L ~rif -fo4-f

Dr Sohl Applied Optics Homework Chapter Pedrottil ~3 A double convex lens has a diameter of 5 cm and zero thickness at its edges A point object on an axis through the center of the lens produces a real image on the opposite side Both object and image distances are 30 cm measured from a plane bisecting the lens The lens has a refractive index of 152 Using th~ equivalence of optical paths through the center and edge of the lens determine the thickness of the lens at its center

-~- -~----~ ---=~~--------------baL-+---------=~lIo--- tSA ~e

1------5-----+1

S - S- 30~

S ~ cit I- t ~Y 141t f-rlt we lt01 +ckc h P

11e -uo or+ic1 pK kLs ~fA$f- b~ e~ lluS

) OPLJ = OPLs

~-----Yt~Q = rt ($- t) + t t amp = 5 ~ t(t1~-I) ~

~ =2 amp-s vtpa--I

J=JSamp+RZ = J~Ooi-)+middot(2rr-) to

J 30107 c--~ I ~

SO_lOVe-- - jfJ(J a-shy

t=-2 S1 -

)

I Dr Sohl Applied Optics Homework Chapter2 Pedrotti1

24 Determine the minimum height of a wall mirror that will permit a 6-ft person to view his or her entire height Sketch rays from the top and bottom of the person and determine the proper placement of() the mirror such that the full image is seen regardless of the person s distance from the mirror

BecGtuse +Ie o-nIe o-F t1cJe~ce QIe 1- re-Plec+-lQ ogtAy p~~e w) aIIDIJ YOfIL fo See f-wmiddotce ~t e ~s ~ Si~ -fe dor (SeetuHy ~IS is 4~tL~f Si~1 I~

up c(lJJE -Hu-f Jce~4 ytAA~r)

o

A- S i WI Itlr $ Ilte+e k WOIA(~ SkocJ ftv- Tk for ( fIe UYo r- IAlA$+shybe ha+ WA)I fpe~e4

1t1( cly ampe eyeS 4J r-f -i-Le-J ce I eS --to -00shy

~~r--~--------------------------

Ier-es 1-Y4 Iy) he flfAce~e+- tgt-P e eyes on lf- JJ-y

Jf)e$tI( ~ j+er Bu+ 1e7S oSStAe ~ ey~$ ca(e = as- J

+rrJW- ~ +or tgt~ +-pounde l-t etJ) os-

Di-Ffe((~L-=--__ ~f Wy to +0 o-f heo s 7~1 r

-

j s 3 I

eXIIldmiddoty amps~1 I1---- If woy -N c_o-r ~ 7S kL ~rs(Jltt

hei k+

) MOJ I( hock p(SM -e ror Allows ~ sMYer o ttA r- -tiel o~ vew Howeler +ie fe(SQII1 wllI If(ar SWlol4r (~ S1AL+II(JleJ Q v Je) h y e yen tJ C Hy -t-ke S Q -e a - 0 -+ +0 tKOiM f-ill 1)(Qlol I bull

Dr Sohl Applied Optics Homework Chapter3 Pedrotti

C) 19 A meter stick lies along the optical axis of a convex mirror of focal length 40 cm with its nearer end 50 cm from the mirror surface How long is the image of the meter stick

Co vtJ W1rrl- h~s ~ kl I~H- 1== -10~

L-J=Js 5 T

~o Ie -Po r k1L --e Jjs +-rtce

o s = Sp5- of

-rhe -Ffr e~J slaquo1- ~=- J~f)~

bull f S I =- 0 tN- (-1IJ~) = - 3 ~ ~ JID~tlO~

lAe ampt+teretllct IS e letfl+A or fi-L M~e

tCli - (-1iraquo)r_ l~C- I o

r-kmiddot Dr Sohl Applied Optics Homework ChapteI1l Pedrottiamp ~O A glass hemisphere is silvered over its curved surface A small air bubble in the glass is located on the central axis through the hemisphere 5 cm from the plane surface The radius of curvature of the o spherical surface is 75 cm and the glass has an index of 150 Looking along the axis into the plane surface one sees two images of the bubble How do they arise and where do they appear

ar- W i rAteJlSpce rl ebullbullfj~~S for ~ flL fwo ~~I ~

------------------~--~~+__--- 0It 4

(Note ~ SlfR=-7 S-~ eort-lert-r )

r1 =- ~ t

I

o

5 12

b) It t frch~ -f -fits tW ---e by -Ile- F IAnt ~((r~ce ))s4-tc-t -10 f (~IIt rvt ce- =- 1 S-~ + 7 ~CAM =IS~ o S =-t S= -t-yen IS-eN- =Eio ~te~ P fle ple s rface

