Focal Lines of Cylindrical Lenses. 79 



line. At the tangential focus all the tangential focal lines 

 due to one of the object bands will together form a horizontal 

 image band the width of which parallel to the axis of the 



lindrical surface of the lens will be — -, and the distance 



-J3 



cv 1 



between the centres of the image bands will be — -, where 



v z is the distance of the image and — u the distance of the 

 object from the lens. 



Consider now what will happen at the axial focus. Every 

 point in one of the horizontal object bands will produce an 



2h v 



axial focal line of which the length = * 2 , where v is the 



h 

 distance of the image from the lens h x the axial semi- 

 aperture, and/i the focal length of the piano-cylindrical part 

 of the lens. The horizontal image bands will have a width 

 parallel to the axis of the cylindrical surface of the lens 



the first term being the width which the image band would 

 have if the lens were spherical. The distance between the 



OLVc\ 



centres of the image bands will be - . Fig. 17 (p. 80) is a 



similar view to fig. 14 but showing the limits of the focal lines 

 due to five narrow object bands, the distance apart being =a 

 and the width iv being small enough to be neglected. The 

 lines A, B, C, D are drawn at distances from the lens by 

 putting a=l, 2, 3, or 4 in the formula 



_ qfxfs 



°9 



«/i - 2/sV 



and the lines I, H, G, F at distances obtained by making 

 a = l, 2, 3, or 4 in the formula 



«A+2/A 



If the object be placed at such a distance from the lens 

 that the image is formed at the line A or I, the edges of 

 adjacent image bands will coincide and there will appear to 

 be no image at all ; a screen placed at A or I will be very 

 nearly uniformly illuminated. If the images are formed 

 nearer to the lens than I or further away than A, five 

 separate and distinct image bands will be formed. Again, if 



