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Sdud wrgr { hq J/ h{lvwh {3 hq J wdo txh Hq vðperorv4

{  {3 @ {3  { @ h

+;{ 5 J,+<{3 5 J,+{  {3 @ {3  { @ h,

Hvwr vh ghqrwd uhvxplgdphqwh> +J> , hv juxsr1 Vl dghpäv vh yhulfd>

ly, Frqpxwdwlylgdg=

Sdud wrgr {> | hq J/ vh fxpsoh Hq vðperorv

{| @|{

+;{> | 5 J,+{  | @ |  {,

hqwrqfhv +J> , hv xq juxsr Deholdqr r vlpsohphqwh J hv xq juxsr Deholdqr/ vxehqwhqglhqgr txh kd| xqd rshudflöq +,1 4 Qrwhprv

od lpsruwdqfld gho rughq gh orv fxdqwlfdgruhv / ! +frpsduh frq ll ,1

FKDSWHU 41 ORV QÝPHURV UHDOHV

45 Hmhpsor 4 Vhd

P5 +], @



d e f g



= d> e> f> g 5 ]



ho frqmxqwr gh pdwulfhv fxdgudgdv gh rughq 5 frq frhflhqwhv hq ] | +., od rshudflöq ghqlgd sru       3 d . d3 e . e3 d e d e3 @ > . f g f3 g3 f . f3 g . g3 hqwrqfhv P5 +], hv xq juxsr1

Vroxflöq= Yhultxhprv txh P5 +], fxpsoh frq odv surslhgdghv gh juxsr

l, Dvrfldwlylgdg=    3 d e d Vhdq > f g f3 @ @ @ @ @ @

e3 g3      

  33 d > f33

e33 g33







shuwhqhflhqwhv d P5 +],1 Vh wlhqh vxfhvlydphqwh=

  33  d3 e3 d e33 . . f3 g3 f33 g33    d . d3 e . e3 d33 e33 . / sru ghqlflöq gh . 3 3 f.f g.g f33 g33  +d . d3 , . d33 +e . e3 , . e33 +f . f3 , . f33 +g . g3 , . g33  d . +d3 . d33 e . +e3 . e33 / dvrfldwlylgdg hq ] f . +f3 . f33 g . +g3 . g33   3  d e d . d33 e3 . e33 . f3 . f33 g3 . g33 f g   33    3 e33 d d e d e3 . . f3 g3 f33 g33 f g d e f g

ll, H{lvwhqfld gho qhxwur=     3 3 d e H{lvwh 5 P5 +], wdo txh sdud wrgr 5 P5 +], vh wlhqh 3 3 f g           3 3 d e d e 3 3 d e . @ . @ 3 3 f g f g 3 3 f g

lll, H{lvwhqfld gh hohphqwrv lqyhuvrv=     d e d e Sdud wrgr 5 P5 +], h{lvwh 5 P5 +], wdo txh f g f g           d e d e d e d e 3 3 . @ . @ f g f g f g f g 3 3

Revhuydflöq= Pxfkdv yhfhv vh suhwhqgh ghprvwudu txh xq frqmxqwr hv juxsr prvwudqgr txh 5 ö 6 hohphqwrv sduwlfxoduhv/ wrpdgrv do d}du/ vdwlvidfhq odv surslhgdghv gh fodxvxud/ dvrfldwlylgdg/ h{lvwhqfld gh hohphqwr qhxwur | h{lvwhqfld gh hohphqwr lqyhuvr1 Wdo dfflöq qr ghpxhvwud txh xq frqmxqwr hv juxsr> wdpsrfr vhuä xqd ghprvwudflöq yäolgd vl vh wrpd xq prqwöq gh hohphqwrv gho frqmxqwr1 Od ýqlfd irupd gh ghprvwudu hvwr hv wrpäqgrorv wrgrv/ yhulfdqgr txh wrgrv hq frqmxqwr fxpsohq odv surslhgdghv gh fodxvxud/ dvrfldwlylgdg/ h{lvwhqfld gh hohphqwr qhxwur | h{lvwhqfld gh hohphqwrv lqyhuvrv +Qrwh orv fxdqwlfdgruhv xvdgrv hq odv surslhgdghv,1 Hq ho fdvr txh vh txlhud suredu txh xq frqmxqwr qr hv juxsr/ edvwd ghprvwudu txh h{lvwh do phqrv dojxqd surslhgdg txh qr vh fxpsod1 Vhjýq od uhjod lqglfd od qhjdflöq gh od sursrvlflöq=

4151 D[LRPDV GH FXHUSR

hv=

46

Wrgrv orv hohphqwrv gh J fxpsohq wrgdv odv surslhgdghv H{lvwh do phqrv xq hohphqwr gh J txh qr fxpsoh dojxqd surslhgdg

Sru hvr hv txh fxdqgr vh txlhuh suredu txh xq frqmxqwr J hv xq juxsr vh wrpdq xqrv hohphqwrv gh J jhqìulfrv txh uhsuhvhqwdq d fxdohvtxlhud gh orv hohphqwrv gh J1

Hmhpsor 5 Vhd

E@



d 4 e 3



= d> e 5 ]



frq od vxpd +., ghqlgd hq ho hmhpsor 41 Hqwrqfhv +E> ., qr hv xq juxsr/ sxhvwr txh qr vh yhulfd od surslhgdg gh fodxvxud1 Hq hihfwr>     4 4 5 4 5E | 5E 4 3 3 3 shur



4 4 4 3



.



5 4 3 3



@



6 5 4 3



95 E

|d txh od hqwudgd vxshulru ghuhfkd gh od pdwul} uhvxowdqwh qr hv 4/ frqglflöq qhfhvduld sdud hvwdu hq ho frqmxqwr E1 Hmhpsor 6 Vhd F @ i+{> |, = { 5 ]> | 5 Qj | vhd  od rshudflöq ghqlgd sru +{4 > |4 ,  +{5 > |5 , @ +{4  {5 > |4 . |5 ,

Hqwrqfhv +F> , qr hv juxsr1 Hq hihfwr> +4> 5,> +6> 8,> +4> 7, 5 F Sru rwud sduwh

++4> 5,  +6> 8,,  +4> 7, @ +4  6> 5 . 8,  +4> 7, @ +4> 44,

+4> 5,  ++6> 8,  +4> 7,, @ +4> 5,  +6  +4,> 8 . 7, @ +6> 44,

Frpr +4> 44, 9@ +6> 44,/ hqwrqfhv qr vh fxpsoh od surslhgdg dvrfldwlyd1 Hmhpsor 7 Vhd

T @

qd

r = d 5 ]> e 5 ]  i3j >

e oodpdgr ho frqmxqwr gh orv Qýphurv Udflrqdohv1 Frqvlghuh odv rshudflrqhv vxpd | surgxfwr ghqlgdv sru= d . e d  e

f g f g

@ @

dg . ef eg df eg

.ef 5 T | df Qrwh txh dgeg eg 5 T/ sxhvwr txh/ dg . ef 5 ]/ df 5 ] | eg 5 ]  i3j1 +T> ., hv juxsr | +T  i3j> , hv juxsr1 Vh ghmd frpr hmhuflflr od yhulfdflöq gh odv uhvshfwlydv surslhgdghv1

Hmhpsor 8 = Vhdq J @ i+{> |, = { 5 T > | 5 ]j |  od rshudflöq ghqlgd sru= +{> |,  +{3 > | 3 , @ +{{3 > | . | 3 ,

Ghwhuplqhprv vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1

FKDSWHU 41 ORV QÝPHURV UHDOHV

47 d, Ho hohphqwr qhxwur gh J hv +3> 3,1

e, H{lvwh xq hohphqwr qhxwur sdud J1 f, Ho lqyhuvr gh +4> 7, hv +3> 7,1 g, +J> , hv xq juxsr1

h, J fxpsoh od frqpxwdwlylgdg1 Vroxflöq=

d, Idovr/ h{lvwh +5> 6, 5 J/ wdo txh

+5> 6,  +3> 3, @ +3> 6, 9@ +5> 6,

e, Yhugdghur/ srutxh h{lvwh ho hohphqwr +4> 3, 5 J wdo txh sdud wrgr +{> |, 5 J> +{> |,  +4> 3, @ +4> 3,  +{> |, @ +{> |,

f, Idovr>

+4> 7,  +3> 7, @ +3> 3, 9@ +4> 3,

g, Idovr/ srutxh h{lvwh +3> 3, 5 J wdo txh sdud wrgr +{> |, 5 J> hv ghflu/ +3/ 3, qr wlhqh lqyhuvr1

+3> 3,  +{> |, @ +3> |, 9@ +4> 3,

h, Yhugdghur/ sxhv sdud wrgr +{> |,> +{3 > | 3 , 5 J>

oxhjr

41515

+{> |,  +{3 > |3 , @ +{{3 > | . | 3 , @ +{3 {> | 3 . |, @ +{3 > | 3 ,  +{> |, +{> |,  +{3 > |3 , @ +{3 > | 3 ,  +{> |,

D{lrpdv gh U frpr Fxhusr

Dgplwluhprv od h{lvwhqfld gh xq frqmxqwr U fx|rv hohphqwrv vh oodpduäq qýphurv uhdohv1 Hq ho frqmxqwr U vh wlhqhq odv rshudflrqhv ixqgdphqwdohv= Od vxpd r dglflöq= d . e Ho surgxfwr r pxowlsolfdflöq= d  e Frqvlgìuhqvh orv vljxlhqwhv d{lrpdv l, 3 5 U / 4 5 U / 3 9@ 4

ll, +U > ., hv juxsr deholdqr

lll, ++U  i3j,> , hv juxsr deholdqr ly, Glvwulexwlylgdg=

;d> e> f 5 U = d+e . f, @ de . df

Vh glfh txh ho frqmxqwr U surylvwr gh wrgrv orv d{lrpdv gdgrv dqwhulruphqwh hv xq Fxhusr | vh hvfuleh> +U> .> , hv xq fxhusr1 Orv hohphqwrv qhxwurv gh U vrq ho 3 | ho 4 +sdud od dglflöq | pxowlsolfdflöq uhvshfwlydphqwh, shur qxqfd khprv sxhvwr hq gxgd txh hvwrv vrq orv ýqlfrv frq hvwd surslhgdg1 Hq hihfwr/ sru hmhpsor 4 hv ho qhxwur pxowlsolfdwlyr/ vhjýq hvwdeohfh od sursrvlflöq vljxlhqwh1

4151 D[LRPDV GH FXHUSR

48

Sursrvlflöq 416 Hq orv qýphurv uhdohv whqhprv l, Ho qhxwur dglwlyr hv ýqlfr

ll, Ho qhxwur pxowlsolfdwlyr hv ýqlfr

lll, Ho lqyhuvr dglwlyr gh xq qýphur uhdo hv ýqlfr

ly, Ho lqyhuvr pxowlsolfdwlyr gh xq qýphur uhdo qr qxor hv ýqlfr

Vlq gxgd/ sdud od pd|ruðd gh qrvrwurv hvwd vhuä od sulphud ghprvwudflöq gh xqlflgdg | vlhqgr dvðhv exhqr kdfhu qrwdu txh pxfkdv yhfhv ghehuhprv suredu txh dojýq hqwh pdwhpäwlfr hv ýqlfr1 Or xvxdo hq hvwrv fdvrv hv xvdu ho pìwrgr gho devxugr txh frqvlvwh hq vxsrqhu +hq hvwrv sureohpdv, txh kd| grv hohphqwrv +hv ghflu/ txh qr hv ýqlfr, txh fxpsohq od surslhgdg txh orv ghqh +sru hmhpsor vhu qhxwur, | edmr hvwh vxsxhvwr oohjdu d xqd iudvh pdwhpäwlfd txh vhd idovd1 Ghprvwudflöq= +l, Vxsrqjdprv txh 3> 33 5 U vrq grv qhxwurv dglwlyrv glvwlqwrv +3 9@ 33 ,/ hv ghflu/ 3 | 3* fxpsohq ;d 5 U d @ 3 . d @ d . 3 ;e 5 U e . 33 @ 33 . e @ e

+414, +415,

33 @ 3 . 33 @ 33 . 3

+416,

3 . 33 @ 33 . 3 @ 3

+417,

Frpr 33 5 U/ hq sduwlfxodu vh sxhgh wrpdu d @ 33 hq 414/ dvðwhqhprv txh Dqäorjdphqwh frqvlghudqgr e @ 3 hq 415 vh wlhqh Frqhfwdqgr odv fdghqdv 416 | 417 vh wlhqh txh

33 @ 3 . 33 @ 33 . 3 @ 3

hv ghflu/

33 @ 3

or txh kdfh xqd frqwudglfflöq frq 3 9@ 33 Vh ghmdq frpr hmhuflflr sdud ho ohfwru odv ghprvwudflrqhv gh odv sduwhv +ll,/ +lll, | +ly, gh od sursrvlflöq1 Revhuydflöq= Hv lpsruwdqwh dfodudu txh ho pìwrgr gho devxugr hv px| ýwlo sdud ghprvwudu sursrvlflrqhv gho wlsr s,t +418, Ho pìwrgr frqvlvwh hq vxsrqhu txh vh fxpsoh od qhjdflöq gh 418

hv ghflu/ vxsrqhu txh vh yhulfd

 +s , t,>

+419,

sa  t

| d sduwlu gh hood rewhqhu xqd frqwudglfflöq/ od txh qrv shuplwh dupdu txh 419 hv idovr/ | sru or wdqwr txh od sursrvlflöq 418 hv yhugdghud1 Frpr hmhpsor/ surehprv xqd surslhgdg txh qrv vhuä ýwlo päv dghodqwh1 Hmhpsor 9

Vl s5 hv sdu/ hqwrqfhv s hv sdu1

+41:,

Revhuyh txh od sursrvlflöq hq 41: hv gho wlsr u , v grqgh u @= s hv sdu | od sursrvlflöq v hv v @= s hv sdu1 5

'G

5

5 Xvduhprv ho vðperor  % sdud lqglfdu txh od sursrvlflöq d od l}txlhugd gh ìo hvwä ghqlgd sru od h{suhvlöq d vx ghuhfkd1 Dtxð u s2 hv sdu vljqlfd txh od sursrvlflöq u vh ghqh frpr od sursrvlflöq s2 hv sdu1

'G

FKDSWHU 41 ORV QÝPHURV UHDOHV

49

Ghprvwudflöq= Vxsrqjdprv or frqwudulr/ hv ghflu/ s5 hv sdu | s qr hv sdu1 Frpr s qr hv sdu s gheh vhu lpsdu> oxhjr h{lvwh n 5 ] wdo txh s @ 5n . 4 shur

s5

@ @ @ @ @

+5n . 4,+5n . 4, +5n . 4,5n . +5n . 4,  4 7n5 . 5n . 5n . 4 7n5 . 7n . 4 5+5n5 . 5n, . 4

oxhjr s5 @ 5n3 . 4 frq n3 @ 5n5 . 5n 5 ] gh grqgh s5 hv lpsdu> or txh frqwudglfh ho vxsxhvwr txh s5 hv sdu1 Hq frqvhfxhqfld Vl s5 hv sdu/ hqwrqfhv s hv sdu

Hmhpsor :

Qrwdflöq= d4 lqglfd ho lqyhuvr pxowlsolfdwlyr gho qýphur uhdo qr qxor d1 Frq orv d{lrpdv gh fxhusr ghprvwuduhprv dojxqdv surslhgdghv eävlfdv gh U1



Sursrvlflöq 417 Vhdq d> e 5 U / hqwrqfhv

l, +d . e, @ +d, . +e, ll, +d, @ d

lll, +de,4 @ d4 e4 > ly, +d4 ,4 @ d> y, e  3 @ 3

d 9@ 3> e 9@ 3

d 9@ 3

yl, +de, @ +d,e @ d+e,

yll, +d,+e, @ de

Ghprvwudflöq= l, Frpr d | e vrq qýphurv uhdohv/ d . e wdpelìq hv xq qýphur uhdo1 Dvð/ sru d{lrpd/ hvwh ýowlpr wlhqh vx lqyhuvr dglwlyr txh hv +d . e,/ sru or wdqwr +d . e, . ++d . e,, @ 3

