TEL 



159 



TEL 



is expressed by f ; the distance from the lens, of the point 

 at which the ray after refraction will meet the axis, is 

 f-f\q+q'J, where / is the focus for parallel rays in- 

 finitely near the axis, and may be found as above, and 

 '(g+g') i s the aberration. Here, neglecting the thick- 

 ness of the lenses and the interval between them, 



(R and S being the radii of the two surfaces of the convex 

 lens,) and 



u'-l 



q' = - 



_ 

 ** 



4iy+m a* , 



F.RW ) 2 : 



where F is the principal focus of the convex lens, and 



R'S' 

 n'ss =-, -57; (R' and S' being the radii of the surfaces 



of the concave lens.) 



It is evident that, in order to correct the spherical aber- 

 ration, the values of the radii of the surfaces must be de- 

 termined from the equation q+q' = Q. This equation is 

 however indeterminate, because it contains several un- 

 known quantities ; but it. may be made subject to certain 

 conditions by which there will remain only one : for ex- 

 ample, 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 q and q' the terms represented by n and n' 

 are respectively equal to half the radii of equivalent isos- 

 celes lenses ; and it has been shown, in the investigation 

 concerning the chromatic aberration, that these are to one 



V 



another as fy to fy' ; consequently n'= n , and there- 

 fore ' 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 curvatures, the value of R may 

 be found from the equation q+q'=Q, in terms of n, by a 

 quadratic equation. 



Sir John Herschel, in a paper on the aberration of com- 

 pound lenses and object-slaves 'Phil. Tranx., 1821), has 

 also investigated formula! for the values of the chromatic 

 and spherical aberrations; and M. Littrow, of Vicnn:i. 

 setting out with Euler's formula for spherical aberration 

 (Diiyptn'rn, torn, iii., 17G9), 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 the four surfaces may be deter- 

 mined by such conditions as may be thought convenient. 

 (Memoirs of the Astrrm. Soc., vol. iii., part 2.) In solving 

 the problem relating to the determination of the four radii, 

 Professor Littrow uses a method which possesses some 

 facilities for computation, and on that account it has been 

 adopted in the followinjr process. 



The radii of the sur!';i<-es of the first lens may be deter- 

 mined on the supposition that the whole refraction of light 

 in passing through the lens is a minimum : that is, that 

 the incident and emergent rays make equal angles with 

 the surfaces, or with those radii. Thus let a ray PQ be 

 incident on the first surface in a direction parallel to th< 

 axis XY of the lens, and infinitely near it ; and RQT being 



and -5- a+a (=T'QF) is the angle of incidence on the 

 second surface : and, by optics, 1 is to /i as this last angle 

 is to ^ + a(/j-l), the angle of refraction (=T'QF') at 



the second surface. But by hypothesis, this angle is 



R 2- 



to be equal to a ; therefore -g- = - . Again, by optics, 



-pa i 



,-r^, . - is equal to the focal length of the lens ; and 

 R+S ft-l ^ 



supposing this to be equal to unity, we obtain -g-= 



~ M + 1 : equating this last term with - - above, we get 

 1 /* 



p 



T"' 



the radius (=R) produced, of that surface let t'.e angle 

 PQT of incidence be represented by ; then p '.\'.',n'.- 



' HQF. the angle of refraction at that surface). Tint i 

 R'QT' be the radius (=S) produced, of the -croud surface 

 then, in the triangle R'QH, neglecting the thickness of the 



R 

 lns and substituting arcs fortheir sines, S ; R :: a -go 



R = 



; whence S = 



Therefore the two 



radii are found on 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. RQF = - -- ; and by trigonometry, 

 n the triangle RQF, 



UP = R . HP 



also, respresenting the thickness MN of the lens by I, 

 sin. RQF 



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

 we get SF+ g S ~* s in. P'QF = sin. T'QF ; 



Q'T' | O / 



Consequently by optics, ^ n sin. P'QF = sin. T'QF', 



or the sine of the angle of refraction at the second surface. 

 Now T'QF'-T'QF+P'QF = QF'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 



Suppose next a double concave lens, the centres of 

 ' surfaces are at R" and R'", and whose radii are R 

 and S', to be applied to the convex lens on the side N : 

 then, neglecting the thickness of the concave lens 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 R"Q* , try 

 IriiTonometry, 



+ sin. P'QF' = sin. T"QF', 



or the sine of incidence on the first surface of the second 

 lens ; and by optics, 



an. P'QF' = sin. T"QF". 



But P'QF' - (T"QF' - T"QF") = P'QF"; and in the tri- 

 angle R"QF", by trigonometry, we have 



/sin T"QF" 

 wherefore NF" = R'"( SSTFJJjP - 1 ) ' and considcrin S 



NR'" to be equal to S', R'"F" will be equal to NF"-S'. 

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



NF" 8' 

 sin. R'"QF"= gr si"- Q F " N 



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

 by optics, 



H^V n- QF"N = sm. R'"QF'", 



>> 



the sine of refraction at the fourth surface : then 

 QF"N-(T'"QF"-T'"QF'")=P / QF'", or =QF'"N; 