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti1 16 A plano-convex lens having a vocal length of 250 em is to be made with glass of refractive index 1520 Calculate the radius of curvature of the grinding and polishing tools to be used in making this lens

fGlt1 0 eon vec L ertS

f 3 ~O GhVl

A _ bull __ 0 00 r - r f Co

YI t1 =520 J A~

L euro r$ W1oke r~o -(n-z I)I Ylz-nc -f il Rz-

R2 -02- 1) + =- (I 5ZO-IDOII) 2 s- OCoM-

R2 - - 3 I 0 ~

+00 Is) 0 M flbull+-Slc I) J0

o-F fI3 0 ~ )

J

~flrlt Ib j-I Dr S~b1 A~~~~ ~ti~ Homework Chapterl PeclrottiJ ~S23 A small object is placed 20 em from the frrst of a train of three lenses with foeallengths in order

of 10 15 and 20 em The first two lenses are separated by 30 em and the last two by 20 em Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive (b) the middle lens is negative (e) the first and last lenses are negative Provide ray diagrams for each case

I II (amp I- J tl tJtI fA tf -fr tAfJe Calc amp( ( +0$ W~ d fII 0 It e s 0( I

If -tkte We -FroVtll 0 e or-he ele 1I4Il1I+ becotlfes ft(L objee+ (Dr

n e e liT fgt f-t e- bull 1 eltf tIM e r1 + middotII l- L ~ -L - S I _sAc($CLA I vJf WI If U ~e s $p s-f

+l 0 tf rJ s el - s wl~re -= z to 3 CA~

A A-I

1 -1

~ S~O~cR-- -= 3() ~ JL3

~OCN-

ct) f = 10 eN- Fa I ~c-- c =2)~ I

s I = ~OtN- Ie eN- = tOc--shyI 2) CfltI4r- I (J GIIA

cR - s- 3CJ c- - 2 0 ~= 1- IOc-shy1

l - ICe- r~ =- - 30~)z shyD -

S3 J -s 2DcAMo-(-30~) S-O~middot _ 23

s -= (33 3 eo- ) ~ -sf Ies

(Yl - M Wt WJ 1J-f~J= L5)L5 y- 1)-t -~ - s l Sl 0 e-- ngt tNS3 IIUI

pe-cil is rfAy frut e Its1 Reci S fDy G-ret roy 0( fer

+nAC~ ~r It 3 leKS ~2-

r-kmiddot Dr Sohl Applied Optics Homework ChapteI1l Pedrottiamp ~O A glass hemisphere is silvered over its curved surface A small air bubble in the glass is located on the central axis through the hemisphere 5 cm from the plane surface The radius of curvature of the o spherical surface is 75 cm and the glass has an index of 150 Looking along the axis into the plane surface one sees two images of the bubble How do they arise and where do they appear

ar- W i rAteJlSpce rl ebullbullfj~~S for ~ flL fwo ~~I ~

------------------~--~~+__--- 0It 4

(Note ~ SlfR=-7 S-~ eort-lert-r )

r1 =- ~ t

I

o

5 12

b) It t frch~ -f -fits tW ---e by -Ile- F IAnt ~((r~ce ))s4-tc-t -10 f (~IIt rvt ce- =- 1 S-~ + 7 ~CAM =IS~ o S =-t S= -t-yen IS-eN- =Eio ~te~ P fle ple s rface

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti1 16 A plano-convex lens having a vocal length of 250 em is to be made with glass of refractive index 1520 Calculate the radius of curvature of the grinding and polishing tools to be used in making this lens

fGlt1 0 eon vec L ertS

f 3 ~O GhVl

A _ bull __ 0 00 r - r f Co

YI t1 =520 J A~

L euro r$ W1oke r~o -(n-z I)I Ylz-nc -f il Rz-

R2 -02- 1) + =- (I 5ZO-IDOII) 2 s- OCoM-

R2 - - 3 I 0 ~

+00 Is) 0 M flbull+-Slc I) J0

o-F fI3 0 ~ )