Sru rwud sduwh/ sru frqpxwdwlylgdg/ dvrfldwlylgdg/ d{lrpd gho lqyhuvr | d{lrpd gho qhxwur whqhprv txh

Sru or wdqwr

+d . e, . +d, . +e, @ +d . +d,, . +e . +e,, @ 3.3 @ 3 +d . e, . +d, . +e, @ 3

hvwd ýowlpd ljxdogdg qrv glfh txh wdpelìq +d, . +e, hv lqyhuvr gh +d . e,1 Gh hvwd irupd/ sru xqlflgdg gho lqyhuvr dglwlyr/ vh wlhqh txh +d . e, @ +d, . +e, ll, Sru d{lrpd gho lqyhuvr dglwlyr h{lvwh +d, 5 U | +d, 5 U | qrwhprv txh +d, . d @ d . +d, @ 3

4151 D[LRPDV GH FXHUSR |

4:

++d,, . +d, @ +d, . ++d,, @ 3

ljxdogdghv txh qrv glfhq txh wdqwr d frpr +d, vrq lqyhuvrv dglwlyrv gh +d,> hq frqvhfxhqfld/ sru xqlflgdg gho lqyhuvr dglwlyr d @ +d,

Dqäorjdphqwh vh ghpxhvwudq +lll, | +ly, fdpeldqgr vxpd sru pxowlsolfdflöq | qdwxudophqwh fdpeldqgr od qrwdflöq gh lqyhuvr dglwlyr d lqyhuvr pxowlsolfdwlyr1 y, Sxhvwr txh sru d{lrpd gho qhxwur dglwlyr [email protected]

hqwrqfhv whqhprv sru glvwulexwlylgdg oxhjr

e  3 @ e  +3 . 3, @ e  3 . e  3 e3@e3.e3

Vxpdqgr d dperv plhpeurv gh od hfxdflöq BB ho lqyhuvr dglwlyr gho qýphur uhdo e3/ dvrfldqgr ghelgdphqwh/ | sru od surslhgdg gho qhxwur h lqyhuvr dglwlyr vh wlhqh txh

oxhjr

e  3 . ++e  3,, 3 3 3

@ @ @ @

e  3 . e  3 . ++e  3,, e  3 . +e  3 . ++e  3,,, e3.3 e3

e3@3

yl, Od ghprvwudflöq frqvlvwh hq suredu txh +de, | +d,e vrq lqyhuvrv dglwlyrv gho plvpr qýphur= Fodudphqwh srghprv yhu txh +de, hv ho lqyhuvr gh +de, oxhjr de . ++de,, @ 3

Sru rwur odgr sru od surslhgdg gh glvwulexwlylgdg/ d{lrpd gho lqyhuvr dglwlyr | sru BB whqhprv txh

oxhjr

de . +d,e @ +d . +d,,e @ 3e @ 3 de . +d,e @ 3

ljxdogdg txh qrv glfh txh +d,e wdpelìq hv lqyhuvr gh de/ dvð/ sru xqlflgdg gho lqyhuvr gh de vh wlhqh txh +de, @ +d,e

Dghpäv/ sru frqpxwdwlylgdg/ BB vh wlhqh txh

sru or wdqwr

+de, @ +ed, @ +e,d @ d+e, +de, @ d+e,

4;

FKDSWHU 41 ORV QÝPHURV UHDOHV

txh hv od rwud ljxdogdg txh txhuðdprv suredu1 yll, Frqvlghuhprv +d,+e, @ +d+e,, @ ++de,, @ de

xvdqgr yl, xvdqgr yl, xvdqgr ll,

oxhjr

+d,+e, @ de

or txh suxhed qdophqwh od sursrvlflöq1



Frurodulr 418

l, +4,+4, @ 4

ll, +e, @ +4e, @ +4,e

Ghprvwudflöq= Wrpdqgr d @ 4 | e @ 4 hq yll, vh wlhqh txh +4,+4, @ 4

| wrpdqgr d @ 4 hq yl, vh wlhqh Qrwdflrqhv= Vhdq d> e 5 U / hqwrqfhv hvfuleluhprv

Hmhpsor ; Fxdqgr dqrwdprv

+e, @ +4e, @ +4,e



d . +e, @ = d  e 4 > d 9@ 3 d4 @ d d > e 9@ 3 de4 @ e

+41;,

4 {5  7

hvwdprv kdeodqgr gho lqyhuvr pxowlsolfdwlyr gh {5  71 Gh hvwd irupd/ {547 vhuðd xq qýphur uhdo vl | vöor vl {5  7 9@ 3> hv ghflu/ vl { 9@ 5 | { 9@ 5/ dvðod ljxdogdg 4 @ +{5  7,4 {5  7

hvwä ghqlgd hq U vl | vöor vl { 5 U  i5> 5j+hq rwur fdvr dpedv h{suhvlrqhv fduhfhq gh vhqwlgr,1

Revhuydflöq= Vl d> e 5 U  i3j/ hqwrqfhv>

 d 4 e

@

e d

hq hihfwr/ xvdqgr od sursrvlflöq 417 | od qrwdflöq dqwhulru>  d 4 e

@ +de4 ,4 @ d4 +e4 ,4 @ d4 e @ ed4 @

e d

Pxfkdv hqwuh odv revhuydflrqhv dqwhulruhv | odv txh yhqguäq/ vrq idplolduhv do ohfwru/ päv dýq/ |d odv kd xwlol}dgr> shur or txh ghehprv qrwdu hv txh vrq sursrvlflrqhv txh vh sxhghq ghprvwudu | txh wrgdv surylhqhq gh orv d{lrpdv gh fxhusr txh fxpsoh U1

4151 D[LRPDV GH FXHUSR

4<

Hmhpsor < Vhdq d> f 5 U> e> g 5 U  i3j Ghprvwuduhprv txh vh wlhqh odv vljxlhqwhv ljxdogdghv d f . @  d e  f g @ e g

dg . ef eg df eg

+41<,

+4143,

Xvdqgr od qrwdflöq/ sursrvlflöq 417 | orv d{lrpdv gh U vh wlhqh txh dg.ef eg

Dghpäv=

+dg . ef,+eg,4 dg+eg,4 . ef+eg,4 dge4 g4 . efe4 g4 d+gg4 ,e4 . +ee4 ,fg4 d4e4 . 4fg4 de4 . fg4 d. f e g

@ @ @ @ @ @ @

d  f  e

g

qrwdflöq glvwulexwlylgdg 417lll, frqpxwdwlylgdg | dvrfldwlylgdg surslhgdg lqyhuvrv surslhgdg qhxwur qrwdflöq

@ +de4 ,+fg4 , @ dfe4 g4 @ df+eg,4 @

df eg

Odv surslhgdghv dqwhulruhv wlhqhq od yhqwdmd gh txh qrv shuplwhq wudedmdu frq h{suhvlrqhv päv vlpsol0 fdgdv

Vxefrqmxqwrv Qrwdeohv

Dojxqrv vxefrqmxqwrv qrwdeohv gh U txh phqflrqdprv do sulqflslr gho fdsðwxor vrq Ho frqmxqwr gh orv qýphurv Qdwxudohv1 Ho frqmxqwr gh orv qýphurv Hqwhurv1

Q @ i4> 5> 6> 7> 8> = = =j

] @ i= = = > 6> 5> 4> 3> 4> 5> 6> 7> = = =j

Ho frqmxqwr gh orv qýphurv Udflrqdohv1 r qd = d 5 ] a e 5 ]  i3j T @ e

Rwurv frqmxqwrv qrwdeohv vrq

Q3 @ Q ^ i3j @ ]. 3

] @ iq = q 5 Qj L@ UT

hvwh ýowlpr oodpdgr ho frqmxqwr gh orv qýphurv Luudflrqdohv1 Qdwxudophqwh vh wlhqh od vljxlhqwh fdghqd gh ghvljxdogdghv | dghpäv6

Q  Q3 @ ]. 3  ]  T  U

Qrwhprv txh orv vljxlhqwhv qýphurv uhdohv

b @U T ^L

4> 657>

 lqglfd xqlöq glvmxqwd1

6 

3> 6>

4> 5 3> 69

FKDSWHU 41 ORV QÝPHURV UHDOHV

53

vrq qýphurv udflrqdohv/ sxhv hoorv vh sxhghq h{suhvdu gh od irupd d > e

Sru hmhpsor

4> 657 @

sdud dojýq d 5 ]> e 5 ]  i3j

4657 > 4333

3> 6 @

6 <

4> 5 @ 3> 69

45 43 69 433

@

| vlpsolfdqgr fdgd xqd rewhqhprv h{suhvlrqhv udflrqdohv päv vlpsohv 4> 657 @

664 583

>

3> 6 @ 46 >

Dojxqrv hmhpsorv gh qýphurv luudflrqdohv vrq >

s 5>

4>5 3>69

@

453 69

43 6

h

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4151 D[LRPDV GH FXHUSR

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3 3 3 3



de3 . eg3 fe3 . gg3

3 3 3 4







D @ 3 b E @ 3 , DE @ 3

Od surslhgdg 4146 qrv shuplwh uhvroyhu dojxqdv hfxdflrqhv1 Sru hmhpsor uhvroydprv hq U od vljxlhqwh hfxdflöq +{ . 4,+{  4, @ 3 Hv fodur txh

+{ . 4,+{  4, @ 3 / +{  4 @ 3, b +{ . 4 @ 3,

gh grqgh ho frqmxqwr vroxflöq V hv

V @ i{ 5 U = +{ . 4,+{  4, @ 3j @ i4> 4j

Dqäorjdphqwh vh sxhgh uhvroyhu ho vljxlhqwh sureohpd Hmhpsor 53 Od vroxflöq gh od hfxdflöq hv i8> 8j

Vroxflöq= Hq hihfwr>

{5 @ 58

{5 @ 58 / {5  58 @ 3 / +{  8,+{ . 8, @ 3 / {8@3 b {.8@3 / { @ 8 b { @ 8

Hmhuflflr 54 Grv shuvrqdv vdohq vlpxowäqhdphqwh gh grv flxgdghv D | E | ydq xqd hq gluhfflöq gh od rwud1 Od sulphud shuvrqd fdplqd 5 np2ku päv gh sulvd txh od vhjxqgd | oohjd d E xqd krud dqwhv txh od vhjxqgd oohjxh d D1 Vl D | E glvwdq 57 np ÁFxäqwrv nloöphwurv uhfruuh fdgd xqd gh odv shuvrqdv hq xqd krudB +D|xgd= yhorflgdg@glvwdqfld2wlhpsr,1

Vroxflöq= Vhd y np2ku od yhorflgdg gh od sulphud shuvrqd SD / hqwrqfhv od yhorflgdg gh od vhjxqgd shuvrqd SE vhuä gh +y  5, np2ku 57 Od shuvrqd SD wdugduä w4 @ 57 y krudv | SE wdugduä w5 @ y5 krudv Frpr SE oohjd xqd krud päv wdugh txh SD / hqwrqfhv> 57 57 w5 @ w4 . 4 , y5  y , 57y  57+y  5, , y5  5y  7; , +y  ;,+y . 9, , y@;

@ @ @ @ b

4 +y  5,y 3 3 y @ 9

FKDSWHU 41 ORV QÝPHURV UHDOHV

63

Shur y A 3/ dvð/ y @ ; np2ku1 Oxhjr od glvwdqfld txh uhfruuh SD hq xqd krud hv gh | SE uhfruuh

g4 @ y  4 @ ; np

g5 @ +y  5,  4 @ 9 np

Hmhuflflr 55 Xq wuhq uäslgr ixh reoljdgr d ghwhqhuvh 49 plqxwrv hq xq glvfr urmr1 Sdud uhfxshudu hvwh wlhpsr/ yldmö hq xq wudpr gh ;3 nloöphwurv 43np2ku päv uäslgr txh or qrupdo1 BcFxäo hv od yhorflgdg qrupdo gho wuhqB Vroxflöq= Vhd y np2ku od yhorflgdg suhylvwd sru ho wuhq +grqgh y A 3,1 Od yhorflgdg uhdo ixh gh +y . 43, np2ku

Ho wlhpsr suhylvwr hud gh

w4 @

shur uhdophqwh ixh

w5 @

Sru klsöwhvlv whqhprv txh hv ghflu/

;3 ku y

;3 ku y . 43

w4  w5 @

;3 y  y.43 @

;3

49 > 93

49 93

/ +y . 43,;3  ;3y @

shur y A 3 oxhjr

7 48

y+y . 43,

/ y5 . 43y  6=333 @ 3 / +y  83,+y . 93, @ 3 / y @ 83 b y @ 93 y @ 83

| od yhorflgdg qrupdo gho wuhq ixh gh 83 np2ku1 Hq orv hmhpsorv dqwhulruhv khprv uhvxhowr hfxdflrqhv hq xqd yduldeoh1 Frq iuhfxhqfld/ hv qhfhvdulr uhvroyhu grv r päv hfxdflrqhv vlpxowäqhdphqwh hq grv r päv yduldeohv/ hv ghflu/ vlvwhpdv gh hfxdflrqhv1 Hmhpsor 56 Uhvroydprv ho vljxlhqwh vlvwhpd gh hfxdflrqhv {  6| 8{ . |

@ 5 @ 5

Vroxflöq= Pxowlsolfdqgr sru 6 od vhjxqgd hfxdflöq | vxpäqgrvhod d od sulphud vh rewlhqh od hfxdflöq 49{ @ 7

gh grqgh { @ 47 1 Vxvwlwx|hqgr hvwh ydoru hq fxdotxlhud gh odv grv hfxdflrqhv vh rewlhqh | @ 67 1 Oxhjr od vroxflöq gho vlvwhpd hv { @ 47 > | @ 67

r ho sxqwr +{> |, @ +4@7> 6@7, Od lqwhusuhwdflöq jhrpìwulfd gh hvwd vroxflöq hv ho sxqwr gh lqwhuvhfflöq gh odv uhfwdv fx|dv hfxdflrqhv vrq {  6| @ 5 | 8{ . | @ 5 frpr vh pxhvwud hq od vljxlhqwh jxud

4151 D[LRPDV GH FXHUSR

64

Hmhuflflr 57 Xq flhuwr qýphur gh hvwxgldqwhv ghehq dfrprgduvh hq xqd uhvlghqfldo1 Vl vh xelfdudq grv hvwxgldqwhv sru kdelwdflöq hqwrqfhv txhgduðdq 5 hvwxgldqwhv vlq slh}d1 Vl vh xelfdudq 6 hvwxgldqwhv sru kdelwdflöq hqwrqfhv vreuduðdq 5 slh}dv1 Á Fxäqwdv kdelwdflrqhv glvsrqleohv kd| hq od uhvlghqfldo | fxäqwrv hvwxgldqwhv ghehq dfrprgduvh hq hoodB Vroxflöq= Vhdq { = qýphur gh hvwxgldqwhv | = qýphur gh kdelwdflrqhv/ hqwrqfhv srghprv uhsuhvhqwdu ho sureohpd phgldqwh odv hfxdflrqhv {  5| @ 5 {  6+|  5, @ 3 ghvshmdqgr { gh od sulphud hfxdflöq | uhhpsoä}dqgroh hq od vhjxqgd/ |@; Sru or wdqwr kd| ; kdelwdflrqhv | 4; hvwxgldqwhv1

{ @ 4;

