, ^O by d^ CO by r, MO by v, and the angle PCO ])y 6. Then from Art. 9, QPC = cp, and ^PC = f. Since QPC is a plane triangle, PQ : QC :: sin PCQ : sin QPC; sin PCQ^ sin d _ PQ sin QPC sin^ QC. But PQ^= PM^ + MQ^; and QC =- D-r. and MQ'^=: (QO-OM)'^= {D-vf^ D' + v'--2Dv; also PM2 = OM (2-OC-OM), Euc. vi. 8, = 2rv-v^ . ;. P]\P + QM'-^ = IP - 2Dv + 2rv = D^ -2v {JJ-r), . sin 6 VU-^ -2v{D~r). 1. e. D- r. sill , and will become so ibr those reliected from that surface by writing —1 for fi. The E(]nation is perfectly symmetrical and easily remem- bered, and lies at the foundation (»f nearly all the particular relations deduced in the course of this work. CHAPTER II. ON TEE REFLECTION OE LIGHT. 18. When light tails upon the rough surface of a non- trans- parent body, part of it is absorbed by the body and totally lost, I 16 PLANE MIRROR. while the other part is reflected in all directions and is said to be scattered. It is by this scattered light that we are '^nabled to see opaque bodies. If, liowever, tlic surface upon whicli the light falls is polished, the greater portion of it is I'eflocted regu- larly according to the law of Art. 7, while a small quantity is still absorbed, and a small cpiantity scattered. By imrreas- ing the polish of the surfac(MV(' may diminish the ([uantity of scattored light until it is scarcely sufHcient to render the sur- face of the body visible, a fact well exemplified in the diflficulty we experience in attenq^ting to examine the sin-face of a well polished mirror. 19. Bodies with polished surfaces used for the reflection of light, are termed mirrors or speGula^ and receive ajq^ropriate names depending upon the forms of the surfaces. Mirrors may be of a great variety of kinds, but only three of these are of any importance in Practical Optics, viz : the plane mirror, the spherical mirror, and the parabolic mirror; these constitute the elementary instruments used in the reflection of liglit. The Plane ITIirror. 20. The plane mirror has a plane for its reflecting surface, and is the sinq:)lest of all Optical instruments. Such is the common looking-glass. 21. Prop. To lind the effect of a plane mirror upon parallel rays of light. Let (Fig. 4,) AB be a section of the reflecting surface; QP, UV incident rays of light, and P<^, ^u their courses after reflection. i)raw the perpendiculars PC and YD ; and from P and V draw PR and YS perpendiculars upon YU and Vq respectively. ISow, the incident ravs beinif; parallel, QPC =UYD;but QPC = ClVand 'UYDr=DY^^ by the law of reflection, .*. CP^=DYw; or, the re-^ fleeted rays are parallel. Again V UYD=QPC=CP<^, .-. ^^PY^UYP. But PRY=PSY both being right angles, and PY is com- mon to the triangles PliY and PSY ; .•. PK,=YS ; or, the reflected rays are eij^ioidistant with the incident rays. 16 OEOMETRIOAL OPTICS. { Lastly, siiico UV i« at tlie right of tlie incident rays, but?^V at the left of the i-eflected rays^ it follows that the order of the rays is iiiverted hy reflection. Cor. If the rays P^ J^n'l ^'^^ were incident upon a second plane mirror it is evi(lent, tliat, their order being inverted, the (lonhly rctlected rays would have the same order as the original ones. 2*2. Prop. To Hnd tlie oUcct of a plane mirror upon divergent ravs of light. "In tig. 5, let AB be the section of the reflect- ing surface ; QP, QV rays of light diverging from the point C^, and PIT, Yu, their paths after reflection from the surface AB. Draw QM perpendicular to AB and produce it to q making M.q ecjual to M(^ Join qV and Since QMP=PM(7, both being right angles ; and QM=:M<7, and MP is common to the triangles QPM and «/PM, .". Z QPM=:oints. Thus Q is said to bo conjugate to 7, and q to Q. 25. Prop. To find the relative ])osition8 of the conjugate points to a })lane mirror. From Art. 22, it api>ears that the points are situated upon the same straight line drawn perpendicularly to the surface of the mirror, and that QJV1=7M. Now taking the point M as an origin, if we consider QM to be positive, we must regard ; as before. This gives the relative positions of the points when the reflecting plane is perpendicular to the axis BQ, and passes through the point M. And since one measure is positive and the otb negative, it follows that the conjugate points must be upon opposite sides of the mirror. 26. Since the point Q, in fig. 5, is in front of the mirror, it is possible for all the rays to pass through it ; Q is therefore a 3 I 18 (iKOMKTltlOAL OPTICS. rt'cd focus. On i\w other luind, tlio })()int a hein^ hehiiul the inuTor, tlic riiys cunnot in reality jtasH tliroujijli it, hut only U])i)U!ir to do 80 ; tor this reason q is called a virtual focus. It follows from E([uation (4) that, in a plane mirror, one of the foci is necessarily real, and the other virtual. A i^ooil illtistration of a virtual focus is had hy holdincrpendicu- lar to the mirror. Then the angle contained between QP and PQ' is the deviation. Kow in order to know whether wo are to take the angle i^W^ or (^P*/, we have recourse to the following consideration, Py the law of Uellection, / QPC = CPQ'; becomes a right angle, CPQ' becomes a right angle. Put QP then coincides with AP, and PQ' withPil; i. e., the deviation then l)L"jomes nothing. Ihit the angle QP^ then becomes nothing; therefore QP theujiven mirror. Find 7 tlio conjui^atc iK)int to (^ ; draw (/(i' and let it out AB in I'. .Join qK Then QP, l*Q' is tlu; path of the ray. For, sinee y is eonjngate to (i, the rays whieh come from Q will after retlection appear to eome from q. IJut the ray PQ' proceeds ^ if comin<^ trom 7; hence PQ' is the reilected ray required, . »'' therefore Ql',rQ' i^^ the ]»athof the ray. Two IMuiic ]?Iirn>rN. The princijde of Itellcction at two i)lane mirroi's enters into the construction of several important ()i)tical instruments, such as the Quadrant, the Sextant, cVc. The liglit is made to un- dergo two successive reflections at the surfaces of the inirrors, they being either parallel or inclined to one another. 30. Pro]). To liiul tlie devijition when a ray of li at tlu; 'lirst mirror ; hut tiinco 2;: i8_ a whole circiunlerenco, it' we measure the an«,'le in the opposite direc- tion, it is evident that we «'et for tlie deviation f = 2e. .? (7) Hence we see that, di8rej,'ardin<^' the direction in which tho ariLde is meat^ured, the deviation is equal to twice the angle con- tamed hetioccn the rejlcct'mtj mrfacis of the inirmn. If the deviation at the second mirror be dif- ferent in directi(»n from that at the first ; then, since VPB = CFi;, botli bcinf,' right angles, it follows that PCP'=. PBF= e. But PCF= IIPP'- PP'C = ifi—ip, And since the whole deviation is the ditlerence of the ]^artial ones, we have from (5), e = - - 2^1 - {7r-2C be two ])lane nurrors meeting at C and Q a ])oint l)etwecn them. From Art. 24, it ai)peart^ that Q will have a conjugate point at li, found by drawing QMli perpendiculur to AC and taking UU = MQ. Again, It, acting as a princ'i])al point to BC, will have a conjugate point at S ; and in ihe same manner, in reference to the mirror AC, 8 will have a conjugate point at T, ifec. But, beginning with tho mirror BC, Q will have a conjugate point at R', H' at S', &c. Now, since QM= MK, uul CM is common to the triangles CQM and CRM, and /.i^rxi^ CMR, /. ZMCR=MCQ, and CR = CQ. . ^^^ TWO PLANE MIBROLS. 21 In tho samo maniior it niav ])o nhowii tlmt CT, CS, CS', iiiwl CK', uvc ttU ciiuiil to CU or CQ. Thorotiorc, 21ie conjuijate poinU cue situated upon ike civ- Qujiijji<:iu:6 of a circle passln(f ihromjh the (jtven pointy and fuMtiiam iM centre upon the line of intersection, of the htirro-rs. Afjjuiii ; I)ci;otiii«^ tlin nuixU) ACD by s, and xVCQ bv a, wo have, BCIl = HCA + ACJ' = t-f « ; But IICR - liClS, S belli*,' conjugato to R, .-. ECS e + «; and ACS = ACB + BCS -- e+e+« =- 2e+a. P)ut T being conjugate to S in respect to the mirror AC, ZACT-AOS; .-. ACT-2e + «. In u Biinilar manner l)y denoting QCB by a' we obtain, BCK' a' and /. ACR'"^£4-«'. But ACS'= ACR', S' being conjugate to R' ; .-. ACS' ==€ + «'. But since a'=- e — a, we obtain, ACT= 2e + a; ACS'=^= £ + (£—«) --= 2e— a; ACR=a; ACQ==a ; ACR'™ s + (e— w) --^ 2e~a ; ACS= 2e + a. But, TCS' ^ ACT= ACS'= 2€ + « — (2e— a) ^ 2a ; RCQ RCA + ACQ-=2a; iSzQ., A:c., Also, S'CR^= ACS' — ACR= 2e— «— « — 2 (e— a) ; QCR'= ACR'— ACQ = 2£— a— «=2 (e—a) &c., &c. Hence, we see that the conjwjate p)oints and the given jpoint form a system of equidistufU pairs. 3Sfc» The angular distance of any point from the bimilarly situated point in the adjacc: t pair, is 2£. For TCR = ACT— A OR - ^ 2 £ + «— a == 2 £. Hence we infer that there will he «v majet/ pair's in the whole circumference as the number of times 2 £ is contained in 360°, or £ in 180°. These principles enter into the construction and operation of tho Kaleidoscope. 33. Prop. Given two points and two mirrors meeting at an angle, to hnd the path of a ray which shall pass from one point to the other, being reliected by both mirrors in its com-se. >i^" 22 GEOMETRICAL OPTICS, Fig. 9*. Let AC, BG be the mirrors, and Q, q, tlie given points. Take Q', q\ conjugate points, toQ and q respectively. Join QV/ and let the joining line cut AC in 'r and BC in F. Join Q*P, Vq, Then QF, BF, V'q is the path of the ray. For Z QFA =^Q'FA -= CPF' ; and Z qV]'\l qVT> CV'V', which satislies the law of Reilection. 34. When a ray of light falls obliquely upon one of two par- allel plane mirrors and is reflected at both mirrors, it sutfers no deviation, but is displaced. For, tiie mirrors being ]xirallel, e== and .*. (7) ^=0. Prop. To And the displacement. Let AB and DE bo the mirrors, and QPP'<^ the path of a ray of light. Draw PM perpendicular to the mirrors, and P'R perpendicular to QP. Denote PM by D; and P'R, the displace- ment by d. Then ^^-=P'R =^= P'P sin'EPP' =FP sin 2 f. But PF= PM sec MPP' - D&ecal focus ; and the distance of the principal focus from the mirror is termed the focal length of the mirror. Prop. To lind the focal length of a concave spherical mirror. By moving the point Q, Fig. 12, to an intinite distance, we make the incident rays parallel. Hence making Z>= ocin (9), and denoting by / the particular value which d assumes, we have, JL 2l and /. / =-^ (lu) SPHERICAL MIKROR. S5 d ^ D (11) Hence, the focal length of a spherical mirror is one half of the radius of curvature. Also, since f depends upon r for its algebraic sign, the focal length of a concame mirror is a positive quantity^ and of a convex one, a negative quantity. 41. By writing 2/ for r in (9) and dividing numerator and denominator of the right hand member by 2, we obtain, 1 1_ / This equation being perfectly symmetrical is easily remem- bered, and it presents no difficulties in its application. From it we obtain, /=^f (12) 42. Prop. To find the relative positions of the conjugate foci when tlie principal focus is taken as an origin. Denote by J the distance of the incident focus from the principal focus, and by d the distance of the conjugate focus from the same point. Then, since D and d, in (11), measure the distances from the optical centre, and since the principal focus is at a distance f in front of the optical centre, we must have, I)= Jl -\-f^clndd=^+f; &nd writing these values of D and d in (11), we have, 1 1 _ 1_ Keducmg, / (J + d) + 2/^=/ (J + d) + Jo^ +/V From which, ^d-r; (13) which expresses the required relation. 43. Using equation (13) we are enabled to examine very readily the positions of the foci. i. V /* is essentially positive, J and d must have like alge- braic signs ; and hence, both foci are iipou the same side of the principal focus. ii. If J > /, ^ f; :. if Q be between the principal jocAis and the centre of curvature^ q 'mil be beyond trie centre of cur- vature. V. If J - = 0, n -_.= o) ; /. if Q be at the prhicipal focus, q is infinitely distant,, or the reflected rays proceed parallel. vi. If J be negative and less than/, ^ will be negative and greater than /; .'. if (^ be between the principal focus and the ojytical centre, q will be behind the mirror, a^>d consequently a virtual focus. vii. If J — /, then o f and the foci coincide at the optical centre. viii. If J >/and negative, rJ D or O (^ > OQ ; i\nd, since tlie rays diverge from the point Q, and after re- iloction, appear to diverge from a point q farther distant from the mii'ror, it follows that tlie reflected rays are less divergent i% SPHERICAL MIRE0R8. 27 tlian the incident rays, i. e., the incident rays have been render- ed more convergent l)y reflection. Case iii. When the incident foeics is virtual. Reverse the light in its course ; then q becomes the incident focus, and Q the conjugate focus. The incident rays fall upon tlie mirror converging towards the point q, but after reflection they ]^ass tli rough Q which is nearer to tlie mirror than q is ; hence the convergency of the rays has been increased. 