130 The Lumimferous Medium, 



and by the relation previously found namely, 

 / 2 m 2 n~ 



1 



By the usual procedure for determining envelopes, it may be 

 shown that the locus in question is the surface of the fourth 

 degree 



x z _ _f _fl_ _ n 



which is called Fresnel's wave-surface* It is a two-sheeted surface, 

 as must evidently be the case from physical considerations. In 

 uniaxal crystals, for which * 2 and c 3 are equal, it degenerates into 

 the sphere 



r 2 = l/ e> , 

 and the spheroid 



^ + fl (tf + 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 substi- 

 tuted 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 



*i x ~ + 6 2^" ~*~ *3~~ = * 



be intersected hy 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. 



