1B24 Waves Optics and Acoustics Lecture 33 (UCL)

Page 1

and the corresponding rays refracted through the focal points. The nodal points are the points of intersection with the axis of rays which pass through the optical centre of the system, that is, rays which pass through the lens without being deviated in angle. In the diagrams below, only one set of the focal and principal cardinal points is shown: a corresponding diagram for rays from the other side will show the other set.

! " # $ %'&)(+* &),.- /10324- 5 245

Nodal point

Focal point

Principal point

Nodal point

6$798':<;>=@?.ABADC9EBF GIH<=KJ LB;NMD7POQLBRTSVU'79OXWZY[UB\$L WNH<7PJ^]"R9= O<Y[Y`_aY`WZ=cbdFe\3UaJ LBRfS)U'7POXWZYcgQS<;N79Oah Jc7PSQLBRTSVU'79OXWZY g<LBO<MiO<UaM<LBRfS)U'7POXWZYcC jk\elm= O<=c= MnWZUoM<=cYNJ ;>7Pp)=qWZH<= 79brLB8B=K\3U';>brLsWN7PU'O pt_ LuWNH<7PJ^]dRP=cO<Y gvU';@LuR9= O<Y!bwLBM<= :<Sx\3;>U'byYN=czB= ;NLBR{J UBbwS)U'O<=cOtWNYi|3l[H<79J^Hxl[7PR9R}:<Y>:QLBR9R9_~pV=wWZ;N= LsWZ=cM LBY€YN= z'=c;ZLBRmWNH<7PO R9= O<Y>= Y^‚ƒgQ79WI79YmWZHD=!J LB;>M<7POQLsRfSVUB7POXWZYILsO<MuWZH<=!;NL„_4YIWZHD;NU':<8BHoWNH<=Â…\3U4J LBRfS)U'79OtWNYml[H<79J^H LB;>=Â…WZH<=@]s=c_"†"LBY L ;>= b 7PO<MD= ;{WZHD=c_oLB;>=@YNH<U„l[O 79Ou‡Q8':<;>=@?.ABADC9EBC ˆ 24-.2t‰‹ŠQ0}ÂŒ ‰Â?2tŠDŒ„(+24-VÂŒ &Â?ÂŽ ŒƒÂ?q&<‘Z0Â’24-.5 5c“ 5cŒ„24( ”N- &QÂŒwŽ•&)‰w2X–vŠQ(x—’-.ŠaŒ‹—’&)˜™=Â…J LsOuM<= ;>79z'=@LK8'= O<=c;ZLBRT= ÂšaS<;N=cYNY>7PU'Ou\3U';eWZH<=Â…\3U4J LBR)RP=cO<8BWNH"UB\ LKW›l}Ush›R9= O<YIY`_aY`WZ=cbdg bwLBM<=!UB\.W›l}UdœžÂ?4Â&#x;ž rR9= O<Y>= Y gQ:DYN7POD8 WNH<=ÂĄ8'= U'b =cWN;>_rYNH<U„l[O 79Ou‡Q8':<;>=K? A'AaC£¢aC ¤ U'O<Y>7PM<=c;!WZH<=€W›l}Ush›R9= O<Y@Y>_4Y`WZ= bÂĽ79O l[H<7PJ^HnWZH<= 79O<J U'b 7POD8"SQLB;NLBRPR9= R$R97P8'HXW!79YÂ…\3Ush Jc:<YN=cM~pX_dRP=cO<Y E€LÂ?WÂŚWZHD=qSVUB7POXW!§Â€C)GIH<7PY l[7PRPRv\3U';>b¨WZH<= z479;>WZ:<LBR UBp4Š>= J WÂ…\3U'; RP= ODY@¢ag LBODM€YNUÂ…WZHD7PYÂŞU'ptŠ>= JcWÂŞ7PY LÂ…M<79Y>WZLBO<J =IÂŤXÂŹX­ ÂŽ!WZUÂ…WNH<={;N798'HXWÂŞUB\)R9= O<Y ¢4C$GIH<=cO WNH<={M<79Y>WZLBO<J = ÂŻ \3;NU'b#R9= O<Y ¢qWZU€WNH<=!\3UaJc:<Y 6$°Il[7PRPRVpV=@8'7Âąz'=cOdpX_ E ­ E E ÂŻ ÂŤXÂŹ$­²Ž Âł ÂŤÂ?°Q´ jk\ÂŞWNH<= 79O<79WN7ÂľLBRf;NL _o= OXWZ=c;NYÂ…R9= O<YKEqLsWÂśLrH<=c7P8'HXWÂ…¡VÂŹ[LBO<M = Âša79WNYÂśR9= O<YÂ…¢wLsWÂŚLwH<=c7P8'HXW ¡<° WZH<=cOwWZHD=Âś79OtWN= ;>YN= J WZ79U'OwS)U'79OtWÂś|38'79z479O<8!WZH<=[YN=cJ U'ODM"S<;N79O<J 79SQLBR<S)U'79OtWe¸ ° Â‚ÂŞ7PYÂŞ\3U':<ODM \3;>U'bšWNH<=@YN7Pb 7PRPLB; WZ;N7PLBO<8'R9= Y ¤ 6$°>º…°œLBO<MdÂť{6$°^¸ ° ¡Q° ¡VÂŹ ÂŻ Âł ÂŤ ? A'A‹h^E Thickl1.mcd

