126 The Luminiferous Medium, 



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 v l9 which may be represented by the 

 radius of Huygens' sphere say, 



Vi = &; 



while the other, the " extraordinary ray," is polarized in a plane 

 .at right angles to the principal section, and has a wave- velocity v 9 , 

 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 



if + z'" x* 



+ ^ = 1 - 



and if (I, m, n) denote the direction-cosines of the normal to the 

 plane of the wave, we have therefore 



v,~ = a*(m* + n*) + ?> 2 / 2 . 



But the quantities 1/Vi and l/t? 8 , as given by these equations, 

 are easily seen to be the lengths of the semi-axes of the ellipse 

 in which the spheroid 



6 2 (?/ 3 4- z~) + arx- = 1 



is intersected by the plane 



Ix + my + nz = ; 



.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 

 <?ase of biaxal crystals could be covered by substituting for the 

 latter spheroid an ellipsoid with three unequal axes say, 



x z if z* 

 _++_ = 



If I/Vi and l/^ denote the lengths of the semi-axes of the 

 .ellipse in which this ellipsoid is intersected by the plane 



Ix 4 my + nz - 0, 



