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,

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

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

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

is intersected by the plane

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

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,

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