1

24/03/99 00:37


Two-Lens Systems General Treatment B

A

C

h1 h2 V1

V2

H2

F2

G

s

d

f1 f

6 7P8':<;>=@? A'ADC¼¢aF GIH<=@8'= UBbw= WZ;>_"UB\ªLqW l}Ush R9= O<Y[Y`_aY`WZ=cbdC LBODMo\3;>U'b ¤ § º °¶LBO<Mi½ §!º¡¬ ·Q° ·V¬ «'¬ª­ ® ³ « l[H<79J^Hd8B79z'=cY[:<YIW lmU = aS<;>= YNY>7PU'ODY¶\3U'; ·<° ¾'·V¬ ·<° ¯ «X¬$­²® ·V¬ ³ « ³ «X¬ 8'7±z47PO<8 :<Y LqzsLBRP:D= \3U'; ¯ WZU Y>:<p<Y>WN79WN:DWZ=@pQLBJ^] «'¬ ­ E E «ª|¿«X¬$­²®4 «X¬ ­ ® ³ « ° l[H<79J^HdbwL _"p)=@;N= Ls;N;ZLsO<8'= MoWZU 8'7±z'= E E E ­ ® « ³ «'¬[À « ° «X¬N« °<´ Á 79bw79RµLB;>R9_'gQUBO<= brL _d8'U"pQLsJ^]dWZU" <O<M WZH<= S)U'YN7±WZ79U'O<Y UB\ WZH<= S<;>7PO<Jc7PSQLsR S)U'7POXWNY C ½¶Y ¯ ³ «X« ¬ | «X¬$­²®4 ? A'A h`¢

Twolens.mcd

1

24/03/99 00:39


WNH<=@SVU'Y>79WN7PU'OuUB\$¸¶° ;N=cRµLsWN79zB= WZU WZH<=@Y>= J UBO<M R9= O<YI79Y{WZH<=cO ¯ ­²« ³ «X« ¬ | «'¬ ­ ®a $­²« ³ ­ «T«X®¬ ´  UBWZ=rWNH<=uYN798'Ofg Y>H<U l[7PO<8ÃWNHQLsW 7±\[pVUBWNH²RP=cO<YN=cY LB;N="JcU'OXz'=c;N8'79O<8 WZH<=cO WNH<="Y>= J UBO<M S<;>7PO<Jc7PSQLsRTS)U'7POXW[RP79= Y{WNU WNH<=@RP= \ WIUB\$WNH<=@YN=cJ U'O<MiRP=cO<Y C Á 79bw79RµLB;>R9_'gXWNH<=! Q;NY`W¶SD;N7PODJ 7PS<LBRTS)U'7POXW[79Y[LsW Ä^ÄkÆÅ ;N=cRµLsWN79z'= WNU WZH<=! Q;>Y>W RP=cO<Y C ÇrImaging ÈeÈ[É`Ê Ë~by ÌxͪaÎBÏ`Ð.Two-Lens Ñ Ò Ò4Ó¶ÒtÔ<Ð.Õ System Î ÐfÖ¦× ÕÙØ!ÏÚÐ 1 h1

