8 CONICAL REFEACTION. 



and it can be easily shown that the tangent of the angle which the 

 optic axis makes with the normal to the circle, or the cusp-ray, is 



Now, this is about half the former, since b* = ac, nearly; and 



consequently the optic axis nearly bisects the angle contained by 



the extreme normals in the plane of xz. Hence if A and B be 



the intersections of the two optic axes with the sphere whose 



centre is at the cusp, and 



N the intersection of one 



of the normals at that point 



with the same (fig. 2), the 



angle NA C ranges through 



every magnitude between 



and 360, the arc NA being 



all the time very small. Let the angle NAC be denoted by a, 



and NPC by w, NP being the arch bisecting the angle N\ then 



in the triangle APN, we have 





or, since AN is very small, and therefore cos^jY = 1, nearly, 

 cos o> = cos (a - i-ZV), and ta = a- ^N, nearly. 



But, when any side of a spherical triangle is very small in com- 

 parison with the other two, the adjacent angles are together equal 

 to 180 q.p. Consequently, 



N = a, and w = Ja, nearly. 



From this it appears that the angle, which the plane of polari- 

 zation of any ray makes with the plane of the optic axes, is half 

 the angle which the plane passing through the normal and the 

 , near axis makes with the same plane. But this latter angle, it 

 may be easily shown, is very nearly the same as that which the 

 plane passing through the emergent ray and the axis of the cone 

 makes with the plane of the optic axes. Consequently the angle, 

 which the plane of polarization of any ray of the emergent cone 



