TELESCOPE. 



fg. 12. P/a/?XXVni.). wf firft determine the focus of B, 

 which becomes in this cafe, if we fubftitute — for p, 

 drR 



a 

 JrR 



F = — X 

 a 



R, 



adR + adr — Rr 



dR-lr dr- 



a 



Now /" the focal diftance of B, thus found, is manifcftly 



the radiant diftance of the fccond or middle lens C ; and as 



the general theorem above referred to involves the radiant 



diftance </, we have only now to apply that theorem again 



to the fecond lens C, fubftituting, as before, h for — , and 



for d, the quantity laft found as the focus of B. This gives 

 the compound focal diftance of thcfe two lenfes B and C, 

 which we will call <p ; and this again becomes the radiant 

 diftance of the lens D : therefore, laftly, the general theorem 



is again applied to this lens, fubftituting c for — , and the 



laft^ found focus (?>) for «/; by which procefs, we arrive at the 

 compound focus (*) of all the three lenfes. In the apph- 

 cation of thefe fucceUive fteps, it will be proper to attend to 

 the figns of the quantities, where one of them, which in our 

 cafe is the middle one, has its focus negative with con- 

 verginsT rays. To exemplify this procefs in a triple objeft- 

 glafs for parallel rays, let B reprefent the outermoft lens, 



. , m — n 



which we will confider as a double convex lens witli 



n 



= a = 0.53, and r and R each = 10 ; let C be the double 



concave of fimilar radii 'r and 'R, and with = h 



n 



-=. 0.6 ; and let D be a plano-convex, and confequently R 



. m — n 



infinite, but r = 10, as before, and = <•=« = 0.53: 



n 



then for the focus of B, putting — = a, we have — X 



drR drR _ drR 



adR + adr— Rr ~ adR— rR 



dr 



dR + dr 



a 



{adr being neglefted, when R is infinite) 



; and 



ad— r 



fince d is alfo infinite with parallel rays, the expreffion be- 

 comes — = F, as in our firft table of theorems for the 

 a 



refrafted foci of lenfes, for the firft lens B. This expreffion 

 is now put for d, when we come to confider the theorem as 



applied to C : here we have — = i, and the expreffion 



P 



I drR drR 

 becomes -7- x —^, or — j-~g _; 



b . . .„ '■R bdr + idR — rR 



dr + dR- 



r 



then, as r is taken equal to R, it will be 



2bd- 



'F 



of the lens C. Now, if in this expreffion we fubftitute — , 



a 



the focus of B, for d, we have for —j—r — x 



2bd— r a 



2 br — r = 



2br — i 



2b- 



for >?, the compound 



focus of B and C, or rather, as the rays fall converging 



on C, and 2 3 is more than a, = -. . Again, 



— 2 b — a 



this quantity will become d for the lens D, and puttf.Tg 

 the fame fubftitution as before, in the general theorem 



for D, where 

 drR 



put for p, we (hall have — x 



drR 



dr + dR - 

 dr 



R' 



:dr + cdR~ rR' 



or, when r =^ R^ 



2cd ■ 



hence we obtain 



2b — a 



lex 



2b~ a 

 — *, or 



2cxr— 2b— axr 2c — 2b + a 



pound focus of all the three lenfes, B, C, and D. Let 

 us take, by way of example, three lenfes as follow ; Wz. 

 B, a double convex lens of crown-glafs, with its refraftive 

 power by experiment =: 0.53 = a, and with equal radii, 

 where r and R are each =: 10; let the fecond lens C be a 

 double concave of flint-glafs, with the fame radii, and of a 

 refraftive power = 0.6 ; and let the third lens D be a plano- 

 convex of crown-glafs, of a refraftive power of 0.53 alfo, 

 with R ^ io likewife ; then, according to our laft expref- 



fion 



10 



10 



2c — 2b + a 



2c 



10 



2b + c' 

 = 25.6 = *. 



3<r - 2b 

 In this way 



1.59 -1. 2 0.39 



the compound focus of any number of lenfes may be deter- 

 mined, and the courfe of the rays might be traced in a geo- 

 metrical figure out of one glafs into another, until they come 

 to their ultimate focus. 



For inftance, let us confider S and /, in £g, 1 2, to be 

 two parallel folar rays incident on the firft lens B, at the 

 points b and b : thefe rays, on entering, are bent towards 

 the axis <if^, and then from the points of emerfion tend to 

 their piincipal focus f; but being intercepted by the double 

 convex lens C, they diverge, after entering at c and c, in a 

 direftion which points backwards to the virtual focus $ ; but 

 in their progrefs, they again become incident on the piano- 

 convex at the points d and d, and are again refrafted to- 

 wards the axis, and meet in a diftant point *, which is, 

 therefore, the compound focus of all the three lenfes. And 

 if we conceive the parallel rays, S and s, to be pencils of 

 folar rays, that difpcrfe on entering the convex lens B, they 

 will difperfe in a contrary direftion on entering the con- 

 cave C, and will again, on entering the plano-convex lens D, 

 have the excefs of difperfion of C counterafted by a fecond 

 oppofing difperfion of D, and, inftead of coming to unite 

 at the diftant points P and Q, to which they tended on 

 entering D, ihey will meet, by virtue of the prevailing re- 

 fraftion of the two lenfes B and D taken jointly, over the 

 refraftion of C taken fepai-ately, at the compound focus <J>, 

 where the image of the fun will be formed ; and if both the 

 focal diftances and radii of curvature of all the lenfes were 

 achromatically adjufted, as we fhall prefently direft, the 

 image would be free from colours, and well defined. 



The fame determination of the foci f, <p, and *, in any 

 combination, mayj however, be obtained more conveniently 



in 