45. Since (Art. 37,) the radius of curvature of a convex mir- ror is a negative quantity, the principal focus must lie behind the mirror ; and hence in order that a focus may be real, it must be at a greater distance from the principal focus than one half the radius of curvature. But, J being this distance, if J > /", f5 < f; Hence, one of the fad must he mrtual. Moreover, when J is negative d is negative ; whence it fol- lows that both foci may he virtual. 46. In a similar manner it may l)e shown that in a concave mirror, one focus must he real, while hoth of them may he so. 47. By reasoning similar to that employed m Art. 44, it may be shown that the effect of a convex mirror upon a pencil of light, is to deG7'ease the convergency, or to increase the diver- gency of its constituent rays. Reflection from two Spherical Mirrors. 48. We meet with successive reflection from two spherical mirrors in the Cassegrainian and Gregorian telescopes, and in some forms of the reflecting microscope. The mirrors have a common axis, and may be both concave, both convex, or one concave and the other convex. The one upon which the rays flrst fall is termed i\\Q primary, and the other the secondary mirror. 49. Prop. To find the position of the resultant focus when licrht has been reflected successively at the surfaces of two spherical mirrors. In Fig. 14, let O O' be the mirrors, and O Q their common axis. In order to have positive (piantities only in our result, let botli mirrors be concave in respect to the rays incident Fig. 14. f 28 OEOMETRIOAL OFTIOS. upon tliem. Let O bo the primary, Q its incident focus, q its conjugate focus, and F its principal focus. Then O' being the secondary and q its incident focus, let q' be its conjugate focus, and W its principal focus. As before, denote QO by A sP by dy FO by F; and also, q O' by D', ^O' by d, and F'Cr by/. Now, let 00' —1= the distance between the mirrors ; and O'^' =A= the distance of the resultant fo< us from the secondary mirror. Then, !)'= qp'= 00' -Oq = l-d ; and, d' = qO' = X. But from (11), 1 + J: = i d' ly f •••4+i^=T ^'*^ an equation from which I may be found in terms of Z, d^ and/. This form applies equally well to all cases, by merely chang- ing the algebraic signs of such quantities as require it. Cor. If the primary rays are parallel, as is the general case with telescopes, d—F; and this gives in (14) , 1.11 nK\ — ''T:^f=7 ^^ For convenience in reference we here collect the results of this chapter. Reflection at a plane mirror, d = -P (4) Deviation at a plane mirror, ^=7t-^ip (5) Deviation at two plane mirrors, or e--2e r. (^) Displacement by two plane mirrors, 'd = 'iDm\

sin <^P1) ; .'. sin QPC > 1, an impossibility. It appears, then, that rays which fall between gP and AP cannot come under the law of refrac- tion ; and by having recourse to experiment in order to ascer- tain what becomes of them, we rind that they are totally reflected from the surface and thrown back into the denser medium. This angle ^I'D when at its maximum is called the critical migle, from being the angle of incidence at which refraction ceases ; or, the angle of total reflection, from being the one at which reflection of this kind commences. 5ft. This kind of reflection, from the internal surface of a dense medium, comes the nearest to pei'fection of any known ; and when glass is the medium employed, this principle enters into the construction of the Camera Lucida, and one kind of diagonal eyepiece. This angle is for glass about 42°, and for water about f 0''. Fit,'. 10. K The principles investigated in Art. 54 give rise to s resting phenomena, of which one of tU^^ niost singuh 56 interestin some ar is liWWArnoN TiiiioiKiii A I'r.ANi-; sruFAOK. 83 Fi'.'. 17. that to a poivuii sitiiatiMl liciieath the ,-urtaee of water the whole liorizoii and all olijccts u|m»ii it ai)[»t'ar to lie in a circle about his zenith liiivinij; an an;j;ular radiiis (d'aWout 50". Let Oil he the biirtace of tja- water, HT Ji tree in the hori/.on, and E the l)Ositlon of the eye. Then the rays (•(iniinin; from jj .'ire bent at 1* and enter the eye as if coni- in- ])ended in the pusiliun ///, .sin-h that llic iniolc //K( ) is about .^O"^; that is, llie crilical allele for water. Anotlier familim- instance is ilic ;i]t|»;iifnt bciidiiiu' of a Btrai_i;'lit ro(| wlicn ]i|i;ii:i'cd ol.ii(jiicly inio water. Let A 111)0 the surface of the water, and (ill th(! rod. Then the rays IIP, 111*' coniini!; from 11, will \h', bent at P and P', and move in tlu' paths IV/, \''q\ as if <'omin^ from a ])oint 11' sitnated above 11; thus the e\iremily II a]) pears to be I'aised into the [xisitioii IT, and with it, the whole innner,-ed part takeb the apparent posi- tieii Xir. 57. i*ro}). if the rays incident np(jn the jdane surface of a medium are parallel, ihe refracted rays are also parallel. For let ^ be the common an^-le of incidence, since the rays are parallel, and f i , f : , the angles of relractioji of any two rays. Thou by the laws uf Kefraction, Sin (f - // sin (fi , and sin ^=- fi sin

^ 1 cos

! }i 1) (18) Cor. If the rays proceed IVoni a point witinn tlie denser me- dium, as is i>-enerally tiie case, we must (Art. 15, II.) write for // in (IS). 60. ]jy (18) we arc enai)led to explain the fact that a river in which the l>ottom may be seen, is deeper than it really ap- pears when viewed i'rom above the surface. Since tlie rays proceed from the bottom of tlie river they pass from dense to rare, and hence we must write the rccip- j. rocal c»f [i in (is). l>ut for water fi = -— ; o lienee d - 1 />, or the river ap}>ears three-fourths as deep as it really is. Rcrractioii at two Plane Surfaces. <*1. Wlien llu; surfaces are ]»arul]el, a ray passing them in succession sufi'crs no doviation, but simply a" displacement. Pro]>. To llnd the displacement. UEFUACI'ION THHoruJl l'l,^N^, rKn-\( K. Let A I), CJI), he tlio juiniUc'l siirljifi;. iinl Ql%/ the; i»!itli of n niy (tt'liu:lit. l*nt(lu('(! yS l»;i('k\v;u-(ls to iricct A 15 itiT; draw I'li, SN, porju'iidicMhirs upon ST ainl AJi ivspectivoly. Denote tluj (lispliiceineiit IMJ l»y I); tlie (listanee SN l>y ^/ iiml tlie uiiifN; of iiKMiloncc TSN l).y SN 'JCC \ I) I sec I'oadily rcdiK f) f ^iii iD sin 'SI, H I'S NSP PSN t rtoc, ^' ip' sin (^ — t^')- '(.'S to ( 1 cos ri>^'in. Tlio lino of intersection of the ])lanes is \\\v. /y/j/^, and the planes vvhicli mec t^ to form tlieedi^'e, are tlie/'cfcc-vot' tlie prism. The axis is any line in tlie ])rism parallel to its ed*?e. Let AEB bo a section of a prism per- pendicular to its axis; then, EA and Ell are the faces ; E, tin; edi!j:;e ; and llu; aiiiji'le AEI>, the refractiiiL'' aiiL;le, oi- .-imply the angle of the i)rism. Let ilV^q he the path of a niy of liirht; draw the normals CF and IXi m('(■till^• in F; also, produce Ql* to meet AE in T, and produce ackwards to meet QT in K. QPC is tht! >' (a) Again, v (a), ' -s —cf', :. sin ' --sine cos ^' —cos ssin^', and fi sin (/>' - --- sin (^'> — /i sin e cos (f' — /j. cos £ sin - - fi sin £ cos - f; But (p' f (/>' = e, by (a) ; ' ... ^ =:(p -i-"

-/^e ~ (f, and this in (22) gives — ^-(/^-l) £ • (23) Hence, in a thin i)rism we may safely assume that the devi- ation vat'ies as the angle of the prism. 65. Equation (23) furnishes a convenient method uf ascertain- inc; the index of refraction of a iriven medium. Let AEB be a -f-hin jnisn.i, and QP a ray of light Avhich wouxd fall u])on a screen ;it q , but by the intci-position of the prism is made to fall ar g^ a ]>oint at a small distance from ^• Then qi^q is the deviati(»n-, ;uid is very near- ly ecpial to qq divided by */'P. ^licnce deimting */'P by d, and qq by J. we ha\ o iVom (23) , ^ (//-1)£; Vvs. i'i. (J d from wliicli, Te PRISM. 4- 1 37 (24) In this inaTincr, by formiiiii; a tliin prism of the i^iven medimn, we are cnal)lc(l to obtain a very close aproximation to its index of refraction. <»6. Prisms are sometimes employed for purposes of Reflec- tion, as stated in Art. 55. When so nsed it is customary to prevent all refraction by makini:: the angles of incidence and emergence each e([ual to zero. In this case the pencil meets the iirst surface at right angles and consequently sntfers no retraction ; it then undergoes one or more rejections, entirely within the prism, and finally emerges at right angles to another surface. Hie ol)ject sought in ap})lying prisms to the reflection of liglit is to cliange the direction of the rays with as little loss as possible. Let (tig. 28) AI3C be a section of a prism, and QPq the course (»f it r;iy of liglit meeting BC and AP> at right angles. Let the angle ABC be de- noted by d. Since the ray is regularly retlected at P, the angle QPq is double the angle of incidence upon AC. But QP^ is the complement of ABC, that is of 6^; hence, Fi^!,'. 23. d rr — 9. (p. Ihit (5) .- = ;r - 2 ^, .-. ^ = 6. From which it apjiears that the angle of the prism must be e(]ual to the deviaiion reijuired. This deviation, and consecpiently the angle of the prism, is generally a right angle ; in which case we iind the angle of in- cidence upon the face AC to be 45^. But from Art. 55, it appears that the ray cannot possibly pass out of glass when the angle of incidence is greater than about 42" ; hence glass answers perfectly in such a prism. <>7. In tlu! ('aiuera Lucida tlui light is twice reflected before leaving the prism, and hence (Art. 21, Cor.) the rays have their original order restored. Let' (fig. 24) A BCD be a section of the prism, and i}W'q the course of a ray of light. The deviation of the ray being given, to find the angle ADC made by the reflecting faces of the prisin. Fig. ^. 38 OEOMETRIOAL OI'TICS. Since the <1eviiitions at t.lic separate surfaces take ])lace in the same direction, we have for tlie wliohi deviation, from ((I), c= 2 7r--2e; •£=. ADC::.. 1(271-0-'- ^- and, ? 1)einii; , qO by d, CO by rand the index of the medium by fjt. Then D and d are the dis- tances of the conjugate ])oints or foci of the surface from the point in the surface. .69. To find the i-elation between JJ and d. This relation is given accui-ately in (3) ; but assuming that the aj^erture is small, we have v a very small quantity in (3) ; and rejecting those terms into which v enters as a factor, as be- ing of no account in comparison with /P and d'^ we reduce (3j to— n-r _ J) d-r ^ ■ d' dr = Dd{fi-~l), FiR. 25. Hence, /Jt j. r and dividing by Ddr, d 1 (24)i D r Since (24)i is independent of the a[)erture of the surface, i.e., of OP, it follows that all rays proceeding from Q, and meeting the surface not far from the axis, proceed after refraction as if coming from q ; hence i} and q ai'e the foci. OV LENSES. 39 or I^CIINCM. "yo. A lens is a portion of a nicdium boiuuled hj two splieri- cal surfaces wliieh have a common axis. This common axis is termed tlie axis of tlie lens. In tliis definition we consider tlie phme as tlie surface of a si)here hav- ing its radius intinitely great; hence, a medium bounded by parallel plane surfaces may l)e considered as a lens, although never used as such in practice. Tiie dilferent forms whicli lenses may take are shown in the t.i'coiiipanying diagram, and these arc sometimes distinguished by particular names, as follows : — I. Convexo-concave, II. Plano-concave, III. Double-concave, IV. ri;uu>-convex, V. Double-convex, VI. CoiH^avo-convex, or Meniscus. It will be readily seen that lenses arc naturally divided into two distinct groups ; those which are thinnest at the centres and whicli receive the common name of concave lenses, and those whicli are thickest at the centi'cs, and which are termed convex lenses. These names are sutHcient to characterize a lens in a general sense ; but optically, a lens is known only when the radius of curvature of each surface is given, and its position with re- spect to the incident light. In order that we may have only ])ositive (puintities in our results, we take as the standard lens that one which has the radius of curvature of each surface a positive cpuuitity, and its principal focal length a positive quantity. Tl. Prop. To find the relative ])ositions of the foci in a thin lens. In iig. 27, let !