120 PHYSICS. 
bi-convex, a; plano-convex, ); concavo-convex, ur meniscus, in which: 
either the convexity is of least radius of curvature, c, or the concavity is of 
least radius, f; bi-concave, d; and plano-concave, e.. In general, all lenses 
that are thicker at the middle than at the edges, are called. convex -or 
collecting lenses; and those which exhibit the greatest thickness at the 
borders are concave or separating lenses: a, b, and c belong to the former, 
d, e, and f to the latter. 
The axis of a lens is that straight line which connects the two centres of 
the sphere, portions of which form the surface of the lens. Lenses are. 
theoretically referable to the prism for their principle. In fig. 83, let abed 
be an elongated rhomb, upon which are placed, above and below, equal 
parallel trapezia. Upon the trapezium abfg,a triangle. fgh, is superimposed, 
a similar one being placed on the Jower trapezium. The two sides not 
parallel of the trapezium might, when produced, form an isosceles triangle, 
whose angle at the vertex is half the size of the angle ghf. If the figure 
thus produced be rotated about the axis MN, a lens-shaped body will be 
produced, which consists of several zones, and whose centre forms a plane 
disk. Ifa ray of light impinge upon this body, passing from a point of the 
axis MN, the deviation produced may be determined according to the laws 
of refraction in prisms. If the point S be so situated that a ray emitted 
from it and striking the surface ag in 2, shall experience the least possible 
deviation in its passage through abfg, then it will cut the axis in a point, R, 
equally distant with S from the lens. A ray of light passing from 8, and 
experiencing the minimum of deviation in passing through the triangle fgh, 
will, if the refracting angle of the upper prism be half that of the lower, be 
diverted twice as much as in abfg from its original direction. Hence it 
follows that the lower ray, Si, forms half as great an angle with the axis MN 
as the upper one; both rays, however, are refracted to R. If we suppose 
the broken lines dbfh and cagh to be replaced by curves whose centres lie 
in the axis MN, we shall obtain an actual (bi-convex) lens. We may 
therefore assume that there is a point, 8, of the axis, all the rays coming 
from which and meeting the lens, are concentrated in one and the same 
point, R, situated at the same distance as 8 from the lens. The curvature 
of the lens from the centre to the circumference must, however, be very 
slight (as will be assumed in what follows), else the above condition would 
be impossible. 
If a bi-convex lens be met by a number of rays parallel to the axis, or 
which come from an infinite distance in this direction, they will all be 
refracted to a point in the axis called the focus. The distance from the 
focus to the lens is the focal length (pl. 21, fig. 34). The focus is always 
half the distance of the points S and R from the lens. If the luminous point 
lie at a finite distance from the lens, on the axis, there is equally a point of 
union of the rays; this, however, is further from the Jens than the focus of 
parallel rays, and will be further as the radiant point approaches nearer. It 
will be at an infinite distance when the radiant is in the focus of parallel 
rays. If the luminous point lie within the focal distance ( jig. 88), the rays 
falling on the Jens will not unite, but will diverge even after emerging 
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