TELESCOPE. 



V = 20.856 



* = 30.04 



X a = 'r ; fo that tlie four radii of the faces and the cor- 

 refponding foci will ftaiid thus ; viz. 



r = 7.50 7 and F = 9.12, the geometrical focus of the 

 R = 11.63 J convex lens. 



'R = 10.428 1 and 'F = 13.9, the geometrical focus of the 

 concave lens, 

 the compound refrafted focus of the tele- 

 fcopt', according to the proper theorem. 



We have now brought the calculations of our firft ex- 

 ample to a conclufion, accompanied by fuch explanatioRS 

 as may render it unnecedary to dwell fo minutely on the 

 following examples ; and when we liave gained thefe radii 

 for a telefcope of 30 inches focus, we have the means of 

 making a telefcope equally achromatic and dillinft of any 

 other length; for the ratios r : R and V : 'R, being once de- 

 termined for crown and flint glafs of given refraftive and 

 difperfive powers, require only to be increafed in equal 

 quantities to fuit the foci of the propofed telefcopes, as in 

 the fubjoined table ; and it may be proper to notice, that 

 though 'the fpcciiic gravity has not been taken into the ac- 

 count in the calculations of this example, yet it is ufeful as 

 an index to point out the ratio of the fuies of incidence and 

 of refraftion, and of the difperfive powers to be ufed, 

 when the fpecimens of glafs are feledled by their fpecific 

 gravities only, without an experimental trial by gnnding. 



The fubjoined table is fuitable for achromatic double 

 objeft-glaffes of various lengths ; where m : n in the crown- 

 glafs is as 1.528 : i, and in the flint as 1.5735 : i ; their 

 difperfive powers being i : 1.524. 



Table I Radii of double ObjeA-glaffes in Inchei. 



In this table, R and 'R arc the faces of the two lenfes 

 which come in contaft, and r and 'r the external faces; and 

 it will be feen that R, being a little longer than 'R, the 

 convex face, will approach the concave one very clofely, 

 but will not touch it, which is a neceflary practical con» 

 difion. When the aberration exceeds 1.666, which is 

 that of /- : R when they are each =: 1, or alike ; then r 

 exceeds R, and the lens muft be reverfed, or put in its 

 worft pofition ; which is the cafe in all our tables for double 

 objeft-glaffes with the flint-glafs ; otherwife the concave 

 would not have had fufficient aberration for the convex. 



In our next example it will not be necefiary to do the 

 work at full length, but only to give fuch an abridgment as 

 will be intelhgiblc to the reader who underftands the procefs 

 minutely explained in the preceding example. The denfity of 

 different forts of crown-glafs feldom varies ; but two fpeci- 

 mens of flint can feldom be found to be ahke. The greater 

 the denfity of flint-glafs, the more fuitable it is for the pur- 

 pofe of making a concave lens of an achromatic objeft-glafs, 

 becaufe the radii of both the lenfes may be longer for the 

 fame compound refraftcd focus ; and, confequently, the 

 fpherical aberration will be lefs than in glafs that requires 

 fliorter radii to produce the fame compound focus. If the 

 form of the concave had been given, the convex would have 

 been determined by a reverfed operation, where the muU 

 tiplier 1.524 would have been a divifor, and z a multi- 

 plier. Sec. 



Example 2. — Let it be required to form a double objeA- 

 glafs of 30 inches focal length, as before, with the fame 

 crown-glafs for the convex, but with the denfeft flint, in 

 which the ratio m:n n as 1.599 • '> ^""^ ^'^^''^ difperfiva 

 powers I : 1.757 ? 



In this example we have 1.056 = 2 a, or the divifor for 



thecrown, asbefore, and 1.599 — 1x2= 1.198 

 I 



divifor for the flint ; then 



= -94697 ■■ 



1-757 



- = 1.466 



1.056 

 'F, alfo refrafted ; and alfo 



2 a, or 

 F refrafted, and 

 F X 'F 



1. 198 "" ' ' 'F - F "~ 



2.67, the ratio of the compound focal length, or what we 

 called the rational compound focus ; confequently, the 

 ratio between F geometrical and the compound focus is 

 in this example i : 2.67 ; and the ratio between the re- 

 frafted foci of the feparate lenfes, to correft the colorific 



rays, is •! ^ ^^ [ • With refpeft to the fpheri- 



' (-94697 : 1-4663 



cal aberrations, which are next to be confidered, we may 

 in the firfl: place determine the quantities x and y, and » 

 the correaing divifor, which is derived from them, thuB ; 



.834, &c. 



{ 



.}' 



V. 946973 = .2914 = .r I ' '"" 1. 198 



gives the refrafted focus of the flint or concave lens, (F geo- 

 metrical being = 1 in this cafe,) ^ .834^ = .2408 =_)>, and 



— = 1-i — = z = .826, the correBixg divifor required for 

 X .2914 



this refraftive power of the flint-glafs. 



In the next place. 



-2— = 11.23 = ^ (geometrical), and 11.23 ^ '-757 = 

 2.67 



ig.73 ^ 'F (geometrical alfo). Let us here alTume r = 9 



r F 



-^ = R, we get 



inches, and then by the theorem 



R := 14.92, and confequently 



9 : 14-92, or I : 1.66 in its loweft term*. 



zr — F 

 the ratio 



r : R will be 



A double convex 

 lens 