*(>, i^'O' be seel ions of the bounding surfaces, O'OQ the common axis, ami V, C the centres of curvature. Let a ray of light (^1* meet the lirbt surface in P, and after refraction proceed to P' u])on the second surface as if coming from Q' ; let it be now refract- ed at the second surface and 40 (GEOMETRICAL Ol'TIGS. P, ii leave the lens as if coining from q. Then for tlic first surface, Q is tlie incident focus and Q' tlic conjugate one ; and for the second surface, Q' is the incident focus and y tlie conjugate one ; finally, for the lens Q is the incident focus and y the con- jugate one. Put QO = D, Q'O = d\ qO = d, CO = r, CO' =r. Since the lens is thin, O' is very near O, and practically we may consider them as coinciding. Then for the first surface we have from (24)i, // _ 1 />«— 1 17 (a) n - r Now, in i)assing the second surface the light moves from a dense medium into a rarer one ; hence, by Art. 15, II, we must 1 write /' for the index ot refraction. Therefore, for tlie second surface, remembering that Q' is now the incident focus, - 1 --^ •J d' -— :?-' Or 'd JL d' 7' f * T .(b) (25) Then, adding (a) and (b), ^disappears, and we have- ^-__.^(^_1) (_—-,>), which is the relation required. Definition. Tlie irrhicipal focus of a lens is the focus con- jugate for pjirallel incident rays ; and the distance of the prin- cipal focus from the lens is called X\\q, focal leugt/i of the lens. ''2. rro]\ To find the focal length of a lens. In order that the incident rays may be parallel, the incident focus must be infinitely distant. Therefore, making I) = x in (25), and writing /for d^ we obtain, A =0-1) (-L ^-X.) (2.1V ''Ji. Prop. The focal length of a concave lens is a positive quantity. RELATIVE POSITIONS OF FOOI, 41 A little consideration will suliice to show that a concave lens is characterized by one of the three following conditions, viz : i. Both radii are positive, and v < r' . ii. Bcjtli radii are negative, and r > t' . iii. T is positive, and r' is negative. Now since n is always greater than unity, (Art. 10), therefore //—I is a positive quantity ; hence the algebraical sign of/, or the focal length, depend;^ upon — 1 1 '/• T' But this expression is positive nnder each of the three condi- tions stated above ; hence, /, or the focal length, is positive. ''4. By a process of reasojiing precisely as in the last article, It may be shown that the focal length of a convex lens is a negative (juantity. . ■ys. From Arts. 73 and 74, it appears that in general the two species ot lenses are fully distinguished by the algebraic sign of their foc-al lengths. Moreover, since the light i^asses through the lens and is not thrown l)iick as in the case of mirrors, the real conjugate focu^ of a lens is upon that side which is opposite to the' incident light. It follows, then, that convex lenses have their principal focus Tcal, and correspcjud to concave mirrors ; while on the other hand, concave lenses have their principal focus virtual^ and conset|uently correspond to convex mirrors. ^6. Pro]). To express the i-elative positions of the foci in terms of the focal ieiigth. Writuig, from Equation (26), the value of/ in equation (25) we obtain — 111 rru^ J (27) Cor, From this we readily obtain — YT. Prop. To trace tlie relative positions of the foci. Taking the concave lens as our standard, since in it m\y is the focal length positive, we may make the equation I> -JI // / to be generally true, by giving proper values to n. Writing this value of D m (27) and reducing, we readily obtain— 6 f lit 42 GEOMETRICAL OrTICB. d=f. n (A) n + 1* I. Concave Lenses. i. Wliilc n is positive, D is positive, and d is (A) positive and less tlian /; hence, when tlic incident focus is real,_ tlic conju^^ate focus is bctveen the lens and the ])rincipal focus, and is conse(|uently virtual. ii. If n is negative and less than unity, D is negative and less than /, while d is negative; lienco, when the incident focus is behind tlie lens at a less distance than the local length, the coniuirate focus is also behind the lens and real. iii. If n is negative and greater tlian unity, B is ly -./c and greater than/, and d is' positive; hence, when the incident focus is behind the lens at a greater distance than the focal length, the conjugate focus is in front of the lens and virtual. IL Convex Lenses. Changing the algebraic sign of / in (A) we obtain — D-—n f. _y;' n ,\ (B) d n+ 1 i. When n is positive, D is negative and d is negative: hence, when the incident focus is br' "nd the lens, the conju- irate focus is there also and is real. When n is negative, D is positive, while d is positive for in 11. all values less than unity, but negative for all values greater than unity ; hence, when the incident focus is in front of the mirror, at a less distance than the focal length, the conjugate focus is also in front and virtual ; but if the incident focus be at a greater distance than the focal length, the conjugate focus is Ijciiind the lens and real. In a similar manner many other relative positions might be traced. ■?*. Prop. To find the deviation caused by a lens. ' Let OP be a semilens, C^P an incident ray of lio'ht, and ' = 12, and /i --= 1*5. From ecpiation (26) — _ 5 ^ Y4' and the lens is convex. K^r- 4g GEOMETRICAL OPTICS. 5 A watch ir\im of five iiiclics (nirvatuR! is tilled witii spirits of tun.entiiie, and a pencil <.f rays cuiuin^' from a |)Oint '20 in- ches above it, is brou^dit to a fu(!U8 at a distunce ot ti8 m.helow it ; determine the index of refraction of the lluid. Taking cpiation (25), , lind ip' when 8. When light falls u])un a (pertain medium, ^ — 50", and (p'.^ 85"; find" its refractive index. 9. If a vessel 12 inches deep be tilled with alcohol how deep will it appear to be ? 10. What must be the refracting angle of a prism of crown •rlass, that the deviation may be 5"" t 11. Find the curvature of a plano-convex lens v^ water when the focal length is 4 inches. li*. One surface of a crown glass lens has a radius of curva- ture of 3 inches ; rind the radius of curvature of the second surfiuo, when rays coming from a ]u)int lo feet in front of the lens are brought to a focus 2 foet behind it. 18. In a combination of two lenses, the focal Icnoth of the primary is — 3 feet, and of the secondary 8 inches, and they are 2 feet apart; if the incident focus be i) feei m tront ot the Drimarv. wtiat is the position of the resultanr, iot;uB. It IMAGEB. 49 14. Find tlio position of tliu ruBultant focus in 13, when the lonbeB are in contuct. ClIAl'TEll IV. ON THE FOIIMATTON AND SIZE OF IMAGES. 85. When a body is viBil)le to us wc fie(> it hy iiiejins of the rays of liglit wiiicli every point of it trHiibniits to our eye, and wo naturally refer tlie body to that position from which the rays appear to come, altlioii^ii It nuiy at the same time be placecl m a position (juite ditfercnt from the apparent one. This is well exeniplilied in the case of the common looking- glass, where an object which is really aiul necessarily in front of the glass appears to be situated behind it. Every point in front of the glass has its conjugate point be- hind the glass, and the rays really come to the eye as if from that conjugate point, and as if thei'e was a second object placed in that position, lience, if by any nu-ans wo cause the rays which , enumate from any point of a body to pass through a given point before reaching the eye, we naturally refer the real point to that through which the rays pass before entering the organ of vision. This is the principle of the formation of an image, and may be illustrated as Ibllows : — Let Al> be a convex lens, and MN an object in front of it. The rays proceeding from M liave M as their incident focus, and, by prin- ciples established in Chapter 111, have a conjugate focus at some point iii where the rays cross before passing on to E *, in like manner N has its conjugate focus at n, and every point between M and N has a conjugate point between ?/i and n ; moreover, these rays enter the eye at E precisely as if they came Irom a body placed at m n / for this reason we call run the imagt of MN. Again, since no ray of light coming from MN bv way of the lens can reach m except those which come from M, it follows that the point m must possess all the peculiarities of brilliancy, colour, ifcc, w Inch characterize the material point M. And the same may be shown for every pair of corresponding points in the object and its image. i'ig. 31, so GEOMETRICAL OPTICS. Ill If a screen be placed at mn, and the light troni the ob.icct be of sufficient intensity, the concentration of rays at m n will produce a picture of the object upon the screen, and this picture becomes visible by having its hght scattered in all directions. 86. Prop To establish an approximate relation between the linear dimensions of an object aud the linear dimensions of its image. Let P (fig. 32) be a portion of surface bounding a dense mcdiuin upon the left, OQ a straight Hue passing not far from P, PC a per ^..,^ ^^ pendicular to the surface at 1 , and PO an arc of a circle described from C as a centre. Then, since QO passes not far from P, PCO is a small angle. Let QP be a ray of light meeting the surface of the medium at P and aUer refraction let it proceed as if coming irom q. ^ From what we saw in the last Cha])tor, Q and q are conju- gate points. Now, if QP be turned through a^ small angle Ql Q , f, then _£_ is an improper fraction, and t/ 1 — ^ is a negative quantity. ?.'< IMAGES BY REFRACTION. 68 Hence, /is negative, or the image is inverted, iv. If i> = 2/, -5- = 2, and 1 - -^-= - 1. :, /= — 0, or the image is equal to the object, and inverted. T) T) V. If Z> > 2/, -^>2, and 1 - ^^ is negative and greater than unity ; hence tlie image is less than the object, and in- verted. Hence, it appears that while the object remains at a less distance than the focal length, the image is erect ; but when the object is at a greater distance from the mirror, the image is inverted. 93. Prop. To trace the ratio between the linear dimensions of the object and image in a convex mirror. Taking e(iuatioii (37), we readily perceive that/+I>is greater than/, and consequently that the object is greater than the image, both being positive ; hence, in a convex mirror the image is less than the object and erect. Images foriiicn this supposition we obtani from (18)— d — liB ; and writing this value of cZ in (34) we readily get— = 1. Hence, the image is equal to the object, and erect. (b) Single Lens. 95. Let I denote the linear dimensions of the image formed by the action of the iirst surface of the lens upon the incident light ; /, of the image formed by the nnitual action of both surfaces, or by the lens ; and 0, of tlie object. Now it is evident that the image formed by the tirst surtace becomes an object to the second surface, and that in passmg the second surtace the light passes from a dense medium into a rarer one. 64 GEOMETRICAL OrTICS. IIgiicc, from (34) — ^ = /f_5. (E) D' beini^ the distance of the Urst image /' from the origin. 1 Also, l_ ^ T^ (F) / d and multiplying together (E) and (F) — ' ('^, (38) Id Now taking the concave lens as our standard, since it has a positive focal length, we obtain from (27)— D -, i> -d-^^7^ hence, j — ^ ,. (^^) 96. Prop. To determine the nature of the image formed by a concave lens. . Taking equation (39) we readily perceive that / and U bQ- ing essentially positive quantities, / + D is greater than /; hence, O is greater than /; or, the image is less than the object, Moreover, the image, being formed at (Art. 77. 1, i.) a virtual focus, is virtual. 97. To trace the variations in the image formed by a convex lens. since the focal length of this species of lens is negative (Art. 7i), we must change the algebraic sign of / in (39), which gives — _ 0-f-^ (40) 7 / From ec^uation (40) it appears : — i. /is positive only when D is less than/; hence, the image is erect whenever the ohject is at a less distance from the lens than its focal length. ii. /is negative when D is greater than/, since is always ])Ositive ; hence, the image is inverted whenever the ohject is heyo7id vhe principal focus. iii. The conditions which render /positive (Art. 77. II. ii.), make it also virtual, while those which render it negative do- it to 00 real. MAGNIFICATION. 66 08. Wo have now considered the ratio of the linear dimen- sions of the image to those of the object, whether that image be formed by means of a mirror or a lens. If, however, D becomes very great, or the object be inac- cessible, or its distance or linear dimensions be unknown, the foiementioned formulae are inapplicable. In this case M^e may resort to angular measurements, and determine the ratio between the angles subtended, at the origin, by the object and by its image. By rejecting the negative sign in (35) it Ijecomes identical witli (38) ; l)ut, as has been stated (Art. 87), the negative sign is useful only in giving the position of tlie image ; hence, re- jecting it, we have for objects and their images the proportion — J - d' But ^ measures, very nearly, the angle subtended, at the D I centre of the lens or mirror, by the object, and — measures the corresponding angle of the image ; hence, at the centre of the lens or mirror employed the object and its image subtend equal angles. This relation is an important one, and is true for all practical values of D and d. 90. Prop. To liud the linear dimensions of tlie image when the angular dimensions of the object are known. Denote the angle subtended by the object by (o. Then, angle su'btended by the image equals co by Ai't. 1>8 — Or, ^=^; .