48 Geometrical Propositions applied 



equation of the surface in its second position, when the centre 

 has moved through a distance equal to / along the axis of s, will 

 be U f + V = ; and these two equations combined will give 

 U' = 0, V' = 0, for the equations of one of the ring-edges. The 

 equations of the other ring-edge are deduced from these by 

 changing the sign of I. 



The projection of each of the ring-edges on the plane xy is 

 the curve traced by the point R on the surface of the crystal 

 (50). This curve may be called a ring-trace. Its equation is 

 obtained by eliminating z between the equations of a ring-edge ; 

 and as the result must be the same whether / be taken posi- 

 tive or negative, the equation of the ring-trace, when found by 

 this general method, will contain only even powers of /. The 

 radii drawn from to the points R of the ring-trace are (50) 

 the sines (to the radius 08) of the angles of incidence or emer- 

 gence of the rays that form an optical ring, the rays that come 

 from this ring to the eye being parallel to the sides of the cone 

 described by the right line S'OS, while the point R describes the 

 ring-trace. 



53. It is evident that tangents to the ring-edges, at the 

 points P and M, are parallel to each other, and therefore parallel 

 to the intersection of two planes touching the surface of refrac- 

 tion at P and M, because these tangent planes pass through the 

 tangents. But the directions OP', OM f , are perpendicular to 

 the tangent planes, and therefore the plane P'OJLT, containing the 

 two rays, is perpendicular to the intersection of the tangent planes, 

 and of course perpendicular to the parallel tangents. Hence 

 the plane P^OM' intersects the face of the crystal in a right line 

 perpendicular to the projection of the parallel tangents on the 

 face of the crystal. As this projection is a tangent to the curve 

 described by jR, it follows that the normal to the ring-trace at 

 the point R is parallel to the line joining the points in widen 

 the two refracted rays cut the second surface of the crystal. 



In like manner, taking any two consecutive rays (P and m), 

 having a common extremity on one surface of the crystal, the 

 line joining the points where these rays cut the other surface is 



