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71.—That the same principles given above for the prism will hold in the achromatic compound lens, is already demonstrated by the comparison of lenses and prisms shown in [Fig. 5]; but for the sake of clearness it may be again shown diagrammatically in Fig. 12 for an actual objective, wherein the parallel rays ab, proceeding from a distant object or star, are shown refracted to a′b′, and coming to a focus at F, although dispersed at the meeting surfaces of the two glasses, as shown diagrammatically, by the internal cone of rays.
Fig. 12.—Showing achromatic objective.
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72.—Practically, the matter is not quite so simple as it would appear to be theoretically, by the above-described conditions, as we actually find the spectrum of a prism of flint glass of equal dispersion to one of crown glass does not give exactly similar extent of separate colours within its spectrum, the medium ray of the spectrum in the flint glass being nearer the blue than in the crown. Thus, this compound lens does not perfectly correct by inversion as it does in the perfect case discussed, and shown in [Fig. 10]. For this reason better definition is found by slight displacement and slight difference of total extent of dispersion of one of the spectra in coincidence on the meeting planes between the lenses, leaving in all cases a certain amount of residual colour, blue or red, uncorrected, by making the glass under- or over-corrected, as it is termed, which does not, however, seriously impair distinct vision. It is quite possible that, by some future improvements in the chemical constitution of the glass, this defect may be remedied. English glass-workers prefer to over-correct, German and French glasses are more often under-corrected.
73.—The measurements of refraction and dispersion being both in one direction, may be taken together within certain angular limits in one term in the construction of a lens as the ratio of dispersive powers, the indices being certain dark lines which are observed uniformly in the spectrum of the sun projected from a narrow slit. These lines or bands in the sun's spectrum are known to be due to metallic vapours which are present in his atmosphere, and can therefore be reproduced by the deflagration of like metals on a small scale. To certain of these lines letters of the alphabet have been applied. Of these letters, a pair of lines due to sodium vapour marked D, and three lines due to hydrogen, marked C, F and G, are commonly taken for reference of dispersion. Achromatism is generally considered duly corrected when the lines C and G are united. The middle of the spectrum between these lines is about E; and chromatic dispersion of optical flint and crown may be taken to be fairly corrected if the spectra are coincident in colour at this line.
74.—Curvatures in the Achromatic Lens.—A large amount of mathematical power has been expended upon this matter, but the perplexity of the subject is due to small differences of the material; and the impossibility of working absolutely true spherical curves has rendered this work of little practical value to the optician, who still resorts to the formulæ of Dollond and Tully. Those who care to follow the subject beyond the scope of this work will find numerous papers in the Phil. Trans., and in the works of Herschel, Barlow, Coddington, Robinson, and Stokes, wherein what is known theoretically of the subject is fully investigated and discussed.
75.—For all small achromatics, such as are employed in surveying instruments with Chance's hard crown and dense flint, the following approximate formula is commonly employed, expressed in terms of the radius of the curved surface into f, the total focus of the finished objective, for first working before trial:—
1st.—Outside surface, f 2 convex,