OF DIOPTRICS AND CATOPTRICS. 



75 



ABC and DEC are similar, and AC : BC : : DC : CE=: 

 -, and AE-—^, AC ; but AE : AD : : AC : AF 



AC 



AD.ACq 



AC 

 caa 



d+it 

 b+'i 

 Tab 



"DC.BC — ACq d.{ru—t)—aa 



; also 1 : r :: CA : BA :: 



— |. If we substitute for EE its ultimate value 

 •a / 



FA ■=. ~ .ABwiU become . ( ru — t — qa. — — 1. 



qb—a {qi—ay- \ i + 20/ 



d.(ru — t) — aa 



(xtY 

 Now, when x' is small, x'zz ' , since (xx)-—2xx, there- 

 DE.ACq 21- 



DC.BC — ACq fore since 'El=^(2ay — yy), or ^/(■2a7/), DE or dz:^ 

 —f. yVheaa—CO,f=rd; whenrf=co, ((Z' + 2/)' + 2a i/)z=i'+ —i/; and since the small angle ACEG 



._ is equal to EDC+ECD, its'sine may be considered as the 



CD : DE=rd; and AE : DE :: AC : CF=: 



rdna 



fc: . If t and u denote the tabular cosines, /~ 



ru — t 



rda , , , /._»■« 



, and when o::i CO , /~ . 



d.{ru—t]—a' ■' ru—t 



433. Theorem. For parallel rays, fall- 



/ a \ cc 



1 of their sines, and srz^(2aj/}.l ' +7 ), sszz^ay .—, 



cc ss 

 ^{a<t—ss)—t—a—^, - = 2ay. 



■~Ti»andti::na rr^Vm 



rrlb rrbb •'' 



-ing obliquely on a double convex or double whence m^ra—^.y, and ru—t=qa+~.y i butd+ 



concave lens, the distance of the radial fo- 

 cus is •; — r-T7-, r, ; w and t denoting the 



(c 2cc\ rf+2< 



qcc 



rbb'' 

 cb — ICC 



lb+2ajbb 



.y=i 



{a + b).{ru — t) 

 cosines corresponding to the radius unity, 



rda 



"ice — cb cc + ca + cr — ca — ci 



{c+a)bb -y— 1— " 

 be 



y='^-—b-y 



This expression is obtained from fzz 



, sub- 



stituting — forr, 



d.[ru — t) — a 

 for d, — b for a, I for u, and u for t. 



{c+aj'jb 

 , ,_ be / qcc qac \ qlc 



(fc + rac\ 

 rbb ) 



_ qc'{ra-\-c') 

 y~rbqb a)^'^" '^^^^> when i=: 00 , becomes 



434. Theorem. The longitudinal aber- X7- "When c la, the aberration vanishes, the point D 



ration of rays refracted at a spherical surface being in the circumference of the outer circle employed for 



qc}(T(l +c) . determining the refraction (425). 



is ultimately— i; ; — .?/, o beina; r—\, a .„ r^. r^, , . ,. , 



^ rbiqb-ay-^'^ =" ' 435. Theorem. The longitudinal aber- 



the radius, b the focal distance of incident ration of parallel rays refracted by a double 

 raySjC-f 6 = c, and 3^ the versed sine of the sire convex or double concave lens of incoiisi- 



of the surface; and, for parallel rays, -^. 



qr 



The focus of the rays ^ ^— i& 



next the axis being A, 



the longitudinal aber-e? 



ration will be AB. Now 



AC is 



b.{qa) — aa 



, and BC=:- 



..„ caa.(d.(ru — 4) — aab) , ^„ rdaa 



andABn- — ; — !— 1 '- — ' but EBz:- 



(qub — aa).{d.{ru — (] — aaj 



and AB=EB.iif^-Z?h:!^z::EB.- 



d.{ru — t) — aa 



■(ru—i 



derable thickness, and of equal radii, the re- 

 fractive density being 1.5, is to the thickness 

 of the lens as 130 to 81. 



The effect of the aberration at the first surface is modified 



by the refraction of the second, and instead of — , or — , 



qr 3 



becomes — y ; for the first focal length is — which may 

 81 q 



be called — d, and — =: — | — ■ , whence — =: , 



e ra rd a ee rrdd 



1 , , "^ ,1 , , , rraa 



nearly, and e'zz-—d'; but rftf— z:oaa and ezza. 



Tad qq 



rd[qab — aa) r{qab — aaj 



h \ I) f/_l_o/ o g 



— qa— ) ; but FD : ED : : GD : HD, and — =-— , whence e'=— d' =~y. Then substituting in the formula 



" / o i'+2a 27 81 



AB : AK, which is ultimately =;EI. 



ri^qab — ao) 



/ r«— t— 



rb{qb — a)' 



— Il6a'(aa + 4a) 16.14 28 



f, 4a, we have — ■■ — ■ .t/ — .1/^ — y, 



2a(,2aj'' ■' 72^0-^ 



