TELESCOPE. 



fifld the point /, or foca! diftai'.cc D/, of the ray D M, 

 coming from tlie radiant B, after being twice refrafted, 

 viz. at M and N, the points of ingrefs and egrofs, is the 

 general problem of d'uiplrics. 



In folving this problem, our original theorem for fimple 

 refraftion gives us m df — m d r - n rf + n df, ( making 

 M F=; (?,) from which equation we deduce this expreflion ; viz. 



■h d 



whxh 



gives 



thi 



_ j3^ _ B C A F 



T ~ o'^r ^ 7" ~'C F ^ A B' 



verfal canon : viz, "the ratio of the fine of incidence to the fine 

 of refraftion, is compounded of the ratio of the dilLinces-of 

 the conjugate foci B and F from the centre C, and of the 

 ratio of their diftances from the vertex A." This rule be- 

 ing general, finds the focus /, after the fecond refraftion 

 at N : for let Df = f, the radius G D = R, and the thick- 

 nefs of the lens A D = / ; then we have for the refraftion 



out of a denfe medium into a more rare one, - — • = — -- 



m /G 



/D (p+R-/ / <• , 



X ■ t._ = i^ — X -^ — : irom whence we jret 



FD I + R (? — I 



f =: = D /", the focal diftance 



m 1(5 — m ^ + m R — B 9 + » / 



required. If we omit the tlu'cknefs of the lens /, as be- 

 ing inconfiderable, we may reduce the equation into a more 



fimple form ; for we fliall have 



this will 



give ip =! 



»jip -(- ni R 



■ n? 



= /; and 



= /• 



n R -)-«_/"— m/ md—nd- 

 which equation reduced gives 



ndrK 



mrd — nrd + mdR — ndR. — nrR 

 But to reduce the number of fymbols, let us put 



= a, and confequently m — n = a, when n is unity, 



drR 



and then this equation becomes 



■=/' 



ard \- aRd— rR 



and this may with propriety be called the vniverfal dioptric 

 iheorem, by which the refraSed focus of a ray may be deter- 

 mined after paffmg through any lens of a given denfity, or 

 refrafting power. 



The theorems in the fubjoined Table I. are all derived 

 from the univerfal theorem thus determined, and will be 

 of great ufe to the optician to determine the refraSed focus 

 of any lens, and for any diftance of the radiant, which re- 

 frafted focu», with parallel rays, will be always equal to 



the true, or nicely meafured/war/oiru/, where d is infinite ; 

 vvhereas the focus determined from the old tht-orems in 

 Table II. where the value of a is omitted, is always the 

 geometrical focus, determined on a fuppofition that the fine 

 of incidence is to the fine of refraftion in all glafles as 3 : 2, 



in which cafe ,1, e. = 



= \ invariably, and \ r. 



in a double convex lens of equal radii, of whatever rcfrac 

 tive power, ;=/. In order, tiicreforc, to dl(lingui(h the 

 focus determined from the theorfms in Table 1., from thofe 

 arifing from the theorems in Table II., we will always call 

 the firll the nfraSed focus ; which is that from which the 

 powers of a telcfcope or of a microfcope are derived ; and 

 the fecond we will denominate the fiomf/nW focus, which 

 is that arifing from the fimple confidcration of the radii of 

 curvature, without reference to the refraftive power of 

 the glafs, otherwife than as we have ftated ; but is not- 

 withftanding ufeful to opticians in the formation of the 

 curved faces of their grinding and polifhing tools ; for when 

 the curves of a lens of a given refraftive power are to be 

 formed, to produce a given refraftcd focus, as is frequently 

 required in the nicer optical inftruments, the rcfrafted 

 focus mull firfl be converted, by means of the value a of 

 its refraftive power, into the geometrical focus, and then the 

 radii of curvature belonging to this calculated geometrical 

 focus, will be proper for the tools of the Ions of a given re- 

 fraftive focus. Hence we confider it as a matter of great 

 praftical importance, to give, in the fame place, two tablet, 

 one for finding the refraSed, and the other for finding the 

 geometrical foci of fuch Icnfes as are ufually applied in either 

 a telefcope or microfcope of the refrafting conftruftion. 

 In all cafes where the glafs has two radii, the firft, as we 

 have faid, will be douominatcd by r, and the fecond by R- 



But before we proceed to tabulate our theorems for both 

 refraSed and geometrical foci of fingle lenfes, we wifii it to be 

 clearly undcrllood by our readers, that the praftical appU- 

 cation of thofe theorems, and of others to be derived from 

 them, to the purpofe of aftual conftruftion of achromatic 

 objeft-glaffes, and of achromatic eye-pieces, is intended to be 

 the leading feature of our article ; for while volumes have 

 been filled with abftrufe calculations, derived from formulae 

 of the moft celebrated mathematicians, the refults of thofe 

 calculations have never produced proper data for the ufe of 

 opticians ; more particularly with rcfpeft to achromatic 

 objcft-glafTes, which cannot be conftrufted from any calcu- 

 lations but what are grounded upon experimental examin- 

 ation of the identical Ipecimens of glafs that are intended to 

 be ufed. And we flatter ourfclves, that the information we 

 have to lay before our readers on this interelling fubjcft, 

 will be ihejirfl that has yet been publifhed in fuch a praSical 

 form as will facilitate the labours of the working opticiai;. 



Gg I 



Tabie 



