594 LIGHT put - s for r and - r for s. This leaves the result un changed. All lenses, therefore, whose sections are of any of the forms in fier. 16, whichever way they are turned, render Fi parallel rays which pass through them divergent. Their characteristic is that they are thinnest at the middle. But is negative for lenses whose sections are of any of the forms shown in fig. 17. Such lenses, whichever way they are turned, render parallel rays convergent. Their characteristic Fig. 17. is that they are thickest at the middle. But these char acters are interchanged when //, is less than 1 ; as, for instance, when the lens is an air-space surrounded by water. The similarity on reversal is not in general true in a second approximation. The formula for a thin lens now takes the form, J__JL_JL IV U f and differs from that for a curved reflecting surface only in the sign of the second term. With the proper allowance for this, then, all that we liave said of reflexion at spherical mirrors holds true of refraction through thin lenses with spherical surfaces. We may now put the whole matter in the excessively simple form which follows : A thin lens increases or diminishes Ijy a definite quan tity the convergence or divergence of all rays ivhich pass through it. This quantity is the divergence or convergence of rays falling on the lens from or passing from it to its principal focus. Or it is the convergence or divergence which the lens produces in parallel rays. Thus, if the distance of an object from a convex lens is twice the focal length of the lens, the image is formed at the same distance from the lens, and is equal in size to the object. Images Figs. 18 and 19 show the production of a real image pro- and of a virtual image by lenses which produce converg- (lucedby ence o f parallel rays along with the rays by which these ses> are seen by an eye placed behind the lens. In either case it is obvious that the sizes of object and image are, respectively, as their distances from the centre of the lens. Fig. 18 shows how a lens produces a real inverted image Real, of a body placed farther from it than its principal focus. This is the case in the camera obscura, in the solar Fig. IS. microscope, and in the object glass of a telescope. Fig. 19 shows how a virtual image is formed of a body Virtual, placed nearer to a lens than its principal focus. This is the case of a single lens used as a microscope. In the former case the divergence of the incident rays is so small that the lens renders them convergent. In the latter the divergence is so great that the lens can only diminish, not destroy it. In using a hand-magnifier in this way, we so adjust it, by practice, that the enlarged image appears to be formed at the distance from the eye at which vision is most distinct. It is obvious that the amount of magnification must, then, be greater as the focal length of the lens is less. We can now understand the working of the ordinary Astrono astronomical telescope (fig. 20). The object glass furnishes Imcal an inverted but real image of a distant body, within our c reach. We can, therefore, place the eye-glass (like the single microscope above spoken of) so as to form a virtual magnified image of this real image treated as an object. It is still, of course, inverted. It is easy to see that the angle subtended at the eye by the virtual image seen through the eye-piece is to that subtended by the object at the unaided eye approximately as the focal length of the object lens is to that of the eye lens. These angles are, in fact, those subtended . at the centres of the two lenses by the real image. This ratio is, therefore, called the magnifying power of the telescope. The compound microscope, in its simplest form, is pre- cisely the same arrangement as the astronomisal telescope. The only difference is that the object, being at hand, can be placed near to the object-glass (still, however, beyond its principal focus), so that the real image formed is already considerably larger than the object, and is then still further magnified by the eye-glass. Com-
