74 



OF DIOPTRICS AND CATOPTRICS. 



focal distance of the incident rays d, the 

 distance of the peripheric focus of refracted 



•n u '■'^"" 



rays will be -^ •, n=^^- 



•' rail— at— tt 



Let AB be d, BC, t, BD, », then CE and DE being the 

 sines of incidence and refraction, are to each other as r to 

 1, or making EFi^r.EB, as EF to EB, and since CEF be- 

 becomes equal to the angle of refraction, or BED, z.BEF= 

 CBDirCED, and BEF, DEC are equiangular, and DEG 

 is also similar to BEC, and FB : BC : : CD : DG, and if 



GH||BC, •.: BC:GH=^; also FB : FC :: CD :CG:: 



FB 



BD : BH=-^:ir-, and AI : BH (=IG} : : AB : BK, or AB 



Let AB and AC be two incident rays infinitely near 



to each other, refracted into the positions BD, CD ; 



then EF, GH, will be the increments of the sines 



EI, GI, which are in the constant ratio of r to l. 



Now the angle at A being to the radius unity as 



EF GH J . 



EF to AE, is =— -,, andtheangleatD=— -,and/.A: 

 AE GU 



^D; 



AE 



— ::r.GD 

 GD 



: AE. But /.A: /.BKCr: BK 



! AB, and z.BKC(=;BLC) : Z.D :: BD : BL, therefore 

 /_K: /_n: : BK.BD : AB.BL :: BE.BD : AB.BG :: r.GD 



: AE, or te : du ■■: r{e—u) : d+t, det+eitzzdurf—duru, 



rdue—dle—ltezzrdiiu, and err— — -. If for t and 



rdu — dl— U 



u we substitute it and au, taking t and u the cosines to the 



rdavu , . , , 



radius unity, we have e=— -, which,wben a— go , 



' rdu — dt — all 



becomes 



rdim 



rami 

 -I 



and if (i=: oo, e — - 



—tl ' r 



429. Definition. The relative centre 

 is the point of intersection of the right lines 

 joining any two pairs of conjugate peripheric 

 foci of pencils of oblique rays, falling on the 

 same point of a curved surface in the same 



direction. 



Scholium. For the radial foci, the relative centre is 

 always the centre of the sphere. 



430. Theoeem. The relative centre is 

 situated in the bisection of that chord of the 

 circle of curvature which bisects the two 

 chqfds cut off from the incident and refract- 

 ed rays, 



A I F E. 



BCq 

 " FB 



FB 

 BD.FC 



.: AB: BK=:- 



AB.BD.FC 



But since 



FB AB.FB— BCq 



FEr:r.BE, FC=r.BD=ru, and FB=ru— «, whence BK: 

 ditru 



-zne. And it is obvious that AI and HK are the 



rdu — dt — tt 



distances of the conjugate foci from the foci of parallel rays 

 coming in a contrary direction, and that their product is 

 always equal to IB.BH. 



431. Theorem. For parallel rays fall- 

 ing obliquely on a double convex or double 

 concave lens of inconsiderable thickness, of 

 which the radii are a and b, thedistanpe of 



the peripheric locus is er: — — r-. -; r 



^ '^ a + o ru—t'~. 



and u being the cosines corresjjonding to the 

 radius unity. 

 This expression is obtained by substitution and reduc- 



tion, from ez:- 

 for u, t; for d, 



raduu 



-, taking for r, — ; for a, — t; 



and for t, u. 



rdu — dt — at t 

 rauu 

 ru — t 



432. Theobem. The radius of a sphere 

 being a, the actual cosine of incidence t, 

 that of refraction, m, the distance of the fo- 

 cus of incident rays from the given point d, 

 from the centre of the sphere c, the distance 

 of the radial focus from the point of inci- 



rdaa . . . 



dence is -r: rr > ^"^d from the centre 



d.{ju—t) — aa 



caa 



Let AB be =:r.AC, thet» 



J" /. BAC will be the angle of 



deviation, and BCmru — (; 



and if DE 1 1 CF, the triangles 



