TELESCOPE. 



rcfrafted yo/ar focus; whereas, by Table II., the geome- 

 trical focus is -7 = >• = 21.5. If the refra&ive power 

 a 



of the glafs, and confequently the value of a, had been given, 

 and it had been required to determine the radius of the tool 

 that will grind the given glafs into fuch equal radii as will 

 give the refrafted folar focus exaftly 17.82 inches, then 



f 

 the theorem — = _/" becomes, hy tranfpofition, 2 a/ = r, 



and 1.298 X 17.94 = 21.5, as before. In a fpecimen of 

 crown-glafs ground to the fame radius, where d was 4I4>75 



inches, 



dr + rf 



gave « = 0.5318, and confequently m : n 



IS 1.531!: 

 21.5 



: I, with which lens the true folar focus was 

 20. 2 1 4 ; and if the lens had been a fingle convex* 



1.0636 



the true folar focus would have been 



40.428, 



r 21.5 



a ".5318 



or double the length of the former, while the geometrical 

 focus for parallel rays, by Table II., would have been 



— — = 2 »• = 43.0 ; fo that for many praftical purpofes, 

 a 



where m — n is known in the particular glafs ufed, the 



advantage of the theorems in Table I., over thofe in 



Table II., muft be evident 



Again, let us fuppofe that the ratio of wz : n is afcertained 



by a prifm of any fpecimen of glafs, or by Dr. WoUafton's 



or Dr. Brewfter's inftruments for this purpofe, and that it 



is known to be 1.599 • ' > then we know that .599 = a, 



as before ; and let it be required to find the refrailed focus 



■with diverging rays, when the radiant is as before at the 



dillance of 417.25 inches, and the radius of curvature 



of each furface 21.5: in this cafe the theorem is 



dr J- • 1 417-25 X 21.5 



= /, or, m numbers, — 



2ad- I- 2 X 0.599 X 417.25 — 21.5 



= 18.75, ^^ before; and in this way the terms given may 



be varied at pleafure, and the theorem made applicable to 



the cafe in queftion. If the rays had been converging in 



— dr 



the laft calculation, the theorem would have been r— 



— 2ad— r 



■= f; or, changing all the figns, (which are here negative, 

 becaufe the diitance is more than infinite, that is, the rays 



more than parallel,) the fame may be taken = f, 



■zad -^ r 



417.25 X 21.5 



or ^^—^ — = 17.204, which IS lefs 



2 X 0.599 X 417-25 + 21.5 



than the folar focus by 0.74 of an inch. In this cafe the rays 

 muft have pafled through fome other glafs, in order that 

 they may proceed in a ftate of convergence before they enter 

 the lens in queftion, and the focus of that other glais is here 

 confidered as the radiant point from which the rays proceed in a 

 ftate of convergence ; and this confideration leads us naturally 

 to inquire into the nature of a focus when two glaffes are 

 employed jointly to produce it, under the different circum- 

 ftances of tigure and diftance. 



Suppofe the parallel rays A N and B M, mjlg. 8, to fall on 

 a plano-convex lens M N, with the curved face turned to the 

 radiant, and to be refrafted to its focus at F ; then if another 

 plano-convex lens be placed in the line of its axis, at any dif- 

 tance lefs than Cr,fo as to intercept the converging rays, tliey 

 will be refrafted ftill more, and will now converge into the 



7 



(hortcr focuey, which it therefore called the eompound focut of 

 both the Icnfes. The angle fubtcnded at/, where the eye is 

 fuppofed to be placed, and which is called the optic angle, is 

 now larger than that formed at F by the fuft lens, and it 

 equal to what would be formed by the imaginary double 

 convex lens E E, the focal diftance of winch would be 

 y/- Now let C F be put = F for the fi.cal diftance of 

 the lens N M ; OP :== ji for the focal diftai.c of lens G H J 

 ^""^ Q/ — * ^°^ llic focal diftance of the imaginary double 

 convex lens EE : alfo let Of, the compound tocal diftance, 

 be = /; and CO, the diftance between the lenfes NM and 

 G H, be = D. As tile rays, which tend to the point F, 

 after leaving the lens NM, fall on the Iciis G 11 converging, 

 let us call O F = ^, and then, by common opticB, we Ih^ 



yf 



have d = = F — D ; from wliich equation we get 



¥ + y — D : Y — D :: y :f; and from this analogy the 

 compound focal diftance O/ is calily obtained. In like 

 manner, the parallel rays LG and SH are rcfradted by the 

 lens GH, now fuppofed to bo the firft lens, to the Icn* 

 NM, as they proceed towards the point i ; but are refrafted 

 to the nearer point (f, which is the compound focus on the 



other fide ; and now we have Cy ^= ■= — ^ := _y — D : 



whence F + y — D : y — D :: F: <^ = C<!>, which ig 

 therefore known again, becaufe of the fimilar triangles F N C, 

 FGO, and/Eg,/GO; and becaufe EQ = NC, we 

 have CF : OF :: NC (^ EO) : GO :: Q/: O/; that 



IL 

 F-D 



is, F : F — D :: K -./; and, therefore, p— ^ = «• E"t 



F — D X y 

 we had above :;:; ;^ — f; which being fubftituted 



for_/", will give — 



F+^- D 



Fy 



= X ; from which theorem ow 



F+^-D 



problem for finding the compound focus of two lenfes, or 

 rather the focus of one lens, that fliall have the fame focal 

 diftance and vifual angle as two given ones placed at a 

 given diftance (hall have together, may be tlius found : wi. 

 " divide the produd of the two focal lengths of tlie given 

 lenfes by their fum, leflencd by their diftance, and the 

 quotient will be the focal length of tlie fingle lens, as re- 

 quired." By way of exemplification, let the focus of N M 

 be put = 6 inches, and that of GH = 4; and then, fup- 

 pofing the curved furfaces turned to the radiant, which 

 is called the beft pofition, as will be foen hereafter, 

 and the diftance z=. 2, we fliall have, by the theorem for 



this purpofe. 



6 X 



24 



= 3 for the focus in quef- 



6 + 4 — 2 8 

 tion ; but if the diftance had been = 3, then the refult 



would have been 



6x4 24 , _ .^ 



^ — = —, or 3.42 nearly. But if 



6+4-3 7 



the diftance had been made 4, equal to the focal diftance 

 of the lens G H, the compound tocus would have been 4 

 alfo ; and if 6, equal to the focal diftance of the lens N M, 

 the compound focus would ftill have been 6, without any 

 gain of magnifying power in either cafe, over what would 

 have accrued from the refpeftive fingle lens ; alfo if the 

 lenfes are brought into contadl, that is, if D = o, then we 

 ftiall have the compound focus the ftiorted poflible, viz. 



6x4 

 6 + 4-0 



24 

 = — = 2.4. 



But dlJimSrefs is an objeft 

 of 



