Crystalline Reflexion and Refraction. 163 



which I have called the surface of indices, or the index-surface* 

 because its radius ON, being the reciprocal of OS, represents 

 the index of refraction, or the ratio of the sine of the angle of 

 incidence to the sine of the angle of refraction, when the wave 

 ROy', to which OM is perpendicular, is supposed to have passed 

 into the crystal out of an ordinary medium in which the velocity 

 of propagation is unity. The angles of incidence and refrac- 

 tion are understood to be the angles which the incident and 

 refracted waves respectively make with the refracting surface 

 of the crystal. 



The wave-surface and the index-surface have the same geo- 

 metrical properties since they are both biaxal surfaces. Let us 

 consider the former, which is generated by the ellipsoid whose 

 semiaxes are a, b, c ; and let us conceive this ellipsoid to be in- 

 tersected by a concentric sphere of which the radius is r. Then 

 the equations of the ellipsoid and the sphere being respectively 



x z + y z + z z 

 L ' ~~ 



we get, by subtracting the one from the other, 



for the equation of the cone A which has its vertex at 0, and 

 passes through the curve of intersection. If OQ be equal to r, 

 it will be a side of this cone ; and a plane touching the cone 

 along OQ will make in the ellipsoid a section of which OQ will 

 be a semiaxis ; so that OT will be perpendicular to that plane, 

 and equal in length to r. Therefore, as OQ describes the cone 

 A, the right line OT describes another cone B reciprocal to A, 

 and the point T describes the curve in which the wave-surface is 

 intersected by the sphere above mentioned ; this curve being a 

 spherical ellipse, reciprocal to that which the point Q describes on 



* See Transactions of the Royal Irish Academy, VOL. xvui. p. 38 (supra, p. 96). 

 I had previously called it the surface of refraction, YOL. xvu. p. 252 (supra, p. 36). 



