TELESCOPE. 



pal or folar tocus of a lens, the rays will emerge and con- 

 tinue parallel, on account of the other conjugate focus being 

 at an infinite diftance ; and for the fame reafon, when an ob- 

 jeft, viewed by a fingle lens, is placed in its principal focus, 

 the rays will enter the eye in a parallel ilate, and will be 

 converged to a point on the retina by the humours of the eye, 

 and a number of thefe rays crofling will form a pitlure be- 

 hind the eye of the objeft viewed : for, what is one of the 

 mod remarkable properties of refrafted rays coming from 

 a luminous objeft, they bring with them not only the figure, 

 but the colours of the objeft viewed, and form a pifture or 

 image of it, in the place where the different pencils of rays 

 crofs one another ; and, what is equally remarkable, this 

 pifture is not vifible until all extraneous light is excluded. 

 We- will not pretend to explain this wonderful property 

 of a lens, that direfts the tranfmitted rays fo as to form 

 a pifture of a diftant objeft in its focus, but merely mention 

 here, that, without it, no telefcope, microfcope, camera 

 obfcura, or magic lantern, ceuld be conflrufted on dioptric 

 principles. 



After having (hewn, by our foregoing theorems, how any 

 focus, folar, proper, conjugate, or virtual, may be determined 

 of a fingle lens, or of a combination of two lenfes w ith the 

 intermediate diftance given, the fame might be done for any 

 number of lenfes, by confidering the compound focus of the 

 firft two lenfes, as the focal diftance of a fingle lens, to be 

 combined with the third lens, and fo on till all the lenfes are 

 included. Dr. Smith has given, in his Optics, chap. v. 

 the application of Cotes's theorem " for determining the ap- 

 parent diftance, magnitude, fituation, degree of diftinftnefs 

 and brightnefs, the greateft angle of vifion and vifible area of 

 an objeft feen by rays fuccefTively reflected from any number 

 of plane or fpherical furfaces ; or fuccefTively refrafted 

 through any number of lenfes of any fort, or through any 

 number of different media, the furfaces of which are plane or 

 fpherical, with an application to telefcopes and microfcopes ;" 

 which account our readers may confult with advantage : 

 hut as the illuftrations and demonftrations demand more 

 plates than can be given to this article, in addition to the 

 eight we have had occafion to introduce, we have been 

 obliged merely to refer to them in this place. 



We propofe, however, to fubftitute fome praSical theo- 

 rems, derived from our tables, which we have been favoured 

 with by Mr. Tulley, that will be found extremely ufe- 

 ful to the working optician, who muft be fuppofed, gene- 

 rally fpeaking, unable to transform the theorems vvhich we 

 have given in our tables, for the purpofe of finding the focal 

 diftance of a lens, or of a combination of lenfes already con- 

 ftrufted ; and which tabulated theorems arc principally ufeful 

 for determining the powers, and for explaining the conftruc- 

 tionofTm inftrument to which they are apphcable. 



Prad'ical Theorems. 



I . When r, the radius of one face of a lens, is given, and 

 r, its principal geometrical focus, to find R, the radius of 



the other face, the theorem is 



rY 



R for a double 



2r— F 

 convex : thus, let r = 9, and F = 10.3 inches, and the cal- 



, . ... , OX 10.5 02.7 , , 



culation will be — — = -— - = 12 nearly, the 



9x2- 10.3 7.7 



truth of which may be proved by our theorem for parallel 



2 r R 



fays with a double convex lens, in Table II. tx'z. _ ,or 



' R + r 



Vol. XXXV. 



9 X 12 X 2 216 , r I 



= = 10.3, as before very nearly, for the 



9 -h 12 21 ^ ^ ' 



required focus ; and when the refraftive power or ratio be- 

 tween the fines of incidence and of rcfraftion is given, this 

 geometrical may be converted into the refrafted focus by 

 the quantity 2 a, ufcd as a divifor ; or, on the contrary, the 

 refrafted focus may be turned into the geometrical focus by 

 ufing 2 a as a multiplier. 



2. With a menifcus lens, where r, the convex fide, is given, 



r F - 



together with F, the theorem is .= for finding R, the 



concave fide. 



3. But when the concave fide of a menifcus is given 

 with the focus, to find the convex, the theorem becomes 



-^ = ^- 

 2r-f F 



4. When the focus of a double convex lens, and the ratio 

 between its two radii, are given, to find the aftual radii rand 



2 r R 



R refpeftively, firft our theorem in Table II. = ■=. F, 



R + r 



will give the focus, on a fuppofition that one fide is unity, 

 and the other any given quantity that forms the other term 



of the ratio ; fuppofe as i : 4 ; thus 



1x4x2 



= 1.6, the 



'+4 



rational focus ; then fuppofe the focus given ^12, and 

 there will be this analogy, as 1.6: i :: 12 : 7.5 = r; and 



1.6 : 4 :: 12 : 30 = R, or otherwife — ^ = 7.5, and 



alfo 



will be the refpeftive 



1.6 

 radii r 



and R, as 



7.5 X 4= 30 

 before. 



5. When the compound focus of two convex lenfes, and 

 the feparate focus of one of them, are given, to find the fepa- 

 rate focus of the other, that fhall be fuitable to form the 

 combination ; if we puty"= the focus of the lens given, F =: 

 the combined focus, and x — the focus of the lens required, 



/F 



for 



the theorem for this ufeful purpofe is 



example, let f= 36, and F = 15, then ^- ^ = 2^.7 



nearly, for the focus of the lens required, which is a pofitive 

 focus, becaufe both lenfes are double convex, and might be 

 plano-convex, or one double convex and the other plano- 

 convex, or even menifcus, as the ratio of the radii r and R 

 may be difregarded when the focus only is the objcft of con- 

 fideration. But whatever be the forms of the curves rela- 

 tively, F, the compound focus of two lenfes, or more, will, 

 in praftice, be the refrafted focus ; and, therefore, in this 

 theorem, y and .v will alfo be the refrafted foci of the fepa- 

 rate lenfes, and, confequently, when the geometrical focus 

 ofyis given, it muft be converted into the refrafted focus 

 by the divifor 2 a, before the calculation is entered upon ; 

 it being necelTary that all the terms be of the fame deno- 

 mination. 



6. If F, the compound focus, be longer than /, the 

 focus of the given convex lens, as is the cafe in the conftruc- 

 tion of a double achromatic objcft-glafs, then the lens re- 

 quired will be concave, of which the focus x is fought, and 



/F 



the theorem becomes — .= 



F-/ 



Hh 



Let us, in this example, 

 rcTcrfe 



