TELESCOPE. 



concave is by calculation .1653, and '. T^" = 2.074. is it« 



.1653 



proportional aberration. But as the thickiiefs, breadth, and 

 geometrical focal length of every lens, of whatever form, 

 mull, from the properties of the circle, be in proportion to 

 each other (fee Martin's New Syftem, art. 705.), 'F may 

 be taken at once, inftead of ufuig T and 'T with tlieir cal- 

 culated values (which require fome operations), and then 

 the work will be greatly facihtated ; thus 1.361 x T x 

 1.524 = 2.074 X 'T = 'A. Now, as this quantity 2. 074 

 bears the fame proportion to 1.361, as the focus of the con- 

 cave does to the focus of the convex, it might be concluded 

 that this would be the proper aberration to correft tlie aberra- 

 tion 1. 361 of the convex lens; but this is not the cafe, for, 

 firft, the longitudinal aberrations ariling from the figure are 

 not in ihe Jtmple proportion to the foci of the lenfes refpeft- 

 ively, neither is the quantity the fame with the flint as with 

 the crown glafs. Martin afferts that the fpherical longi- 

 tudinal aberrations are to each other, in like lenfes of dif- 

 ferent focal lengths, inverfely dL?,\.\\e fquares of tiie foci refpeft- 

 ivcly ; confequently, in our example, thefe aberrations would 



• 2C7C 'A 



- ■ = .883 =r z, the correfting divifor, and alfo — = 

 .2914 •' ° » 



^■074 

 .883 



= 2.348 = 'A corrcdcd. 



Having now afcertaincd the aberration 2.348 x 'T of the 

 concave lens, that will balance the aberration 1.361 x T of 

 the convex, we muft proceed to determine the ratio 'r : 'R 

 of the concave, that Aiall have exaftly thie aberration : to 

 be able to do this without a table of aberration, requires an 

 acquaintance with quadratic equations ; for the proportion 

 of the radii '/• and 'R muil be invciligated from the correded 

 aberration which we have now afcertained. 



27'-' + 6 rR + 7R' 



X T = A (by 



I ft. We have 



6 X rr^"- 



the general theorem) = 2.348 x 'T, as before found ; but 

 we make no diilinftion between r, R, T, A, and V, 'R, 

 'T, 'A, that we may fimplify the fymbols : this equation, 

 by evolution of »- + R.1^ in the denominator, becomes 



27 r^ -I- 6rR -I- 7 R' 



be inverfely as F' -. 'F^ ; or as 13.9 x 13.9^: 9.12 x 9.12'!; r^ + 2 r R -j- R' 



X g- = 2-348 



X T : now by put- 



that is, as 193.21 : 83.174, or as •) , _ « f ! '^"'^ *'"S R — i, there will be 



27r^ -f 6r + 7 



X — = 2.348 



when TuUey took 0.585 ^ 'A, this aberration was found 

 much too little ; for when he had gronnd tke lenfes with 

 curves to produce this aberration, he found that the eye -tube 

 required to be drawn outwards more than inwards by the 

 fcrew, from the true focal point, before the image difap- 

 peared, which is a proof that the concave had lefs than its 

 fhare of aberration ; it being confidered as a tell of good 

 correftion, when the image difappears at points of the tube 

 equally diftant from the point of diftinft vifion, accordingly 

 as the tube is pulhed in or drawn out from its focal point. 

 And here was probably the difficulty that Martin experi- 

 enced between his theory and praftice. Neither was 

 the aberration thus obtained in due proportion, when cor- 

 refted by the fimple ratio of the two divifors 2 a : 2'a, or 

 1.056 : 1.147, for the difference of the refraftive powers; 

 for as 1.147 : 1.056 :: 1.161 : 1.253 ; but 1.253 X 'T ^ 'A 

 was ftill too little for due corredlion. Though the telefcope 

 was achromatic by virtue of the ratio of the foci of the 

 crown and flint lenfes, yet there was a want of perfeft dif- 

 tinSnefs, owing to the deficiency of aberration attaching to 

 the concave lens. After a multiplicity of inveftigations, 

 calculations, and praftical trials, TuUey at length difcovered 

 a method of balancing the oppofite aben-ations, which he 

 has continued to pradlife with fuccefs for years, and which 

 is therefore no new projeft. The method is this : the value 

 of 'A (2.074 X 'T) being firft determined from A, in the 

 ratio of F : 'F, as above explained, the correQing number is 

 thus obtained ; if we call the fquare root of the cube of the 

 refrafted focus of the convex r= x, the geometrical focus 

 being taken = i ; and put alfo y for the fquare root of the 

 cube of the refrafted focus of the concave, when its geo- 



metrical?focus is = i ; then — = z is the correfting number, 



X 



by which the proportional aberration, before determined, 

 muft be divided, to gain the proper or correded aberration, 



'A 

 now exprelfed by the fymbols — . In the inftance before 



us, the calculation will be .947' = .8492781, and its fquare 

 root = .2914 = x; and ^/ .872' = -2575 — y ; then 



+ 2r -f I 



T 



X T ; or, dividing both lidts by — , 6 x 2.348 = 14.088, 



27r^ -f- 6r + 7 . , . , . . . 



or 1 4. 1 = _, ; and multiplymg both by tJie 



* -r 2 i -f- I 



denominator, 14. 1 r^ 4- 28.2 /■ 4- 14. i = 27 /•' -(- 6 r + 7 : 

 then fubtradling equal quantities from both, there remains 

 22.2 r -f- 7.1 =; iz.gr^i and by tranfpofition, I2.9r'— 22.2 

 r = 7.1 for the quadratic. Now to find the root, we have firft 



12.9 



7-1 

 12.9 



; and adding the fquace of half the 



_ . , 22.2 ii.il' 



co-emcient, r r 4- 



12.9 12.9 



7.1 II. IP 



h ; there- 



12.9 12.9 



fore the root r — 



II. r 



\/ 



12.9 

 II. I 



= x/ 



7-1 _^ iJj^V 

 12.9 12.9' 



and r = 



4- 



have 



7.1 II. ii 

 12.9 12.9 12.9 

 Laftly ; to colleft the aggregate of the values of r, we 

 7.1 1 1 . iV 



12.9 



•55' 



^/i.29= 1. 135; 



12.9 



likewife 



.74, and V -55+^74) = 



I T 



860 ; therefore 1.135 



12.9 



-r .860 = 1.995 ^^ '"' which was defired ; and the ratio 

 r : Ri, which we now put ag.iin 'r : 'R =■. 1.995 • ' > ''"'1 

 which in TuUey 's Table ftands 2:1. After having thus 

 determined the ratio of the radii 'r and 'R to be 2 : i very 

 nearly, we muft now find the rational geometrical focal dil- 

 tance of this concave by the fourth of our pradical theorems 



above exemphfied ; viz,, from ^„ , we firft have 



r -f- K 



= 1.333 ! •'""'' *' *'''■ geometrical focus is known 



2x2x1 

 "14-2 



to be 13.9, we 



1 3.0 

 have alfo —^-^ = 10.428 = 'R, and 10.42S 



••333 



I i 2 X 2 



