FRESNEL'S THEORY OF DOUBLE REFRACTION. 203 



The equation of the wave surface is of the fourth order ; 

 it has been thrown into the following symmetric form by Sir 

 William Hamilton, 





(215) The form of the wave-surface being known, the 

 directions of the two refracted rays . are determined by tan- 

 gent planes drawn to the two sheets of the surface, ac- 

 cording to the construction of Huygens. Conceive three 

 surfaces, having their common centre at the point of incidence, 

 and representing, respectively, the simultaneous positions of 

 three waves diverging from that point, the first in air, 

 which is a sphere ; and the other two within the crystal, 

 which are the two sheets of the surface we have been con- 

 sidering. Let the incident ray be produced to meet the 

 sphere, and at the point of intersection let a tangent plane 

 be drawn. Through the line of intersection of this plane 

 with the refracting surface, let two planes be drawn touch- 

 ing the two sheets of the refracted wave ; the lines connect- 

 ing the centre with the points of contact are the directions 

 of the two refracted rays.* 



* If, in place of the ellipsoid mentioned above, we take that whose semi- 



axes are,., the three principal refractive indices of the medium, the 

 a b c 



surface derived from it by the same construction will represent the normal 

 slowness of the waves, and is connected with the wave-surface by a remark- 

 able relation of reciprocity. The properties of this surface lead to the fol- 

 lowing elegant construction for the directions of the refracted rays, a con- 

 struction which is, in many cases, more convenient than that given above : 

 " With the point of incidence, as a common centre, construct tbe surfaces 

 of wave-slowness belonging to air and to the crystal, respectively. Let the 

 incident ray be produced to meet the sphere, which represents the normal 

 slowness of the wave in air ; and from the point of intersection let a per- 

 pendicular be drawn to the refracting surface. This will cut the surface of 



