at a Cylindrical Surface. 55 



All the light which falls on the cylinder will therefore 

 virtually pass through the area in the horizontal plane 

 hetween arcs of two circles of radius cV and d^' having the 

 radiant-point as centre. The width of the focal area on the 

 axis of x = d" — d! and this width increases as the vertical 

 aperture h increases, and becomes infinite when It is infinite. 



The focal lines or areas we have discussed are produced by 

 the intersections of symmetrical rays. Besides these the 

 refracted rays produce two caustics which are the loci of 

 intersections of consecutive rays. The first is the locus of 

 the intersections of consecutive refracted rays in the plane 

 containing the radiant-point and the axis of the cylinder, and 

 is the same as that which would be produced by rays in one 

 plane refracted at a plane surface dividing any two media of 

 different refractive indices. The second is the locus of the 

 intersections of consecutive refracted rays in the plane con- 

 taining the radiant-point and normal to the axis of the 

 cylinder, and is the caustic of the circle. Incident rays in 

 oblique sections of the cylinder do not produce caustics since 

 the corresponding refracted rays are not in the same plane. 



If we suppose light from a radiant-point to fall on a 

 cylinder the radius of which gradually increases to infinity, 

 it will be seen that ultimately, when the curvature is zero, 

 the two focal areas will coincide and be reduced to a short 

 piece of the line through the radiant-point and normal to the 

 surface. Similarly, if we suppose the cylinder to become a 

 semi-ellipsoid which gradually becomes a hemisphere, then 

 the two focal areas will ultimately coincide and become a 

 portion of the line joining the radiant-point and the centre of 

 the hemisphere. 



Instead of having caustics in two planes only we now have 

 a caustic in every plane passing through the radiant-point 

 and the centre of the hemisphere. 



If light proceeding from or to a point a fall on the plane 

 or spherical surface of a piano- or sphero-cylindrical lens of 

 small aperture it will pass on to the cylindrical surface as if 

 it proceeded from a point a\ a and a' being conjugate with 

 respect to the first surface. The focal areas produced by a 

 piano- or sphero-cylindrical lens are therefore, for small 

 apertures, identical with those produced by refraction at a 

 single cylindrical surface ; and if we define two optical 

 systems as "equivalent''^ when they produce identical focal 

 areas, then we can say that a piano- or sphero-cylindrical lens 

 with the radiant-point at a is equivalent to a single cylin- 

 drical surface with the radiant-point at a ; a and a being 

 conjugate with respect to the first surface. 



