LENSES. 171 



each ray ; or each curved surface may be regarded as 

 consisting of a number of minute plane surfaces 

 placed at right angles to the perpendicular. Thus, in 

 PI. XII. fig. 5, the ray a, incident at the point b of 

 the curved surface, is refracted towards the perpen- 

 dicular p, as if it had fallen upon the plane surface 

 represented by the tangent t. The forms of the most 

 common lenses are represented in PL XII. figs. 6-10; 

 fig. 6 being doubly convex, or both surfaces being 

 convex; fig. 7, plano-convex, or one surface plane, 

 the other convex; fig. 8, doubly concave, or both 

 surfaces being concave ; fig. 9, plano-concave, one sur- 

 face being plane, the other concave ; and fig. 10 is a 

 meniscus, in which one surface is convex and the other 

 concave. The curved surfaces of lenses are usually 

 portions of spheres. 



The manner in which the course of a ray may be 

 traced through a lens is illustrated by PL I. fig. 11, 

 which requires no explanation after what has been 

 already stated. 



To facilitate the comprehension of the general ac- 

 tion of lenses, they may be regarded as composed of 

 two triangular prisms, with their bases in contact in 

 a convex lens, as in fig. 12 ; their apices being op- 

 posed in a concave lens, as in fig. 13. 



The point to which the rays converge after passing 

 through a convex lens is called the focus (PL XII. 

 fig. 14/), the distance of which from the centre of the 

 lens, called the focal length, obviously depends upon 

 the direction of the incident rays. When these are 

 parallel, which those coming from distant objects 

 may be considered to be, the focus at which they 

 meet is called the principal focus, or the focus for 

 parallel rays : thus, in PL XII. fig. 14, the parallel 

 rays meet at/, which is the principal focus. 



If the incident rays are convergent, as in PL XII. 

 fig. 15, the focus o will be situated nearer the lens 

 than the principal focus,/. If, on the other hand, 



