In order to diminish the spherical aberration, the object-glasses of achromatic telescopes frequently consist of three lenses, of which the first and third are of the kind called double convex, and are formed of crown gleam, while the second is double concave, and mado of flint glass. In this case, sines the index of refraction is the same for the third lens as for the first, if the radius of each surface of the third lens be R', the reciprocal of the principal focal lengths of the separate lenses for red rays will be 2(a -1) 2(µ-1) 2(µ-1) , and le ; 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, µ+dµ for a, and n' + firs' for n', in order to obtain the reciprocals of the focal length for violet rays, we shall have, when the chromatic aberration is corrected, dis 1 1 R )- R R + ; R = 0, or le( + = R /1 But - 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 formula: for the correction of the spherical aberration is a process of some labour, and is scarcely a fit subject except for a mathematical work it is treated with great perspicuity in llobison's Mechanical Philosophy,' vol. ill, from which the subjoined theorem is borrowed, the notation only being changed for that which has been adopted above; and also in the articles Lass 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'- (q + V), where f is the focus for parallel rays infinitely near the axis, and may be found as above, and f-(q + q') is the aberration. Here, neglecting the thickness of the lenses and the interval between them, n+2 12'(4 as - an' it-a + - nod n ; (11 and a being the radii of the two surfaces of the convex lens), and - 1 11.0 240 + p'+2 — • /.2 n R n •ft • F . F . n where F is the principal focus of the convex lens, and n'= - (a' and s' being the radii of the surfaces of the concave lens.) It is evident that, in order to correct the spherical aberration, the values of the radii of the surfaces must be determined from the equation q + q' = O. 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 q and q' the
terms represented by a and a' are respectively equal to half the radii of equivalent isosceles lenses ; and it has been shown, in the investi gation concerning the chromatic aberration, that these are to one ' another as aµ to 2114'; consequently te=a SAL , and therefore re is known in terms of is. 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 a may be found from the equation q + =0, in terms of a, by a quadratic equation.
Sir John Herschel, in a paper on the aberration of compound lenses and object-glasses (' Plul. Trans.,' 1S21), has also investigated formulas for the values of the chromatic and spherical aberrations ; and M. Littrow, of Vienna, setting out with Euler'e 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 the four surfaces may be determined by such conditions as may be thought convenient. (' Memoirs of the Astro!). 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 fur computation, and on that account it has been adopted in the following process.
The radii of the surfaces of the first lens may be determined on the supposition that the whole refraction of light in passing through tho is a minimum ; that is, that the incident and cfnergent rays make equal angles with the surfaces, or with those radii. Thus let a ray PQ.
jig. 4, be incident on the first surface in a direction parallel to the axis x Y of the lens, and infinitely near it; and R Q T being the radius (=n) produced, of that surface let the angle P Q T of incidence be repre sented by a ; then n : 1 : : a : (= R Q F, the angle of refraction at that surface). But if 11'Q T' be the radius (=s) produced, of the second surface ; then, in the triangle n'g It, neglecting the thickness rz of the lens and substituting arcs for their sines, a and - a+ a- a - (=T'Qr) is the angle of incidence on the second surface : and, by optics, 1 is to n as this last angle is to + a(p.-1), the angle of refraction (=eis s") at the second surface. But by bypothesis, it 2-n this angle is to be equal to a; therefore ; = . Again, by optics RS 1 R+S.A is equal to the focal length of the lens; and supposingR-A+ 1 this to be equal to unity, we = : equating this last 2-n 2(n-1) term with above, we get It = - whence a = . Therefore the two radii are found on the supposition that the focal distance of the lens is unity.