FOCAL LINES OF A EEFRACTED PENCIL. 



335 



and for C infinite the tangent of the angle must be 



cos ! cos 



,(15). 



COS #jCOt 0j COS #2 COt ft, ' ' 



If the plane of refraction cuts the surface along a line of curvature, 

 C= so , and if one of the focal lines of the incident pencil is in the plane of 

 refraction, C l = oo . The points A t , B l then coincide with the focal lines of the 

 incident pencil, and A*, B 3 with those of the refracted pencil. 



That A! may coincide with B v and A 3 with B lf the line joining both pairs 

 of points must be on the line AB. There is therefore one, and only one, point 

 on the axis of the incident pencil from which a pencil may diverge so that, 

 after refraction, it still diverges from or converges to a single point. 



4. When the quantity C has a finite value, the plane of refraction is not 

 a plane of symmetry, and we have to deduce the quantities a, 6, < from 

 A, B, C. The following construction enables us to pass from either of these 

 systems to the other : 



Let OA, OB, OC be the values of A, B, C. 



Draw A A' perpendicular and equal to AO. Join BA', and produce to D, 

 a point on the perpendicular to OA through 0. Cut off OD' equal to OD, 





