ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 



1063 



towards the axis, and is at one part of its path parallel to the axis. 

 Consider next a spherical wave m n (fig. 235) proceeding from x ; after 

 entering the cylinder, as at wjj n,, it will be gradually altered in shape, 



as shown by m 2 n i} m 3 n 3 , m 4 n 4 , successively (the velocity being least 

 along the axis), and will emerge as a concave wave at m 5 n b , so that 

 rays diverging from x will converge to y. The figure also indicates 

 that if the cylinder be cut through at m 3 n 3 , where the wave-front is 

 plane, the beam will emerge parallel to the axis ; in other words, 

 x is the focus of the cylinder a c m 3 n 3 , and y is the focus of the cylinder 

 b d n 3 m 3 . The form of the curve m 5 n 5 will depend upon the law by 

 which the index varies, but in any case it will be a surface of revolu- 

 tion about the axis, and consequently the portion in close proximity 

 to the axis may be replaced by its sphere of curvature ; hence, if 

 central pencils only be taken into account, it is clear that an image 

 of x will be produced at y. 



It may be proved that the focal length is inversely proportional 

 to the length of the cylinder ; this, however, is only true within 

 certain limits, since with a long cylinder the course of a beam of 

 parallel rays a b may be periodically re-entrant, as shown in fig. 236, 



Fig. 236. 



and there will be a succession of foci within the cylinder ; the latter 

 should therefore be shorter than the distance between two con- 

 secutive foci. 



It may be shown also that the ordinary lens formula — ( — = 



u v f 

 is equally true for the cylinder, u and v being the distances of the 

 object and image respectively, and / the focal length. Such a 