J

~flrlt Ib j-I Dr S~b1 A~~~~ ~ti~ Homework Chapterl PeclrottiJ ~S23 A small object is placed 20 em from the frrst of a train of three lenses with foeallengths in order

of 10 15 and 20 em The first two lenses are separated by 30 em and the last two by 20 em Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive (b) the middle lens is negative (e) the first and last lenses are negative Provide ray diagrams for each case

I II (amp I- J tl tJtI fA tf -fr tAfJe Calc amp( ( +0$ W~ d fII 0 It e s 0( I

If -tkte We -FroVtll 0 e or-he ele 1I4Il1I+ becotlfes ft(L objee+ (Dr

n e e liT fgt f-t e- bull 1 eltf tIM e r1 + middotII l- L ~ -L - S I _sAc($CLA I vJf WI If U ~e s $p s-f

+l 0 tf rJ s el - s wl~re -= z to 3 CA~

A A-I

1 -1

~ S~O~cR-- -= 3() ~ JL3

~OCN-

ct) f = 10 eN- Fa I ~c-- c =2)~ I

s I = ~OtN- Ie eN- = tOc--shyI 2) CfltI4r- I (J GIIA

cR - s- 3CJ c- - 2 0 ~= 1- IOc-shy1

l - ICe- r~ =- - 30~)z shyD -

S3 J -s 2DcAMo-(-30~) S-O~middot _ 23

s -= (33 3 eo- ) ~ -sf Ies

(Yl - M Wt WJ 1J-f~J= L5)L5 y- 1)-t -~ - s l Sl 0 e-- ngt tNS3 IIUI

pe-cil is rfAy frut e Its1 Reci S fDy G-ret roy 0( fer

+nAC~ ~r It 3 leKS ~2-

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti1 16 A plano-convex lens having a vocal length of 250 em is to be made with glass of refractive index 1520 Calculate the radius of curvature of the grinding and polishing tools to be used in making this lens

fGlt1 0 eon vec L ertS

f 3 ~O GhVl

A _ bull __ 0 00 r - r f Co

YI t1 =520 J A~

L euro r$ W1oke r~o -(n-z I)I Ylz-nc -f il Rz-

R2 -02- 1) + =- (I 5ZO-IDOII) 2 s- OCoM-

R2 - - 3 I 0 ~

+00 Is) 0 M flbull+-Slc I) J0

o-F fI3 0 ~ )

J

~flrlt Ib j-I Dr S~b1 A~~~~ ~ti~ Homework Chapterl PeclrottiJ ~S23 A small object is placed 20 em from the frrst of a train of three lenses with foeallengths in order

of 10 15 and 20 em The first two lenses are separated by 30 em and the last two by 20 em Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive (b) the middle lens is negative (e) the first and last lenses are negative Provide ray diagrams for each case

I II (amp I- J tl tJtI fA tf -fr tAfJe Calc amp( ( +0$ W~ d fII 0 It e s 0( I

If -tkte We -FroVtll 0 e or-he ele 1I4Il1I+ becotlfes ft(L objee+ (Dr

n e e liT fgt f-t e- bull 1 eltf tIM e r1 + middotII l- L ~ -L - S I _sAc($CLA I vJf WI If U ~e s $p s-f

+l 0 tf rJ s el - s wl~re -= z to 3 CA~

A A-I

1 -1

~ S~O~cR-- -= 3() ~ JL3

~OCN-

ct) f = 10 eN- Fa I ~c-- c =2)~ I

s I = ~OtN- Ie eN- = tOc--shyI 2) CfltI4r- I (J GIIA

cR - s- 3CJ c- - 2 0 ~= 1- IOc-shy1

l - ICe- r~ =- - 30~)z shyD -

S3 J -s 2DcAMo-(-30~) S-O~middot _ 23

s -= (33 3 eo- ) ~ -sf Ies

(Yl - M Wt WJ 1J-f~J= L5)L5 y- 1)-t -~ - s l Sl 0 e-- ngt tNS3 IIUI

pe-cil is rfAy frut e Its1 Reci S fDy G-ret roy 0( fer

+nAC~ ~r It 3 leKS ~2-

~flrlt Ib j-I Dr S~b1 A~~~~ ~ti~ Homework Chapterl PeclrottiJ ~S23 A small object is placed 20 em from the frrst of a train of three lenses with foeallengths in order

of 10 15 and 20 em The first two lenses are separated by 30 em and the last two by 20 em Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive (b) the middle lens is negative (e) the first and last lenses are negative Provide ray diagrams for each case

