OPTICS. 121 
from the lens; less, however, than after refraction from the first 
surface. 
-In.a bi-convex lens whose two surfaces are of equal radius of curvature, 
the focal length is equal to the radius. Plano-convex and convex meniscus 
lenses have likewise foci: in a plano-convex lens of glass (when the index 
of refraction for air and glass is assumed to be 8) the focal length will be 
twice as great as the radius of the curved surfaces. 
Concave lenses have no true focus, but rather a focal point of divergence. 
If the rays incident on such a glass are parallel to the axis, they diverge 
alter emergence as if they came from one and the same point called the 
negative focus. If the incident rays be divergent, as if coming from a point 
on the axis at a greater or less distance from the lens, they will be made 
still more divergent; and the focal point of divergence will be nearer the 
glass the nearer the luminous point. If the incident rays be convergent 
( pl. 21, fig. 84), all these cases will be possible. If they converge towards 
the focal point of divergence, they will emerge parallel on the other side; 
if they converge still more than this, they emerge convergent. If they 
converge less, they diverge after emergence, as if they came from a point 
betore the glass. 
The preceding observations apply in general to rays coming from a point 
elsewhere than in the main axis of the lens, provided the line drawn from 
this point through the centre of the lens (the secondary axis) forms but a 
small angle with the principal axis. All rays proceeding from this point 
and incident upon the (convex) lens, are united in a point of the secondary 
axis, which is at the same distance from the lens as if the luminous point 
were situated in the principal axis. 
We shall now be able to examine the formation of images of objects by 
lenses. In fig. 37, let AB be an object placed before the convex lens VW, 
and at a greater distance from it than the focus F. In this case, an actual 
but inverted image, ab, will be formed, which will be of the same size as the 
object, or greater, or less, as the distance of the object from the lens is equal 
to, greater, or less than twice the focal distance. In fact, image and object 
are always to each other in the ratio of their respective distances from the 
lens. If the object lie within the focus of the lens (fig. 38), no actual or 
convergent image will be formed, but an eye situated on the other side of 
the lens (to the right in our figure) will see the object, AB, magnified in ab ; 
ab is therefore to be considered the image of AB. Concave lenses afford 
images of this latter kind; they are, however, diminished instead of being 
magnified ( fig. 39). It thus appears that convex lenses alone magnify : 
concave lenses always diminish. 
In order that all rays coming from a luminous point shall unite actually in 
one point, the aperture of the lens must not exceed 10°—15°. If the aper- 
ture be larger, as in the Jens VW (fig. 40), only three rays near the axis 
will unite in the focus of parallel rays: the exterior ones will unite at points 
nearer to the lens. 
Fig. 42 represents a Fresnel or Polyzonal Lens, by means of which the 
light of a light-house may be cast to a distance of many miles. It consists 
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