TELESCOPE. 



mm 

 nn 



whofe diameter is '^^ x ^rir: very nearly, if they were all 



fine of incidence out of glafs into air be to tlie fine of re- 

 fraftion as n to m ; the rays which come parallel to the axis 

 of the glafs ftxall, in the place where the imago of the objed 

 is moa diftindly made, be fcattcred all over a little circle, 



?1 

 DD 



equally refrangible. As for inftance, if the fine of in- 

 cidence n be to the fine of rcfraftion m as 20 to 31, and 

 if D, the diameter of the fphere to which the convex fide 

 of the glafs is ground, be loo feet, or 1200 inches, and 

 confequently the telefcope about 100 feet long, and S, the 

 femi-diamcter of the aperture, be two inches ; the diameter 



, . m m S^ -111 



of this circle of aberrations, that is x =r^, will be 



nn IJL) 



3 ' ^ 3' X ^^ tttSj'Aw parts of an inch. 



20 X 20 X 1200 X 1200 



But the diameter of the little circle through which thefe 

 rays are fcattered by unequal refrangibility, will be about 

 the 55th part of the breadth of the aperture of the objeft- 

 glafs, which is here four inches. And therefore the aber- 

 ration arifing from the fpherical figure of the glafs, is to 

 the aberration arifing from the different rciiangibility, as 

 TTs'AVo-o to ^f , that is as l to 5449 ; and therefore, being 

 if) comparifon fo very Uttle, defervcs not to be confidered in 

 the theory of telefcopes. If we fuppofe the little circle of 

 aberrations arifing from unequal refrangibility, to be 250 

 times narrower than the circular aperture of the objeft-glafs, 

 it would contain all the orange and yellow, and would per- 

 mit the other fainter and darker colours to pafs by it, which 

 perhaps may fcarcely affeft the fenfe; yet even in this cafe, the 

 aberration caufed by the fpherical figure, would be to the 

 aberration caufed by the unequal refrangibility, in a loo-feet 

 telefcope, but as tttVtWtj- to -^, or only as I to 1200, 

 which fufSciently proves the propofition. Q. E. D. 



Carol. I. — If the focal diftances and apertures of arefleft- 

 ing concave and a plano-convex glafs be both the fame, the 

 diameter of the circle of aberrations, caufed by their figures, 

 will be above 30 times lefs in the refleftor than in the re- 



fraftor. For thefe diameters are ^ -^_- and =— - X 



16 CF' T^^iry- 



which 



to = 



, 31 X 31 

 II X n' 



Hence, if the length of each telefcope be 100 feet, the 

 lateral aberrations in the refleftor wonld be 30 x 5449, or 

 163470 times lefs than the lateral aberrations caufed by un- 

 equal refrangibility in the refraftor. 



Carol. 2. — The number of pencils, fome of whofe rays 

 are mixed together in every point of a confufed pifture, is 

 as the area of the circle of aberrations of the rays in any one 

 pencil ; and confequently the mixture of the rays of dif- 

 ferent pencils, caufed by the fphericalnefs of the figure of 

 an objeft-glafs, if they were all alike refrangible, would be 

 to their mixture caufed by their unequal refrangibihty, as 

 1 to 5449 X 5449, or 29691601 in the prefent inftance. For 

 conceiving any point in the confufed pidlure to be a centre of 

 a circle of aberration, it is manifeft that all other equal circles 

 of aberration, whofe centres fall upon the firft-mentioned 

 circle, will cover its centre, that is, fome rays of as many 

 pencils will be mixed in this centre as there are points in the 

 circle iti'elf ; or, which is the fame thing, the number of 

 pencils mixed in this centre is as the area of the circle of 

 aberrations." 



Double achromatic OljeH-glaJfes. — From thefe five propoji- 

 t'tons, and the corollaries deduced from them, in all of whicli 

 tlie ratio of the fines of the angles of incidence and of re- 

 fraftion out of air into glafs :s taken as 3 : 2, (which 

 anfwers nearly to the French platc-glafs,) our readers will 

 fee, that when any fingle lens is ufed as the objeft-glafs of a 

 refrafting telefcope, there will be not only fringes of colour, 

 but indiftinftnefs in the image formed at its focal point, 

 arifing rcfpeftively out of tiie two kinds of aberration, the 

 prifmatic and the fpherical. But Dollond has Ihewn, that 

 thefe aberrations are not the fame in all forts of glafs : the 

 former depends on the difperfive power of the glafs ufed, 

 and the latter on the ratio of the radii of curvature of the 

 two furfaces of the lens. The difperfive power of a prifm 

 of any fpecimen of glafs will be to that of another like prifm 

 of a different fpecimen, as the lengths of the prifmatic 

 fpeflra, formed by them, are refpeftively to each other ; 

 and if the foci of two lenfes of different difperfive powers; 

 one convex, of crown-glafs for inftance, and the other 

 concave, of flint, be made direftly as their difperfive 

 powers, and be placed contiguous, fo that the convex lens 

 may receive the rays firft, and be of the (horter focus, or 

 thicker, its difperfive power will be fo counterafted by 

 the oppofite difperfive power of the other thin lens of 

 longer focus, that the extreme or prevailing colours of 

 the primary fpcdlrum, being reverfed, will both difappear ; 

 and a fecondary fpeftrum, compofed of the remaining in- 

 termediate colours, will be very inconfiderable in a good 

 achromatic objedl-glafs thus compofed. If the refrafted 

 focal diftances of the two lenfes remain unaltered, when 

 duly proportioned, as i : 3, or nearly fo, the proportion 

 of the radii of the furfaces may be altered at pleafure, fo as 

 to produce their due proportions of fpherical aberration. 

 To effedt the defirable purpofe of baniftiing the fpherical 

 aberration as much as poflible, the optician is obliged to 

 calculate the aberrations belonging to convex lenfes of dif- 

 ferent unequal radii, in order to make the contrary aberra- 

 tions of the concave as equal thereto as may b* ; and for 

 this purpofe the general theorem of Huygens is peculiarly 

 adapted, which we (hall, therefore, introduce and ex- 

 emplify here, before we proceed to the conftruftion of an 

 achromatic objeft-glafs. According to this theorem, if we 

 put r for the radius of the firft furtace of any lens, or that 

 which firft receives the incident rays ; R for the fecoud fur- 

 face ; and T for the thicknefs of the lens : then the aberra- 

 tion arifing from the figure of any lens, concave or convex, 



•ill, 27>-'-h6rR4- 7R' T • fii /o 

 will be = — ^ , X 1 univerfally. ( See 



6 X rT^' 



Martin's New Elements of Optics, part vi. chap. iii. and 



Dr. Smith's Optics, book 2. chap, xiii.) When the 



centres of the curves are on oppofite fides of the lenfes, the 



figns are as here put down ; but if thefe centres are on the 



fame fide as in a menifcus, then the fign of r, or of R, muft 



be negative, as the cafe may require. For inftance, let us 



firft put r and R equal, and each = i ; then, as unity is 



not altered by multiplication or divifion, we (hall have the 



27 4-6-4-7 



fimpleft cafe, viz. — ^^ = 44 = !, or 1.66 of 



^ 6x2x2 



T, for the longitudinal aberration, and it will make no dif- 

 ference which face of the lens is turned to the radiant. 

 Secondly, let us take r = i, and R =: 2, in which cafe 



-2 = ^4, or 4 of T, very nearly. 



0x9 ' 



But if we reverfe the fides in pofition, by making r = 2, 



and 



we (hall have 