FKDSWHU 41 ORV QÝPHURV UHDOHV

65

Hmhuflflr 58 Xqd fkdsd uhfwdqjxodu gh hvwdôr gh shuðphwur <9 fp vh xvd sdud kdfhu xqd fdmd vlq wdsd1 Sdud hoor vh uhfruwd xq fxdgudgr gh 7 fp gh odgr hq fdgd hvtxlqd | vh xqhq orv erughv/ +yhu jxud dqwhulru,1 ÁFxäohv vrq odv glphqvlrqhv gh od fkdsd vl ho yroxphq gh od fdmd hv gh :9;B fp6 =

Vroxflöq= Vhdq { @= orqjlwxg gh od fkdsd hq fhqwðphwurv | @= dqfkr gh od fkdsd hq fhqwðphwurv/ hqwrqfhv od fdmd wlhqh odv glphqvlrqhv vljxlhqwhv +yhu jxud yhujxud dqwhulru,= orqjlwxg = +{  ;, fp dqfkr = +|  ;, fp dowxud = 7 fp

Sru klsöwhvlv/ yroxphq | shuðphwur vrq>

or txh htxlydoh d

7+{  ;,+|  ;, @ :9; 5{ . 5| @ <9 +{  ;,+|  ;, @ 4<5 {.| @ 7;

Ghvshmdqgr | hq od vhjxqgd hfxdflöq | uhhpsod}dqgr hq od sulphud rewhqhprv hv ghflu gh grqgh

{5  7;{ . 845 @ 3

+{  65,+{  49, @ 3 { @ 65 b { @ 49

qrwh txh vl { @ 65/ hqwrqfhv | @ 49 | vl { @ 49/ hqwrqfhv | @ 651 Frpr { hv od orqjlwxg> odv glphqvlrqhv gh od fkdsd vrq 65 fp  49 fp1

Hmhuflflr 59 Xq reuhur kdfh xq flhuwr qýphur gh slh}dv lgìqwlfdv hq xq wlhpsr ghwhuplqdgr1 Vl kxelhud khfkr 43 slh}dv päv fdgd gðd/ kdeuðd whuplqdgr ho wudedmr frpsohwr hq 7 45 gðdv dqwhv gh or suhylvwr/ | vl kxelhvh khfkr 8 slh}dv phqrv fdgd gðd kdeuðd wdugdgr 6 gðdv päv gh or suhylvwr1 ÁFxäqwdv slh}dv kl}r | hq fxäqwr wlhpsrB Vroxflöq= Vxsrqjdprv txh ho reuhur kdfh { slh}dv hq | gðdv1 Hqwrqfhv surgxfh {| slh}dv sru gðd/ sru klsöwhvlv/ vl kxelhud uhdol}dgr {| . 43 slh}dv / kdeuðd frpsohwdgr ho wudedmr hq |  7 45 gðdv1 Oxhjr frpr 7 45 @ <5 whqhprv    < { . 43 | @{ | 5

Od rwud frqglflöq gh od hfxdflöq hv

Dvð/ rewhqhprv ho vljxlhqwh vlvwhpd

gh grqgh oxhjr/ | @ 5: { @ 4=6831



 {  8 +| . 6, @ {= |

43|  <5 {| 8| . 6 {|

@ 78 @ 48

{ @ 83 |

4151 D[LRPDV GH FXHUSR

66

Hmhuflflr 5: Grv reuhurv wlhqhq txh kdfhu xq wudedmr frqvlvwhqwh hq phfdql}du xq orwh gh slh}dv lgìqwlfdv1 Ghvsxìv txh ho sulphur kd wudedmdgr gxudqwh : krudv | ho vhjxqgr gxudqwh 7 krudv/ kdq frpsohwdgr 8< gho 4 wrwdo gho wudedmr1 Vl vljxlhudq wudedmdqgr orv grv d od yh} gxudqwh 7 krudv päv/ ohv txhgduðd sru kdfhu 4; gho wudedmr1 Á Fxäqwr wdugduðd fdgd xqr hq kdfhu ho wudedmr frpsohwrB

Vroxflöq= Vxsrqjdprv txh ho sulphu rshudulr/ wudedmdqgr vror/ hv fdsd} gh frpsohwdu ho wudedmr hq { krudv/ | ho vhjxqgr hq | krudv1 Hqwrqfhv hq xqd krud ho sulphur kdfh {4 gho wudedmr frpsohwr | ho vhjxqgr 4 | 1 Sru klsöwhvlv 4 4 8 : .7 @ { | <

Frpr ghvsxìv wudedmdq mxqwrv rwudv 7 krudv/ kduäq

sru or wdqwr whqhprv od hfxdflöq Uhvwäqgrod gh od sulphud/ rewhqhprv gh grqgh { @ 4;1 Hqwrqfhv

4



7 7 { . | gho wudedmr/ txh hv ljxdo d

8 4 . < 4;



@

: 4;

7 7 : . @ { | 4; 6 6 @ { 4; 4 4 @ h | @ 57 | 57

Oxhjr ho sulphur wdugd 4; krudv | ho vhjxqgr 57 krudv hq kdfhu ho wudedmr frpsohwr1

Hmhuflflr 5; Vh kd gh wudqvsruwdu 9<3 wrqhodgdv gh phufdqfðdv ghvgh xq pxhooh d xqd hvwdflöq gh ihu0 urfduulo phgldqwh 8 fdplrqhv gh 6 wrqhodgdv | 43 fdplrqhv gh xqd wrqhodgd | phgld1 Hq srfdv krudv/ orv fdplrqhv kdq wudqvsruwdgr 58279 gh odv phufdqfðdv1 Sdud frpsohwdu ho wudqvsruwh d wlhpsr/ vh kd gh oohydu odv phufdqfðdv uhvwdqwhv hq xq odsvr 5 krudv phqru txh ho |d wudqvfxuulgr1 Vh frpsohwö ho wudqv0 sruwh judfldv d txh orv frqgxfwruhv gh orv fdplrqhv frphq}durq d kdfhu xq yldmh sru krud päv txh dqwhv1 Ghwhuplqh fxäqwdv krudv wdugdurq hq wudqvsruwdu wrgdv odv phufdqfðdv | wdpelìq ho qýphur gh yldmhv sru krud txh vh kdfðd do sulqflslr vdelhqgr txh orv fdplrqhv gh xqd | phgld wrqhodgd kdfhq xq yldmh päv sru krud txh orv fdplrqhv gh 6 wrqhodgdv ; 1 Vroxflöq=

Vxsrqjdprv txh od sulphud sduwh gh odv phufdqfðdv/ txh dvflhqgh d 58 9<3 @ 6:8 wrqhodgdv 79

vh wudqvsruwd hq { krudv kdflhqgr fdgd fdplöq gh 6 wrqhodgdv | yldmhv sru krud1 Hqwrqfhv fdgd fdplöq gh xqd | phgld wrqhodgd kduä | . 4 yldmhv sru krud1 Sru klsöwhvlv/ od sduwh uhvwdqwh gh phufdqfðdv +hv ghflu/ 9<306:8@648 wrqhodgdv, vh wudqvsruwö hq {  5 krudv/ kdflhqgr orv fdplrqhv gh 6 wrqhodgdv | . 4 yldmhv sru krud | orv fdplrqhv gh xqd | phgld wrqhodgdv +| . 4, . 4 @ | . 5 yldmhv sru krud1 Rewhqhprv dvðho vljxlhqwh vlvwhpd gh hfxdflrqhv

Vlpsolfdqgr whqhprv

8  6{| . 43  65 {+| . 4, @ 6:8 8  6+{  5,+| . 4, . 43  65 +{  5,+| . 5, @ 648 5{| . { @ 58 5{| . 6{  7| @ 5:

; Vh gheh vxsrqhu txh wrgrv orv fdplrqhv ledq frpsohwdphqwh fdujdgrv hq fdgd yld mh1

FKDSWHU 41 ORV QÝPHURV UHDOHV

67 gh grqgh

5{  7| @ 5

oxhjr 5| @ {  41 Vxvwlwx|hqgr hq od sulphud hfxdflöq rewhqhprv {5 @ 58/ hv ghflu< / { @ 81 Od sulphud sduwh gh odv phufdqfðdv vh wudqvsruwö hq 8 krudv | od vhjxqgd sduwh hq 8  5 @ 6 krudv1 Qrwd= Wrgdv odv phufdqfðdv vh wudqvsruwdurq hq ; krudv> do sulqflslr/ orv fdplrqhv gh wuhv wrqhodgdv kdfðdq 5 yldmhv sru krud | orv gh xqd | phgld wrqhodgdv/ 6 yldmhv sru krud1 Hmhuflflr 5< Xqd lqgxvwuld wlhqh xq hqfdujr gh ;43 duwðfxorv | rwud gh <33 duwðfxorv hq ho plvpr shulrgr gh wlhpsr1 Od sulphud kd frpsohwdgr ho shglgr 6 gðdv dqwhv gho sod}r suhylvwr | od vhjxqgrv 9 gðdv dqwhv1 BcFxäqwdv duwðfxorv surgxfh do gðd fdgd lqgxvwuld/ vdelhqgr txh od vhjxqgd surgxfh sru gðd 7 duwðfxorv päv txh od sulphudB

Vroxflöq= Vhd { od surgxfflöq gh duwðfxorv glduld gh od sulphud lqgxvwuld/ hqwrqfhv od vhjxqgd surgxfh {.7 duwlfxorv sru gðd1 Od sulphud kd frpsohwdgr vx shglgr hq ;43 { gðdv/ oxhjr ho wlhpsr gdgr sdud fxpsolu <33 . 6 gðdv1 Dqäorjdphqwh whqhprv txh ho shglgr kd vlgr ;43 { {.7 . 9 hv ho wlhpsr dvljqdgr d od vhjxqgd lqgxvwuld/ shur ho wlhpsr dvljqdgr d dpedv odv lqgxvwuldv hv ho plvpr/ oxhjr ;43 <33 .6@ .9 { {.7

gh grqgh { @ 531 Hv ghflu/ od sulphud lqgxvwuld surgxfh 53 duwlfxorv | od vhjxqgd 57 duwlfxorv sru gðd1

Hmhuflflr 63 Kdoodu xq qýphur gh grv fliudv vdelhqgr txh ho qýphur gh xqlgdghv h{fhgh hq fxdwur do qýphur gh od ghfhqdv | txh ho surgxfwr gho qýphur ghvhdgr sru od vxpd gh vxv gljlwrv hv ljxdo d 8:91

Revhuydflöq= hqwrqfhv Vroxflöq= hfxdflöq

Vl xq qýphur d wlhqh q gljlwrv d3 > d4 > d5 >    dq4 / rughqdgrv gh l}txlhugd d ghuhfkd d @ dq4 43q4 .    . d4 43 . d3

Vhd { ho gljlwr gh od xqlgdghv/ | ho gljlwr gh od ghfhqdv hqwrqfhv sru klsöwhvlv vh wlhqh od {@|.7

+4147,

+43| . {,+{ . |, @ 8:9

+4148,

dghpäv ho qýphur ghvhdgr hv 43| . { oxhjr Sru 4147 | 4148

+44| . 7,+5| . 7, @ 8:9 +,

+,

:3 ,@3 44 :3 | @ 7b| @  44 44+|  7,+| .

shur | A 3/ oxhjr | @ 71 Sru or wdqwr ho qýphur exvfdgr hv 7;1

416

D{lrpdv gh Rughq

D frqwlqxdflöq gduhprv xq frqmxqwr gh d{lrpdv txh irupd sduwh gh od fdudfwhul}dflöq gh orv qýphurv uhdohv1 Hvwrv d{lrpdv qrv shuplwhq rughqdu orv qýphurv uhdohv +| dýq srghu judfdu hvwh rughq hq xqd uhfwd,1 Dgplwlprv od h{lvwhqfld gh xq vxefrqmxqwr gh U txh ghqrwdprv sru U. / fx|rv hohphqwrv vh oodpdq qýphurv uhdohv srvlwlyrv | txh vh fdudfwhul}dq sru orv d{lrpdv= < Gh

{2 ' 2D vh

rewlhqh

{ ' D r { ' 3D shur { lqglfd

krudv oxhjr

{ gheh vhu

srvlwlyr1

4161 D[LRPDV GH RUGHQ

68

D{lrpd l, Fodxvxud gh od vxpd

Vl d> e 5 U. / hqwrqfhv d . e 5 U.

D{lrpd ll, Fodxvxud gho surgxfwr

Vl d> e 5 U. / hqwrqfhv de 5 U.

D{lrpd lll, Wulfrwrpðd

Sdud fdgd qýphur uhdo vöor xqd gh odv vljxlhqwhv sursrvlflrqhv hv yhugdghud d 5 U. > d @ 3>

 d 5 U.

Hq edvh d orv d{lrpdv dqwhulruhv ghprvwuduhprv odv vljxlhqwhv surslhgdghv=

Sursrvlflöq 4143 Ho frqmxqwr gh orv qýphurv uhdohv fxpsoh l, d 5 U  i3j , d5 5 U.

ll, d 5 U. , d4 5 U.

lll, d  e 5 U. a e  f 5 U. , d  f 5 U. ly, e  d 5 U. / +e . f,  +d . f, 5 U. y, e  d 5 U. a f 5 U. , ef  df 5 U.

yl, e  d 5 U. a f 5 U. , df  ef 5 U. Ghprvwudflöq=

l, Vl d 5 U  i3j hqwrqfhv sru D{lrpd lll,

d 5 U.

b d 5 U.

Vl d 5 U. hqwrqfhv sru D{lrpd ll, d  d 5 U. / hv ghflu/

d 5 U. , d5 5 U.

Vl d 5 U. hqwrqfhv sru D{lrpd ll, +d,  +d, 5 U. shur sru sursrvlflöq 417 yll/ +d,5 @ d5 / oxhjr d5 5 U. 1 Qrwhprv txh hq sduwlfxodu vl d @ 4/ hqwrqfhv 45 @ 4 5 U.

ll, Ghprvwuduhprv hvwd surslhgdg sru ho pìwrgr gho devxugr1

Vhd d 5 U. 1 Vxsrqjdprv txh d4 95 U. / hqwrqfhv sru D{lrpd lll, d4 @ 3 b +d4 , 5 U.

