5g4 PHILOSOI'HICAL TRANSACTIONS. [aNNO ISQS. 



prevent the cdincidence of so many lines,) the radius of the segment towards 

 the object CBorCA = r, and the radius of the segment from the object 

 K ^ or K a = p, and let B b the thickness of the lens he = t; then let the 

 sine of the angle of incidence D A G be to the sine of the refracted angle 

 HAG, or C Ay, as to to n ; and in very small angles the angles themselves will 

 be in the same proportion. Whence it will follow that, as d to r, so the angle 

 at C to the angle at D, and hence d-^r will be as the angle of incidence 



GAD: again, as m to n, so c? + r to , which will be as the angle 



GAH = CAy; this being taken from A CD, which is as d, will leave 



?*""' — ~ - analogous to the angle KfD; and the sides being in this case 



proportional to the angles they subtend, it will follow, that as the angle A/D 

 is to the angle A D/, so is the side A D or B D, to A/ or B/: that is B/ will 



be = ^j- > which shows in what point the beams proceeding from D 



would be collected by means of the first refraction. But if n r cannot be sub- 

 tracted from m — n. d, it follows that the beams after refraction do still pass 

 on diverging, and the point /is on the same side of the lens beyond D: or if 

 n r be equal to w — n. d, then they proceed parallel to the axis, and the point 

 y is infinitely distant. 



The point/ being found as before, and Bf—Bb being given, which we 

 call i ; it follows, by a process like the former, that b F, or the focal dis- 

 tance sought, is equal to _ i, , ^ =J- ^"^ instead of J substituting 



B /•— Bb = '^- t, putting p for -^—, after due reduction this 



^ m — n,d—nr ^ ° ' m — n 



followmg equation will arise, ^^r + mdf -mprf-m-«.dt + nrt =/• Which 

 theorem, however it may seem operose, is not so, considering the great num- 

 ber of data that enter the question, and that one half of the terms arise from 

 our tak ng in the thickness of the lens, which in most cases can produce no 

 great effect, however it was necessary to consider it, to make our rule perfect. 

 If therefore the leris consist of glass, whose refraction is a 3 to 2 it will 



^^ uf+Z7-6r,-T T:^Yrt = f- 'f °^ "'^*^'' ^hose refraction is as 

 4 to 3, the theorem will stand thus -—'^^^pli^fi+flil— = /. If it 

 could be made of diamond, whose refraction is as 5 to 2, it would be 

 ^drlidf-^^T^- idhTTt = /. And this IS the universal rule for the 

 foci of double convex glasses exposed to diverging rays. But if the thickness 