-. 1= do) (ioy Cor. If the object be very distant, as the sun or moon, we have d =f, and hence— '^ I = f 'J very distant, "B the heavenly bodies, we obtain, hy dividing l)y />, and then making D equal to infinity — M=^ -i— (44) E—f ^ ^ 103. Prop. To find an expression for the magnification when the image is formed ])y refraclion. In the case of refraction the liglit is not tlirown back but passes through tlie surface of the medium ; hence the eye and the object must 1)0 upon o])posi>;e sides of that surface. Let, tlieu, () he a ])ositive or concave surface or lens, SQ its axis, and S the position of the eye, or the origin. Denote, as in the last article, SO by E . Then, c ^ SQ --- SO + OQ =^ ^ + i> ; and, d ^ Sy == SO -f O^ ^^ E -f d. And tliese values in (-12) give — / E + D Fig. 34. M -- — - . ^ — O E+ d In the case of lenses we obtain from (38) ■ O D ' d K -t- D (45) .'. M .(46) B E-\-d 104. If the rays come iVoni below the surface of water, which is a very common case, we have from Art. 59. Cor., and from Art, 94, 0^1; these relations in (45) give— E+D m M E^ D- D Since /j. is greater than unity, jE' + — is less than E ^ D ,* hence M is greater than unity. From this^it appears that bodic.^ at the bottoni of a vessel of water are magnified ; and, upon the same principle, if we lay 8 m 58 oEOMErrmcAr. orTUJB. \ ii a t}iitk pifito of glass iipijii the pa.i^^e of u l)ook the letters ap- pour Ini'ger than before. 105. If ill (46) D becomes indetinitely u;re}it, we obtain by a reduction similar tu that employed in Art. 102, Cor., M^ f- ' ("iS) ^' +"' ... 1 which is an expression adapted to concave or positive lenses. Of tlic Image formed by a combination of ]!Itrror§ or lOii. Ill Articles 49 and SP. we determined the position of the resultant focus when light is successively retlected from two mirrors, or refracted through two lenses. We now propose to find an expression for the size of the resultant image, or the huage formed at the resultant focus. Jn addition to the notation of Arts. 4S and 82, we wdl lor convenience, term the combination a comimmd mirror or lens, tluM.iiage formed by the iirst mirror or lens the i>rwiary image, and that formed by tlie coml)inatioii the rcmltani image. 107. J»rop. To determine the linear dbnensions of the resul- tant iniage fornied by a conipoiind inirror. Let C>be the linear dimension ot the object; 1 , ol Uie primary image ; and /, of the resultant image. ^ Abo let i) aud d l)e the distances ot the object and primary ima-'o 'respectively from the ])rimary mirror; and 1/ the distance of the i)riiuary image from the secondary mirror. Then, remembering that the primary image becomes an . object to the second mirror, we obtain from (30) And bv multiplication — O __ r _F~JD^ f-J>i 1-T'-1~ '"F ' f But, as in Art. 41), we have, 1)' — I — d, _ F- D f - 1 + d "'IF' ' .r f-D' and — =:''- > -. ,(Cx) f- F ■ f And writing for (/ the viiIhu givuu in equation (12) , we tiiiiilly vuducu (b) to— t^i-1 -i^~i! + ^. m 7 ' F f TMA0E8 AT RK8ULTANT FOOUfl. 69 108. When the iiundciit ruys are [mrallel, the sinj^le mirror, which will form !ui image in every respect equal to the resul- tant one, is ternie*! the cquinalcut mirror. 109. Prop. To find the focal length of the equivalent. Divide both members of (-10) hy D, and we have — 5/ I)\ o f / F But — = the angular size of the object — u) ; and making /> = CO , since the incident rays are parallel — il=: L _i. - 1 (II) Again, if /' be the focal length of the equivalent, we must have (41), / = f'o) ; and writing this in (II) we have — 1 - -L - 1-1 (50) 110. Pro]). To determine the linear dimension of the resul- tant image formed by a compound lens. Taking positive or concave lenses, and using the notation o^f Art. 107, we have from (39) — r __, and Y- f • () O Hence, -^ ^ But from Art. 82, and from (27), / F^D f^-D' F f D' = d + I; and these relations iinally reduce our equation to — (51) £-1 + 1 + ^ + ^+^ 1-'^ /"" F^ f^ Ff 111. Prop. The elements of a compound lens being given to find the focal length of the equivalent lens. O Dividing both members of (51) l)y D, noticing that and making D = cc, we obtain — T~ F'^ f^ Ff 1) 6>; I \\] w m Hi 00 GEOMETRICAL OrTICS. And denoting' tlic focal length of tho equivalent by /', we have, Buico / =/'<<>? 1 - 1 4- J_ + J_ (52) It must be remembered that tliis i^* an expression for concave lenses, and that in applying it, the focal ^-r'^ths must be taken with their proper signs. . Equations (50) and (52^ .find an important ai)plication m treatmff of the telescope. . . For convenien(;e in reference, we here collect the prmcipal relations of this Chapter. Spherical mirror; ^ = l - ~„ (^^) Lens ; j — — y^ '^ ' / = >... (41) Magnification; W—D Reflection, J/ = - ~ • -YZTd ^^^^ D=oo, M:=^ (44) Refraction, if = -^ • -^-^i ^^^^ D = 00 , M= -gfj^ (48) Compounds; ^ ^ j, j) j,i Mirrors, -^ = 1 - j— - jr -y ^ ly" ' '^ ^ 1 _ ^ __ i _ _1 (50) 7 " Ff F f Lenses, .£.= 1 +y +-^ + j + jy- W^) 1 -i_ + J- 4- -1 (52) Examples C. 1. The focal length of a mirror is G inches, and an object 3 inches in length it placed at a distance ot 0-2d inches trom ^1- - ,„:-^^^ . rir^ff^v^pine *^1'<' notiu'c and size oi tlie iinai»;e. EXEROTSEB. $1 Taking equation (36), wo lmve/= 6, D z= 6*25, = 3. - -JL 24 72 iichcs. Tf 3 6 6 /. 1 = /. TliG image is 72 inches in length, and inverted. 2. A mirror phiced in front of an object 1 inch in diameter forms an image liaving its diameter one-fifth of an inch ; the distance of tlic object fnjm the mirror is inches ; determine thenurror em])loyed. Taking ecination (36), (> = 1, /=. ^, i) =^ 6. Hence, 1 -^ -— ^- ~ /-« / and .•. 5/=/-(! .'. The mirror is convex, and its focal length is 1.6 inches. 3. If in the preceding exa!nple the diameter of the image were / = — - what would be the nature of the mirror ? "" 5' Ir this case we have, 1 - o~ f ' .-.-6/= -6, or/=l. And the mirror would be concave. 4. While looking at the page of a book from a distance of 12 inches, I place upon it a ])latc of glass 3 inches thick ; deter- mine tlie magniiication which take*^ place, the value of// being 1.5. In ecpmtion (47) we have, £=9, D=S and fx =1.5. •• 3 11 ~ 11 9 + 1-5 Ilcncc tlio letters appear one-eleventh larger than before. 5. The focal leiiu'th of a convex lens is 3 inches, and an object one inch in length is i "laced at a distance of 2 inches Fll I I t! 'il 3 since the lens is OEOMETIilOAL OPTICS. fwir. tlio Ions: dotcrniino tliu length of the iiiiuge, and whether it is real or virtual. In cfiuation ^30) wo have, 0'=-\J convex, and D - 2. 1 -3+2 1. and / =• 3 inches. And from Art :»7, iii, it appears that the image is virtual. fi WhPn I Dlaco a convex lens of 6 feet focal length between J cyTS trmoo^^^ at a dista.^e of ^ feet fron. the eye ; determine the magnitication producea. In Equation (48) we havc,/=-0, E -Ti -6 _ i --4- Or the imago appears four times as large as the moon. 7 An eve placed feet above the surface of a pond sees a J'afthe'botton.; how much larger does it appear than it would if no water were interposed i K While lookim? at an object IS inches distant, I bring a comx^S, hav ng a focal length of 8 inches, between the e^'O anT^he Ob ect .o al to be equally distant from each ; do I gam or lose by so doing, and to what extent ^ Q Tho obiect ^'•lass of a microscope consists of t\yo convex image. Use Equations (30) and (51). in A -»Prson in reading a book at a distance of 12 iuchcs, .i.;tnvereyya^ focal length being 20 inches ; ho holds tW^^^^^^^ ^'^^'"^'^'^ ^^^-^- iication. 1 1 A certain lens held at a distance of 3 feet from the eye, caiesthe mo"n to appear double its former size. Deternune the lens. AllERRATION. CIIAPTEU V. ON AIJEJUIATION. lis. It waa discovcrotl by Sir Isaac Newton, tliat when a small pencil of white light, sucli ay may be received from tlio Bun, pasaes oblicpiely through the Hiirfuce of a medium it is nut only bent out of its former courne, but it is alwo decomposed or separated into riiyn posBeisHing dill'erent degrees of rcfrangi- bility and being highly coloured. The diagram will illuBtrate tliis. Let EF be a white screen, and 1ST a nmall pencil of liglit coming Irom the Hiin and gaining ad'.nission l)y a hole in the ishutter of a darkened room. The rays will pro- duce a snudl bright spot at T where they meet the screen. If now the ])rism A I '.(J be interposed in a i)osition as shown in the diagram, the pencil is bent out of its former course ; but in- stead of making a single bright s])ot as might l)e ex[>ected, its rays are scattered or dispersed in a dii'cction at right angles to the axis of the prism, and fo'-ni ujioji the screen an oblong spot which exhibits a regular gradation of colour.s trom red, whieli is the lowest oi' least refracted, to violet, which is the most refracted. Tliis coloured obh)ng sj)ot is called t)ie sukw iipectrum^ and the phenomenon of its fornuitioJi, is called the chromatic aber- ratluii or difc'/'(:twu ol light. BSiJ. The solar spectrum contains seven principal colourts disj».i,s((l in the foUowing ordor — commencing with the least rtfiaugii)le — red, orange, yellow, green, blue, indigo, violet. TliL'iie'colours, however, are so nicely blended at their adjacent bonders, and the gradation is so exceedingly regular that it is lUihoult to say where one colour inds and another begins. JJe^ides the solar spectrum, every luminous or incandescent body gives oil' rays of light which form its own peculiar spec- trum. liy careful observations upon different spectra it has been discovered that they are crossed by many line dark and bright lilies, and that the positions and characters of these lines fully determine the nature of the light experimented upon. This discovery gave rise to the science oisjjtcirum analym-— a science which, though in its infancy, has done wonders in solving many physical proldems. ;1 i^ g4 GEOMETEICAL OPTICS. 114. From what has been said, it foHows that in the hame inediuni the differently coloured rays cannot liave the same index of retraction ; and, altlumgh the d.lierence is m al cases small the index will he greatest tor the violet rays, and least for the red. The mean hidex given in the table is an aritii- metical mean between these two. Def. The angle (VPR, iig. 35) contained between the direction of the red ray and that of the violet one is called the disj}ersio7i. 115. Prop. To find the dispersion when a ray of white light passes through a thin prism. . Pj^j^ u = the dispersion ; f/ == the index for red rays ; and fi" = the index for violet rays. Then, we have from (23), (^/ — 1) e — the deviation of the red rays, and (/i" •— 1) s " the deviation of the violet rays. But the dispersion is the diiference of the deviations ; hence, ic - (//' - 1) £ — (/^' — ^) « ; Or ..^ (//"//) e (53) 116. Prop- In one and the bame medium the dispersion varies as the mean deviation. From (53) u - {]/' — r') « ; and from (23) c ^ (// — 1) £ ; and eliminating £, wo obtain, u --^ (//" — 1^) ■ -j—-^- But in the same medium, //, [^\ and // are constants ; hence u varies as ^. 117. From Art. IIG, it appears that for any one medium, u = a constant quantity, fi-l This constant is called the Disperswe Power ot the medium , and it is, like the index of refraction, dilierent m ditterent media, but constant for the same medmm. A table of Dispersive Powers is appended to this treatise. CHROMATIC DlSl'ERSION. 65 It*. liepresL'iitiniJ!; the di spurs ive power by ^'', wo have fVtMii tlie last Ai'tic'le— ^r it ^ T' /. u -- U^ (54) i, e, T/if' dispersion is equal to the jyroduct of the dispersive poioer and the deviation. 119. Prop, To find tlie dispersion produced by a tliin prism wlieii its rct'ractiniii; angle and dispersive i)Ower are known. From (2-^) we obtain for the deviation — c -- (/i - 1) e . And writiiiii: tliis in (54), >r [feQjt—l) (55) 120 i*i'(»p. To find the dis])ei-si()n when a ray (»f white liii'lit jKissus tlii'ou^'h a thin lens. If the nij i>asses centrally there is no deviation, and conse- (picntly ?io dispersion ; but if it does noi. pass centrally, put a for the distance from the axis at which it meets the lens. Then m'c have from (28), ^ - -j.