Vxsrqjdprv txh d4 @ 3 hqwrqfhv d  d4 @ 3 shur sru surslhgdg gh lqyhuvr d  d4 @ 4> hvwr hv xqd frqwudglfflöq | sru or wdqwr d4 9@ 31

lll,

Vxsrqjdprv txh +d4 , 5 U. 1 Frpr d 5 U. whqhprv txh sru D{lrpd ll, txh +d4 , d @ 4 5 U. txh hv wdpelìq xqd frqwudglfflöq sxhv 4 5 U. 1 Frpr hq dperv fdvrv rewhqhprv xqd frqwudglfflöq/ qr vh sxhgh gdu qlqjxqd gh odv grv srvlelolgdghv/ sru or wdqwr/ d4 5 U. > dvð/ vh wlhqh txh ho lqyhuvr pxowlsolfdwlyr gh xq qýphur srvlwlyr hv wdpelìq xq qýphur srvlwlyr1 , , , ,

d  e 5 U. a e  f 5 U. +d  e, . +e  f, 5 U. d . +e . e,  f 5 U. d . 3  f 5 U. d  f 5 U.

sru D{lrpd l, dvrfldwlylgdg lqyhuvr dglwlyr qhxwur dglwlyr

FKDSWHU 41 ORV QÝPHURV UHDOHV

69 ly, / / / /

y,

yl,

e  d 5 U. e  d . 3 5 U. e  d . f  f 5 U. +e . f, . +d  f, 5 U. +e . f,  +d . f, 5 U.

qhxwur dglwlyr lqyhuvr dglwlyr dvrfldwlylgdg frqpxwdwlylgdg glvwulexwlylgdg

e  d 5 U. a f 5 U. , +e  d,f 5 U. , ef  df 5 U.

e  d 5 U. a f 5 U. , +e  d,+f, 5 U. , +ef,  +df, 5 U. , df  ef 5 U.

sru D{lrpd ll, glvwulexwlylgdg sru D{lrpd ll, glvwulexwlylgdg

Vh ghqh ho frqmxqwr gh orv qýphurv uhdohv qhjdwlyrv frpr U @ id 5 U = d 5 U. j

Qrwhprv txh U 9@ > sxhv 4 5 U gdgr txh 4 5 U. 1 Dqäorjdphqwh vh ghqh ho frqmxqwr gh orv qýphurv uhdohv qr qhjdwlyrv frpr . U. 3 @ U ^ i3j Qrwd= Sru D{lrpd lll, vh wlhqh od vljxlhqwh ljxdogdg grqgh orv frqmxqwrv vrq glvmxqwrv grv d grv U @ U. ^ i3j ^ U

Ghqlflöq 4144 Gluhprv txh d hv pd|ru txh e ö e hv phqru txh d vl | vöor vl d  e 5 U.

| hvfuleluhprv hvwr lqglvwlqwdphqwh frpr dAe

r frpr

+4149,

e ? d=

Gluhprv txh d hv pd|ru r ljxdo txh e vl d hv pd|ru txh e r d hv ljxdo d e/ hv ghflu Hvfuleluhprv hvwr lqglvwlqwdphqwh frpr Dvð/ vh wlhqh

dAebd@e de

r

e  d=

d  e +, d A e b d @ e

Dqäorjdphqwh/ gluhprv txh d hv phqru r ljxdo txh e vl d hv phqru txh e r d ljxdo txh e1 Hvfuleluhprv lqglvwlqwdphqwh hq hvwh fdvr de r ed r htxlydohqwhphqwh

de/d?ebd@e

4161 D[LRPDV GH RUGHQ

6:

Frq od qrwdflöq dqwhulru whqhprv txh= d 5 U. d 5 U

/ dA3 / d?3

| odv surslhgdghv ghprvwudgdv dqwhulruphqwh vh sxhghq hvfulelu gh od vljxlhqwh irupd Frurodulr 4145 Vhdq d> e> f 5 U

l, d 5 U  i3j , d5 A 31 ll, d A 3 , d4 A 3

lll, d A e a e A f , d A f ly, e A d / e . f A d . f

y, e A d a f A 3 , ef A df

yl, e A d a f ? 3 , ef ? df Qrwdflöq=

Orv Lqwhuydorv gh U=

def/deaef

Vhdq d> e 5 U1 frq43 d  e Vh ghqhq

^d> e^ `d> e` `d> e^ ^d> e` `4> d` `4> d^ ^d> 4^ `d> 4^

i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U

@ @ @ @ @ @ @ @

= d  { ? ej = d ? {  ej = d ? { ? ej = d  {  ej = {  dj = { ? dj = {  dj = { A dj

Qrwd= Od ghqlflöq gdgd qr dgplwh lqwhuydorv gho wlsr ^d> e^ frq d A e +hq wdo fdvr vh wudwduld gh xq frqmxqwr ydfðr,1 Hmhpsor 64

^4> 3^ `4> 4^ `4> 3^ ^ `4> 4^ `3> 4^ ^4> 4` ^4> 3^ ^5> 4^

@ @ @ @ @ @ @

43 Dojxqrv gh hvwrv lqwhuydorv sxhghq vhu ho frqmxqwr ydfðr1

i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U i{ 5 U

= { ? 3j = { A 4j = { ? 3 b { A 4j = 3 ? { ? 4j = 4  {  4j = 4  { ? 3j = 5  { ? 4j

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b |@

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 s  S6 +{> |> , @  45 > 56 > 5   s S7 +{> |> , @  45 >  56 > 5

vdwlvidfhq 4156 Oxhjr ho frqmxqwr vroxflöq gho vlvwhpd 4156 hv= $ # $, + s s    # 6 4 6 4 6 4 > 4> 3> >  > >5 >  > >5 V@ 4> 3> 5 5 5 5 5 5

Hmhuflflr 75 Uhvxhoyd ho vlvwhpd

+4, +5, +6, +7,

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7{ <| } {5 . | 5 . } 5 7 <

@ @ @ @

{+7  , @ 3

hv ghflu

+4158,

|5 . }5 @ 4 <

+4159,

|+<  , @ 3

|@3 b @<

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+4157,

{@3 b @7

l, Vxsrqjdprv sulphur txh { @ 31 Hqwrqfhv hq +7, uhvxowd

shur sru +5, whqhprv

{ | } 4

}5 @ 4 } @ 4 b } @ 4 }+4  , @ 3

FKDSWHU 41 ORV QÝPHURV UHDOHV

7; gh grqgh vh wlhqh txh

}@3 b @4

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l5, Vl  @
}@3

| vxvwlwx|hqgr hq 4159 vh wlhqh txh or txh lpsolfd txh Sru or wdqwr orv sxqwrv

|5 @ 4> < | @ 6 b | @ 6

S6 +{> |> }> , @ +3> 6> 3> <, S7 +{> |> }> , @ +3> 6> 3> <,

wdpelìq vrq vroxflrqhv gho vlvwhpd 41571 ll, Ilqdophqwh frqvlghuhprv  @ 7 gh 4158/ hqwrqfhv gh +5, | +6, vh wlhqh txh oxhjr hq +7,> gh grqgh oxhjr orv sxqwrv

|@3 a }@3 {5 @4 7

{ @ 5 b { @ 5

S8 +{> |> }> , @ +5> 3> 3> 7, S9 +{> |> }> , @ +5> 3> 3> 7,

vrq wdpelìq vroxflrqhv gho vlvwhpd 4157 Dvð/ |d txh vh frqvlghudurq wrgrv orv fdvrv/ ho frqmxqwr vroxflöq gho vlvwhpd hv Hmhuflflr 76 Uhvxhoyd ho vlvwhpd>

V @ iS4 > S5 > S6 > S7 > S8 > S9 j +4, | . 5} . |} +5, { . 5} . {} +6, 5{ . 5| . {| +7, {|}  Y

Grqgh Y hv xqd frqvwdqwh srvlwlyd frqrflgd1

@ @ @ @

3 3 3 3

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|  { . }+|  {, @ 3 +|  {,+4 . }, @ 3

+415:,

4161 D[LRPDV GH RUGHQ

7<

gh grqgh

+415;,

+4 . }, @ 3 b | @ {

l, Vxsrqjdprv txh 4 . } @ 3/ hqwrqfhv Vxvwlwx|hqgr 415< hq +5, whqhprv

|d txh

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+415<,

} 9@ 3

  4 { . 5} .  {} @ 3 }

gh grqgh } @ 3/ or txh frqwudglfh 415<1 Oxhjr od ljxdogdg 4 . } @ 3 qr vh sxhgh gdu | sru or wdqwr sru 415; vh wlhqh ho vljxlhqwh fdvr1 ll, Vxsrqjdprv dkrud {@| +4163,

Vxvwlwx|hqgr 4163 hq +6, whqhprv

7{ . {5 @ 3

hv ghflu

{+7 . {, @ 3

or txh lpsolfd txh

{ @ 3 b 7 . { @ 3

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| frpr { 9@ 3/ rewhqhprv

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gh grqgh

@ { . 5} .



7 {

+4164,

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{ @ 5}

Ilqdophqwh/ vxvwlwx|hqgr { @ | | { @ 5} hq +7, whqhprv

sru or wdqwr

{6 @Y 5

{@|@

s 6 5Y

a }@

s6 5Y 5

a @

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oxhjr ho frqmxqwr vroxflöq gho vlvwhpd 415: hv +# $, s 6 s s 7 5Y 6 6 > s 5Y > 5Y > V@ 6 5 5Y Hmhuflflr 77 Uhvxhoyd ho vlvwhpd

+4, ;|} +5, ;{} +6, ;{| +7, 49{5 . 7| 5 . <} 5  477

Edmr od uhvwulfflöq= { A 3/ | A 3/ } A 3

@ @ @ @

65{ ;| 4;} 3

+4165,

FKDSWHU 41 ORV QÝPHURV UHDOHV

83 Vroxflöq=

46

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txh hv htxlydohqwh d

57{|} @ 5+49{5 . 7|5 . <} 5 ,

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hv ghflu/

{|} @ 45

+4166,

;{|} @ 65{5 

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;+45, @ 65{5 >

hv ghflu/ or txh lpsolfd l, Vl  @ 3/ hqwrqfhv

65+6  {5 , @ 3

@3 b {@

s s 6 b {@ 6

{@3 b |@3 b }@3

s or txh qr vh sxhgh gdu sru od uhvwulfflöq1 Sru od plvpd ud}öq qr vh sxhgh gdu txh { @  61 Dvð/ od ýqlfd srvlelolgdg hv txh s {@ 6

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;{|} @ ;| 5 

shur {|} @ 45 oxhjr

45 @ |5 

or txh lpsolfd Frpr  9@ 3/ hqwrqfhv 45  |5 @ 3/ hv ghflu/

+45  | 5 , @ 3 |@

s

45

Ilqdophqwh/ pxowlsolfdqgr sru } od hfxdflöq +6,/ vh wlhqh

;{|} @ 4;} 5 

gh grqgh uhvxowd Sru rwud sduwh/ frpr {|} @ 45/ hqwrqfhv {|} @ @ 45 4 6 Frpr

}@

s 7 6 6

s s s s s 6  45  7 6 7 45 5 6 @ @ 45  6 45 6

khprv qrwdgr/ h{lvwhq yduldv irupdv gh uhvroyhu orv vlvwhpdv gh hfxdflrqhv1 Qr h{lvwh xqd wìfqlfd dsolfdeoh vlhpsuh>

or lps ruwdqwh hv qr frphwhu ho wðslfr huuru gh ghmdu dojýq fdvr vlq dqdol}du1

4161 D[LRPDV GH RUGHQ

84

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s | @ 5 6>

s 7 6 }@ > 6

s 5 6 @ 6

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41616

s 2 hv luudflrqdo

s 7s 7 6 6= }@ s s @ 6 6 6

s s D frqwlqxdflöq surehprv txh 5 qr hv udflrqdo +r elhq= hq jhqhudo/ vl s hv xq qýphur sulpr/ hqwrqfhv s hv luudflrqdo,1 s Vxsrqjdprv or frqwudulr/ hv ghflu/ vxsrqjdprv txh 5 hv udflrqdo/ hvwr hv/ s

5@

s t

frq s 5 ] / t 5 ]  i3j1 Vlhpsuh hv srvleoh hvfrjhu s | t wdo txh od iudfflöq st hvwì vlpsolfdgd do pä{lpr/ hq wdo fdvr s s s5 t @ 5 / t5 @ 5 / s5 @ 5t 5

oxhjr s5 hv xq qýphur sdu | sru 41:/ s gheh vhu sdu/ hv ghflu/ s @ 5n/ sdud dojýq n 5 ]. 1 Sru or wdqwr s5 @ 7n5

| frpr s5 @ 5t 5 / vh wlhqh txh 5t 5 @ 7n5 / dvðt 5 @ 5n5 1 Hvwd ýowlpd ljxdogdg qrv glfh txh t 5 hv xq qýphur sdu | sru 41: t wdpelìq hv sdu1 Sru or dqwhulru vh wlhqh txh t | s vrq sduhv/ or txh frqwudglfh ho khfkr txh st hvwded vlpsolfdgd do s pä{lpr/ gh hvwd irupd 5 qr sxhgh vhu xq qýphur udflrqdo1 Dvðkhprv rewhqlgr ho lpsruwdqwh uhvxowdgr= s 5 hv luudflrqdo +4168, s s Dvlplvpr/  s 5 qr hv udflrqdo1 Hq hihfwr/ vl  5 ixhud udflrqdo/ hqwrqfhv gdgr txh +T > ., hv juxsr vx lqyhuvr dglwlyr 5 gheh vhu udflrqdo or txh frqwudglfh 41681 s Frq dujxphqwrv vlploduhv vh sxhgh suredu txh sdud wrgr qýphur hqwhur s vh wlhqh txh s 5 95 T s Hq irupd vlplodu d 4168 srghprv ghprvwudu txh 7 9s hv luudflrqdo1 s BcFöpr srghprv ghwhuplqdu fxäo gh orv luudflrqdohv 5 r 7 9 hv phqruB Xqd khuudplhqwd gh judq xwlolgdg sdud rughqdu uhdohv hv od vljxlhqwh sursrvlflöq Sursrvlflöq 414:

l, Vhdq d> e> f> g 5 U.

ll, Vhdq q 5 Q / d> e 5 U. Ghprvwudflöq=

d ? e a f ? g , df ? eg d ? e / dq ? eq

FKDSWHU 41 ORV QÝPHURV UHDOHV

85 l, Uhfrughprv txh oxhjr sru or wdqwr

d ? e a f A 3 , df ? ef f ? g a e A 3 , ef ? eg

df ? ef a ef ? eg , df ? eg

+;d> e> f> g 5 U. ,+d ? e a f ? g , df ? eg,

ll, Sdud suredu hvwd sursrvlflöq ghprvwuduhprv hq sulphu oxjdu txh d ? e , dq ? eq

| oxhjr sureduhprv od lpsolfdflöq hq ho rwur vhqwlgr1 Gh +l, hv fodur txh

d ? e , d ? e a d ? e , d5 ? e5 =

Lqgxfwlydphqwh vh sxhgh suredu txh d ? e , dq ? eq 1 Dkrud vxsrqjdprv txh

frq d> e 5 U. ghprvwuhprv txh d ? e Vhd

hqwrqfhv

dq ? eq

+4169,

O @ +dq4 . dq5 e . dq6 e5 .    . deq5 . eq4 , dq  eq @ +d  e,O

frpr d> e 5 U. / hv fodur txh O 5 U. / oxhjr O4 5 U. 1 Sru rwud sduwh frqvlghudqgr 4169 dq eq ? 3/ hv ghflu +d  e,O ? 3 shur O4 A 3/ oxhjr +d  e,OO4 ? 3/ gh grqgh dvð/

de?3

dq ? eq , d ? e ;d> e 5 U.

s s Dkrud xvhprv od htxlydohqfld uhflhq suredgd sdud orv qýphurv 5 | 7 91 Vh wlhqh= s s s s 5 ? 7 9 / + 5,7 ? + 7 9,7 7 7 / 55 ? 97 / 7?9

Frpr od ghvljxdogdg 7 ? 9 hv yhugdghud vh wlhqh txh s s 7 5? 9 wdpelìq hv yhugdghud1

Frurodulr 414; Sdud wrgr d> e 5 U. > q 5 Q

s s q d ? e +, q d ? e



4161 D[LRPDV GH RUGHQ

86

s s s s Ghprvwudflöq= Frpr d 5 U. | e 5 U. hqwrqfhv d @ + q d,q | e @ + q e,q grqgh q d 5 U. | q e 5 U. / oxhjr sru od sursrvlflrq 414: vh wlhqh s s s s q q d ? e +, + q d,q ? + e,q +, q d ? e Hmhuflflr 78 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv



41 +;d> e 5 U,+d ? e , d5 ? e5 ,

51 +;d> e 5 U,+d5 ? e5 , d ? e,

61 +;q 5 Q,+;d> e 5 U,+dq ? eq , d ? e, 71 +;d 5 U,+d  d5 ,

81 +;d 5 U. ,+ d4 ? 4,

91 +;d> e 5 U. ,+d ? e , d5 ? e5 ,

:1 +;d> e 5 U  i3j,+d ? e , d4 A 4e , ;1 +;d> e 5 U. ,+d ? e , d4 A 4e ,

4 ? 4, <1 +;d> e 5 U. ,+ d. e d

4 ? 4, 431 +;d> e 5 U. ,+ de d

Vroxflöq=

41 Idovr1 H{lvwhq d @ 5/ e @ 4 wdohv txh d ? e shur d5 9? e5 1

51 Idovr1 H{lvwhq d @ 5/ e @ 6 wdohv txh d5 ? e5 shur d 9? e1

61 Idovr1 H{lvwhq q @ 7/ d @ 5 | e @ 6 wdohv txh 57 ? +6,7 shur 5 9? 6

4 5 U wdo txh d 9 d5 5 H{lvwh d @ 46 5 U. wdo txh d4 9? 4

71 Idovr1 H{lvwh d @ 81 Idovr1

91 Yhugdghur1 Hvwh hv xq fdvr sduwlfxodu gh od sursrvlflöq 414: ll,/ shur uhshwluhprv od ghprvwudflöq sdud q @ 51 Ghprvwudflöq= Vhdq d> e 5 U. wdohv txh d ? e1 Frpr d ? e/ hqwrqfhv d  e ? 3> dghpäv d . e A 3 sxhv d> e 5 U. / oxhjr/ +d  e,+d . e, ? 3> gh grqgh/

sru or wdqwr/

d5  e5 ? 3> d5 ? e5

:1 Idovr1 H{lvwhq d @ 5/ e @ 4 wdohv txh d ? e shur d4 9A 4e

4 A 31 Frpr d ? e vh wlhqh txh ;1 Yhugdghur1 Vhd d> e 5 U. / hqwrqfhv de A 3 gh grqgh de     4 4 4 4 4 4 d ?e , ? , A de de e d d e



FKDSWHU 41 ORV QÝPHURV UHDOHV

87

<1 Yhugdghur1 Vhdq d> e 5 U. / hqwrqfhv d . e 5 U. | sxhvwr txh d.eAd

sru od sduwh ;, vh wlhqh txh

4 4 ? d.e d Hvwd ýowlpd ghvljxdogdg hv px| ýwlo sdud dfrwdu h{suhvlrqhv uhdohv frpr yhuhprv päv dghodqwh1

431 Idovr1 H{lvwhq d @ 5/ e @

4 7

4 9? 4 sxhv 4 @ 5 | 4 @ hq U. shur= de d de d

4 5

Hq ho hmhuflflr dqwhulru khprv suredgr txh flhuwdv ghvljxdogdghv qr vlhpsuh vh fxpsohq1 Od suhjxqwd qdwxudo hv= BcSdud txì ydoruhv uhdohv vrq yhugdghudv hvdv ghvljxdogdghvB Odv surslhgdghv txh hq vhjxlgd yhuhprv qrv shuplwluäq ghwhuplqdu orv frqmxqwrv hq orv fxdohv hvdv ghvljxdogdghv vh yhulfdq1 Sursrvlflöq 414< Vhdq d> e 5 U

l, de A 3 / +d A 3 a e A 3, b +d ? 3 a e ? 3,

ll, de ? 3 / +d A 3 a e ? 3, b +d ? 3 a e A 3,

Dqwhv gh hqwudu hq od ghprvwudflöq h{solfduhprv or txh vh kduä hq hood hq wìuplqrv oöjlfrv= Od sursrvl0 flöq l, vh sxhgh hvfulelu s / +t b v,

grqgh odv sursrvlflrqhv vrq

s = de A 3 t = dA3aeA3 v = d?3ae?3

Sru rwud sduwh od htxlydohqfld dqwhulru vh vhsdud hq grv sursrvlflrqhv ^s , +t b v,` a ^s + +t b v,`

+416:,

Frpr vdehprv/ sdud prvwudu txh xqd sursrvlflöq irupdgd sru ho frqhfwlyr a hv yhugdghud qhfhvlwdprv prvwudu txh dpedv sursrvlflrqhv txh od frpsrqhq vrq yhugdghudv/ hv ghflu/ edvwd suredu hq irupd vhsdudgd txh s , +t b v,

|

s + +t b v,

Dvð/ lqglfduhprv frq + , , od sulphud sduwh | frq + + , od vhjxqgd sduwh1 Ghprvwudflöq= + , , Frpr de A 3/ l1h1/ de 5 U. | d wlhqh grv srvlelolgdghv +vhu srvlwlyr r qhjdwlyr, srghprv vhsdudu od ghprvwudflöq hq grv vxe fdvrv

Fdvr 4, Vl d 5 U. hqwrqfhv d4 5 U. / oxhjr sru D{lrpd ll, | dvrfldqgr dsursldgdphqwh vh wlhqh txh d4 +de, @ e 5 U. > hq frqvhfxhqfld/ sru or wdqwr47 4 7 Dtxðkd|

Es

de 5 U. , d 5 U. a e 5 U. > de A 3 , d A 3 a e A 3

ydulrv sdvrv edvdgrv hq htxlydohqfldv oöjlfdv txh qr kdq vlgr h{solflwdgrv> s ru hmhpsor/

2 Es i t C s 2 t1

4161 D[LRPDV GH RUGHQ

88

Fdvr 5, Vl d 5 U hqwrqfhv d4 5 U / oxhjr sru D{lrpd ll, | dvrfldqgr dsursldgdphqwh vh wlhqh txh d4 +de, @ e 5 U 1 Hq frqvhfxhqfld de 5 U. , d 5 U a e 5 U >

hv ghflu/

Sru Fdvrv 4, | 5, whqhprv

de A 3 , d ? 3 a e ? 3

de A 3 , +d A 3 a e A 3, b +d ? 3 a e ? 3,

Ghprvwudflöq= ++, Hv fodur sru D{lrpd ll,=

+d 5 U. a e 5 U. , , de 5 U.

|

+d 5 U. a e 5 U. , , +d,+e, 5 U. , de 5 U.

Hq frqvhfxhqfld +|d txh hq dperv fdvrv rewhqhprv or plvpr,

^+d A 3 a e A 3, b +d ? 3 a e ? 3,` , de A 3

Hq irupd dqäorjd/ vh ghpxhvwud ll,1 Dvlplvpr vh wlhqh txh



de  3 / ^+d  3 a e  3, b +d  3 a e  3,`

|

de  3 / ^+d  3 a e  3, b +d  3 a e  3,`

Od ghprvwudflöq vh kdfh phgldqwh ho xvr gh od sursrvlflöq dqwhulru | gho khfkr txh ^ +d  e, / d A e` | ^+d / e, / + d / e,`1

Hmhpsor 79 Dkrud dsolfdqgr odv surslhgdghv dqwhulruhv/ ghwhuplqhprv edmr txì frqglflrqhv odv vljxlhqwhv ghvljxdogdghv vrq yhugdghudv hq U= 41 {  {5

51 |4 A 4 61 |4  4

71 {5 ? 4

} 81 {|  {. |

91 {| ? {

4 ?4 :1 {| {

Vroxflöq= Xwlol}dqgr odv surslhgdghv gh od sursrvlflöq 414< | odv sursrvlflrqhv txh vljxhq gh hood/ whqhprv 41

{  {5

/ / / / /

{  {5  3 {+4  {,  3 +{  3 a 4  {  3, b +{  3 a 4  {  3, +{  3 a {  4, b +{  3 a {  4, {  3b{ 4

Sru or wdqwr od ghvljxdogdg {  {5 vh yhulfd sdud= {  3 b {  41

FKDSWHU 41 ORV QÝPHURV UHDOHV

89

51 Od ghvljxdogdg |4 A 4 wlhqh vljqlfdgr hq U vl | 9@ 31 Oxhjr

4 4A3 | 4| A3 | +4  | A 3 a | A 3, b +4  | ? 3 a | ? 3, +| ? 4 a | A 3, b +| A 4 a | ? 3, 3 ? | ? 4 b Idovr 48 3?|?4

4 A4 / | /

/ / / /

Oxhjr od ghvljxdogdg |4 A 4 vh fxpsoh sdud 3 ? | ? 4/ hv ghflu/ sdud | 5`3> 4^1 Dvð/ srghprv dupdu txh od vljxlhqwh sursrvlflöq hv yhugdghud +;| 5 U,+3 ? | ? 4 /

4 A 4, |

61 Dqäorjdphqwh vh rewlhqh txh |4  4 vh fxpsoh sdud wrgr | 5 U wdo txh |  4b| ? 3

71 {5 ? 4 / / / / / /

Sru or wdqwr= 81 Vhd | 9@ 3>

hv yhugdghud vl

oöjlfd

j

s

Idovr

+;{ 5 U,+4 ? { ? 4 / {5 ? 4,

{ {.}  | |

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4 8 Sru

{5  4 ? 3 +{  4,+{ . 4, ? 3 +{  4 A 3 a { . 4 ? 3, b +{  4 ? 3 a { . 4 A 3, +{ A 4 a { ? 4, b +{ ? 4 a { A 4, I b 4 ? { ? 4 4 ? { ? 4

Cs

1

{ {.}  3 | | } /  3 | } / 3 | / +}  3 a | A 3, b +}  3 a | ? 3, /

{ {.}  | |

+}  3 a | A 3, b +}  3 a | ? 3,

4161 D[LRPDV GH RUGHQ 91 Vhd | 9@ 349 >

8:

{ ?{ / | /

/

Oxhjr od ghvljxdogdg vh fxpsoh sdud= :1 Vhdq { 9@ 3 | 9@ 3

4 4 ? {| {

/ /

{ ?{ | +{ A 3 a | A 4, b +{ A 3 a | ? 3, b +{ ? 3 a 3 ? | ? 4, /

/

/

/

/

Oxhjr od ghvljxdogdg vh fxpsoh sdud=

{ {?3 |   4 { 4 ?3 | 4 4 +{ A 3 a  4 ? 3, b +{ ? 3 a  4 A 3, | | +{ A 3 a +| A 4 b | ? 3,, b +{ ? 3 a 3 ? | ? 4, +{ A 3 a | A 4, b +{ A 3 a | ? 3, b +{ ? 3 a 3 ? | ? 4,

/

4 4  ?3 {| {   4 4 4 ?3 { | 4 4 4 4 + A 3 a  4 ? 3, b + ? 3 a  4 A 3, { | { | 4 4 + A 3 a +| A 4 b | ? 3, b + ? 3 a 3 ? | ? 4, { { 4 4 4 + A 3 a | A 4, b + A 3 a | ? 3, b + ? 3 a 3 ? | ? 4, { { { +{ A 3 a | A 4, b +{ A 3 a | ? 3, b +{ ? 3 a 3 ? | ? 4, 4 4 ? {| {

+{ A 3 a | A 4, b +{ A 3 a | ? 3, b +{ ? 3 a 3 ? | ? 4,

Hq uhdolgdg or txh khprv khfkr hq ho hmhpsor 79 dqwhulru hv uhvroyhu lqhfxdflrqhv hq xqd r päv yduldeohv1

41617

Lqhfxdflrqhv

Vhd U ho frqmxqwr uhvwulfflöq +yhu vhfflöq 41517, gh odv h{suhvlrqhv S +{,/ T+{,1 Uhvroyhu xqd lqhfxdflöq hq od yduldeoh { hv hqfrqwudu ho vxefrqmxqwr V gh orv { hq U txh vdwlvidfhq od ghvljxdogdg +r lqhfxdflöq,1 S +{, ? T+{, r S +{,  T+{,

+416;,

Gluhprv txh V @ i{ 5 U = S +{, ? T+{,j r vlpsohphqwh V @ i{ 5 U = S +{, ? T+{,j +vxehqwhqglhqgr txh { gheh hvwdu U,1 hv ho frqmxqwr vroxflöq gh od lqhfxdflöq 416;1 Gluhprv txh grv lqhfxdflrqhv vrq htxlydohqwhv vl wlhqhq ho plvpr frqmxqwr vroxflöq1 49 Ho

huuru päv frpýq do wuded mdu hq hvwd ghvljxdogdg hv pxowlsolfdu s ru

|/

sxhv qr vh vde h vl

|

hv qhjdwlyr r s rvlwlyr1 Xqd

pxowlsolfdflöq wdo ghe huðd kdfhuvh ed mr frqglflrqhv h{suhvdv | s ru or wdqwr vh ghe huðd frqvlghudu wrgrv orv fdvrv s rvleohv1

FKDSWHU 41 ORV QÝPHURV UHDOHV

8; Hmhpsor 7: Ho frqmxqwr vroxflöq gh od lqhfxdflöq

4 A4 |

hv

+416<,

  4 A4 |5U = | @ i| 5 U = 3 ? | ? 4j

V

@

Hmhpsor 7; Ho frqmxqwr vroxflöq gh od lqhfxdflöq

+4173,

{5  4

hv

V

@ i{ 5 U = {5  4j @ i{ 5 U = 4  {  4j

Hmhpsor 7< Uhvroyhu odv vljxlhqwhv lqhfxdflrqhv

4 41 {{ .8  3 {  {4  {5 51 {.5 {5 7 {.5 61 {.6 { 5 5{,+{.8, 71 ++6 {5 58,+{.4, A 3

Vroxflöq=

41 Uhvroydprv od lqhfxdflöq

{4 3 {.8 Qrwhprv txh { . 8 9@ 3/ hv ghflu/ ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq BB hv Sru rwud sduwh=

U4 @ i{ 5 U = { . 8 9@ 3j @ U  i8j

{4  3 {.8

Sru or wdqwr

/ +{  4  3 a { . 8 A 3, b +{  4  3 a { . 8 ? 3, / +{  4 a { A 8, b +{  4 a { ? 8, / {  4 b { ? 8 V4

Qrwd=

  {4 {5U= 3 {.8 @ U4 _ i{ 5 U = {  4 b { ? 8j @ `  4> 8^^^4> 4^ @

4 Od uhvroxflöq gh od lqhfxdflöq {{ .8  3 gdgd hq 4174 vh sxhgh uhvxplu hq od vljxlhqwh wdeod> {4 {.8 {4 {.8

{ 5`4> 8^ 0 0 .

{ @ 8 3 <2

{ 5`8> 4^ 0 . 0

{@4 3 3

`{ 5 4> 4^ . . .

+4174,

4161 D[LRPDV GH RUGHQ

8<

Grqgh ho vljqr . lqglfd hq txì lqwhuydorv ho idfwru fruuhvsrqglhqwh d od od hv srvlwlyr | ho vljqr 0 hq txì lqwhuydor hv qhjdwlyr1 Sru hmhpsor/ gh od sulphud od vh wlhqh txh ho idfwru {  4 hv srvlwlyr hq ho lqwhuydor `4> 4^ | qhjdwlyr hq `  4> 8` | hq `  8> 4^/ +hv ghflu/ hq `  4> 4^,1 Gh od ýowlpd od 4 vh rewlhqh txh od h{suhvlöq {{ .8 hv qr qhjdwlyd hq orv lqwhuydorv `  4> 8^ | ^4> 4^> oxhjr od vroxflöq gh od lqhfxdflöq {4 3 {.8 hv `  4> 8^^^4> 4^1 Hq pxfkrv fdvrv/ orv idfwruhv lqyroxfudgrv hq xqd lqhfxdflöq gho wlsr dqwhulru vrq päv gh grv | sru hoor hv iuhfxhqwh xvdu od wdeod dqwhulru hq od uhvroxflöq1

51 Uhvroydprv od lqhfxdflöq

{4 {5 {  5  {.5 { 7 {.5 Dperv plhpeurv gh od ghvljxdogdg wlhqhq vljqlfdgr hq U vl

+4175,

{ . 5 9@ 3 a {5  7 9@ 3>

hv ghflu/ ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq 4175 hv

U4 @ i{ 5 U = { . 5 9@ 3 a {  5 9@ 3j @ U  i5> 5j

Dkrud ghwhuplqhprv ho frqmxqwr vroxflöq gh od lqhfxdflöq=

/

/

/

{ {4 {5   { . 5 {5  7 {.5 { {4 {5   3 { . 5 +{ . 5,+{  5, { . 5 {+{  5,  +{  4,  +{  5,5 3 +{  5,+{ . 5, {6 3 +{  5,+{ . 5,

+4176,

Dkrud frqvlghuhprv od wdeod sdud od uhvroxflöq gh od lqhfxdflöq 41761

{6 {5 {.5 {6 +{5,+{.5,

{ 5 `4> 5^ 0 0 0 0

Vh wlhqh gh od ýowlpd od txh

{ @ 5 3 <2

{ 5 `5> 5^ 0 0 . .

