TELESCOPE. 



TELESCOPE. 



be R", the reciprocal of the principal focal lengths of the separate 

 lenses for red rays will be 



these being added together, their sum will be the reciprocal of the 

 focal length of the compound lens for one kind of light. On sub- 

 stituting in the above terms, n + Sf*. for p, and n' + $n' for /*', in order 

 to obtain the reciprocals of the focal length for violet rays, we shall 

 have, when the chromatic aberration is corrected, 

 M 



But j is known from tables of the refractive indices for different 



kinds of glass : therefore if any convenient relation between the radii 

 of two of the lenses be assumed, the values of all the radii, and con- 

 sequently the focal lengths of the several lenses, may be found. 



The investigation of formulae for the correction of the spherical 

 aberration in a process of some labour, and is scarcely a fit subject 

 except for a mathematical work : it is treated with great perspicuity in 

 Robison's ' Mechanical Philosophy,' vol. iii., from which the subjoined 

 theorem is borrowed, the notation only being changed for that which 

 baa been adopted above ; and also in the articles LENS and SPECULUM. 

 If a compound object-glass consists of one double convex lens of crown 

 glass and a double concave lens of flint glass, and a ray of light be 

 incident upon the anterior surface of the former in a direction parallel 

 to the axis, at a distance from thence, which is expressed by e ; the 

 distance from the lens, of the point at which the ray after refraction 

 will meet the axis, is f'-f l (i + <i)> where / is the focus for parallel 

 rays infinitely near the axis, and may be found as above, and /-'(</ + '/') 

 is the aberration. Here, neglecting the thickness of the lenses and the 

 interval between them, 



M-l f y' V + M M + 21 e B 



M l> ~ + R* / 2 -' andM= RTi ; 



(R and s being the radii of the two surfaces of the convex lens), and 



r (. n- i:-,,- r.n? r.tff 



R'S' 

 where F is the principal focus of the convex lens, and '= , ; 



(R' and s' being the radii of the surfaces of the concave lens.) 



It a evident that, in order to correct the spherical aberration, the 

 values of the radii of the surfaces must be determined from the 

 equation </ + q' = 0. This equation is however indeterminate, because 

 it contains several unknown quantities ; but it may be made subject 

 to certain conditions by which there will remain only one : for 

 example, the different radii of the lenses may be made to have any 

 given relation to one another, so that the values of all, in terms of any 

 one, may be substituted for them. In the values of 7 and >/' the 

 terms represented by and ' are respectively equal to half the radii 

 iiralent isosceles lenses ; and it has been shown, in the investi- 

 gation concerning the chromatic aberration, that these are to one 



another as S/i to 8/1'; consequently ' = r , and therefore ' is 

 known in terms of n. If again it be supposed that R' = s, or that the 

 nearest surfaces of the convex and concave lenses have equal cur- 

 vatures, the value of R may be found from the equation q + j' = 0, in 

 terms of , by a quadratic equation. 



Sir John Herschel, in a paper on the aberration of compound lenses 

 mi object-glasses < Phil. Trans./ 1821), has also investigated formulae 

 for the v.ilues of the chromatic and spherical aberrations; and 

 M. I.ittrow, of Vienna, setting out with Euler's formula for spherical 

 aberration (' Dioptrica,' torn, iii., 1769), and introducing in it the 

 values of the focal lengths of two lenses so that the former aberration 

 may be corrected, has obtained two equations from which the radii of 



Professor Littrow uses a method which possesses some facilities for 

 computation, and on that account it has been adopted in the following 



pi - 



r.wlii of the surfaces of the first lens may be determined on the 

 supposition that the whole refraction of light in passing through the 



Fig. 4. 



;i minimum ; that u, tiut the incident .-unl emergent ray in;ike 

 inglcn with the xurfaccx, or with those radii. Thus let a ray i-o 

 AHT.t ASD SCI. DIV. VOL. VIII. 



fy. 4, be incident on the first surface in a direction parallel to the axis x Y 

 of the lens, and infinitely near it; and RQT being the radius ( = u) 



produced, of that surface let the angle PQT of incidence be repre- 





 sented by a ; then /*:! : : a : - ( = RQF, the angle of refraction at 



that surface). But if R'QT' be the radius ( = s) produced, of the 

 second surface; then, in the triangle R'QH, neglecting the thickness 



of the lens and substituting arcs for their sines, s : R : : a : a ; and 



s 



R a 



- a + a - ( = T'QF) is the angle of incidence on the second surface : 



0MB 

 and, by optics, 1 is to /i as this last angle is to - + a(/u 1), the 



angle of refraction ( = T'Q v 1 ) at the second surface. But by hypothesis, 



R 2 n 

 this angle is to be equal to a ; therefore - = - . Again, by optics 



as 1 

 R + g . _! is equal to the focal length of the lens ; and supposing 



this to be equal to unity, we obtain = - - : equating this last 



, - 

 term with -- above, we get R 



whence s = 



Therefore the two radii are found ou the supposition that the focal 

 distance of the lens is unity. 



Now PQT being the angle of incidence as above, and QF the direction 



of the ray after one refraction, we have by optics, sin R Q F = - ; 

 and by trigonome in the triangle RQF, 



sin RQF 



/sin UQF 

 - 



also, representing the thickness UN of the lens by (, 

 sin RQF 



Then, by trigonometry, in the triangle R'QF, 



SF + g t 

 we get sin p / QF = siu T'QF; 



consequently by optics, - /i sin P'QF = sin T'QF' or the nine of 



o 



the angle of refraction at the second surface. 



Now T'QF' T'QF + I >/ QF = QP'M, or the angle which the second 

 refracted ray makes with the axis of the lens : but by trigonometry, 

 in the triangle R'QF', we have 



sin T'QF 



sin T'QF 



Suppose '>ext a double concave lens, the centres of whose surfaces 

 re at R" and n'". and whose radii are R' and s', to be applied to the 

 convex lens on the side N : then, neglecting the thickness of the con- 

 cave lena and the distance between the two, and supposing QF", QF'" 

 to be the directions of the ray of light after the third and fourth 

 refractions respectively, we have in the triangle H"QF', by trigo- 

 nometry, 



n' + sV 



,., sin p / Qr'=sin T'QF', 



or the sine of incidence on the first surface of the second lens ; and by 

 optics, 



R' + S'F' 



QF". 



But P'QF'-(T"QF'-T"QF")=P'QF"; and in the triangle R"QF", by 

 trigonometry, we have 



K'V'=K' 



sin 



,/sinT"QF" \ 

 wherefore NF"=R ' (^ p , Qp ,,- _^ ; and considering NR'" to be 



equal to s', R'" r" will be equal to NF" g'. 



Again, in the triangle H"'QF", we have by trigonometry, 



y j,// _ -f 



sin R'"QF"= -, gi n QF j, 



for the sine of incidence on the fourth surface ; therefore, by optics, 



HP" -B' 



~, M'in QF"N = sin R'"QF'", 



the sine of refraction at the fourth surface ; then 



Qr"K-(T'"QF"-T'"QF'") = P'QF'", or =<JF"'jf ; 