- And this in (54) gives ^ U-- ^' (56) 121. The general elfect of chromatic dispersion upon the nature of the image may be shown as follows : — Let OP 1)0 a convex lens, GO' its axis, and MN an object in front of it. ^ Let NOR be the ray which passes centrally, NP a ray meeting the lens at P, and after refraction, let PR be the course of the red ray, and PV FiR. 36. that of the violet ray. Since PV is more refracted than PR, it meets the ray NR at a point V nearer to the lens than that in which the red ray meets it. And the same holds good for the red and violet constituents of every ray which goes to fonu the image. The violet rays form their image at V V, and the red form theirs at RR ; and since all the other colours are intermediate l)etween these two, they ^orm their several images between VY and RR. Those images increase in size from VV to RR, so that when viewed frcrn the side opposite the lens the red image appears 8 66 GEOMETRICAL OPTICS. i to overlap the orange one, the orange the yelluw, tl-c. ; tlie {reneral effect being to render the image somewhat contused, and to fringe its borders with the prismatic colours. On the means of Correetlns: Chromatic Aberration. IW. The ill effects of Chromatic Aberration in the working of lenses being far greater in magnitude than those arising from all othe? sources, it was found liighly imi>ortant in tlie construction of the better class of ''''':il''''''^''^' ^ N^^^^ possible, some means of correcting it To ottect this, Ne^^ton tal)Oured long and unsuccessfully, and at length gave it up m despair, turmng liis atlentioii to the use of mirrors lu the con- struction of his toloscopos. ,., , , 4, A i-r. Newton arrived at the conclusion, winch has been tound to be erroneous, that the dispersion was in all case8_ proportional to the deviation ; and hence, that the construf^tion ot a lens which should correct the dispersion was impossible Many years later Mr. Dollond, an optician ot London, by repeating the exiun-iments of Newton, hut with different media, tound that tlui dispersive ])Ower has no connection with the index of refraction : l.ut that two media having the same index of refraction may have .piite diU'erent dispersive powers, and vice ver-m. , . i i^... This discoverv led to the constnicti.m <>! a compound leub consistinu; of two, or sometimes three, sim]>le ones couiiK^sed ot different' media, and so related that while their .lom. ettect prodnced a certain amount of deviation, it destroyed very nearly the whole of the dispersion. Such a connx^und l*ms is said to be ae/iro,Hah<\ and it con- sists jyenerally of a convex lens of crown glass united with a convex one of tliut glass ; or, in the case of a compound consist- ing of three simple lenses, of two convex lenses ot crown glass having a concave one of Hint glass between them. 123. Prop To Hnd the ratio between the refracting angles of two prisms formed of different media, so that a ray ot white light passing through both in close succession may not suiter dispersion, , i .^i Let ABC, DEF, be two prisms placed with the two faces AC and FD very close to one another, and having theii- refracting angles turned in opposite directions. A ray of white light meeting thetirst prism at P undt rgoes a certain amount of dispersion ; il Achromatic lenr. 61 but on passiiijL? the second prism the dispersion takes place in tlie oj)pusite direction, and if the prisms I'o properly propor- tioned, the whole dispersion may be reduced to zero. Denote by e, /i, 17 the retracting angle, index of refraction, and dispersive })ower of tlie first prism; aiid by e', //, U' the like (piantities in the second prism. Then, by (55), dispersion at first ])rism = - U & {[i — 1); dispersion at second prism = U' e (//' — I). But these dispersions being in o])po8ite directions, when they are eM ETKIC AL O I'TIUW. ¥ (59) But since the lenses are iu contact, tlie ray meets each at very neai'ly tlic same distance from tlie axis ; ;. a — a ; und since the (liBpcisions are in opposite directions, we mnst have— f^[± f^'^ in order that tlie wliole dispersion may be zero. " F ^ U '" . . ;. The foml lengths mmt he proportional to the (hsj)ersive poiners of the media cm2)loycle lens of equal dimensions, y^t in i s <.cneral form, as bein- comi)Osed of lenses of crowii and liiit glass, it is not periect. The cause ol this impertection may bt exphuned as follows:-- ,.,.;c.v, nf if two spectra of eiiual dimensions, one lormed by a pusm ot crown o-lais and the other l>y one of tlint g-lass, be placed side by side, it will be seen, upon carelul observation, that the positions occupied by the various colours do not cxaciy cor- respond ; but that wliile the extreme rays hold the sa me relative positions in each spectrum, the intermediate and cent al ones are a little higher or lower in one spectrum than in tlie ""^ On account of this, that proportion between the tell lengths of the constituent lenses which totally corrects the extreme rnys can not fully do so in the case of the central ones. Hence in the best lenses consisting of two simple ones ot crown and lliut glass, the images of bright objects are always slightly tingel abiut the borders with green or blue, they being the central colours of the spectrum. . i i , To correct this slight imperfection, recourse is had to a com- pound composed of a concave lens of iliiit glass V^^f^]^^^'J.^^^^ two convex ones of crown glass, they being made to d Iter slicrhtly in their effects by varying the constituents ot the ghass. Sm4i a lens is termed a ^r?;^;^^^- it is not however generally used, since the gain in the perfection of the i mage docs not compensate for the extra loss ot light .— =— ==== i^=" arising from an addition in the number of surfaces, and the increased errors ol worhnianship. The two kinds of achromatic lenses are shown iu the margin. j?'jg. at», A ( ' 1 1 R( )M ATIC C( >M1 '< )UNI ). 69 I'-if. An aclirouiutic coiubinatioii uuiy l)c Ibriiicd by two convex lenses of tlie same nuiterial sei)arate(l by a given inter- val. Siieb combinations are \\^va\ as eye-pieces, as objectives for common micvoscopes. S:c. Let A 13, (JD re])resent sections of semi-lenses liaving the c(mimon axis QR. Let a ray of light coming from Q meet tlie tirst lens at A, and there snf- fering i»ectively. Then the red constituent, meeting the lens CI) at a jtoint more remote Irom the axis tliaii that at which the violet portion meets it, undergoes a greater deviation ; and if the lenses be sei)arated by a proper interval, the red and violet [»orti()n,> alter [»assing CD berome parallel, and are thns deprived oTauy eil'eclive colour. Pro}.. To iind th- interval 1)1.5. Tut Qli • />, 111) /, Aii ~ a; also, denote the focal length of All for violet rays by F, and for red rays hy n F ; then, since the lenses :ire of 'tliesanu! material , if/ denote the focal length of CD ibi- violet rays, iif will be its focal length for red ones. The dispersio?! beijig in all cases a small «pnmtity, n is very nearly unity, and F and / are very nearly the mean focal length of the lenses. {Since the red and violet constituents are parallel before meeting AL, and also parallel after leaving CD, they must have undergone the same amount of deviation. By (28) we have, deviation of red at AB — deviation of red at CD — a .-. for the total deviation of the red ray we have- a :- + CD nF nf Sinularly, for the total deviation of the violet we obtain— a ED F^-J' And since these are equal, we have— !|. 70 GEOMpyriiioAf. optics. _a_ . CJ) ^ 6^ , ED nF ' ' nf F f ' which rcMhicon to nnd from siniihu' triaiigUjs CD _ <'I^ ^ 1>1'^ ^ AUl - \M) a ~ \\\ Jm Bir llcuce, .(K) =: 1 BR' CD ^ ^ _ J_ _, i a n F D El) _ , _ / D Simihirly, Writing these vuhiesin (Iv), ciiul reducing, we obtain — But n heing very nearly unity, and F and / very nearly the focal leiio-th^^ of the lense.-^ for the i-entnil rays, we may intro- duce those approximate values without material error, and hence obtain — ^'-— -/ (60) 2 — _ I) Cor. if D is great in comparison with F, we have very nearly, l = l{P+f) (61) :. Tlie i/Uc/'val hciicccii the lemof iimd he one half the sum oj tketr focal knythfs. SplicrUrul Aberralioii. I'iS. Chromatic Aberration arises from the nature of light and media, and is present only when refra^jtion takes place ; but there lo another kind known as ^yhtneai aberration, winch depends upon the form of miirors or lenses, and which is com- mon 10 b'.'th rejection and refraction. SPHERICAL ABERRATION. 71 For any ^iven position of the iiicidoiit focus tlioro is soino certain form whicii, if it wcih; ])(>ssiblu lo apply it, would totally overcome this 8])eoit'9 of aberi'ation ; l»iit o'wiiii;- to tlie me- chanical difficultios wliieli i)resent themselves, attempts at the formation of such surfaces have not succeeded well ; and the form generally used for the siii-faces of mirroi's and lenses is the spherical, as heiu^ the (me wliich admits of tlic greatest accu- racy in its formation. Hence tlie name of s])li(Mical aherration. 129. The ^i-neral ctlccts of splu'i'ical aberration inay bo shown as follows : - Let OP be a convex lens, COc its axis, and MN an object in fi-ont of it. Let 7n7i be^he image of MIS' foi'med by those rays which ]>;iss centndly. The points m, <*, and //, will oc nc;ii ly '''i.^'' i"- e isses through the extreme por- tions of the lens has its conjugate focus at e', a i)oint nearer to the lens than c: hence, at ;/?.' n there will be an image formed by the extreme rays, within the one formed by the central rays: and the lens being continnous, there v ill be a continuity' of images filling up the space betwc'on ???. n and 7n n ; and these images increasing in si?:e in jiroportion to their distances from the lens, the outer ones overlap the inner ones. Hence, the effect of spherical abcrratioi'. is to make the inia<>-e curved, and to produce an indistinctness or confusion of outline. This species of aherration is many times less in amount than chromatic aherration, and in lenses of small size and great focal length it becomes almost im])erceptil)le. Ihit when the diameter of a lens or mirror is large in comparison with its focal length, the ill effects of this aI)eiTation are so decided as to render these instruments entirely useless in the formation of images ; and even with small lenses it becomes necessary, in fine instruments, to correct the aberration as far as possible by using combinations. A combination wlnMi corrects the effects of spherical aberra- tion is said to be aplaimth-. The C(mdition of pei'fect aplanatism is never fulfilled in practice, for the same compound must at the same time be achromatic, and it usnally ha])]Huis that both aberrations are only partially, although nearly, coi-rectcd. I M It :iJ:i Y2 GEOMETRICAL OPTICS. 130 For a full invcHtiication into the nature and an^.unt of H)heri*cal aberration the' student must consult home of tlu; lar«'er works upon tliis subject, since all ouv liniits wdl allow is to i;-ive a mere abstract of some of the most unportant results. MlURORS. i. Tlie aberration of concave mirrors is o]»positc to tliat ot convex ones ; so that whenever these two kinds occur in com- l)ination the aberration of one tends to correct that of the other. ii The aberration decreases in amount as the incident focus approaches the centre of curvature, and at tluit ^wint it be- comes zero. iii When the incident ravs are parallel, the aberration is nearly e(pnd to the S(iuare of" the radius c»l' the mii-ror divided by four times the radius of cur\ature. Lenses. iv. The aberration of concave lenses is o])posed to that of convex ones, so that a combination of these tends to correct the resultant aberration. v. The aberration in lenses varies w^ith the focal length and also with the radii of curvature ; hence, lenses of tlie same focal length may give rise to very different amounts of aber- ration. For convenience in reference, we here collect the principal forms of this chai)ter. Dispersion ; u = (//' — /^') £ i^^) u=: f/e (5^) tc = ire{fjt — l) (55) u= U-^ (56) V f U' Achronuitic lens ; -■^„- = jj i^^) exercises. 73 Examples D. 1. The dispersive power of crown jjjIrbs being about one- twenty-cighth, dcternilne the deviation of a ray wlien its dispersion is 10'. Taking (54) we have, u = 10', U — one-twenty-eightli. • 10' = ^ X — • .-. $ = 280', = 4° 40'. 2. A prism of crown glass produces a dispersion of 15', what iw its refracting angle, // ])eiug 1*5'^ 1 8 In (55) we have, ?« = 15', f/ = __ , /y = Jin <■> ... W =1- (^ -l\e 28 V 2 / £ X 56 .-. e = 840', = 14°. 