TEL 



appears to be indistinct and surrounded by a coloured 

 fring i vncs; Dibi-KRsioN.] h may i 



that the principal locus of anv lens, with respect I 

 colovi MIII the formula' in LK.NS I 



stituting in them the value of ft the imK . .ciion. 



for the given kind of light. 



Thus, in an optical instrument, in addition to the distor- 

 tion of the image arising lioni the sphciicily uf th 

 there is an im; 1 by the dispersion of the 



different colour-making rays; and, in a good U-K -. . p, . it 

 is icquisalc that both of these impel, mid us far 



us possible be rcinovid. The chromatic aberration, as the 

 dispersion of the colour u called, , ,r the 



greater evil of the two, for New; ,.vn that it 



.Is the former nearly in the latio 01 ."ii4J to 1; but 

 fortunately it is that wliicli, to an extent sufficient for 

 .cal purpose*, admits of being easily corrected. 



Since different kinds of glass have different degn 

 dispersive power, it is evident that the chromatic aberra- 

 tion may be diminished, if not wholly removed, by causing 

 the light to pass through two lenses of different kinds ol 

 glass, aiid of such forms that they may refract the rays in 

 pencil in opposite directions. Ihe object-glass of a 

 telescope when so formed is said to be achromatic, and 

 the manner in which the ett'ect is produced may he under- 

 stood Irom the following description. Let PQ be the 

 direction of a pencil of compound light incident on the 



A C 



1 V 



B I) 



first surface of the convex lens AB, in a direction parallel 

 to the common axis, XV, of the two lenses. By the refrac- 

 tive power of this lens (crown glass) the red rays in the 

 pencil would, if no object were interposed, proceed In the 

 direction Q6, meeting XY in r, and the violet ray in the 

 pencil would proceed in the direction Qr. meeting the 

 axis in r. But the refractive power of the concave lens 

 CD (flint glass) acts, from its form, in a direction contrary 

 to that of the convex lens, causing the rays either to 

 diverge from the axis XY, or to meet it in points beyond 

 c and r, towards Y : suppose the curvature of this lens to 

 be such that the red rays in the pencil PQ would, after 

 refraction in both lenses, meet the axis in F ;the ray Q6r 

 taking the direction bV) ; then the dispersive power of 

 this kind of glass exceeding that of the other kind, the 

 violet rays iti the refracted pencil will tend farther away 

 from the axis than the red rays do, and thus will tend 

 towards the latter; the ray Qn>, for example, taking the 

 direction cV. It is conceivable, therefore, that thecmva- 

 tuivs of the surfaces of the lenses may be such that, in 

 each incident pencil, the red and violet rays (the extreme 



>f the spectrum) shall after roll-action unite at the 

 place of the image ; and thus the fringe due to these two 

 colours may be destroy ed. 



the t\vo kinds of glass dispersed the different eolour- 

 makiiig rays in the same proportions, their contrary i 

 Cons would i-aiise all the colours to be united on tin- imacc 

 formed at K: no U\o kinds of glass h.i\c Imv, . 

 yet discovered which possess tin ty; and therefore 



the red and violet images only are united : fortunaidv in 

 uniting the extreme rays of the spectrum, the otln 

 brought so nea. . that for ordinary purpose^ the 



image is as i . . limn colour as can be desired. 



From the description just given it will he evident that 

 the place F, of an image in which tin: dispersion of the 

 :d violet rays is corrected, may be dc tciimned on ob- 

 taining, from tin- common theorems of optics, algebraic 

 expressions &* I nirtlis uf tin- compound leu- lor 



each of those kinds of lisrht, and making the < 

 equal to one another. Tims, supposing Iv ami S to he the 

 radii of the < < e of a d. 



crown glass, and i> the index of refra. II.H, . one 



kind , red. lor r 

 the pencils of incident light are paralK I to one another and 



through the lens very near the n\is ; then, liv a fun- 

 damental theorem iu optics we have, F being the distance 



T t: L 



from the focu* to the len, which i, m ;ihout 



thiokn. 



but since, in the present case, the lens may be supposed 



to be isosceles ( R=S>, we have V = . 



- (* - 1 / 



and 



In like manner the focal length F', of a double c 



us of flint glass, It' being the ladiu-, 



id M Mhe index of refraction for red ual to 



.1TY7' " lu ">'* being incident near the axis. 

 Hence, by a fundamental theorem in optics, 

 K' It H' H HR' 



and this last term is the local length of the compound 

 lens for red rays. Its re. . e.jnal to "' M ~ ^- 



2T/-1) 

 j^ , which, in the al^< )>i aie 'lie sum of the 



reciprocals of the focal lengths of the separate lenses. 



On writing p+cp, and n'+fp' in place of u and /' in 

 the litst expression, we have for the reciprocalof the focal 

 length of the compound lens for violet, rn 



R 



In an achromatic telescope the focal lengths of the 

 compound lens for red and s iolet rays are to be eipial to 

 one another; and it is evident that tlus condition will In 



tu ta' 



fulfilled when ^- --^-, = 0. From this equation we ha\e 



R : R':ii> I fy': then, dividing the antecedents by ul 

 and the consequents bv/i' 1, we have [DlsPBRSIONj the 



ratio of the focal lengths of the two len.-es equal to that ol 

 the dispersive powers of the two kinds of class; and L 

 the focal length of the compound lei 



ire, those of the sepaiatc K-n- .enlly the 



radii of their surfaces, may be obtained. 



In order to diminish the spherical aberration, the object- 

 classes uf achromatic telescopes lVei|ucntl\ 

 lenses, of which the first and third are of the kind ci 

 double convex, and are formed of crown glass, whiK 

 second is double com-a\c, and made of flint clas,. la (bis 



lince the index of refi action is the same for the third 

 lens as for the first, if the radius of each surface of I he third 

 lens be H". the reciprocal of the principal focal len-t 

 the separate lenses for red rays will be 



2d*-l) 2C/-1) SOi-1) 



R ' ~ R' ~ ' a ' 



these being added together, their sum will be the reci- 

 procal of the focal leiicth of the compound K: 

 kind of light. On substituting in the above leinis. ,i + i'/, 

 for /j. andi/-|-(V for /', in oiderto obtain the reciprocals of 

 the focal length for \iolet ray-, we shall ha\e, when the 



chroma 



** V ,,, ( l , 1 \ V 



R ~ Ti' + R" = ' " r H (H + H") = ST- 



tu! 

 But 7 is known from tables of the refractive indict - 



dill'crenl kinds of glass: tlu-refore if any convenient re- 



lation between the radii of two of the leiV. . med, 



the values of all (lie mdii, and coilsec|ueiitly the 

 K iigllis of the scM-ial lenses, nmy be found. 



The investigation of formula; for the correction of the 



spherical aberration is a p .d is 



V a lit subject except for a mathematical work : it 



-itcd with great p. n's Me- 



>l Philosophy,' vol. ni., from \\hieh the -nbjoim-d 



n is borrowed, the notation only hi i ,| for 



that which USD in t!i 



i. r:\saud S . If a compound object-glass 



-s and M 



Hint cla-s, ;u.d a i.iy <ii hcht be in- 



.'idenl, \ipon ;ner in a direc- 



tion parallel to the axis, at a distance from thence, which 