I II (amp I- J tl tJtI fA tf -fr tAfJe Calc amp( ( +0$ W~ d fII 0 It e s 0( I

If -tkte We -FroVtll 0 e or-he ele 1I4Il1I+ becotlfes ft(L objee+ (Dr

n e e liT fgt f-t e- bull 1 eltf tIM e r1 + middotII l- L ~ -L - S I _sAc($CLA I vJf WI If U ~e s $p s-f

+l 0 tf rJ s el - s wl~re -= z to 3 CA~

A A-I

1 -1

~ S~O~cR-- -= 3() ~ JL3

~OCN-

ct) f = 10 eN- Fa I ~c-- c =2)~ I

s I = ~OtN- Ie eN- = tOc--shyI 2) CfltI4r- I (J GIIA

cR - s- 3CJ c- - 2 0 ~= 1- IOc-shy1

l - ICe- r~ =- - 30~)z shyD -

S3 J -s 2DcAMo-(-30~) S-O~middot _ 23

s -= (33 3 eo- ) ~ -sf Ies

(Yl - M Wt WJ 1J-f~J= L5)L5 y- 1)-t -~ - s l Sl 0 e-- ngt tNS3 IIUI

pe-cil is rfAy frut e Its1 Reci S fDy G-ret roy 0( fer

+nAC~ ~r It 3 leKS ~2-

0 1lt ~~~ Dr Sobl APPlied~ti~ Homew~rk Chapter3 PedrottP - ~23 A small object is placed 20 cm from the first of a train of three lenses with focal lengths in order

of 10 15 and 20 cm The first two lenses are separated by 30 em and the last two by 20 em Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive (b) the middle lens is negative (e) the first and last lenses are negative Provide ray diagrams for each case

0 V cR SJ d - s Wfre ) -= -ltta3 11

+t CQSe )-1 1 ~-I

J

S 20~J- = 30 ~ JL3 ~O~ ~

a) f -= IO~ ft I ~c- + =- 2tJ~ I

5 = ~OCoN-IO eN- = -o~ I 2(J) tAfJ4-- I 0 ~

s cR - S 30 c- - ~ 0 ~ = 1- 10 c-shy1 11 I

I

53 J - S 2 D~ - (- 3 ot-) ~o~middot _ ~3

S [33 3 c- 7f~ ~+ )~S

1- (-be--) A 0 c- I 0 c-shy

J LJ IpllClt IV

h) + = +0 c +~ =- - ~ c~ f -= + l 0 c~-I l

5 - ~ ) c-- ( 30 crv- J =2 0-I ~ - 23

= (tllCe ~ )0 eM JI

_ I -- ~O~ (-rro ~ c3 0)I

Sl = 3 0 c - l 0 e- = I 0 c ( J ~ ~ t5 1 GI) s~ =- ( -5C- - o e- f~ - 6 c-A (+ro- tl (j))e

5 shy3 shy

t shy-J shy

yY1 = _

s- t-J f

~7 gt c- 6 ~tC + -s Ggt-~

~~1 ~ ft1- (3I -shy deg4

-t3 r _ $E ~

(W - 10 ~

c) + -Ocw +) = +1- cshy

5 - 20 ~ J =- 30 CAM1 12 z 0 c

I -I

iO~ ) =- - b~ G 7 VMA

Sl- 30~ - -b 67c- ) - 3t7~ (cru1 t ~ copy) rt

I (J I )-1 (r rcmiddot e (j))15- = tS-~ - 3b67c = ~ 53

5 = 0 ~ - lt S 3 k c =- -5 3K r-fr e -1 (i))3 1

51 =(I-Ov- - 1-S-~3lc )-1=[7 37~ J e~(2))( FrOM

[~ O 36 )

(=rro ~ ( ~ vt 9) I f3

rY7 =

S c c Ie I

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti3

2~24 A convex thin lens with refractive index of 150 has a focal length of 30 cm in air When immersed in a certain transparent liquid it becomes a negative lens with a focal length of 188 cm Determine the refractive index of the liquid

