Focal Lines of Cylindrical Lenses. 11 



(b) The tangential focal line. 



Fig. 15 shows a plan view of the system. A pair or! rays 

 symmetrical with respect to the plane of fig. 15 diverging 

 from a point a x at a distance a l u = a 1 from the axis ou and at 



F I G. J6 



a distance ou— —u from the lens, and incident at a point h 2 

 at a distance oh 2 = h 2 from the axis ou will, after refraction 

 by both surfaces, intersect in the plane of the paper fig. 15, 

 at a point b the distance of which from the lens =ov s = v s . 

 A similar pair of rays incident at the corresponding point 

 — h 2 will intersect at c and be will be the tangential focal 

 line. Let the length bc = 2l 2 . 

 Now 



1 1___1 



u v 2 fz 



_1 \_ 1 



u v 3 f 2 



Subtracting one equation from the other we get 



1_ _ 1_ __ 1 _ 1 . v s -v 2 _ i _ i = I 



v 2 v 3 ~f 3 f 2 ' ' v 2 v 3 ~/ 3 f 2 fx 



From flo-. 15 we have 



2/, 



«3-«2 



2/l< 



I, 



h f 



= -5r or 



/ _ "ah 



2 " A ' 



that is the length of the tangential focal line is the same as 

 when the source of light is on the axis ou. 



The equations to the two lines which form the upper and 

 lower limits of the tangential focal line when its distance 

 from the lens varies are 



