TELESCOPE. 



lens ground with its radii in this ratio, will have its fpheri- 

 cal aberration = 1.325 x T ; to counteraft which, the con- 

 cave miift; have its proper aberration determined ; and then the 

 ratio of its radii muft be invclliaaled, that (hall make a lens 

 ■with this determined quantity ot aberration. We have fcen 

 alrcadythat A — 1.325 x T, therefore 1.325 x i.757=z 2.328 

 X 'T — 'A, the proportional aberration for F, confidcred 

 as having the fame refrac\ive pov\'er as F ; but the correH- 



ing divifor muft now be applied, and — = -7> :r = 2.818 



a .820 



is the correfted aberration, for which the radii 'r and 'R are 



now to be inveiligated. By putting 'R = i, as before, and 



by working out the root of the quadratic arifmg from 



27r' + 6rR + 7R' 



ratio 'R 



Table II. — Radii of double ObjcA-glaflcs in Inches. 



r + 



Rl^ 



2.818, we fliall have tiie 



3-075- 



And, laftly, for the aftual radi 

 of the concave, we get, by our praftical theorem 



r + 



= F, 



2 X I X 3.075 



= 1.51 = 'F rational, and 



R 



'9-73 

 1.51 



+ 3-075 



= 13.06 = 'R; asalfo 13.06 X 3.075 = 40-15 = 'o the 

 fecond fide of the concave. Whence we now have 



r 

 R 



'R 



* = 



9.00 

 14.92 



13.06 

 40.15 



29.81 



[■ and F = 11.23 geometrical. 

 > and 'F = 19.73 geometrical. 



according to the proper theorem. 



It, may be fatisfaftory to prove, that the geometrical 



quantities F and 'F, which we have here determined, will 



make <J>, the compound focus of the telefcope, = 30 inches. 



But it will be requifite firft to turn the geometrical foci 



F and 'F into the refrafted foci, by their refpeftive divifors, 



1 1.23 

 denominated 2 a and 2 'a, viz.. 1.056 and 1.198 : thus, 



:= 10.634 = F refrafted, and 

 frafted ; then by our theorem 



1.056 

 -^-^ = 16.479 = F' rf- 



1.198 

 F X 



'F- F 



= *, we ha 



= 29. 



! I = * : and if 



10.634 ^ '6-479 _ 175-237686 

 16.479 - 10.634 ~ 5-845 

 the decimal had been carried farther in the geometrical foci, 

 the compound focus would have been quite 30, as required. 

 It may be for the benefit of praftical men to fubjoin a 

 table fimilar to our preceding one, derived from the radii of 

 curvature determined in this fecond example. And let it be 

 underftood by our readers, that in all our tables for the radii 

 of curvature, the length of the telefcope in inches is de- 

 noted by the figures in the firft vertical column ; and that 

 the numbers in the fame horizontal column with any given 

 length, fhew the proper geometrical radii of curvature for 

 convex and concave lenfes to conftruft fach telefcope. 



The following table is fuitable for double achromatic ob- 

 jeA-glafTcs of various focal lengths, where m : n in the crown- 

 glafs is as 1.528 : i, and in the flint as 1.599 : 1 ; and their 

 iiifperfive powers as i : 1-757- 



Example 3. — We fliall now take the fame crown-glafs, 

 with a flint-glafs between the two extremes, wiiich we have 

 ufed, •viz. in which m : n is as 1.584 : i, and their difpcr- 

 five powers as I : 1.59; and let it be required to calculate 

 a double achromatic objeft-glafs of 30 inches focal length, 

 as before ? 



Having already the divifor (2«) of the crown equal 

 1.056, we begin with getting that of the flint thus, 



1.584— 1x2= 1.168 = 2 'a, or proper divifor; thea 



— = .04607 = F refrafted, as before ; and — '-^^r 



i.o;6 1-108 



F X 'F 

 — 1-3613 = 'F refrafted. Alfo ——— = 3. in, the 



rational compound focus ; and I : 3.111 is the ratio between 

 F geometrical and *. We have x = .2914 from our former 



examples, and to get y, we have — — = .856 = 'F re- 



frafted, when 'F geometrical is = I ; therefore / .856* 



V , .2512 „. 



= .2f 12 = y ; but — = 2; hence -^ — = .b62 = %, 

 ■> ■' ' X -2914 



the correding divifor. Again, - — ^ = 9.643 = F geo- 



metrical, and 9.64 x 1.59 = i5-3?7 = 'F >" t'le fame 

 denomination. In this example we will take r = 8 inches ; 



then, by the proper theorem =, = R, we luve 



Jl2<_?i64 _ ^ ,,.„ ^ ,F, and i^ = 1.515 ; con. 

 2 X 8 — 9.64 9.04 



fequently the geometrical ratio r : R = 1 '• i-5'S- A"°i 



from 



