804 OPTICS posing the lens. The best image will be formed at a posi tion midway between the two foci, and the diameter d of the circle over which the rays are spread bears the same ratio to the semi-aperture of the lens (y) that S/ bears to /. Hence The diameter of the circle of chromatic aberration is thus proportional to the aperture and independent of the focal length and, since the linear dimensions of the image are proportional to the focal length, the confusion due to chromatic aberration may be considered to be inversely as the focal length. Before the invention of the achromatic object-glass this source of imperfect definition was by far the most important, and, in order to mitigate its influence, telescopes were made of gigantic length. Even at the present day the images of large so-called achromatic glasses are sensibly impaired by secondary chromatic aberration, the effect of which is also directly as the aperture and inversely as the focal length. Achromatic Object-glasses. It has been shown in vol. xiv. p. 595 that the condition of achromatism for two thin lenses placed close together is 5/u 1 5/j. 1 _f. ,-, in which /, / are the focal lengths of the two lenses, and BfjL/ (fjL - 1), VV^ ~ 1) ^ ne dispersive powers of the two kinds of glass. In practice crown and flint glass are used, the dispersive power of the flint being greater than that of the crown. Thus / is negative and numerically greater than /, so that the combination consists of a convex lens of crown and a concave lens of flint, the converging power of the crown overpowering the diverging power of the flint. When the focal length F of the combination is given, the focal lengths of the individual lenses are determined by (1) in conjunction with .(2). The matter, however, is not quite so simple as the above account of it might lead us to suppose. In consequence of what is called the irrationality of spectra, the ratio of dis persive powers of two media is dependent upon the parts of the spectrum which we take into consideration. What ever two rays of the spectrum we like to select, we can secure that the compound lens shall have the same focal length for these rays, but we shall then find that for other rays the focal length is slightly different. In the case of a single lens the focal length continually diminishes as we pass up the spectrum from red to violet. By the use of two lenses the spectrum, formed as it were along the axis, is doubled upon itself. The focal length is least for a certain ray, which may be selected at pleasure. Thus in the ordinary achromatic lens, intended for use with the eye, the focal length is a minimum for the green, and increases as we pass away from the green, whether towards red or towards blue. Stokes has shown that the secondary colour gives a sharp test of the success of the achromatizing process. " The secondary tints in an objective are readily shown by direct ing the telescope to a vertical line separating light from dark, such as the edge of a chimney seen in the shade against the sky, and covering half the object-glass with a screen having a vertical edge. So delicate is this test that, on testing different telescopes by well- known opticians, a difference in the mode of achromatism may be detected. The best results are said to be obtained when the secondary green is intermediate between green and yellow. This correspond s to making the focal length a minimum for the brightest part of the spectrum. "To enable me to form a judgment as to the sharpness of the test furnished by the tint of the secondary green, as compared with the .performance of an object-glass, I tried the following experiment. A set of parallel lines of increasing fineness was ruled with ink on a sheet of white paper, and a broader black object was laid upon it as well, parallel to the lines. The paper was placed, with the black lines vertical, at a considerable distance on a lawn, and was viewed through two opposed prisms, one of crown glass and the other of flint, of such angles as nearly to achromatize each other in the positions of minimum deviation, and then through a small telescope The achromatism is now effected, and varied in char acter, by moving one of the prisms slightly in azimuth, and after each alteration the telescope was focused afresh to get the sharpest vision that could be had. I found that the azimuth of the prism was fixed within decidedly narrower limits by the condition that the secondary green should be of such or such a tint, even though no attempt was made to determine the tint otherwise than by memory, than by the condition that the vision of the fine lines should be as sharp as possible. Now a small element of a double object-glass may be regarded, so far as chromatic compensation is concerned, as a pair of opposed prisms ; and therefore we may infer that the tint of the secondary green ought to be at the very least as sharp a test of the goodness of the chromatic compensation as the actual performance of the telescope. " 1 In the case of photographic lenses the conditions of the problem are materially different. It is usually considered to be important to secure " coincidence of the visual and chemical foci," so that the sensitive plate may occupy the exact position previously found by the eye for the ground glass screen. For this purpose the ray of minimum focus must be chosen further up in the spectrum. If, however, the object be to obtain the sharpest possible photographs, coincidence of visual and chemical foci must be sacrificed, the proper position for the sensitive plate being found by trial. The middle of the chemically -acting part of the spectrum, which will vary somewhat according to the photographic process employed, should then be chosen for minimum focus. When the focal lengths of the component lenses have been chosen, it still remains to decide upon the curvatures of the individual faces. Between the four curvatures we have at present only two relations, and thus two more can be satisfied. One of these is given by the condition that the first term in the expression for the aberration that proportional to the square of the aperture shall vanish for parallel rays. As to the fourth condition, various pro posals have been made. If equal and opposite curvatures are given to the second and third surfaces, the glasses may be cemented together, by which some saving of light is effected. Herschel proposed to make the aberration vanish for nearly parallel, as well as for absolutely parallel, rays. This leads to a construction nearly agreeing with that adopted by Fraunhofer. The following results are given by Herschel 2 for the radii of the four surfaces, corresponding to various dispers ive powers, and to mean refractive indices 1*524 (crown) and T585 (flint). The focal length of the combination is taken equal to 10, and, as well as the radii, is measured in arbitrary units ; so that all the numbers in the table (with the exception of the first column) may be changed in any proportion. Ratio of Dispersive Powers. Radius of First Surface. + Radius of Second Surface. Radius of Third Surface. Radius of Fourth Surface. Focal Length of Crown Lens. + Focal Length of Flint Lens. 50 55 60 6-7485 6-7184 6-7069 4-2827 3-6332 3-0488 4-1575 3-6006 3-0640 14-3097 14-5353 14-2937 5-0 4-5 4-0 10-0000 8-1818 6-6667 65 TO 6-7316 6-8279 2-5208 2-0422 2-5566 2-0831 13-5709 12-3154 3-5 3-0 5-3846 4-2858 75 7-0816 1-6073 1-6450 10-5186 2-5 3-3333 The general character of the combination is shown in fig. 16. The radii of the first and fourth surfaces within prac tical limits are so nearly constant that Herschel lays down 1 Proc. Roy, Soc., June 1878. 2 Phil. Trans., 1821.
