Now, if the object A B be so remote that, in each pencil, the rays incident upon a lens may ho considered as parallel to one another, the point F is called the principal focus; and in the article LENS, there will be found a collection of formula for the reciprocals of the focal lengths of lenses of all the different kinds; it being understood that the diameter of the lens is small, which is generally the case with telescopes, and that the light is homogeneous. But, since all light is not of oue kind, and a lens acts like a prison in causing in each pencil the rays of the differently coloured light to diverge from one another, it follows that each of the coloured lights will form its owe image at its proper focus ; and the image formed by light of one kind being seen by the eye along with the images formed by light of the other kinds, the representation of an object when formed by a single lens appears to be indistinct and surrounded by a coloured fringe. [AcilitOMArtc ; Lton Dispensiose] It may be observed that the principal focus of any lens, with respect to each colour, may be obtained from the formula in LENS, by substituting in them the value of p (the index of refraction) for the given kind of light Thus, in an optical instrument, in addition to the distortion of the image arising from the aphericity of the lens, there is an indistinctness caused by the dispersion of the different colouranaking rays; and, in a good telescope, it is requisite that both of these imperfections should as far as possible be removed. The chromatic aberration, as the dis persion of the colour is called, constitutes by far the greater evil of the two, for Newton has shown that it exceeds the f<.mer nearly in the ratio of 5449 to 1 ; but fortunately it is that which, to an extent sufficient for practical purposes, admits of being easily corrected.
Since different kinds of glare' have different degrees of dispersive power, it is evident that the chromatic aberration may be diminished, if not wholly removed, by causing the light to pass through two lenses of different kinds of glees, and of such forms that they may refract the rays in each pencil in opposite directions. The object-glare of a tele. scope when so formed is said so be arAromatic, and the manner in which the effect is produced may be understood from the following description. Let r q be the direction of a pencil of compound light incident on the first surface of the convex lens • 0, fig. 3, in a direction parallel to the common axis, x r, of the two lenses. By the refractive power of this lens (cresol glass) the red rays in the pencil would, if no object were interposed, proceed In the direction g b, meeting x T in r, and the violet ray in the pencil would proceed in the direction g c, meeting the axis in r. But the refractive power of the concave lens D (]Gat glass) acts, from its form, in a direction contrary to that of the convex lens, causing the rays either to diverge from the axis x Y, the direction b F.); then the dispersive power of this kind of glass exceeding that of the other kind, the violet rays in the refracted pencil will tend farther away from the axis than the red rays do, and thus will tend towards the latter ; the ray g er, for example, taking the direction c F. It is conceivable, therefore, that the curvatures of the
surfaces of the lenses may be such that, in each incident pencil, the red and violet rays (the extreme rays of the spectrum) shall, after refraction, unite at the place of the image ; and thus the fringe due to these two colours may be destroyed.
If the two kinds of glass dispersed the different colour-making rays in the same proportions, their contrary refractions would cause all the colours to be united on the image formed at F : no two kinds of glass have, however, been as yet discovered which possess this property ; and therefore the red and violet images only arc united : fortunately, in uniting the extreme rays of the spectrum, the others are brought so near together, that for ordinary purposes the image is as free from colour as can be desired.
From the description just given it will be evident that the place F, of an imago in which the dispersion of the rod and violet rays is corrected, may be determined on obtaining, from the common theorems of optics, algebraic expressions for the focal lengths of the compound lens for each of those kinds of light, and making the expressions equal to one another. Thus, supposing R and s to be the radii of the curve surfaces of a double convex lens of crown glass, and g the index of refraction for light of one kind (red, for example); supposing again that the rays in the pencils of incident light are parallel to one another and pass through the lens very near the axis ; then, by a fundamental theorem in optics we have, F being the distance from the focus to the lens, which is moreover without thickness, n. s — • — n + s 1.1 — but since, in the present case, the lens may be supposed to be isosceles (a= s), we have F= In like manner the focal length r', of a double concave lens of flint glass, n' being the radius of each surface, and p' the index of refraction it' for red rays, is equal to rays being incident near the axis.
Bence, by a fundamental theorem in optics, it' me n ke —I p—i 11-1 2(14—l) 2{n'(a—l)—n(p'— I)} and this last term is the focal length of the compound lens for red 20.4— I) 2(se-1) rays. Its reciprocal is equal to R which, in the algebraic sense, is the sum of the reciprocals of the focal lengths of the separate lenses.
On writing p + 5p, and p' + ag , in place of p and p' in the last expression, we have for the reciprocal of the focal length of the com pound lens for violet rays, 2(.t + Op—l) 2(g+ 41-1) me — R' • In an achromatic telescope the focal lengths of the compound lens for red and violet rays are to be equal to one another ; and it is evident that this condition will be fulfilled when R 'e 0. From this equation we have n : a' : : : Sp'; then, dividing the antecedents by p-1 and the consequent., by p'-1, we have [Diseritstoe] the ratio of the focal lengths of the two lenses equal to that of the dip. persive powers of the two kinds of glees ; and hence, the focal length of the compound lens being assumed at pleasure, those of the separate lenses, consequently the radii of their surfaces, may be obtained.