LIGHT 595 The magnifying power of a single lens, when used as a hand microscope, is to be measured by the ratio of the angle under which the virtual image of an object is seen (at the distance of most distinct vision) to that at which the object itself would be seen (at that same distance) ; i.e., it is the ratio of 10 inches to the focal length of the lens. Lensesin Combinations of Lenses in Contact. From the formula contact. 1 / 1 1 we see that the focal length of a simple lens is less as /* is greater. Thus all that we have just said is true for homo- Acln-o- geneous light alone. But if we combine two thin lenses, matic placing them close together, we may arrive at an approxi- lenses. ma t e iy achromatic arrangement. For we have, for the first lens J__JL = .L w u f For a second, close to it, we have .!_ J-.-L x w f For the two, considered as one, we have Now and there is an infinite number of ways in which r and s can be chosen, when r and s are given, such that the values of the right hand side shall be equal for two values of p. and the corresponding values of //. Any one of these gives an achromatic combination, of the necessarily imper fect kind described above in considering prisms. But, as we have now the curvatures of four surfaces to deal with, we can adjust these so as not only to make the best attainable approximation to achromatism, but also to reduce the unavoidable spherical aberration to a minimum. These questions, however, are beyond the scope of this article. We can remark only that the adjustment for two rays, for which the refractive indices are p. and p. + S/u, in the first medium, and //, and //, + S/u, in the second, requires the one relation which involves only the ratio of the focal lengths of the two lenses leaving their forms absolutely undetermined. But, if both p. and // be greater than unity, the signs of /and/ must be different; i.e., in an achromatic combination of two lenses one must be convex and the other concave. The reader must, however, be reminded that we are dealing with a first approximation only, and that spherical aberration does not come in till we reach a second. The details for a proper achromatic combination will be given in OPTICS (GEOMETRICAL). Aclu-o- Before leaving this subject, we must find the behaviour of two thin lenses which are placed at a finite distance from one another. For the first lens we have, as before, matic eye pieces. w u f If the second lens be placed at a distance a behind the first, the rays which fall on it appear to come from a distance iv + a. Hence, for the light emerging from the second lens, we have _1 1_ = _1_ x w + a, f When u is infinite, we have from the last two equations It is obvious that a combination of this nature offers the same kind of facilities for the partial cure of dispersion and of spherical aberration as when the lenses are in contact, with one additional disposable constant. Thus we have compound achromatic eye-pieces, which can be corrected for spherical aberration also. Formation of a Pure Spectrum. We may now go back Pure to the formation of an image by a prism, and inquire spectrum, how, by the use of an achromatic lens, we can project a pure spectrum on a screen. We have seen that a thin prism, for rays falling nearly perpendicular to it, forms a virtual and approximately rectilineal image of a lumi nous point, in which the colours are ranged in order of refrangibility. Suppose the light which passes through the prism to fall on an achromatic lens, placed at a distance greater than its focal length from the virtual image above mentioned. These rays after passing through the lens will proceed to form, at the proper distance, a real linear coloured image of the luminous point, in which (as in the virtual image) the colours do not overlap. Instead of a luminous point, rays diverging from a very narrow slit parallel to the edge of the prism are employed. It is usual to place the lens at double its focal distance from the virtual image, and thus the real image is formed at an equal distance on the other side of it, and is of the same It may now, if required, be magnified by means of an achromatic eye-piece. Or, in other words, it may be examined by means of a telescope. In fact a telescope, whose object glass is covered by a thin prism, has been usefully employed during a total eclipse in examining the light of the solar corona. A similar arrangement, made to have an exceptionally large field of view, is employed to find the nature of the spectra of meteorites or falling stars. Refraction at a Cylindrical Surface. A very simple, Cylin- but interesting, case of refraction at a cylindrical surface drical re is furnished by a thermometer tube. It is easily seen that fractor - the diameter of the bore appears, to an eye at a distance large as compared with the diameter of the tube, to be greater than it really is, in the proportion of the refractive index of the glass to unity. Thus in flint glass it appears magnified in about the ratio 5 : 3. Hence the mercury appears completely to fill the external surface of such a tube, if the bore be only 5ths of the external diameter. But a far more interesting case is that of parallel rays Rainbow, falling on a solid cylinder of glass or water. Its interest consists in the fact that by its aid we can explain the phenomena of the rainbow. We, accordingly, devote special attention to it. The problem, without losing any of its applicability to the rainbow, is much simplified by supposing the rays to be incident in a direction perpendi cular to the axis of the cylinder ; for in this case the whole course of each ray is in a plane perpendicular to the axis. We need not treat i here of rays which pass close to the axis of the cylinder. For such the cylinder acts as a lens, and its focal length (to j the usual first ap proximation) can easily be obtained by methods such as those given above. What we are mainly concerned with is the behaviour of the rays which escape into the air, after one or two re flexions at the inner surface of the cylinder. Suppose first that we consider a ray once reflected in the interior of the cylinder. Let SP (fig. 21) be one of the set of incident parallel rays, and let its path be SPQP S . This involves refraction at P, reflexion at Q, and again refrac-