2 I1= O 2

O1

I2

4 3 f1 u1

f2 v1

d

u2

v2

6 7P8':<;>=@? A'ADCÛADF ½¦Oi= DLBb S<RP=!Us\ªL Y>_4Y`WZ= b¨l[7±WZHuW lmU RP=cO<YN=cY C j OwbwU'Y`WeJ LBY>= Y g4WZH<UB:<8'Hfg479We7PY = LBYN79= ; WZU@lmU';>] WZH<;>U':<8'H WZH<=¶LBOQLBR±_4YN7PY UB\VWZH<=¶Y>_4Y>WN= b \3;>U'bÜ Q;>Y>WeS<;N79O<J 79S<RP=cY C ¤ U'O<YN79M<= ;¶| Q8':<;>=¦? A'ADCÛAX ªLBO U'p4©>=cJcWN79z'= RP=cO<Y UB\T\3U4J LsR<RP=cO<8BWZH E ÝBÝ b bdgD=c_'=cS<7P=cJ =KUB\$\3U4J LBRvR9= O<8sWZH Þ'ß b bigQàtßBÝ bwb¨LsSQLB;>W CmGIH<=cO 7±\$WZHD=KU'pt©>= JcW 79Y!E ßBExample: Ý bwbÙWZU WZHD=¡f R9=c=\ W[100 UB\ RP=cO<mm Y@E!WZH<=@7PbwLB8'v=¦17POu=RP=c300 O<Y@E!l[mm 7PRPRTpV=!8'7±z'=cOdpX_ 1 E E E E u ­ = -E 150 E mm f2 = 75 mm ßBÝ ³ ABÝ'Ý á ¬ ³ «'¬ À â ¬ ³ E Ý'vÝ 2 =E 150mm WNHQLsW 7PY¶A'Ý'Ý bwb¹uWZU 1 WZ=H<=¡- ;N150 798'HXW¶Usmm \ RP=cO<Y@E'g U';K2E ßBÝ b bãWNU WNH<=KR9=c\ W UB\ R9= O<Y¶¢aCmGIH4:DY WNH<=! QOQLBRv79brLB8B=¶l[d7PR9Rf=p)=!450 8'79zB= Oimm pt_ M=2 E E E E ­ E E á ° ³ « ° À â ° ³ ÞBß E ßBÝ ³ E ßBÝ ´ GIH<=¡U z'= ;NLBRPRTbrLB8BO<79 QJ LsWZ79U'O"79YI8'79zB= Oipt_ ä ³ âá ¬¬ âá °° ³ ­¡A'E ÝBßBÝ Ý ­KE E ßBßBÝ Ý ³ ¢ ´ ? A'A hÚA Lines 1 and 2 are used to construct the image I1 of O1 in lens 1; lines 3 and 4 are used to construct the image I2 of I1, treated as the object of lens 2.

