602 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779- 



That the solution may be applicable to telescopes, it is proper that ad = az 

 <7B = flz. Then put ad = f the focal distance of the lens a; ob =/ the focal 

 distance of the lens a; qa ^ f — /; ab = oa — ab = p — 2f; bc = x; cD 

 =zy; (p the focal distance of the lens c. Hence bd = ad + ab = 2 (f —/), and 

 BD = BC -|- CD = r -j- y. 



The two values of bd evidently give, 1st, x -\- y z= i [f — /). 



That the image b, given by the lens, a, may be seen at the distance bc; and 

 that the direction of the ray bhd may form a relative focus in d, whose distance 

 may be equivalent to ad, it is necessary that aB X cd = ad X bc, namely, 2dly, 

 fy = ¥x. 



That the object b seen in the direction bh, may form a focus in d, it is neces- 

 sary that the focal distance of the lens c (viz. the distance tp) have this condition, 

 (p X BD = BC X CD, viz. 3dly, 2p X (f — /) = xy. 



From these f 1° x + j/ = 2 (p — /), ) we easily and incon- 

 3 conditions, s 1° fy = F;r, > testably find what 



viz. (.3° 2f> X (f — /) = xy, ) follows: viz. 



BC = ar ^ — — Vt-^" the distance from the focus b to the lens ch, 



2f X (F — /) ( the distance of the relative focus d, with respect to the 



CD — 2/ — J, j^j ^ ^ lenses a and c, 



_ 2f/x (LzvJ the focal distance of the lens c. 



This solution is general: but to adapt it to a particular case, which may be 

 proper for practice, Mr. J. investigates what relation ought to take place between 

 the distances f and / when ^ is =/. This supposition gives ^ = ^ ,^ ''^fZ/^ 

 z=f; from which we easily extract the relation sought, viz. f =/(v/5 + 2); or 

 this, which comes to the same thing, /= f (^5 — 2). 



^ ^5 + 2 = 4.2361 j-Therefore for tlie^^ ^ 4.2361/ /The relation of 

 B"t| :>? I 2 = 0.236l| ;^1^/JSJ/ = 0-'^36.p|the focal dist. 

 The Application of the general Formula to the particular case of the equal 



lenses a and c- 

 Let AD = p, the focal distance of the lens a, 



-az=f= 0.236 If, the focal distances of the lenses a and c, 

 aA = F — /= O.7639F, the distance between these lenses, 

 AB = p — 1f= 0.5278F, distance, 

 BD = 2(f — /) = 1.5278P, distance, 



The relation 

 /• = 0.2361 p, 

 found for the ■< 

 focal distances 

 gives 



2/" X (y — f) 

 BC = Vv-^ = 0.20 18f, distance. 



gP_ 2F X (t^ -/) _ ,.236of, distance. 