3. It is required to make an achromutic lens of crown and flint glass which shall have a focal length of 10 feet ; deter- mine the focal lengths of the constituents. This is solved by combining (33) with (50). The whole focal length is A = 10 feet ; and for the dispersive powers we have from the table U' = ' 048, U = • 036. "^ (■^^) 4> = V+ r and (59) u •048 ^030 4 3 and • /" = 1 F ' 10 i^ 4 F' whence, F = 17i feet, and/ = 23^ feet. 4. Find the dispersion produced by a lens of rock salt 1^ inches aperture and 10 inches focal length. 5. Determine the lens of crown glass which will achromatize the lens of problem 4. 6. Determine the anglet^ of the constituents in a compound prism composed of flint glass and diamond when, being achro- matic, it causes a deviation of 5°. e W . Jl^ 74 GEOMETHHJAL ^>l'TICS. CHAPTEK VI. I'*' ON ILLUMINATION AND BRIGHTNESS. 131. BodieH are sometimofl divided into luminoup, which shine by their own inhere.it light, and non-hiniinou8, whicli become visible by reflecting or Bcattering the light received from a luminous body. This division, though convenient ut, time: 's not adapted to our purpose, tor ii is immaterial, as far as the light itself ih concerned,' whether that light be .(65) From (65) W(^ infer that the illumination is directly^ propor- tional to the illuminating imoer ; and also, that the illumina- ting power is m^asw d by the product of the illumination into the square of the distance. 136. Article 135 furnishes the means of comparing two lights as to their value for illuminating purpohcs. '^The lights being so placfd as to produce thi ^am > amount oi illumination, let Z and (Z be the illumiuatinu ])o\\er and dis- tance of the iirst, and L\ d\ of the second. Then (65) ill lination given = 3 ; and it also equals Or . L _ L : £' :: dJ' d'y (66) H il m lli 76 OEOMETRIOAL OPTICS. That is, the illuminating powers wrc to one atwther directly as the squares of the (Usta?uies from the illuminated screen. 137. Prop. To find ii i)oint of equal illmnination between two luniinoiw bodies. LetZ, L be their ilhiniinatin^ powers, and J) tlieir dintanee apart. Denote by iji tiic distance of the point from the first body; then J) — x\% its distance from the second. From (0(1) we obtain, L : /; :: .t'^ : (D - x)i, ;. |/Z : ^IT '.'. X : D —x; From whicli D |/Z («7) Cor. Since the surds may take eitlier })his or minus, vo may consider \^ L' as])ositive or ne: hctmecn the luminous bodies. 1S§. rro\). To find how the illumination varies according to angle of incidence. In 'fig. 42, let S •• A15 l.e a i»encil of vayd meeting the ]>hine AC at right angles, and let the illumination produced be denoted by /. Now let the })lane be turned through an angle ^ so as to take the position AP). Then, since the same number of rays is distribiited over a greater s])ace, the illumi- nation of AB is less than that of AC. Denote this second illumination by /'. Then, if;? be the number of rays in the pencil, /'•/•• -i^ ■ JL • • AB • AC AC = AB cos

77 the Incidener. eo-etllrlentH of TrBiiNiiiiHNloii nii«l Rt^nvelloii. 139. Rays oflij^lit incident upon u niediinn are partly trans- mitted, partly refloctcd, and partly ahHorhod ; in hkc manner, those M-hieh'tall ni».)n tiie surface of a ri'He(!tor are partly absorbed and partly retlcctcd. Hence, neither lenses nor niirna's render iivailabK! the whole (|nantity of liji;ht receivetl. The ratio of tluMunnber of rays rendered available t«> the number received, is called, in the case of a lens, the co-vjfieient of tramnmsion, and in that of mirrors, the vo-effic'ient of rejiection. Denote this co-efKciont by c. Then it follows that c is always less than imity. In the best s]>ecula it is scarcely ever greater than -C while in a well polished lens fornied of a highly trans- parent and liomogeneons medium it nuiy be as high as -9, althougli it varies to a considerable extent with ;the nature of the light. It appears, then, that in the construction of Optical Instru- ments it is always desirable to employ as few lenses or spectnla as possible consistently with the object to be attained; and that in the choice of these, lenses have a preference over specula from rendering available a greater pro]>ortion of the incident rays. llliiiiiiiialioii l»> a I^oiin. 140. lA't AP> (tig. 42'^-), be a lumi- nous body, PQ a lens enclosi'd in an aperture in a non-transparent wall, and MN a screen placed in tlie conju- gate focus. The image of AB falls ui)on the screen at DE and illuminates the portion upon which it is depicted. Denote the area of AB by A, of the lens PQ by a, and of the image DE l)y A; also, denote CO by ut witli tiie same eye and the same object, the only element which we can sui>])ose to undergo any change is the size of the puj.Ml. Hence, denot- ing the radiUi^ of the ])Ui)il I'V^a we must \\v\[v j>-7: Un- a, and we thus obtain. B J 1 Now, since our units are i)erfectly arbitrary, we simpliiy this ex})ression by making the iirst factor our unit of measure ; and we thus obtain, B ^^ .p' (72) Hence, w(! inter that the Jn'iiihl ncss of a yhicn ohjcct U- iuile pciulciit of Its (li'duncd, a)i(l naris as ihr square of the radius of the imp'd of the. cije. 143. When we view an oliject with the naked eye, the pupil takes in a pencil of rays which completely tills it; ami althou<;h by the interposition of various instruments we may reduce tJie size of the pencil which enters tl -' eye, yet we cannot increase it beyond the size of the })uih]. Hius, by viewing an object through a small hole i)ici'ce(l in a plate of metal, we reduce the pencil which enters the eye 1<> the size of the ai)erture, and the etfect U])'on the brightness is ihe same as if the pu])il of the eye itself sulfered alike reduction. ' Denoting by b this brightness, and by t the radius of the aperture, we obtain frt)m (72) , A A "■ 2) Or /> = .//. 4 (^^) ' f 80 GEOMETRICAL OPTICS. Now, the etfectivc value of t can never be greater than that of 2), altlio-agh it ?nay be indefinitely less ; tor if t becomes .n-eater tlian^^, the extr^ light admitted is totally lost, smce the largest pencil which can possilily enter the eye lias a radius equal to ^. When t is very small in comparison with p, b is very small in comparison with B. Tliis explains the cause of our being able to gaze at the noon-day sun through a small hole m a card. 144. Prop. To determine the size of the pencil transmitted from any point of an image. Let AB (tig. 43) 1)C a lens forming an imago Mt P of a given object in front, and let ED be' the position of the pupil of the eye. Denote CA, the radius of the lens, by R; PF, the distance of the eye from the image, hy e ; i^D, the radius of the pencil proceeding from a point hi the image, oy «, and PC l)yi (Kig. 44) be a lens, and PQ the positiofii iu'il coming from the point i'*. Now, it is evident that the pencil EPD enters the eye, as also pencils coming from every point between P and Q. Biit from any point beyond i\ part o*f the penciL at least, must tali without' the pupil and be lost, thus produciiig a corre.p.jiidmg diminution m the bi-igiitue.^ of that part of tlie anage trom which it proceeds. A.l points of the image, then, between 1 andtheaxis will appe:u-e(pu.lly bright, and iW^ part of the image will thus prebeiit a uniform brightaee* ; W% beyonci i^ Fig. 44. S2 (i KOMETRIC A L OPTIC 8. |ii »■ i ■ji tho l)rl'>-litness of tliu iinai^^o grows gradually less until it finally d isupi)curs. meet up. a.u And the radius of the circle of uniform brightness, trough C draw CR parallel to BE, and produce GE to _. it ill K, and QP to meet it in S. In addition to the former notation, put PQ, the radius of the circjle of uniform brightness, = V. Then, UFC and SQC l)eing similar, RF : FO :: 8Q : QC ; i.e., Ji \ i> '. d 1 a :: R I V : d; from which we obtain, ih) - R e ,^^x -^ + e ' Writing the value of d p as obtained from (76), we reduce (77) to the more convenient form, V=-j^L^.{h-R) (78) III instruments, where a second Iciis takes the place of the eve ill li"' tti it is usual to mark out this circle by an annular stop termed the ^%Vi/'rt^?/i, thus cutting olf the edges orm^- ged part of the image. For convenience in reference w^e here collect the principal formulae of tliis Chapter. Illumination ; ^^ '^^ ^^^^ V^^^ (68) L

e double rotiectioii is fuHv explained in \rt 07 Were it not tor this arrangement the inbtrument would he nearly uwelesB, as every picture would he reversed. The , which is the sui)plement of the ande made by the faces. The instrument is ])rovided with a graduated circle by which the angle through which it has been turned is mea- sured, and denoting it by /9 we have for the dihedral angle 180° - ^■ The Raleidoicope. 156. The Kaleidoscope is an Optical toy, by means of which an endless variety of pleasing Geometrical ligures can be formed. It consists essentially of two plane mirrors, Ab and BC, nicely joined at the angle 0, which is generally made to be some integral part of the whole circle. , . ^ i • Between the mirrors, as at D, any object being placed, its successive images (Art. 31) arrange themselves in pairs in the circumference of a circle having its ^9^ ^2 THE SEXTANT. 89 centre at C; thus •oii.stituting a ti<,aire of geometrical regu- larity. . I. 1 The instrument gone, .illy onipobed of three stri ^ ol glass ofeciu.il widtli aicJy joined at t^ ?ir edges, t^ is i ming a tube the Bect'on ofwliich is an equiiatend triangle. The ol>- U-.T 3Bc jects are ' r, of coloured glass enclosed between two gh plates :it t lie ^nd uf the tuhe. Wh" 1 1 ming the tub(^ around its axis while looking tlirou.,'h it, the ' 'ts l)y fallhiir from de to side give rise to some o " the mo levant patte ^ wMch i< '= possible to imagine. Ii itscomiion or tr-au' it follows from Art. 32 that th' figure will be lex. i. . The Optical »»quaro. 157. The 'ptii scpiare is an instrument souietiuies em- ployed in surveying, for the purpose of setting olf right angles. ' It consists of a [.lane horizon- tal iJ)\) are two plane mirrors having their reflecting surtaces accurately per- pendicular to the plane of the instrument. I, which is denominated the index ylas.s, is fully silvered, and, by the motion of the arm IL to which it is iixeil, may be turned about an axis at right angles to the instrument's plane. The amount of its motioix is read oti* froui the mm Fig. 60. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 ^ 1^ III 2.0 IIIW U 11.6 % <^ a ^pi ^^\ ^v- 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 m o M/a '"/, 90 GEOMETRICAL OTTIU8. graduated arc MN. The mirror IT, termed the horizon glass, is silvered only upon its lowei' half, and is fixed immovably, except for minute adjustments, to the body of the instrument. Rays of light coming from any object at are successively reflected froni the two mirrors and enter the eye as if coming from O'. But through the unsilvcred portion of the horizon glass the eye sees at the same time any natural object as at P. The angular distance of any two objects, which are thus ap- parently brought together, is evidently equal to the deviation of the reflected ray ; but tluit is, from Art.30, equal to twice the inclination of the mirrors, and this inclination being measured upon the arc MN, the deviation is known. Usually the arc MN is so numbered that every degree of angle at its own centre is counted as two degrees upon the divided limb, thus giving at once the whole angle of deviation, or the angular distance of the objects brought into apparent coincidence. To improve the action of the instrument and render it more delicate, it is often furnished with a small telescope directed towards the horizon glass, by means of which the eye sees two images superimposed in the field of the instrument, and can thus bring them into coincidence with a considerable degree of accuracy. The L and these relations reduce our equation to, p = —£-L ^ (/-D)6+/i>- But since for distinct vision (Art. 162) B must be eq^al to f or very nearly so, we may witnout sensible error put j = D, which gives, P-| (80) From (80) it appears that under the most favourable con- dition, that is when the object is placed in the focus, the rndg- nifying power is independent of the distance of the eye from the lens^ and is equal to the minimum limit of distinct vision divided hy the focal length of the lens. From this it appears that the magnifying power is less for a short-sighted person than for one who is not so. 1ound lens from which the rays emerge parallel, and we thus get for its magnifying power, as in eijiuition (SJ), 1 " =- ^ (7' + .. - -T^. (83) / /r A final negative sign in this result is of no account, and may be rejected, since magnifying power cannot be considered as a negative q^uantity. 