Lens13.mcd

1

23/03/99 22:46


ÇrĂˆeĂˆ[ÉkĂĽ

ĂŚ+Ă?}Ă•šĂ˜ Ă?}ç!Ă‘Â…è¼Ă•Ă™ĂŠ`ĂŞvĂŤQĂ?eĂ’DĂŞvĂ?}Ă˜ Ă?Ă­ĂŹĂŻĂŽ~ðœù[ò

Two-Lens Systems Compound Microscope

Îą2

h' h fE fO

fO

fE

L d

$6 7P8B:<;N=!? A'AaCÛà <F GIHD=KJcU'bwS)U':<ODMib 7PJc;NU'Y>J U'S)=BC (+Š ˆ -.—žóª%'ŠDŒ„—’&)-þôuĂś!á.øÂ?Ăš =@b 7PJ ;>U'YNJcU'S)=!7PY[mm) YN= WÂś:DSdl[79WNHuWZH<=ÂĄ7PbwLB8'=!LÂ?W[7POD‡QOD79W›_'gQ7ÂąW[7PY{WNH<= LsO<8':<RPLB;{bwLB8sh =jk\ -WZH<(h'/f O<7¹‡QJ LÂ?WZ7PUBOwE)/(h/250 l[H<79J^H"7PY};N= R9=czsLBOXW C 6QU';}LBO"79brLs8'=ÂŚLÂ?W{7POa‡QO<79W›_Bg4WNH<=ÂŚ79OXWZ= ;>bw=cM<7ÂľLÂ?WZ= 7PbwLB8'= =79Y[-LsWI (L/f WZH<=!O\3)(250mm/f U4J LBRTSVU'79OXW[UB\ EWN)H<=@=c_B= S<79= J =sC}jkWNY LBO<8':<RPLB;IH<=c7P8'HXW gaWZH<=cOfgQ7PY Ăť H<=c7P8'HXW[UB\{7POXWZ=c;Nb = M<standardize 7PLsWZ=!79brLB8B= ¡Dthe Note that many manufacturers ‹ú of the objective Âł ‹ú distance from the second focus to LBODMotheWNH<7PY[first 79OtWN= focus ;>bw=cM<7ÂľLsWNof=Â…7Pbwthe LB8'=ÂŚeyepiece H<= 798'HXW[7PYI;N=cRÂľatLsWN= L=160 MuWZU WZHD=ÂĄmm. UBp4Š>= J WÂśHD= 7P8BHtW[pX_ H<= 798'HXW UB\I7POXWN= ;Nb = MD7ÂľLsWN=Â…7PbwLB8'= ­ýß HD= 7P8BHtW[Us\IU'p4Š>=cJcW Âł ÂŤÂ?ޅ´ GIH<=eLBO<8B:<RÂľLB;TH<= 798'HXW UB\aWNH<= U'p4Š>=cJcW gs:<ODbrLB8BO<79‡Q=cMfg‹7PYTWNHQLsW Y>= = O LsWvWZHD= Y>W^LsO<MQLB;>M O<= LB;[SVUB7POXW gaYNU ä™ÿ Âł ­ ÂŤÂ?Ăź Ăž ¢'Ă&#x;BĂ?€‹ú bwb ´ Ă‚ÂśUsWZ=qWZH<LsW@brLsOt_dbwLBOt:D\Â’LBJcWN:<;N=c;NY@Y>WZLBO<MQLs;NM<7 =qU'O Ăź gfWZHD= MD7PY>WZLBO<Jc=q\3;NU'b WZH<= Y>= J UBO<M~\3U4J :DYÂĄUs\}WNH<= U'p4Š>= J WZ7Âąz'=€WZUuWZHD= ‡<;NY>W@\3U4J :<Y!Us\}WNH<= =c_'=cS<7P=cJ =sg$LsW E 'Ă?ubwbiC GIH<79Yr=cOQLBp<R9= Y"= _'= SD7P= Jc= YoLBODM1U'p4Š>= J WZ7Âąz'= Y"WNUxp)= bwLB;N]s= M1YN=cSQLB;NLsWZ=cR9_ l[79WNH WZH<=c7P; U„l[O²bwLB8'O<7¹‡QJ LsWN7PU'ODY g l[7ÂąWZHxWNH<="bwLB8'O<7¹‡QJ LsWN7PU'O™UB\IWZHD=rl[H<U'R9="bw79J ;>U'YNJcU'SV=wpV=c7PO<8 WNH<=@S<;NU4M<:<J WÂśUs\ WNH<=!W›lmU<C With the final image at infinity, angular magnification is - Îą2/(h/standard near point)

