane through its centre, and on the perpendicular to that plane take lengths equal to the semi-axes of the section. The locus of these extremities is the wave-surface.

[1]