174. Prop. To find the magnifying power by exi)eriment. Direct the eye-end of the Justrument towards a strong light, and receive the circular image, formed near the other extrem- ity, upon a white paper screen. When this image is sharply defined, measure its diameter with a delicate scale, and let v be the number of divisions over which it extends. With the same scale measure the diaphragm, and let its diameter be V. Then we have, — _ .-= — . Now let f be the focal length of the eye-glass in tlie case of a Huyghenian Ocular, or of the whole eye-p:*ece in the case of a Ramsden's ; then, ' I s F = l-^-- = Or, / V 12 s 7 (84) 'V^ I ! Ill n^'l 98 OEOMETUIOAI. UPllca. 1T5 Ut It be the radius of the Objective ; I its distance from be tn^e, wbich will be nearly the nucrosoope's length; and f the focal length of the Ocular. . ^ ,, Then, for the radius of the pencil transmitted to the eye ^v^ readily obtain, Tinf f and R beino; both small quantities in coinparison with I, f i^ a^so a smaU quantity, and tt.e brightness of the held is from (75), taking B as a unit, ^^ ^^ ^ = ^-'/T^ ' which is a small quantity. Hence the necessity, in u.iiiL; the micro^^^^^^ of illuminating the object strongly. If, for example, wc hayc / = 1, {[= 'J!:^ = ^'' ^" we obtain for the magnitymg power (Ait. lid), 5; ^=-(M-l) 90; and for the brightness of the field, h — c . 6^ X ~f p^ J. 141 And taking i^ as one-tenth of an inch, which is about its r.iean value, and g as -8, we get, nearly, I»« compound Kcllccilng MIcro.copc. Of this class of „,i -roscoues a "i-eat numy forms have been iuvOHtod trom tune ot me Maiw of these ai-o hut uiodiiieatious of oue another, ™d rffer oiUv^ in the u>ethod of iHununatuig the objeet or sol uc > u eehauical details. lu all, however the >umge > b rmo 1 by u.eaus of a speculum ...stead ot a lens, and this for.ns the distinctive characteristic ot the group. This class of microscopes is not at preseut in common use, hav . " been ahuost entirely superseded by the common or IlStin-rMieroscope, which is neater in appearance, more compact, and more easily managed. Among the instruments belonging to this class, wo may n„ticc /i/rMsfcr's and iimitKs. 1 ,4! 1l )■ — ' L. . - — : 1 l| 1 . II r ! fik. m. THE COMPOUND TELESCOPE. 99 ITT. BronTHtcr'N Mlcroicope. In fiff. 54, AB 18 the tube of tlio microscopo. At one extremity a sintill tube ncrews in, whicli contains a coneavc Bpeculuni S, und a Bninll nlaiu; flpccnlum s. rh])- ported n])on a thin arm. The iar<;e 8])0culum in y>ierced with a hole at itn centre, and tl'.o rayt«, coming from an object at ( ), after reflection at tlic plane mirror and ajj^aln at the concave one, form an image at I, where it is viewed by the Ocular. The rays evidently meet the Objective as if coming from an object at r, situated as far l>eliind the plane mirror as the object is in front o/it. 178. Smltli'N microNcopc. h, fij.. 55, AJ? is [ the tube of the microscope, S is a concave mirror pierced with a hole, and S' is a convex mirror also pierced at its centre. liays of light coming from an object at O, near the centre of curvature of S, woul(i, after reflection from the surface of S, bo brought to a focus at some point near O ; but, meeting the mirror S', they are reflected l)ack and caused to converge more slowly so as to form th'.'ir image at I, in the focus of the Ocular. A stop s, placed in the body of the tube, i)revents the direct rays from passing from the opening in 8' to the eye-piece. Although these instruments have the advantage of forming B. perfectly achromatic image, yet they labour under the insu- perable disadvantage of wasting a great ])ortion of the light which enters them, thus efl'ectually prohibiting the use of very high powers with any degree of satisfaction. 1T9. Compound Refracting^ Telescope. The Objective in this instrument consists either of a single lens, or of an achromatic compound, as described in Art. 122. Being intended for distant objects, the object lens is always of large size and long focal distance as compared with the lenses of the Ocular ; and in its general ai)plication the image is formed at the focus of the object lens. The Ocular employed is one of those already described ; the Huyghenian for distinct view, the Ramsden't* for inicromet- rical measurements, the terrestrial for conmion land or soa telescopes, and the diagonal form of any of these when con- venience requires it. ii 100 OEOMKTRIOAL OITICS. ISO. Pnm. To find the nm«ijnifyinjr powor. Ilowovor (.,.a,,.M,n.l tli(« purtn n.a.y bo in the teh'Bcopo, B.nco the avrt enter it parallel and emer^^o ].ural el, M'e may roplm-o othTeObjecti^ arc! the Ocuhw by tlur equivalent lenHe.. Ut, then, i^^be the focal le..-th of the Objective, and /of the Ocular. I The angle un^vcr we have, / I F _ F /gpA f / ^ / Or the maanifving power is the quotient c.Tising from dimd- ^ tejSLgKof the Olrjecthe hy the focal length of the Ocular. . . 181. Prop. To find the magnifying power by cxperiTnerit rv.Lf IhL obiect end of the instrument towards a bright sky Vir.iv^nS a p er screen the small ring of light formed ^ear tritbTe^^^^^ When sharply defined, measure Us 'tt^ bliS ^d;;:!^lu^;b:nd:t?i R the ramus of the clear opening at tlie object end of the instrument. /> "■ Then, V' For the small circle is but the image of the objective aper- ture formed by means of the Ocular. Hence, we have from Z^^m (27), by noticing that the Ocular is convex. d^D- f' and But from (38), d d f 1 R . t^=^-l THE COMPOUND TELE8cal length of the equivalent to the objeetive. But we obtain an identical result if we take F to denote the f >cal length of the object glass, and / of the equivalent lo the eye-piece considered as a coni])ound lens. For, writing - 3/ for F, - f for/, and 2/ for ^. in (52), and reducing, we obtain. ■/' P- That is, the eye-piece is equivalent to a convex lens whose focal length is one-and-a-half times as great as that of the eye- glass But we have for the magnitying power, 2 F _ F lF ],- f 3 f f~ ^ Or, the two forms give the same result. Hence, when the focal length of the eye-piece is known, we may most con- veniently make use of the focal length of the object lens. To illustrate, let us take the following example. In a compound Refracting Telescope the object lens is 6 inches in diameter, and has a focal length of 6 feet. It is fur- nished with a Huygh' uian eye-piece in wiiich the focal length of the eye lens is '3 irch, and the diameter of the diaphragm is '2 inch, iissumejt? = -1 inch. :!■ -j^Q^. GEOMETRICAL OFIICS. E.iuivalent to the objective, (Art. 183), F = 48 inches. Magnifying power, (86), Field of view, (8T), Limit of efficiency, (90), k = IGO x '1 = 16 inches. ^ =: _£- nearly ; p - i5 = 160. •3 Briglitness, («1), h^ 162 28 and if c be -8, which we may suppose, 028 or about — . Tliis .'reat reduction in brightness is upon tlie suppositi^on tliat tl!e°iiupil remains continually of tlie same size; but tlu» no tnC for the eye itself endeavours, by dilatn>g and con- tractina the pupil, to equaU.e as tar as possible the brightness rf bodPes Thus, if the object upon which the telescope was empfo vcd was very bright, the pupil would probably have Se he dkmeter,\he.T viewing' t through the jstrumen^ that it lias when looking at t without, ^vhlch won d crease the brightness to about one-eighth. Hut it the object he lamt the pu?il undergoes a very sliglit change, llence we see the reason'why in viewing such objects '^V'^^te^'and Ke are unaer the necessity of employing low power» and laige apertures. With the iixed stars, on the other hand, we have an exception to our formula for brightness. For, since with the IdglLt powers available they undergo no perceptible mag- nification, tLr brightness, or, as it is l"-"-^!- tff'-t^rease a^ is independent of the magmty.ng power, "'"'^ '\ ""«;^^,ff ^ the square of the radius of the object glass until the limit ot effic incy is reached. Hence, in viewing these objects he highest pow<>''^ "6 employed with advantage; "'j f '"" ^- neously, the brightness of the ground-work ot the sky is diiun- Xd and the intensity of the light of the star is increased. Hence 2o, high powei-s reveal stars perfectly invisible to the naked eye?anl even bring to view moderately bright ones at noon-day. 1S6 Compound Refleciiiig Tcle§cope. In instruments of this class the Objective consists of a single or compound m rror (Art. 106), and the image being t ms formed without any refraction te king place, is perfectly achromatic (Art. IJb). THE COMPOUND TELESCOPE. 105 Hence, these telescopes, when well constructed, have a de- cided advantage over ilefractors in the sharpness of the image produced and the magnifying power which they will conse- quently bear ; but on the other hand, thev yield to Refractors in the proportion of light rendered usefnl, in stability and convenience in working, and in their liability to become de- ranged. They are now seldom constructed except upon a gigantic scale, but even then they are not so generally useful as good Refractors. 187. Gregory's Telescope. In this telescope the Objec- tive consists of a compound mirror, composed of two concave ones. In fig. 57, A is a large con- cave speculum pierced at its centre. Rays of light, coming from a distant object in the direction of O, after reflection at the surface of the mirror. ,.,_ .„,_,, _j-^5^fnB A \ ---^ o Fjg. 57. converge and form an inverted image at I ; the rays after leaving this image meet the second conclave speculum Ji, and arc again converged to a focus and caused to form a second image at I', in the focus of the eye- piece. Let i'" be the focal length of the large mirror, F' of the small one, and / of the eye-piece. For the distance from the small mirror B, at which the image r is formed, we may without material error put X = F +F' ; and this relation in (16) gives for the distance between the mirrors, l^ F + F' + F .- (92) and writing this value of I in (50), and reducing, we obtain for the focal length of the equivalent to the objective. (93) But since the magnifying power is^ , we have for it in the Gregorian telescope, P== ^ (94) The field of view, limit of efficiency, brightness &c., will be 13 li!Li m 106 GEOMBTRIOAL OFTIOS. the same as that given for the refracting telescope, hy nsing the equivalent of (93) for the Objective. 188 Cassegraiii'^ Telescope. The Objective in this instru- ment is compound, and consists of a large concave mirror and a small convex one. In fig. 58, A is a large concave speculum pierced at tlie centre. Kays of light coming from a distant object in the direction of O would, after reflection at the surface of the mirror, converge and form an image at the focus ^. But, before reaching this focus, they are intercepted by the f^^^^^jf ;\^^;,^ae which their convergence is decreased, and they are th s made to form their image at I, which is then viewed by the eye- ^Tsing the same notation as before, and noticing that the focal leSgth of the small mirror is negative (Art. 40), we obtam by a method as in the last article, i==.F-r + r.L; (95) f =^E1 ; (rejecting the negative sign) F P=IL (96) and F'f 189. It will be noticed that these two telescopes come under the same general formulae, and are therefore but modihcations of one another. The Gregorian has the advantage tor terres- trial objects of presenting them erect; for the first image bemg inverted, the secon (99) These are true for both inBtniments by taking F' with the proper sign. 190. Mewton's 'Teteieope. The Objective in this tele- scope consists of a single concave speculum. A (fig. 59) is a concave mirror. Rays of light coming from a distant object in the direction of O would, after reflection at the mirror, form an image at i in the focus ; but be- fore reaching this point they meet rig. 59. the small piano mirror B, inclined at an angle of 45" to the axis of the instrument, by which they are turned out of their course and caused to form the image at I, without the tube of the instrument, and in the focus of the eye-piece. The mirror B has no influence over the convergency of the rays (Art. 23), and hence, B ^ is equal to BI, or the plane mirror must be placed at a distance in front of the focus equal to the radius of the tube. Frequently a prism, as described in Art. QQ, is employed instead of the mirror B to change the direction of the pencil, and it has the advantages of wasting less light and of retaining its polish much better. 