Lens16.mcd

1

23/03/99 22:50

? A'A‹h›Ă


245 &)0 v ¿- * &4 q2t dôuö!÷.ø ù GIH<= ;N=cYNU'R±za79O<8 SVU l}= ;d7PYiM<=cWN= ;Nb 7POD= Mfg!LsYd79O LBR9R!U'SDWN7PJ LsR!7PO<Y`WZ;N:Dbw=cOtWNY g!pX_ WZH<= LBS)= ;`WZ:<;>=BC j OoWZH<79YIJ LBY>=@WNH<=@RP79bw7±WmUB\ ;N=cYNU'R9:DWZ79U'Oo7PYI8'7±z'=cOdpX_ Ý ´ DE ³ ¬aYN7POv| ª¬> 4´ ¸ = ;N= ¬¡7PY WNH<=";>=c\3;ZLsJcWZ7±z'=u79O<M<= +UB\[WZHD="brL WZ= ;>7µLBR 4 ¡WZHD="U'p4©>=cJcWZ7±z'=sg ª¬¡7PY WNH<= LBO<8'R9=qYN:DpDWZ=cO<M<= M+LsW!WZHD= U'p4©>= J WZ7±z'=BC GIHD= UBp4©>= J WK79Y UB\ WN= O z'=c;>_nJ R9U'YN=qWNU"WZH<= R9= O<YcgDYNU HD= ;N= l}=!M<U HQL zB= WZU l}U';N;`_uLBp)U':DW{LBO<8'R9= Y{RPLB;N8B=¦= ODU':<8'H"\3U';eWZH<=!S<LB;ZL a7PLBR LBSDS<;NU a7PbwLsWZ79U'O WZU¡pV= t:<= Y`WZ79U'OQLBp<R9=BC [= Y>U'RP:DWN7PU'O bwL _ WZH<=c;N=c\3UB;N=¦p)= 7Pb S<;NU z'=cMrpX_ LBOoU'79Rf7Pb b = ;NY>7PU'OrU'p4©>= J WZ7±z'=BC ÇrÈeÈ[É È Ë Ð ÔDÐ.Ï`Ð ÒaêvÍmØ ÐýÎ ìõî ð¶ñ Two-Lens 2 <5c &)-.&)Systems (x %' Q0} 2t k Q-%B &

Astronomical Telescopes

h h" fE

fO d

$6 7PLinear 8B:<;N=@? A'ADmagnification C¼ßaFªGIHD= LBY`WZ;>U'O<U'b =7PJ LBfRVEWN/f= ORP=cYNJcU'SV=sF RP7POD= LB;{bwLB8'OD79 QJ LsWZ79U'OfC GIH<=¡LBY`WZ;>U'O<U'b 7PJ LBRT;>=c\3;ZLsJcWZUB;¡|3 Q8':D;N=¡? A'ADC¼ß' mS<;NU4M<:<Jc= Y LBOo7POXzB= ;>WN= Md7PbwLB8'=BCª»mUBWZH WNH<=!U'p4©>=cJcW¶LBO<MuWZHD=¡79brLs8'= LB;>=@LsW[7POa QO<79W _BgDY>U WNH<=!YN=cJ U'O<MiS<;>7PO<Jc7PSQLBR)\3UaJc:<Y[UB\ WZH<= U'pt©>= JcWN79zB="7PYq79O+WZHD="YZLBb =rSDRµLBJc=uLBY WNH<=r Q;>Y>W S<;>7PO<Jc7PSQLsR \3U4J :DYqUB\[WZH<=r=c_'=cS<7P=cJ =rh WNHQLsWqα79Y gªWNH<="YN=cSQLB;NLsWZ79U'O²UB\IWZHD=rW lmUnR9= O<Y>= Y 7PαYK= t:QLBR}WZU WNH<=uYN:<byUB\[WZHD= 7P;¡\3UaJ LBR R9= O<8BWNH<Y C 2

1

fE

fO d

? A'A h`ß

Angular magnification = fO/fE Lens14.mcd

1

23/03/99 22:47


fE

fO d

Linear magnification = fE/fO

α2

α1

fE

fO d

6$798':<Angular ;N=@? A'AaC DF GImagnification HD=KLsY>WZ;>U'O<U'b 7PJ LBRV=WN= fR9= OYNJc/fU'ES)=BFeLBO<8':DRµLB;IbwLB8'O<7± QJ L WZ7PUBOfC !

