72 



OF DIOPTRICS AND CATOPTRICS. 



I 141 JL,_ 1 " 



— , or — — ; and when dzzco, _— ::.. 



e e a d e a 



and (!= — \a. 



418. Theorem. When diverging rays 

 fall on a concave mirror, the reciprocal of 

 the distance of the focus of reflected rays is 

 the difference of the reciprocals of the prin- 

 cipal focal length and the distance of the 

 focus of incident rays ; and the same is true 

 whea converging rays fall on a convex 

 mirror ; and in either case, when the focus 

 of incident rays is within the principal focal 

 distance, the focus of reflected rays is on the 

 convex side of the surface. 



J 1 Q 



The distance iJ being negative,—— — — —, and when 

 e d a 



— > — ,— being positive, the focus is on the convex 

 a fi e 



side. 



419- Theorem. When converging rays 

 fall on a concave mirror, or diverging rays 

 on a convex mirror, the reciprocal of the 

 focal distance of reflected rays is the sum of 

 the reciprocals of the principal focal length, 

 and of the focal distance of incident rays ; 

 and the focus of reflected rays is in either 

 case within the sphere. 



Here d remains positive, and — (JL.4.1.\, 



e \ a d / 



420. Definition. A lens i« a detached 

 portion of a transparent substance, of which 

 the opposite sides are regular polished sur- 

 faces, of such forms as may be described by 

 a line revolving round an axis. In general, 

 one of the sides is a portion of a spherical 

 surface, and the other, either a portion of a 

 spherical surftice, or a plane ; whence we 

 have double convex, double concave, plano- 

 convex, planoconcave, and meniscus lenses. 

 It is simplest to suppose the lens of evanes- 

 cent thickness, and denser than the surround- 

 ing medium. 



421. Theorem. The reciprocal of the 



principal focal length of any lens, is equal to 

 the sum or difference of the reciprocals of 

 the radii, multiplied by the index lessened 

 by unity : and when diverging rays fall on 

 a convex lens, or converging rays on a con- 

 cave one, the reciprocal of the principal 

 focal length is equal to the sum of the reci- 

 procals of the distances of the conjugate 

 foci ; but to their difference, when converg- 

 ing rays fall on a convex lens, or diverging 

 rays on a concave one. 



For the focus after the first refraction wc have — — _L j. 



e ra 



— , and changing the signs, on account of the chanre 



ru a ° 



of direction of the convexity, — — - 



ra rd 



to be sub- 



stituted for — in the second refraction, where the radius is 

 .a 



I', and the index — j hence — : 

 T e 



._r_ , j; t I 1 _ 



l> a a d b 



(r — i)-f-7-H — j — j> and when d= oo,-t- vanishes, and 



1 /i i\ 1 



— •— ('■ — l)'l "T"H )--~f' '" "'^ concave lens, d be- 



1 _ 1 

 '7 



ing negative, ~—-7+—- In the meniscus, the signs 



not being changed, —c=l.-I-+^+2- — L—(r—i\. 



\l a J^ d 



Scholium. Iftheindex be J, as in some kinds of glass, 

 the focal length of a double convex or a double concave 

 lens, will be equal to the common radius ; and of a plano- 

 convex or planoconcave, equal to the diameter : if the in- 

 dex be |, as in water, the focal length will be to that of an 

 equal lens of glass, as 3 to 2. 



422. Theorem. The joint focus of two 

 lenses is found by adding or subtracting the 

 reciprocals of their separate focal lengths, 

 accordingly as they agree or differ with re- 

 spect to convexity and concavity ; or by di- 

 viding their product by their sum or differ- 

 ence. 



For it may be sboivn in the same manner as for two sur- 