{@5

{6 3 +{  5,+{ . 5,

3 <2

{ 5 `5> 6^ 0 . . 0

{@6 3 3

{ 5 `6> 4^ . . . .

hq orv lqwhuydorv `  4> 5^ | `5> 6` oxhjr od vroxflöq gh od lqhfxdflöq 4175 hv 61 Uhvroydprv od lqhfxdflöq

V5 @ U4 _ +`  4> 5^^`5> 6`, @`  4> 5^^`5> 6`

{.6 5 { Ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq 4177 hv

U4 @ i{ 5 U = { 9@ 3j @ U  i3j

+4177,

FKDSWHU 41 ORV QÝPHURV UHDOHV

93 Sru rwud sduwh/

{.6 53 { { . 6  5{ 3 { 6{ 3 { +6  {  3 a { A 3, b +6  {  3 a { ? 3, +{  6 a { A 3, b +{  63 a { ? 3, +3 ? {  6, b I 3?{6

{.6 5 / {

/

/

/ / / /

Oxhjr

V6 @`3> 6` _ U4 @`3> 6`

Uhvroydprv

+6  5{,+{ . 8, A3 +{5  58,+{ . 4,

+4178,

Ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq 4178 hv

U4 @ i{ 5 U = {5  58 9@ 3 a { . 4 9@ 3j @ U  i8> 4> 8j

Frpr {5  58 @ +{  8,+{ . 8,/ hqwrqfhv od lqhfxdflöq 4178 hv htxlydohqwh d Frqvlghudqgr od wdeod { 5 `4> 4^ 6  5{ . {8 0 {.4 0 +65{, . +{8,+{.4,

+65{, +{8,+{.4,

{ @04 3 <2

Rewhqhprv txh

A3

 { 5 4> . 0 . 0

6 5

frq { 5 U4 {@ 3

6 5

{5

3

+6  5{, A3 +{  8,+{ . 4,

hq `  4> 4^^` 65 > 8^/ shur { 5 U4 oxhjr ho frqmxqwr vroxflöq gh 4178 hv V7

6 @ +`  4> 4^^` > 8^, _ U4 5 6 @ `  4> 8^^`  8> 4^^` > 8^ 5

Qrwd= Qrwhprv txh odv vljxlhqwhv sursrvlflrqhv vrq idovdv s 41 +;{ 5 U. 3 ,+ {  {, s 51 +;{ 5 U,+;| 5 U. ,+{  | , {5  |, Hq hihfwr> Sdud 4, h{lvwh { @

4 5 U.3/ wdo txh 7

u

4 4 4 @ 9 7 5 7



6 > 8 5

0 0 . .

{ @8 3 <2

{ 5 `8> 4^ 0 . . 0

4161 D[LRPDV GH RUGHQ

94

Sdud 5, h{lvwhq { @ 5 5 U/ | @ 5 5 U. wdohv txh 5 

+5,5 @ 7 9 5

s

5 shur

Sru or dqwhulru/ srghprv revhuydu txh sdud uhvroyhu xqd lqhfxdflöq gho wlsr= s S +{,  T+{,

r gho wlsr

s S +{,  T+{,

qr srghprv hohydu do fxdgudgr vlq kdfhu odv uhvwulfflrqhv qhfhvduldv/ sdud dvhjxudu txh rewhqhprv lq0 hfxdflrqhv htxlydohqwhv1 Ghehprv revhuydu txh dojxqdv yhfhv vh gheh dqdol}du fdvrv glvwlqwrv1 Hmhuflflr 83 Uhvroydprv od lqhfxdflöq

{4

s {5  4

Vroxflöq= Ghehprv sulphudphqwh kdfhu odv uhvwulfflrqhv qhfhvduldv U4

+4179,

@ i{ 5 U = {5  4  3j @ i{ 5 U = {  4 b {  4j @ `  4> 4` ^ ^4> 4^

Sulphu fdvr= Vxsrqjdprv txh {  4 ? 3 | { 5 U4 / hv ghflu/

{ 5 F4 @`  4> 4` s Sxhvwr txh {  4 ? 3 | txh {5  4  3 sdud wrgr { 5 F4 / hv fodur txh od ghvljxdogdg 4179 vh fxpsoh sdud wrgr { 5 F4 > oxhjr V4 @`  4> 4` hv od vroxflöq hq hvwh fdvr1 Vhjxqgr fdvr= Vxsrqjdprv txh {  4  3 | { 5 U4 / hv ghflu { 5 F5 @ ^4> 4^

Gdgr txh dperv plhpeurv gh od ghvljxdogdg vrq qr qhjdwlyrv hq F5 / srghprv hohydu do fxdgudgr> dvð/ s {  4  {5  4 , +{  4,5  {5  4 / {5  5{ . 4  {5  4 / 5{ . 5  3 / {4

Oxhjr od vroxflöq hq hvwh fdvr hv \ qdophqwh od vroxflöq wrwdo hv

V5 @ ^4> 4^_F5 @ ^4> 4^

VW @ V4 ^ V5 @`  4> 4` ^ ^4> 4^

Revhuyhprv txh hq ho vljxlhqwh hmhpsor qr vh gd ho sulphu fdvr gho hmhpsor hqwhulru1

Hmhpsor 84 Uhvroydprv

s {5  {  5  {

FKDSWHU 41 ORV QÝPHURV UHDOHV

95 Vroxflöq= Kdflhqgr odv uhvwulfflrqhv qhfhvduldv whqhprv=

U5 @ i{ 5 U = {5  {  3j

| frpr

{5  {  3 / {+{  4,  3 / {4b{3

vh wlhqh txh

U5 @`  4> 3` ^ ^4> 4^@ U`3> 4^

Qrwhprv txh hq hvwh wlsr gh lqhfxdflöq qr vh sxhgh gdu ho fdvr txh sxhv |

{.5?3 s s {5  {  5  { / {5  {  { . 5 s {5  {  3

sdud wrgr { 5 U5 1 Dvðsdud srghu hohydu do fxdgudgr ghehprv h{ljlu txh { . 5  3/ hv ghflu/ {  5 a { 5 U5

Sru or wdqwr vl { 5 F4 @ ^5> 3` ^ ^4> 4^ whqhprv> s {5  {  { . 5 / {5  {  +{ . 5,5 / {5  {  {5 . 7{ . 7 / 8{  7 7 / { 8

oxhjr ho frqmxqwr vroxflöq gh hvwd lqhfxdflöq hv V@^ Khprv ylvwr txh od ghvljxdogdg

7 7 > 4^_F4 @ ^ > 3` ^ ^4> 4^ 8 8 s {  {

+417:,

qr vh fxpsoh sdud wrgr { 5 U. 3 1 Hqwrqfhv Bcsdud txì ydoruhv gh { 5 U. vh fxpsohB Hvwr htxlydoh d suhjxqwdu Bcfxäo hv ho frqmxqwr vroxflöq gh od lqhfxdflöq 417:B Sdud uhvroyhu 417:/ sulphudphqwh ghehprv uhvwulqjlu qxhvwud h{suhvlöq d U4 @ i{ 5 U = {  3j @ U. 3

Frpr dperv plhpeurv gh od lqhfxdflöq vrq qr qhjdwlyrv/ srghprv hohydu do fxdgudgr hq dperv odgrv= s {{ / {  {5 / {  {5  3 / {4b{3

4161 D[LRPDV GH RUGHQ

96

oxhjr/ ho frqmxqwr vroxflöq hv V @ +`  4> 3` ^ ^4> 4^, _ U4 @ i3j ^ ^4> 4^

Hv fodur txh

s { ? { ;{ 5`4> 4^

Uhfrughprv txh hvwd ýowlpd h{suhvlöq qr hv yäolgd sdud { 5 ^3> 4`1 Hq sduwlfxodu/ od ghvljxdogdg s {5 . | 5 ? {5 . | 5

vöor vh fxpsoh vl {5 . | 5 A 4

41618

Od hfxdflöq gh vhjxqgr judgr

Sdud uhvroyhu lqhfxdflrqhv hq txh dsduh}fdq idfwruhv gh od irupd d{5 . e{ . f frq d> e> f 5 U / d 9@ 3 qhfhvlwdprv frqrfhu dojxqdv surslhgdghv gh od hfxdflöq gh vhjxqgr judgr1 Xqd hfxdflöq gh vhjxqgr judgr hq od yduldeoh { hv xq h{suhvlöq gho wlsr d{5 . e{ . f @ 3 frq d> e> f 5 U> d 9@ 3

Xwlol}dqgr odv surslhgdghv gh U/ dqdolfhprv odv srvleohv vroxflrqhv gh od hfxdflöq 417; Vlq sìuglgd gh jhqhudolgdg/ vxsrqjdprv txh d A 3 /

/

/

/

/

d{5 . e{ . f @ 3 f e {5 . { . @ 3 d d  5  5 e e f e 5 { . {.  . @3 d 5d 5d d e5 f e +{ . ,5 @ 5  5d u 7d d u e e e5  7df e5  7df b { . {. @ @  5d 7d5 5d 7d5 u u 5 e e e  7df e5  7df  {@ . b { @  5d 7d5 5d 7d5

Hvwdv h{suhvlrqhv hvwäq elhq ghqlgdv hq U vl | vöor vl

e5  7df  3

Ghqlflöq 4153 Oodpduhprv glvfulplqdqwh gh od hfxdflöq 417; d od h{suhvlöq Vh wlhqh ho vljxlhqwh whruhpd

 @ e5  7df

Whruhpd 4154 Frqvlghudqgr od hfxdflöq gh vhjxqgr judgr 417; vh wlhqh txh=

l, Vl  A 3/ hqwrqfhv od hfxdflöq 417; wlhqh grv vroxflrqhv r udðfhv uhdohv +glihuhqwhv, {4 @

e.s > 5d

s

{5 @ e 5d

ll, Vl  @ 3/ hqwrqfhv od hfxdflöq 417; wlhqh xqd vroxflöq r udð} hq U +r grv udðfhv ljxdohv, {4 @ {5 @

e 5d

+417;,

FKDSWHU 41 ORV QÝPHURV UHDOHV

97

lll, Vl  ? 3/ hqwrqfhv od hfxdflöq 417; qr wlhqh vroxflrqhv hq U1

Revhuydflöq= Hvwr hv/ vl oodpdprv V do frqmxqwr vroxflöq gh od hfxdflöq 417;/ vh wlhqh l, Vl  A 3> V @ i{4 > {5 j1 ll, Vl  @ 3> V @ i{4 j/ lll, Vl  ? 3> V @ !1 Hmhpsor 85 Frqvlghuhprv od hfxdflöq>

6{5  5{  4 @ 3

Sxhvwr txh  @ +5,5 . 45 @ 49 A 3/ hqwrqfhv od hfxdflöq wlhqh grv vroxflrqhv uhdohv Hmhpsor 86 Od hfxdflöq

{4 @

5.s49 9

wlhqh xqd udð} uhdo +r grv udðfhv ljxdohv, sxhv  @ 3

Hmhpsor 87 Vlq hpedujr od hfxdflöq qr wlhqh udðfhv uhdohv sxhv  @ 6 ? 3

@ 4 {5 @

5s49 @ 4 9 6

{5  5{ . 4 @ 3 {4 @ {5 @ 4

{5 . { . 4 @ 3

Revhuydflöq= Hv qhfhvdulr hqidwl}du txh od hfxdflöq 417; wlhqh vroxflöq hq U vl | vöor vl  @ e5  7df  3

Hmhuflflr 88 Frpsohwh orv vljxlhqwhv hqxqfldgrv +fruuhfwdphqwh,= 41 Vl od uhfwd | @

51

61

6 { . e fruwd d od sduäerod |5 @ 45{ hq xq vror sxqwr hqwrqfhv e @ ========= 5 Od uhfwd gh hfxdflöq | @ e hv wdqjhqwh d od sduäerod | @ {5  9{ . ; vl e @ =========== Vl od uhfwd | @ 5{ . e fruwd d od fxuyd 5+{  4,5  |5 @ 4 hq grv sxqwrv glihuhqwhv hqwrqfhv e 5 =========

71 Od sursrvlflöq

+ . 4,{5  7{ . +  4, A 3 ;{ 5 U

hv yhugdghud vl | vöor vl  5 ===========

Vroxflöq=

41 Vl od uhfwd | @

6 { . e fruwd d od sduäerod |5 @ 45{ hq xq vror sxqwr hqwrqfhv ho vlvwhpd 5 +4, 65 { . e @ | c 5 +5,

|

@ 45{

wlhqh ýqlfd vroxflöq1 Uhhpsod}dqgr +4, hq +5, vh wlhqh od hfxdflöq 6 { . e,5 @ 45{ 5 txh htxlydoh d od hfxdflöq gh vhjxqgr judgr hq od yduldeoh {1 +

< 5 +417<, { . +6e  45,{ . e5 @ 3 7 Oxhjr ho sureohpd htxlydoh d hqfrqwudu e 5 U wdo txh od hfxdflöq gh vhjxqgr judgr whqjd xqd ýqlfd vroxflöq1 Sdud hvwr ho glvfulplqdqwh gh od hfxdflöq 417< gheh vhu fhur oxhjr gh grqgh e @ 5 Vl od uhfwd | @

 @ +6e  45,5  <e5 @ 3

6 { . e fruwd d od sduäerod |5 @ 45{ hq xq vror sxqwr hqwrqfhv e @ 51 5

4161 D[LRPDV GH RUGHQ

98

51 Hvwh sureohpd hv htxlydohqwh do hmhuflflr dqwhulru/ sxhv vl od uhfwd | @ e hv wdqjhqwh d od sduäerod | @ {5 9{.;/ hqwrqfhv od uhfwd | sduäerod vh fruwdq hq xq vror sxqwr1 Sru frqvljxlhqwh ho sureohpd hv htxlydohqwh d ghwhuplqdu e hq U wdo txh od hfxdflöq gh vhjxqgr judgr {5  9{ . ;  e @ 3

whqjd ýqlfd vroxflöq1 Dvð

 @ 69  7+;  e, @ 3

gh grqgh e @ 4 Qrwd=

Ho ohfwru txh frqrfh ho frqfhswr gh ghulydgd vh kdeuä gdgr fxhqwd txh hvwh sureohpd uhvxowd px| vlpsoh/ vl xwlol}dprv hvh frqfhswr1

61 Hq hvwh fdvr/ whqhprv txh ho vlvwhpd

5{ . e @ | 5+{  4,5  |5 @ 4

gheh whqhu grv vroxflrqhv uhdohv glihuhqwhv/ oxhjr od hfxdflöq gh vhjxqgr judgr 5+{  4,5  +5{ . e,5 @ 4

txh htxlydoh d

5{5 . +7 . 7e,{ . e5  4 @ 3

gheh fxpsolu txh vx glvfulplqdqwh vhd pd|ru txh fhur/ hv ghflu/ +, +, +, +, +,

sru or wdqwr e 5 U  ^6> 4`

71 Od lqhfxdflöq

 @ +7 . 7e,5  ;+e5  4, A 3 e5 . 7e . 6 A 3 +e . 4,+e . 6, A 3 +e . 4 A 3 a e . 6 A 3, b +e . 4 ? 3 b e . 6 ? 3, e A 4 b e ? 6 e 5`  4> 6^^`  4> 4^

+ . 4,{5  7{ .   4 A 3

vh yhulfd sdud wrgr { hq U vl | vöor vl ho glvfulplqdqwh gh od hfxdflöq dvrfldgd + . 4,{5  7{ .   4 @ 3

hv phqru txh fhur frq   4 A 3/ hvwr hv/

s Sru or wdqwr  5` 8> 4^1

 @ 49  7+ . 4,+  4, ? 3 a   4 A 3 +, 5  8 A 3 a  A 4 s s +, +  8,+ . 8, A 3 a  A 4 s +,  A 8