5 fo~t w i-k tJe Ie Vs ~ur ~ uJ~ -

V1~- Yl l _ 1) Y1 ( R Rz

1he sf04- of bafr ~ere is +t+- cJ-t- Ji~+ rOIN R IJr 7lt~ W~ cOfAJ tlstlL-e a s1tMo-r leMJ RRlJ Io+- we tioI-IshyVI eeJ fD J ~+- i flu (Jkbullheor- ~ I lAtA- R- Q~) 7Lyr~ COV$rrA+middot

t ~ (1 -tJ == t == S4pe Cgl+-f(a MJe yen ae)

us e 2 J i rt It fi fbulli ) () -()l-------- y - rlt - a_3 _ - shyS- OJshy~ shy

~l bull shymiddottJ~~

1- (-be--) A 0 c- I 0 c-shy

J LJ IpllClt IV

h) + = +0 c +~ =- - ~ c~ f -= + l 0 c~-I l

5 - ~ ) c-- ( 30 crv- J =2 0-I ~ - 23

= (tllCe ~ )0 eM JI

_ I -- ~O~ (-rro ~ c3 0)I

Sl = 3 0 c - l 0 e- = I 0 c ( J ~ ~ t5 1 GI) s~ =- ( -5C- - o e- f~ - 6 c-A (+ro- tl (j))e

5 shy3 shy

t shy-J shy

yY1 = _

s- t-J f

~7 gt c- 6 ~tC + -s Ggt-~

~~1 ~ ft1- (3I -shy deg4

-t3 r _ $E ~

(W - 10 ~

c) + -Ocw +) = +1- cshy

5 - 20 ~ J =- 30 CAM1 12 z 0 c

I -I

iO~ ) =- - b~ G 7 VMA

Sl- 30~ - -b 67c- ) - 3t7~ (cru1 t ~ copy) rt

I (J I )-1 (r rcmiddot e (j))15- = tS-~ - 3b67c = ~ 53

5 = 0 ~ - lt S 3 k c =- -5 3K r-fr e -1 (i))3 1

51 =(I-Ov- - 1-S-~3lc )-1=[7 37~ J e~(2))( FrOM

[~ O 36 )

(=rro ~ ( ~ vt 9) I f3

rY7 =

S c c Ie I

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti3

2~24 A convex thin lens with refractive index of 150 has a focal length of 30 cm in air When immersed in a certain transparent liquid it becomes a negative lens with a focal length of 188 cm Determine the refractive index of the liquid

5 fo~t w i-k tJe Ie Vs ~ur ~ uJ~ -

V1~- Yl l _ 1) Y1 ( R Rz

1he sf04- of bafr ~ere is +t+- cJ-t- Ji~+ rOIN R IJr 7lt~ W~ cOfAJ tlstlL-e a s1tMo-r leMJ RRlJ Io+- we tioI-IshyVI eeJ fD J ~+- i flu (Jkbullheor- ~ I lAtA- R- Q~) 7Lyr~ COV$rrA+middot

t ~ (1 -tJ == t == S4pe Cgl+-f(a MJe yen ae)

us e 2 J i rt It fi fbulli ) () -()l-------- y - rlt - a_3 _ - shyS- OJshy~ shy

~l bull shymiddottJ~~

c) + -Ocw +) = +1- cshy

5 - 20 ~ J =- 30 CAM1 12 z 0 c

I -I

iO~ ) =- - b~ G 7 VMA

Sl- 30~ - -b 67c- ) - 3t7~ (cru1 t ~ copy) rt

I (J I )-1 (r rcmiddot e (j))15- = tS-~ - 3b67c = ~ 53

5 = 0 ~ - lt S 3 k c =- -5 3K r-fr e -1 (i))3 1

51 =(I-Ov- - 1-S-~3lc )-1=[7 37~ J e~(2))( FrOM

[~ O 36 )

(=rro ~ ( ~ vt 9) I f3

rY7 =

S c c Ie I

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti3

2~24 A convex thin lens with refractive index of 150 has a focal length of 30 cm in air When immersed in a certain transparent liquid it becomes a negative lens with a focal length of 188 cm Determine the refractive index of the liquid

5 fo~t w i-k tJe Ie Vs ~ur ~ uJ~ -

V1~- Yl l _ 1) Y1 ( R Rz

1he sf04- of bafr ~ere is +t+- cJ-t- Ji~+ rOIN R IJr 7lt~ W~ cOfAJ tlstlL-e a s1tMo-r leMJ RRlJ Io+- we tioI-IshyVI eeJ fD J ~+- i flu (Jkbullheor- ~ I lAtA- R- Q~) 7Lyr~ COV$rrA+middot

t ~ (1 -tJ == t == S4pe Cgl+-f(a MJe yen ae)

us e 2 J i rt It fi fbulli ) () -()l-------- y - rlt - a_3 _ - shyS- OJshy~ shy

~l bull shymiddottJ~~

Dr Sohl Applied Optics Homework Chapter 2 Pedrotti3

2~24 A convex thin lens with refractive index of 150 has a focal length of 30 cm in air When immersed in a certain transparent liquid it becomes a negative lens with a focal length of 188 cm Determine the refractive index of the liquid