+( -. óª%' D &)-õôuö!÷.ø GIH<=wWZ;ZLsO<Y>zB= ;NY>=ubrLs8'O<79 <J LsWN7PU'O ä ³ « ú.¾B«sþ 79YKO<UsW L :<Y>=c\3:<R t:QLBOXWZ7±W _+79O WNH<7PY J LBYN=sC{GIH<= ;>= R9=czsLBOXW brLB8BO<79 QJ LsWZ79U'O~|3 <8':<;N=K? A'ADC X {7PYIWZHD= LBOD8':<RµLs;IbrLB8BO<79 QJ LsWZ79U'O l[H<79J^H J LBO+p)= M<= MD:<J =cMx;N= LBM<7PR±_ |3YN= = M<7PLB8';ZLsb" [:<YN79O<8iWN;N7PLBO<8'R9= Y!= LsJ^H+HQL z47PO<8iLBY 7±WZYIH<=c7P8'HXWIWZHD=¡HD= 7P8BHtW UB\.WZHD=¡79OXWZ= ;>bw=cM<7µL WZ= 79brLB8B=BF ä ÿ ³ . °¬!³ ·D· û±û ¾'¾'« « úþ ³ «s« ú!þ ´

Lens14.mcd

1

23/03/99 22:47

"

$#

%

&

245 &)0 v ¿- * &4 q2t dôuö!÷.ø GIH<=¦;N= Y>U'R9z479O<8 S)U lm=c;{UB\fWNH<=¶WN= RP=cYNJcU'SV=¦M<= S)= O<MDYI= OXWZ79;N= R±_ U'OrWNH<=¦R97PO<= LB;}MD7µLBb =cWN= ; UB\TWZH<= UBp4©>= J WZ79zB=BC GIH<=!LsO<8':<RPLB; YN= S<LB;ZLsWN7PU'OrUB\fW l}U SVU'79OXWZYmUBOuLBO"U'pt©>= JcW}WZHQL W{J LsO ©>:<Y`W pV=!;>= YNUBR9z'=cM 79Y E ´ ¢'¢ ³ l[H<=c;N= ¨7PY{WZHD=¡UBp4©>= J WZ79zB=@M<7µLBb =cWN= ;cC 2d 24 245c 3 Q0 2a03245 %'& * 2 j O L WZ=c;N;N=cY>WN;N7µLsR WN= R9= YNJcU'S)=BgTUBO<=KlmLBOXWZY¦LBOÃ:<S<;>7P8'HXW¶7PbwLB8'=sC O<=qLBSDS<;NUXLsJ^HÃ7PY[WZH<= §@LBRP79RP= LBOrU'OD=r| Q8':<;>=@? A'ADC¼Þ' C ¸ = ;N=¡LB8'LB7PO"WNH<=@YN= S<LB;ZLsWN7PU'OoUB\.WZHD=¡R9= O<Y>= Y[7PYIY>=cW[WZU WNH<=IYN:<b Us\)WNH<=I\3UaJ LBR<R9= O<8BWNH<Y | pD:DW WZH<=I\3U4J LBRDRP= OD8BWZH UB\)WZHD= = _'=cS<7P=cJ = 7PYªO<=c8XLsWN79z'= C dU';>=wUs\ WZ= OTg$LibwU';>= J U'b S<R9= ~LB;>;ZLBOD8'= b = OXWKUB\{R9= O<Y>= Y¡7PY@:<YN=cM+WZUibwLB]B= WZH<= 79brLB8B= = ;>= JcW C ? A'A h

('

)