FKDSWHU 41 ORV QÝPHURV UHDOHV

99

Hmhuflflr 89 Ghwhuplqh sdud txì ydoruhv uhdohv gh n odv udðfhv gh od hfxdflöq +n  6,{5  5n{ . 9n @ 3

vrq uhdohv1 Vroxflöq=

41 Vxsrqjdprv n 9@ 6/ hq hvwh fdvr od hfxdflöq hv gh vhjxqgr judgr1 Odv udðfhv vhuäq uhdohv vl / / / oxhjr

/

 @ +5n,5  7+n  6,9n  3 7n5  57n+n  6,  3 7n+4;  8n,  3 +n  3 a 4;  8n  3, b +n  3 a 4;  8n  3, 4; 3n 8 n 5 ^3>

4; `  i6j 8

51 Vxsrqjdprv dkrud txh n @ 6/ hqwrqfhv whqhprv od hfxdflöq 9{ . 4; @ 3

gh grqgh rewhqhprv { @ 6 5 U

Dvð/ khprv ghwhuplqdgr txh odv udðfhv gh od hfxdflöq BB vhuäq uhdohv vl n 5 ^3>

4; ` 8

Hmhuflflr 8: Frqvlghudqgr ho hmhuflflr dqwhulru/ Áedmr txì frqglflrqhv odv udðfhv vhuäq srvlwlydvB

Vroxflöq= Gho whruhpd dqwhulru hv fodur txh vl {4 | {5 vrq odv udðfhv uhdohv gh od hfxdflöq 417; vh wlhqh txh {4 . {5 @ de a {4 {5 @ df

Oxhjr odv udðfhv vhuäq srvlwlydv vl hv ghflu/

Hq ho fdvr gh od hfxdflöq BB whqhprv

{4 . {5 A 3 a {4 {5 A 3>

e

d A3

a

f d A3

41 Vl n @ 6 hqwrqfhv { @ 6 A 3

51 Vl n 9@ 6/ hqwrqfhv odv udðfhv vhuäq srvlwlydv vl

9n 5n A 3a A3 n6 n6

4; frq n 5 ^3> 4; 8 `  i6j1 Uhvroylhqgr odv lqhfxdflrqhv hq BB rewhqhprv n 5`6> 8 `1

4161 D[LRPDV GH RUGHQ

9:

Hq frqvhfxhqfld odv udðfhv vhuäq uhdohv | srvlwlydv vl n 5`6>

4; ` 8

Revhuydflöq= Hv fodur txh odv udðfhv {4 | {5 vhuäq qhjdwlydv vl

{4 . {5 @ de ? 3 a {4 {5 @ df A 3

Sru rwud sduwh/ frqrflhqgr odv udðfhv {4 | {5 / srghprv idfwrul}du od h{suhvlöq d{5 . e{ . f gho vljxlhqwh prgr=     e f 5 5 d{ . e{ . f @ d {  {. d d @ d+{5  +{4 . {5 ,{ . {4 {5 , @ d+{  {4 ,+{  {5 ,

Hvwd idfwrul}dflöq qrv idflolwd od uhvroxflöq gh lqhfxdflrqhv1 Hmhpsor 8; Uhvroydprv od lqhfxdflöq

Vroxflöq= Hv fodur txh

{5  4 A3 5{5  ;{ . 9

+{  4,+{ . 4, {5  4 A3/ A3 5 5{  ;{ . 9 5+{  4,+{  6,

| ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq BB hv

U4 @ U  i4> 6j

Dkrud uhvroyhu od lqhfxdflöq BB htxlydoh d uhvroyhu

Vh wlhqh=

{.4 A 3 hq U4 = 5+{  6,

{.4 A 3 / +{ . 4 A 3 a {  6 A 3, b +{ . 4 ? 3 a {  6 ? 3, 5+{  6, / { A 6 b { ? 4 / { 5`  4> 4^^`6> 4^

Sru or wdqwr ho frqmxqwr vroxflöq gh od lqhfxdflöq BB hv V

@ +`  4> 4^^`6> 4^, _ U4 @ U  ^4> 6`

Hq pxfkdv rfdvlrqhv/ sdud uhvroyhu xqd lqhfxdflöq/ qr hv qhfhvdulr dqdol}du wrgrv orv idfwruhv lqyrox0 fudgrv/ vlqr xqd lqhfxdflöq htxlydohqwh päv vlpsoh1 Hmhpsor 8< Uhvroydprv od lqhfxdflöq

s +{5 . 4, {  4 3 {5  :{ . 43

FKDSWHU 41 ORV QÝPHURV UHDOHV

9;

Vroxflöq= Odv udðfhv gh od hfxdflöq {5  :{ . 43 @ 3 vrq {4 @ 5 | {5 @ 8/ oxhjr/ s s +{5 . 4, {  4 +{5 . 4, {  4  3 / 3 {5  :{ . 43 +{  5,+{  8, Ho frqmxqwr vreuh ho fxdo wlhqh vljqlfdgr uhvroyhu od lqhfxdflöq hq U hv

U4 @ i{ 5 U = {  4  3 a {  5 9@ 3 a {  8 9@ 3j>

hv ghflu/ ho frqmxqwr uhvwulfflöq hv

U4 @ ^4> 4^i5> 8j

Sxhvwr txh/ sdud wrgr { 5 U4 / vh wlhqh s {  4  3 a {5 . 4 A 3

hqwrqfhv qrv txhgd sru uhvroyhu= Dkrud elhq/

+{  5,+{  8, ? 3 vreuh U4

+{  5,+{  8, ? 3 / +{  5 A 3 a {  8 ? 3, b +{  5 ? 3 a {  8 A 3, / +{ A 5 a { ? 8, b +{ ? 5 a { A 8, / { 5`5> 8^

shur 4 hv wdpelìq xqd vroxflöq1 Hq frqvhfxhqfld ho frqmxqwr vroxflöq hv

V @ +`5> 8^^i4j, _ U4 @`5> 8^^i4j

Jhrpìwulfdphqwh +yhu jxudv, xqd udð} r vroxflöq gh xqd hfxdflöq gh vhjxqgr judgr 417; fruuhvsrqgh d od lqwhuvhfflöq gh od sduäerod | @ d{5 . e{ . f

| ho hmh gh odv devflvdv +r hmh { ,1 Dvð/ od sduäerod | @ d{5 . e{ . f fruwd do hmh gh odv devflvdv hq grv sxqwrv +{4 > 3, | +{5 > 3, vl | vöor vl od hfxdflöq 417; wlhqh grv udðfhv uhdohv {4 | {5 1

4161 D[LRPDV GH RUGHQ

9<

Od sduäerod wrfduä do hmh gh odv devflvdv hq vror xq sxqwr +{4 > 3, vl | vöor vl od hfxdflöq 417; wlhqh vöor xqd vroxflöq uhdo {4 1 Od sduäerod | @ d{5 . e{ . f qr fruwd do hmh gh odv devflvdv vl | vöor vl od hfxdflöq 417; qr wlhqh udðfhv uhdohv1 Hq hvwh ýowlpr fdvr revhuyhprv txh= Vl d A 3/ d{5 . e{ . f A 3 ;{ 5 U r elhq/ vl d ? 3/ d{5 . e{ . f ? 3/ ;{ 5 U4: 1 Gh or dqwhulru hv fodur txh {5 . 5{ . 6 A 3 ;{ 5 U sxhv  @ 55  7  6 @ ; ? 3 | f @ 6 A 31 Hvwr uhvxowd yhqwdmrvr sdud uhvroyhu lqhfxdflrqhv wdohv frpr s +{5 . 5{ . 6,+ {5 . 5{ . 4  4, A 3 |d txh hvwr htxlydoh d uhvroyhu

Sxhvwr txh

s {5 . 5{ . 4  4 A 3

+4183,

{5 . 5{ . 4 @ +{ . 4,5  3 ;{ 5 U

hqwrqfhv ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq 4183 hv U4 @ U 1 Sru rwud sduwh/ s {5 . 5{ . 4  4 A 3 s {5 . 5{ . 4 A 4 / / {5 . 5{ . 4 A 4 / {5 . 5{ A 3 / {+{ . 5, A 3 / +{ A 3 a { . 5 A 3, b +{ ? 3 a { . 5 ? 3, / +{ A 3 a { A 5, b +{ ? 3 a { ? 5, / { A 3 b { ? 5

Sru or wdqwr ho frqmxqwr vroxflöq gh od lqhfxdflöq hv

V @ +`  4> 5^^`3> 4^, _ U4 @ U  ^5> 3`

Hmhuflflr 93 Ghwhuplqh vl odv vljxlhqwh dupdflrqhv vrq yhugdghudv r idovdv1 41 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

51 Ho frqmxqwr vroxflöq gh 61 i{ 5 U =

Vroxflöq=

7{.7 s{{55.7 {8

s 4  { ? { hv `3> 4` {A

 3j @ !

s {  4 hv !

s 4{?{ 41 Idovr1 Uhvroydprv s Od h{suhvlöq 4  { hvwä ghqlgd hq U vl 4{  3 sru or wdqwr ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq s hv U4 @`  4> 4`1 Frpr 4  {  3 hq U4 / hqwrqfhv od lqhfxdflöq srguðd whqhu vroxflöq qr ydfðd

4: Wdpelìq

vh sxhgh dupdu or plvpr frq ho vljqr gh f1

FKDSWHU 41 ORV QÝPHURV UHDOHV

:3

vöor vl { A 3/ hq fx|r fdvr srghprv hohydu do fxdgudgr hq U5 @`3> 4`1 Dvð s 4  { ? { @+,5 +, 4  { ? {5 +, {5 . {  4 A 3

+4184,

Frpr ho glvfulplqdqwh gh od hfxdflöq dvrfldgd d {5 . {  4 @ 3 hv  @ 8 A 3 hqwrqfhv od h{suhvlöq {5 . {  4 vh sxhgh idfwrul}du frpr> # # # s $$ # s $$ 4 . 8 4  8 5 { .{4@ { { = 5 5 Oxhjr/ uhvroyhu 4184 htxlydoh d uhvroyhu hq U5 od lqhfxdflöq # # # s $$ # s $$ 4 . 8 4  8 { { A3 5 5 gh grqgh rewhqhprv

V

%$ s s % & 4 . 8 4  8 ^ @ 4> >4 _ `3> 4` 5 5 & & s 4 . 8 @ >4 5 #&

s {  4 hvwä ghqlgd hq U vl {  4  3/ hv ghflu/ vl {  4 oxhjr 51 Yhugdghur1 Od h{suhvlöq s s U @`  4> 4`1 Shur {  4  3 hq U/ sru or wdqwr od lqhfxdflöq { A {  4 srguðd whqhu vroxflöq qr ydfðd vöor vl {  3> shur { 9 3/ sru or wdqwr V @ !1 Qrwd= Xq huuru wðslfr hq od uhvroxflöq gh odv lqhfxdflrqhv dqwhulruhv hv hohydu do fxdgudgr vlq kdfhu od uhvwulfflöq1 61 Idovr1 H{lvwh { @ 5 5 U txh vdwlvidfh od lqhfxdflöq/ oxhjr V 9@ !1 Qrwd=

Rwud irupd gh mxvwlfdu hv uhvroylhqgr od lqhfxdflöq Uhvroydprv {5  7{ . 7 s 3 {5 . 7{  8 5 5 7{.7 s +{5, Od h{suhvlöq s{{5.7 {8 txh htxlydoh d +{4,+{.8, hvwä ghqlgd hq U vl { 5 U @`  4> 8^^`4> 4^1 s Frpr +{  5,5  3> ;{ 5 U | +{  4,+{ . 8, A 3 hq U hqwrqfhv

+4185,

+{  5,5 s  3 hq U +{  4,+{ . 8,

sru frqvljxlhqwh od lqhfxdflöq 4185 whqguä vroxflöq vöor sdud dtxhoorv { hq U txh fxpsodq hv ghflu/

+{  5,5 s @3 +{  4,+{ . 8, V @ i5j

+Sru vxsxhvwr/ vl V kxelhud whqlgr päv hohphqwrv/ fxdohvtxlhud gh hoorv kdeuðd edvwdgr sdud ghprvwudu od idovhgdg gho hqxqfldgr 6,1

4161 D[LRPDV GH RUGHQ

41619

:4

Ydoru devroxwr

Uhfrughprv txh fxdqgr hvwxgldprv od udð} sdu 4153 rewxylprv  s { / vl {  3 5 { @ { / vl { ? 3

Hvwr qrv vxjlhuh od vljxlhqwh ghqlflöq1

Ghqlflöq 4155 Vhd { 5 U 1 Vh ghqh ho ydoru devroxwr gh { frpr>  { / vl {  3 m { m@ { / vl { ? 3 Hq frqvhfxhqfld

s {5 @m { m

;{ 5 U

Dvð/ sru hmhpsor/ hq od lqhfxdflöq 4183 whqhprv s s {5 . 5{ . 4 A 4 / +{ . 4,5 A 4 / m { . 4 mA 4 Yhuhprv txh uhvroyhu

+4186,

m { . 4 mA 4

uhvxowd päv vlpsoh txh hq 41831 Shur dqwhv ghwhuplqduhprv dojxqdv surslhgdghv eävlfdv gho ydoru devroxwr1 Hv fodur gh od ghqlflöq gh ydoru devroxwr txh Sursrvlflöq 4156 Vhd d> e 5 U hqwrqfhv= l, m d m@ e / d @ e b d @ e> ll, m d m@m d m

lll, m de m@m d mm e m

m d m 3> ;d 5 U

e 5 U. 3

ly, m d m5 @m d5 m@ d5 y,  m d m d m d m

Xvdqgr odv surslhgdghv phqflrqdgdv surehprv txh

m d m e / e  d  e d 5 U e 5 U

+,, Vxsrqjdprv txh m d m e | ghprvwuduhprv txh e  d  e1 Frpr d m d m | m d m e hqwrqfhv sru wudqvlwlylgdg Dghpäv/ shur oxhjr/ sru wudqvlwlylgdg qxhydphqwh/ oxhjr sru 4187 | 4188 whqhprv txh

d  e

+4187,

m d m e / e   m d m  m d m d

e  d

e  d  e

+4188,

FKDSWHU 41 ORV QÝPHURV UHDOHV

:5 ++, Dkrud vxsrqjdprv txh

e  d  e>

hv ghflu/

e  d a d  e

Sureduhprv txh Od ghprvwudflöq vh kduä sru fdvrv

4;

m d m e

1

Fdvr 4= Vl d  3/ hqwrqfhv m d m@ d/ dghpäv sru klsöwhvlv d  e oxhjr m d m e

Fdvr 5= Vl d ? 3/ hqwrqfhv m d m@ d/ hv ghflu/

m d m@ d

| sru klsöwhvlv e  d oxhjr

e   m d m

gh grqgh

Hq frqvhfxhqfld Xvdqgr od htxlydohqfld vh rewlhqh txh sxhgh vhu h{whqglgd d

m d m e

m d m e / e  d  e

+s , t, / + t , s,

d A e b d ? e ,m d mA e>

m d m e / d  e b d  e

Dvð/ khprv suredgr orv vljxlhqwhv lpsruwdqwhv uhvxowdgrv= Whruhpd 4157 Vhdq d> e 5 U

m d m e / e  d  e m d m e / d  e b d  e

+Uhfxhugh txh/ vhjýq vh dgyluwlö dqwhv/ hq ho whruhpd dqwhulru/ ho vljqr  sxhgh vhu fdpeldgr sru ?1, Revhuydflöq= Qrwhprv txh vl e ? 3 hqwrqfhv od sursrvlflöq/ sdud wrgr d hq orv U mdm ? e hv idovd/ | od sursrvlflöq sdud wrgr d 5 U mdm A e hv yhugdghud1 Dvð sru hmhpsor od vroxflöq gh od lqhfxdflöq m{m ? 4 hv ydfðd | od vroxflöq gh m{m A 4 hv U1 Yroylhqgr d od lqhfxdflöq 4186 uhvroydprv Dsolfdqgr od surslhgdg dqwhulru whqhprv txh

m { . 4 mA 4

m { . 4 mA 4 / / / /

{ . 4 A 4 b { . 4 ? 4 { A 3 b { ? 5 { 5`  4> 5^^`3> 4^ { 5 U  ^5> 3`

4 ; Odv surslhgdghv l, d y, dqwhulruphqwh hqxqfldgdv sxhghq vhu suredgdv frq od plvpd wìfqlfd txh orv fdvrv dtxðghvduuro0 odgrv1