5 fo~t w i-k tJe Ie Vs ~ur ~ uJ~ -

V1~- Yl l _ 1) Y1 ( R Rz

1he sf04- of bafr ~ere is +t+- cJ-t- Ji~+ rOIN R IJr 7lt~ W~ cOfAJ tlstlL-e a s1tMo-r leMJ RRlJ Io+- we tioI-IshyVI eeJ fD J ~+- i flu (Jkbullheor- ~ I lAtA- R- Q~) 7Lyr~ COV$rrA+middot

t ~ (1 -tJ == t == S4pe Cgl+-f(a MJe yen ae)

us e 2 J i rt It fi fbulli ) () -()l-------- y - rlt - a_3 _ - shyS- OJshy~ shy

~l bull shymiddottJ~~

Dr Sohl Applied Optics HOIll~work ChapterI PedrottiJ ~27 A lens is moved along the optical axis between a fixed object and a fixed image screen The object and image positions are separated by a distance L that is more than four times the focal length of theo lens Two positions of the lens are found for which an image is in focus on the screen magnified in one case and reduced in the other If the two lens positions differ by a distance D show that the focal length of the lens is given by f =(L2 - ~ )4L This is Bessel f s method for finding the focal length of a lens

I~ 5 )( S )

I $cLe~lts pos 1

- LC~ pos-ot-lt [) 2

lt S ~E

~(-----------------~~-------__ L ________________S~~_______~~gt

f+- ~f = + + ~ ~

No-fe sy -t+ry

o fA) e kr (lit) ~+ 5 - ~ ~

11()~ -Ike r-ampe(e ~be tJL sec ~+ S-t-b+~ = L n f ~ JI p(l J SIlIta raquo2 =S we Ie -Z5 +1gt =L

or 5 =(L-))~

L ke wise S _7gt s1 = SJ ~ I

I 1- -1) r_ L+D s = S +1gt - ~ +jIshyI I L

s ~ S I p~-s ~ 5 s )- =--

I 2f= S+-Si Now rrlf itt L-b

Kt L-))) (LfP)f= 2 lL-D +- ~ (L+iraquoCj ~ ~lL-V

s 5

L= s s

L -I-- J =- 1 s s (

s = sectp hlAf- S- L- S s- f

no t) So Ie to L tU Cl +-uit~tIt _ (L-s)f of s ~ ~ A JI+ JlrllS - l- s-ft -Ie+- -10 -AJ fIte ~r ~14-

zs-f-s=O

s =- ~f +0r L f t rI - bull

L ~ S -t S l-F --1 ~f - Iff I

(L= fiJ )$ J +e II u hypes wk s tp

9-1

s

9- Solaquo=shy -thoeuroftvo ~lAJitmS fwo ullco~ltS (ampcS

13 u + )

We CA ~ ke hfe et Y J are_

cl eo--r -hl al s raquo -P= 6 SiX -f

50 -yVt --

_ (

S shyT-

of-shys

s 5

L= s s

L -I-- J =- 1 s s (

s = sectp hlAf- S- L- S s- f

no t) So Ie to L tU Cl +-uit~tIt _ (L-s)f of s ~ ~ A JI+ JlrllS - l- s-ft -Ie+- -10 -AJ fIte ~r ~14-

zs-f-s=O

s =- ~f +0r L f t rI - bull

L ~ S -t S l-F --1 ~f - Iff I

(L= fiJ )$ J +e II u hypes wk s tp

9-1

s

9- Solaquo=shy -thoeuroftvo ~lAJitmS fwo ullco~ltS (ampcS

13 u + )

We CA ~ ke hfe et Y J are_

cl eo--r -hl al s raquo -P= 6 SiX -f

50 -yVt --

_ (

S shyT-

of-shys

9-1

s

9- Solaquo=shy -thoeuroftvo ~lAJitmS fwo ullco~ltS (ampcS

13 u + )

We CA ~ ke hfe et Y J are_

cl eo--r -hl al s raquo -P= 6 SiX -f

50 -yVt --

_ (

S shyT-

of-shys