191. Herschel's Tele§cope. This telescope is a modifica- tion of Newton's, in which the small speculum is dispensed with, and the image formed at i (fig. 59) is viewed by the eye-piece. This saves the loss of light at the second mirror, but is applicable only to instruments of such dimensions that the diameter of the speculum is large in comparison with that of the human head. To bring the image into a convenient position the speculum is slightly inclined to the axis of the tube, thus throwing the focus close to one edge of the opening. This is the form of telescope given by Sir William Herschel and Lord Eosse to their mammoth instruments. In regard to magnifying power, field of view, &c., it is mani- fest that Newton's and Herschel's telescopes are measured precisely as in the Refri*c\ .»r (Art. 179), making the Bpeculum to represent the Object lens, 108 ^1 GEOMITBIQAL OPTIOB. The Magic Lantern. Fig. 60. 199. This is an instrument by which a small object or painting may have its magnified image thrown upon an illu- minated screen so as to be exhibited to an audience. It consists of a light, L (fig. 60), a large lens, C, of short focal length called the con- denser^ and a small lens, M. termed the ma^/ii/^er. Tlio object is placed at O, a I little without the focus of M and its magnified image is received upon a screen b b . The condenser, by collecting a large pencil of rays from the lamp, L, transmits, through the mstrument, a strong ligW which serves to illuminate the screen, thus causing the image to appear as a darkish or coloured painting upon a light ground. The objects are generally pictures painted upon glass slides in transparent colours, or those formed by means of the 1 hoto- grapher 8 Camera. Since the imago is inverted in regard to the object, it is necessary to place the slides in the instrument in an inverted position, in order to have an erect picture upon the screen. ^ The lenses and the lamp are enclosed in a box from which no light can escape, except by way of the magnifier, and the exhibition is conducted in a darkened room. 193. Solar Microscope. This is but a Magic Lantern hi which the magnifier is accurately constructed and compounded so as to be achromatic and aplanatic, or nearly so. ^ The resul- tant focal length is short, and in order to get sufficieiit hght, the rays of the sun, rendered horizontal by a plane mirror, are condensed by a lens and transmitted through the instrument. The same instrument when illuminated by the oxy-hydrogen light is known as the oxy-hydrogen microscope. In these microscopes it is not easy to arrive at any definite conclusion in regard to the magnifying power, for the image will not generally admit of a very close inspection. We can, however, obtain an approximate Value as follows. Denote the distance from the magnifier to the screen by d, the focal length of the magnifier by f, and the minimum distance at which tlie picture upon the screen will bear satisfactory inspection, by E. I-"! can. PHOTOMETRY. 109 Then from (27) , since d and /are negative, wo obtain, J _d-f But the angle under which the image is seen is =- *, aud that under which the object may be seen is — .. Hence, from (38), Therefore, Or, S w E- f Photometrj* (100) Photometry is the application of instruments, called photo- meters, to measuring or rather comparing the illuminating powers of two or more luminous bodies. The methods of photometry most commonly employed are known as Ritchie's, Kumford's, and " the extinction of shadows." 194. Bitchie's method. This method consists in causing two luminous bodies to give equal amounts of illumination. Tlie triangular prism EFG- (fig. 61) is nearly cove) od with white paper, and so placed that its faces, EG and EF, have the same inclination to the horizon. The two bodies, whose illu- minating powers are to be compared, are placed in a horizontal line with the centre of the prism, and in such positions. A, B, that they may illuminate equally (as detected by the eye looking down from S) the respective faces EG and EF. Denote the illuminating powers of the bodies by L and L\ and their respective distances from the instrument by d and d. Then, the angle of inclination being the same in each, we reject it, and obtain from {^%\ L \ L wd : d"' .....(101) 195). RumfoT'd^s method. This method differs from Eitchie's in details and in simplicity of apparatus, although it employs the same principle. Fig. 61. !■; 110 QEOMETRIOAL OPTIOB. Tlie two luminous bodies, which we may denote by A and B are made to cast shadows of the same non-transparent body upon a white screen, and the lights are so placed that the shadows may be side by side, and appear of the sanie depth ot Z\ou7. Now it is evident that the illumination ot the shadow ca t by A is due to B, and of that cast by B to A, while the niumination of the screen is due to both. Hence in this method as in the other, the luminous bodies are caused to give the same amount of illumination, and we consequently com- pare them as before. 196. Method of extinction of shadows This depends upon the principle that the eye is not able to detect the effects of a light when brought into the presence of one about sixty-lour times as powerful. One of the luminous bodies. A, is made to cast a shadow upon a Xto screen, and the body, B, is then made to approach the L7een until the shadow can no longer be distinguished. In Ekte we infer that the illumination given by B is about 64 times as great as that given by A. Hence denoting the illuminating powers of A and B by Z and r respectively, and their distances by D and d, we have from (65), ^ A — _ illumination given by and illumination by B = d^ £ r--64. Or, dz i>3 The numerical factor of 8 ^ is for normal eyes, but as its value ma^vary slightly in different persons, it must be found IxpTrimenWby Bome of the other methods. If, however, we wish to compare a third body, 0, with B, we eliminate this factor entirely, For, L" : Z:: (S dy : D«, and, .-. Z' : r :: d^ : d'% in which the factor, being constant for the same eye, has dis- id. TABLB OF fi AND JJ. Ill ▲ TABLE OF THE VALUES OF fJL AND U 8UB8TAN0E9. FOR TUB MOST COMMON Alcohol 1-870 0-029 Alum 1-457 0-036 Beryl 1-598 0-037 Canada Balsam 1-545 0-045 Crown Glass 1-530 0030 Diamond 2-440 0-038 Ether 1-366 0-037 Feldspar 1-536 0-043 Flint Glass 1-580 0-04H Fluor Spar 1-485 0-032 Iceland Spar... Nitric Acid Oil of Turpentine... Plate Glass Rock Crystal Rock Salt Sapphire Sul])hide of Car])on. 655 409 •470 •510 560 •557 •780 768 Sulphuric Acid 1-435 Water 1.336 0040 0-045 0042 033 0.03C 0053 0-036 0-115 0031 0035 MISCELLANEOUS PROBLEMS. 1. Wliat is tlie relative index when liglit passes from water into flint glass ? 2. Find the deviation when the angle of incidence upon a plane mirror is 22° 30'. 3. At what angle must two plane mirrors be inclined so that a ray incident parallel to one of them may, after reflection at both, be parallel to the other ? 4. Rays falling upon a mirror from a distance of 10 feet are brought to a focus at a distance of 6 feet in front ; determine the mirror. 5. Parallel rays fall upon a concave mirror having a focal length of 3 feet, and thence upon a convex one of 3 inches focal length. If the mirrors be 38 inches apart, find the position of the resultant focus. 6. In Problem 5,'find the distance between the mirrors when the resultant focus is at the primary. 7. What is the critical angle for diamond ? 8. An equilateral triangular prism is to be employed for the purpose of total reflection without producing refraction \ deter- mine the lowest index necessary for the substance forming the prism. 9. A stone at the bottom of a pond is seen obliquely at an angle of 40°, and appears to be 3 feet below the surface ; deter- mine the depth of tlie pond, 112 QEOMKTRIOAL OPTIOB. lli 10. Ill a double-convex leuB of crowii glass the radii of cur- vature are 3 and 4 inches respectively ; Hud its focal length when used under water. 11. The primary lens of a compound has a focal length of 20 inches, and receives rays from a point 12 feet distant; determine the secondary which, being placed at a distance of two feet from the primary, may have the resultant focus two inches behind itself. 12. A convex lens is placed at a distance of 2 feet from an object 1 inch long, and the hnagc is found to be 2-25 inches in length ; determine the focal length of the lens. 13. An object 5 inches in diameter is placed 18 inches in front of a convex lens of 7 inches focal length ; hnd the posi- tion and size of the image. 14. Compare the size of the image with that of the object in Problem 11. 15. Find the dispersion in a lens of crown glass 3 feet in focal length, and 4 inches in diameter. 16. Determine the distance between the focus for red rays and that for violet oneri in the lens of rroblem 15. 17. Determine the constituents of an achromatic prism of water and sulphide of carbon, when causing a deviation of 5°. 18. The first face of a Hint glass lens has r = 12 inches ; what must be the radius of curvature of the second face to achromatize a convex crown glass lens of 3 feet focal length ? 19. A simple microscope consists of two lenses, the first be- ing 1 inch focal length, and the second 2 inches. What must be their distance apart in order to be achromatic when viewing an object 6 inches from the first lens? 20. A lamp is placed 6 inches from a plane wall. At a point on the wall 12 inches from the lamp compare the illumination with the greatest received by any point on the wall. 21. The least distance at which a person can see distinctly is 45 inches ; determine the lens he should use. 22. In viewing a small object with a convex lens of 1 inch focus, the lens is one-half inch from the eye, and seven-eighths of an inch from the object. Determine the magnifying power under these conditions, r MISCELLANEOUS PROBLEMS. 113 23. Determine the inagnifyinf^ power of a Ramsden's eye- piece. 24. In a given eompoiind microscope vso have, J^ = • 8. y= 1*5, li = '\ and L = 10; lind the magnifying power and the brightness of the field. 25. In a ^iven telescope the radius of the object glass is 1'6 inches, and its focal length 2 feet. Required the highest power which can bo used without dhninishing the brightness of the field. 26. If in the instrument of Problem 25 a power of 120 be used, and the diaphragm have a radius of "1 inch, what will be th field of view and its brightness \ i:, / riow much shorter would Cassegrain's telescope be than Gregory's, if in each the focil length of the large speculum were 4 feet, of the eye-piece 2 Indies, and if the magnifying power were 100 ? 28. A luminous body. A, extinguishes the shadow cast bv another, B ; compare their illuminating powers, the distances from the screen being, for A 8 inches, and for B 10 feet. ¥ ii I' hi GLOSSARY OF TERMS AND PHRASES. Tho number refers to the article m wliich the term is found expluined. Aberration, — Chromatic — Spherical Achromatic Amplifying lens Angle of emergence — Incidence ... ... — Reflection — Refraction ... ... — Total reflection Aperture Aplanatic ... ..• .•' Brightness ■ Centre of curvature Chromatic Aberration €o-eflicient of transmission — Of reflection Compound Instruments .. Condenser Ooujugate points Critical angle Deviation Diagonal eye-piece Diaphragm Dispersion Dispersive power Displacement Equivalent Eye-lens Eye-piece, Diagonal ■ — Huyghenian — Negative • — Positive — Ramsden's — Terrestrial Field Lens Field of view Focal length Focus, Conjugate .. — Incident — Principal 112 128 122 169 62 7 7 8 54 35 129 142 35 112 139 139 163 192 24 54 27 171 147 114 117 34 108 169 171 169 168 168 170 172 169 182 40 6 6 40 FocuH, Heal ... ... — ReHultant — Virtual Ilori/on glass Huyghenian eye-piece Illuniinaticm Illuminating power .. Image — Xirect ..• ... — Inverted ... — Real ... ... — Resultant — Virtual ... Incidence, angle of — Point of Incident ray Index glass Index of refraction ... — Absolute — Relative Iris ... ... Lens ... ... — Achromatic ... — Amplifying — Ap}anatic — Compound ... — Concave ... — Convex — Eye ... .•• — Field — Optical centre of — Power of ... Magnification Magnifying power . . . Medium Mirror — Concave — Convex — Compound — Optical centre of — Plane 26 49 26 158 109 133 134 85 87 87 87 106 87 7 7 8 158 9 12 12 150 70 122 169 129 106 70 70 169 169 ;9 8a 164 5 19 35 35 106 35 20 1^1 GLOSSARY. ll.") 164 5 19 85 35 106 35 20 Mirror. — Primary ... — Secondary Myopia Ncgutivt! eyepiece Objective Ocular Optical centre Pencil — Convergent — Divergent — Parallel ... Plane Mirror Positive eye-piece Preshvopia ... Principal focus Prism — Edge— Face Pupil 85, 48 48 152 108 H7 107 70 n 6 20 108 158 40 62 02 150 Ray, Incident — Direct — Ilertected — Kefracted IladiuH of curvature Hi;al focuH Resultant focus Ramsden' • eye-')icco Retina Spectrum Speculum (sw Mirror.) Simple Instrument ... S[)hericHl Aberration Terrestrial eye-piece Total reflection, angle of Triplet Virtual focus Virtual Image • • • 6 • •• 6 • • • 6 • « • 6 • • • 85 • •• as • •• 49 • •• 170 • • • 150 • • • 113 • • • 168 • « • 128 • • • 173 • • • 54 • • • 126 • •• 36 • • • 87