*

+

,+

.-

/

0


Two-Lens Systems GalileanSystems Telescopes Two-Lens Galilean Telescopes

h

h"

h

h" fE

fE fO

fE

fE

fO

d

6$7P8B:<Linear ;N=!? A'AaCÂŁĂžaF magnification GIHDd=q§@LBRP79RP= LBOwWN= RP==cYNJcfU'ESV/f=sOF RP7POD= LB;{bwLB8'OD79‡QJ LsWZ79U'OfC Linear magnification = fE/fO

Îą1 Îą1

Îą2 Îą2

fE fO

fE

fE

fO

d

Lens15.mcd

fE

Angular magnification = fO/fE d Angular!1 magnification = fO/fE

6$798':<;N=@? A'AaC DF$GIH<=q§@LBR97PRP= LBOwWN= R9= YNJcU'S)=BF}LBO<8':<RPLB;{bwLB8'O<7¹‡QJ LÂ?WZ7PUBOfC 1

23/03/99 22:48

ÂŒ 2dÂŒ 2a032t* &QŒ„&x0324- 5 GIH<79Y€7PY€79O²YNU'b ="l{L _4Y YN7Pb 7PRPLB;!WZU WZHD= §@LBR97PRP= LBO™WN= R9= YNJcU'S)=Bgep<:DW€l[79WNH²WZH<=uR97P8'HXW p<;>U':<8'HXWIWNU L ;>= LBRV\3UaJc:<YÂĄ|Â’79O"WZH<=!S<RPLBO<=Â…Us\ WNH<=…‡QRPbr‚ C$GIH<=!RP=cO<YIYN=cSQLB;ZLÂ?WZ7PUBOfgDWZHD= Ofg 79YIRÂľLB;>8'= ;mWNHQLBOi7POuWZHD= §@LBR97PRP= LBOwWN= R9= YNJcU'S)=@l[7ÂąWZHoYN79bw79RÂľLB;}RP=cO<YN=cY C *.‰Â?—¿5 (™ŠDŒ‹—’% $—’-.&f%X, 03Š<‰Â?5 j›OĂƒSD;N7PY>brLÂ?WZ7PJ@p<79O<U4J :<RPLB;NYcgÂ?WNUBW^LsR 79OtWN= ;>OQLBR.;N= <= JcWN7PU'O 7PO SD;N7PY>bwY 7PY :<Y>= M WZUw= Âš4WZ=cO<M WNH<= SQLÂ?WZHĂƒR9= O<8BWNH p)=cW›l}= =cO U'p4Š>= J WZ7Âąz'=€LBODMĂƒ=c_B= S<79= J =sgf= V= J WZ7Âąz'= RÂą_ Ă›\3U'R9M<7POD8 <WZH<=KU'Sah WN7PJ LsRfSQLsWNHfC

Lens15.mcd

1

23/03/99 22:48

2

43

65

67

? A'A‹h`Þ

$8

:9


2 <5c &)-.&)(x %' Q0} 2 2a%B &) GIH<=i Q;NY`W";>= Q=cJcWZ79O<8 WN= R9= YNJcU'S)=dlmLBY"M<=cYN7P8BO<= M pX_ j YZLBLBJà=clIWZUBOfg{WZU+L zBU'7PM WZH<= S<;>U'p<R9= b YIUB\$J^HD;NU'bwLsWZ79J!LBp)= ;>;ZLsWN7PU'Ou79Oi8BRµLBY>YIRP= ODYN= YcC , $032 T*. Q%'2 m24032a5 %'&)* 2~ôuö ø ù ø 7±WZHuWZH<=@¸ :<p<p<R9=!WZ= R9= Y>J U'S)=BgQH<79Y>WNU';>_rHQLBY LBRPb U'Y`WIJ U'b = \3:<R9RTJ 79;NJcRP=¶ wWZH<UB:<8'HuWZH<= M<=cYN798'Oo7PY[LBJ WZ:QLBR9R9_ ¤ LBYNY>= 8';NLB7PO<7PLBOo;ZLsWNH<= ;{WNHQLsW¶Â¶= lIWZU'OD7µLBOfC

<;

=

65

>3>3

@?

BA

(C

ED (CF"

? A'A h

01


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.