=Formation of Real Images.=--Let A B (Fig. 13) be an object in front of a lens, at a distance less than the principal focal length. It will have a real image on the other side of the lens. To determine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting the lens in A′. The ray represented by this line will, after refraction, pass through the principal focus, F, and its intersection with the secondary axis, A O, determines the position of _a_, the focus conjugate to A. We can in like manner determine the position of _b_, the focus conjugate to B, another point of the object; and the joining line _a b_ will then be the magnified image of the line A B. It is evident that if _a b_ were the object, A B would be the image. [Illustration: Fig. 13.--Real and Magnified Image.] The figures 12 and 13 represent the cases in which the distance of the object is respectively greater and less than twice the focal length of the lens. The focal length of a lens is determined by the convexity of its surfaces and the refractive power of the material of which it is composed, being shortened either by an increase of refractive power, or diminution of the radii of curvature of the faces of the lens. The increase or decrease of spherical aberration is determined by the shape or curvature of the lens; it is less in the bi-convex than in other forms. When a lamp or other source of light is placed at the focus of the rays constituting that portion of its light which falls upon the lens, the light is so refracted as to become parallel. Should the source of light be brought nearer to the lens than the focus the refracted rays are still divergent, though not to the same extent; on the other hand, if the source be beyond the focus, the refracted rays are rendered convergent so as to meet at a point which is mathematically related to the distance of the luminous source from the focus. The former arrangement is that with which we are most familiar, since it is the ordinary magnifying glass. Concave Lenses. The refracting influence of a _concave_ lens (Fig. 14) will be precisely the opposite of that of a convex. Rays which fall upon it in a parallel direction will be made to diverge as if from the principal focus, which is here called the _negative_ focus. This will be, for a _plano-concave_ lens, at the distance of the diameter of the sphere of curvature; and for a _double-concave_, in the centre of that sphere. [Illustration: Fig. 14.--A Virtual Image formed by Concave Lens.] In Fig. 14 A B is the object and _a b_ the image. Rays incident from A and B parallel to the principal axis will emerge as if they came from the principal focus F; hence, the points _a b_ are determined by the intersections of the dotted lines in the figure with the secondary axis, O A, O B. An eye on the other side of the lens sees the image _a b_, which is always virtual, erect and diminished.