Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/146

crystal can be resolved into two plane-polarized components; one of these, the "ordinary ray," is polarized in the principal section, and has a velocity v₁, which may be represented by the radius of Huygens' sphere—say,

$v_{1}=b$;

while the other, the "extraordinary ray," is polarized in a plane at right angles to the principal section, and has a wave-velocity v₂, which may be represented by the perpendicular drawn from the centre of Huygens' spheroid on the tangent-plane parallel to the plane of the wave. If the spheroid be represented by the equation

${\frac {y^{2}+z^{2}}{\alpha ^{2}}}+{\frac {x^{2}}{b^{2}}}=1$,

and if (l, m, n) denote the direction-cosines of the normal to the plane of the wave, we have therefore

$v_{2}^{2}=\alpha ^{2}(m^{2}+n^{2})+b^{2}l^{2}$

But the quantities 1/v₁ and 1/v₂, as given by these equations, are easily seen to be the lengths of the semi-axes of the ellipse in which the spheroid

$b^{2}(y^{2}+z^{2})+\alpha ^{2}x^{2}=1$

is intersected by the plane

$lx+my+nz=0$

and thus the construction in terms of Huygens' sphere and spheroid can be replaced by one which depends only on a single surface, namely the spheroid

$b^{2}(y^{2}+z^{2})+\alpha ^{2}x^{2}=1$

Having achieved this reduction, Fresnel guessed that the case of biaxal crystals could be covered by substituting for the latter spheroid an ellipsoid with three unequal axes—say,

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

If 1/v₁ and 1/v₂ denote the lengths of the semi-axes of the ellipse in which this ellipsoid is intersected by the plane

$lx+my+nz=0$