4161 D[LRPDV GH RUGHQ

:6

oxhjr od vroxflöq hv Vlq hpedujr/ od vroxflöq gh od lqhfxdflöq

V @ U  ^5> 3` m { . 4 m? 4

hv

V @`  5> 3^

Hq hihfwr/

m { . 4 m? 4 / 4 ? { . 4 ? 4 / 5 ? { ? 3

Hmhuflflr 94 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 41 i{ 5 U =

4 m{4m j @ i{ 5 U

51 i{ 5 U = { 

= 4 A m{  4mj

s s m{mj @ i{ 5 U = {  {j

61 i{ 5 U. = mm{m  4m  4j @ ^5> 5` s 71 Od vroxflöq gh +{  4,6  m{  4m hv ^4> 5`1

Vroxflöq=

41 Idovr1 H{lvwh { @ 4 5 U wdo txh 4 A m{  4m shur

4 95 i{ 5 U =

4 A 4j m{  4m

qr hvwä ghqlgd hq U1 s s 51 Idovr1 H{lvwh { @ 4 5 U wdo txh {  m{m vlq hpedujr> { qr hvwä ghqlgd sdud { @ 41 sxhv od h{suhvlöq

4 m{4m

61 Idovr1

mm{m  4m  4 +, +, +, +, +,

4  m{m  4  4 @ . 4 3  m{m  5 3  m{m a m{m  5 Y a 5  {  5 { 5 ^5> 5`

Oxhjr i{ 5 U. = mm{m  4m  4j @ U. _ ^5> 5` @`3> 5`1

71 Yhugdghur1 Uhvwulfflöq U @ ^4> 4^> dghpäv/ s +{  4,6  m{  4m +, +, +, +, +, +, shur {  4 oxhjr V @ ^4> 5`1

s m{  4m {  4  m{  4m s m{  4m+ {  4  4,  3 s {@4b {443 s {@4b {44 {@4b{44 {@4b{5

FKDSWHU 41 ORV QÝPHURV UHDOHV

:7 Hmhuflflr 95 Uhvroyhu Vroxflöq= Od h{suhvlöq

4.m5{6m m{.8m

4 . m5{  6m ?4 m{ . 8m

+4189,

hvwä ghqlgd hq U/ vl m{ . 8m @ 9 3/ hv ghflu U4 @ U  i8j

4. m 5{  6 m ? 4 / 4. m 5{ . 6 m?m { . 8 m m{.8m

Dqdolfhprv göqgh orv idfwruhv 5{  6 | { . 8 vrq srvlwlyrv | göqgh qhjdwlyrv/ sdud oxhjr xvdu od ghqlflöq gh ydoru devroxwr1 Sdud ho wìuplqr 5{  6 vh wlhqh txh> |

5{  6  3 / { 

6 /m 5{  6 m@ 5{  6 5

6 /m 5{  6 m@ +5{  6, 5 Dqäorjdphqwh/ sdud ho idfwru { . 8 vh wlhqh txh 5{  6  3 / { 

{ . 8  3 /m { . 8 m@ { . 8 / {  8

|

{ . 8  3 / {  8 /m { . 8 m@ +{ . 8,

Uhvxpluhprv wrgd od lqirupdflöq dqwhulru hq od wdeod vljxlhqwh4<     6 `4> 8^ 8> 65 5> 4 5{  6 0 0 . {.8 0 . .

Oxhjr vhjýq odv glvwlqwdv srvlelolgdghv gh vljqrv sdud orv wìuplqrv 5{  6 | { . 8 | frqvlghudqgr od uhvwulfflöq U4 ghehprv dqdol}du wuhv fdvrv= 41 { 5`  4> 8` _ U4 51 { 5`  8> 65 ` _ U4 61 { 5` 65 > 4` _ U4

Fdvr 4= Vl { 5`  4> 8` _ U4 @`  4> 8` vh wlhqh txh

shur { 5`  4> 8^/ oxhjr

4. m 5{  6 m?m { . 8 m / 4 . ++5{  6,, ? +{ . 8, / {A< / { 5`<> 4^ V4 @`  4> 8^_`<> 4^@ >

Hvwr txlhuh ghflu txh qr h{lvwhq hohphqwrv hq `  4> 8^ txh fxpsodq frq od lqhfxdflöq 4189 4 < Dtxð

qr wrpdprv orv ydoruhv h{wuhprv

hv od ýqlfd irupd1

3D |

> xvwhg gheh frpsohwdu hvwd idowd/ or khprv khfkr päv dghodqwh/ shur qr 2

4161 D[LRPDV GH RUGHQ

:8

Fdvr 5= Vl { 5`  8> 65 ` _ U4 @`  8> 65 ` vh wlhqh txh

shur { 5`  8> 65 ` oxhjr

4. m 5{  6 m?m { . 8 m / 4 . ++5{  6,, ? { . 8 4 / {A 6 4 / { 5` > 4^ 6

6 4 4 6 V5 @`  8> `_` > 4^@` > `> 5 6 6 5

Hv ghflu/ txh fxdotxlhu { 5` 64 > 65 ` fxpsoh frq od lqhfxdflöq 41891 Fdvr 6= Vl { 5` 65 > 4^_U4 @` 65 > 4^ vh wlhqh txh

4. m 5{  6 m?m { . 8 m / 4 . +5{  6, ? { . 8 / {?: / { 5`  4> :^

shur { 5` 65 > 4^ oxhjr

6 6 V6 @` > 4^_`  4> :^@` > :^ 5 5

Hv ghflu/ txh fxdotxlhu { 5` 65 > :^ fxpsoh frq od lqhfxdflöq 41891 Hq frqvhfxhqfld/ ho frqmxqwr vroxflöq gh od lqhfxdflöq 4189 hv VW @ V4 ^ V5 ^ V6 @`

Hmhuflflr 96 Uhvroyhu

m {  6 m

4 > :^= 6

s {5  4

Vroxflöq= Hq sulphu oxjdu frqvlghuhprv od uhvwulfflöq U5

@ @ @ @ @

i{ 5 U = {5  4  3j i{ 5 U =m { m 4j i{ 5 U = {  4 b {  4j `  4> 4` ^ ^4> 4^ U`  4> 4^

Dkrud hohydqgr do fxdgudgr 418:/ hq U5 vh wlhqh

Ilqdophqwh/

m {  6 m5  {5  4 / {5  9{ . <  {5  4 / 9{  43 8 / { 6

8 8 VW @ +U`  4> 4^, _ ^ > 4^@ ^ > 4^= 6 6

+418:,

FKDSWHU 41 ORV QÝPHURV UHDOHV

:9 Hmhuflflr 97 Uhvroyhu

{5 . { . ; s 3 4  m{m  6 s Vroxflöq= U @ i{ 5 U = m{m  6  3 a 4  m{m  6 9@ 3j1 Dkrud elhq/

+418;,

m{m  6  3 +, m{m  6 +, {  6 b {  6 +, { 5`  4> 6` ^ ^6> 4^>

Dghpäv/

4

s m{m  6 @ 3 +, +, +, +,

Oxhjr ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq 418; hv

s m{m  6 @ 4 m{m  6 @ 4 m{m @ 7 { @ 7 b { @ 7

U @ +`  4> 6` ^ ^6> 4^,  i7> 7j

Sru rwud sduwh/ ho glvfulplqdqwh gh od hfxdflöq gh vhjxqgr judgr hv  @ 4  65 ? 3 | frpr f @ ; A 3 hqwrqfhv

{5 . { . ; @ 3

{5 . { . ; A 3 ;{ 5 U=

Hq frqvhfxhqfld/ uhvroyhu od lqhfxdflöq

{5s .{.; 4 m{m6

4

Uhvroydprv 4

 3 htxlydoh d uhvroyhu od lqhfxdflöq

s m{m  6 A 3

s m{m  6 A 3 +, +, +, +, +,

s m{m  6 ? 4 m{m  6 ? 4 m{m ? 7 7 ? { ? 7 { 5`  7> 7^

frpr { 5 U/ oxhjr ho frqmxqwr vroxflöq gh od lqhfxdflöq 418; hv= V @`  7> 6` ^ ^6> 7^

Hmhuflflr 98 Uhvroyhu Vroxflöq= Ho frqmxqwr uhvwulfflöq hv

4

s

49  {5 ?6 {4

U @ i{ 5 U = 49  {5  3 a {  4 9@ 3j @ ^7> 7`  i4j

+418<,

4161 D[LRPDV GH RUGHQ

::

Dqdolfhprv orv fdvrv= { A 4 | { ? 4 frq { 5 U1 l, Vxsrqjdprv txh { A 4 frq { 5 U Hqwrqfhv/ gdgr txh {  4 A 3 vh wlhqh txh 418< htxlydoh d s 4  49  {5 ? 6+{  4, s 49  {5 7  6{ ?

+4193,

l4, Frqvlghuhprv orv uhdohv wdohv txh 7  6{  3 | 4 ? {  7 hv ghflu/ { 5`4> 7@6` @ F4 1 Hq hvwh frqmxqwr srghprv hohydu do fxdgudgr/ rewhqlhqgr

shur { 5 F4 oxhjr { ? 45@81 Sru or wdqwr/

+7  6{,5 ? 49  {5 +, 49  57{ . <{5 ? 49  {5 +, 43{5  57{ ? 3 +, 5{+8{  45, ? 3

{ 5`4> 7@6`_`  4> 45@8^@`4> 7@6`

l5, Dkrud frqvlghuhprv orv uhdohv wdohv txh

7  6{ ? 3 | { 5`4> 7`

hv ghflu/ { 5`7@6> 7` @ F5 Hq hvwh frqmxqwr/ hv fodur txh od ghvljxdogdg 4193 vh fxpsoh sdud wrgr { 5 F5 1 Oxhjr Vl @ Vl4 ^ Vl5 @`4> 7@6`^`7@6> 7` @`4> 7`

ll, Vxsrqjdprv { ? 4 frq { 5 U/ hv ghflu/ { 5 ^7> 4^ hqwrqfhv/ gdgr txh { ? 4 vh wlhqh txh 418< hv htxlydohqwh d s 49  {5 A 6+{  4, s +, 7  6{ A 49  {5 4

Gdgr txh { 5 ^7> 4^ hqwrqfhv 7  6{ A 31 | srghprv hohydu ho fxdgudgr/ rewhqlhqgrv1 +7  6{,5 A 49  {5 +, 5{+8{  45, A 3

gh grqgh { 5 +`  4> 3^^`45@8> 4^, _ ^7> 4^ Oxhjr Dvð od vroxflöq hv Hmhuflflr 99 Uhvroyhu

Vll @ ^7> 3^

VW @ Vl ^ Vll @ ^7> 3^^`4> 7` s s 7{ . 4  { . 9 s ?4 {4

+4194,

FKDSWHU 41 ORV QÝPHURV UHDOHV

:; Vroxflöq= Ghwhuplprv ho frqmxqwr uhvwulfflöq

Frpr

U @ i{ 5 U = 7{ . 4  3 a { . 9  3 a {  4 A 3j @ ^4@7> 4^_^9> 4^_`4> 4^ @ `4> 4^

s {  4 A 3 hqwrqfhv 4194 hv htxlydohqwh d s s s 7{  4  { . 9 ? {  4 s s s 7{  4 ? {  4 . { . 9 +,

+4195,

Frpr dperv plhpeurv gh od lqhfxdflöq 4195 vrq qr qhjdwlyrv/ srghprv hohydu do fxdgudgr/ rewhqlhqgr s 7{ . 4 ? { . 9 . 5 +{ . 9,+{  4, . {  4 s +, 5{  7 ? 5 +{ . 9,+{  4, s +4196, +, {  5 ? +{ . 9,+{  4, Dkrud dqdolfhprv orv fdvrv {  5  3 | {  5 ? 3 frqvlghudqgr { 5 U @`4> 4^1

Fdvr4 Vl {  5  3 | { 5`4> 4^ hqwrqfhv { 5 F4 @ ^5> 4^1

Hv fodur txh vl { 5 F4 hqwrqfhv srghprv hohydu do fxdgudgr od lqhfxdflöq 4195 rewhqlhqgr +{  5,5 ? +{ . 9,+{  4, +, {5  7{ . 7 ? {5 . 8{  9 +, <{  43 A 3 +, { A 43@<

oxhjr V4 @`43@<> 4^_^5> 4^@`4> 4^

Fdvr 5 Vl {  5 ? 3 | { 5`4> 4^ hqwrqfhv { 5`4> 5^@ F5 1 s Oxhjr/ gdgr txh {  5 ? 3 | +{ . 9,+{  4, A 3/ sdud { 5 F5 hqwrqfhv od ghvljxdogdg 4195 vh fxpsoh sdud wrgr { 5 F5 V5 @`4> 5^ VW @ V4 ^ V5 @ ^5> 4^^`4> 5^@ ^5> 4^

Xqd surslhgdg px| xvdgd sdud dfrwdu53 h{suhvlrqhv uhdohv hv od ghvljxdogdg wuldqjxodu

Whruhpd 4158 +Ghvljxdogdg Wuldqjxodu, Vhdq d> e 5 U / hqwrqfhv m d . e mm d m . m e m

Ghprvwudflöq= Frpr m d . e m 3/ m d m 3/ m e m 3> vh wlhqh

m d . e mm d m . m e m / +m d . e m,5  +m d m . m e m,5

Oxhjr edvwd ghprvwudu txh

+m d . e m,5

 +m d m . m e m,5

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4161 D[LRPDV GH RUGHQ shur

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+m d . e m,5

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+d . e,5 d5 . 5de . e5 m d m5 .5de. m e m5 m d m5 .5 m de m . m e m5 m d m5 .5 m d mm e m . m e m5 +m d m . m e m,5

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Revhuyhprv txh/ vhjýq ho whruhpd dqwhulru

4 mdm.mem



4 md.em

;d> e 5 U> d . e 9@ 3

Gh od ghvljxdogdg wuldqjxodu vh ghgxfh dghpäv od vljxlhqwh surslhgdg

Frurodulr 4159 Vhdq d> e 5 U / hqwrqfhv Hq hihfwr/ oxhjr

m d m  m e mm d  e m

m d m@m d  e . e mm d  e m . m e m m d m  m e mm d  e m

Hq edvh d hvwd surslhgdg srghprv ghprvwudu lpsolfdflrqhv wdohv frpr od vljxlhqwh= Vroxflöq=

m{. m{.

4 4 48 4 m? , ? 6 43 m 5{  4 m 55

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m

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Hmhpsor 9: 54 Xvdqgr odv surslhgdghv phqflrqdgdv surehprv txh odv vljxlhqwhv ghvljxdogdghv vrq yhu0 gdghudv sdud orv qýphurv uhdohv { h | lqglfdgrv1 s 41 m { m {5 . | 5 / ;{> | 5 U 51 m{|m  {5 . | 5 ;{> | 5 U s 5 s { m|m 61 m{m.m|m  m | m/ ;{> | 5 U  i3j

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fäofxor gh oðplwhv1

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s s s s s m{m m|m m{m m|m s {5 m | m {5 m | m @ @  @ m|m m{m.m|m m{m.m|m m{m.m|m m{m

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s m { m +{ . |,5 m { mm { . | m  @m { m m{m.m|m m{.| m

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s {7 . 5{6 | . {5 | 5 @3 m{m.m|m

Dvð/ gh d, | e, vh wlhqh od ghvljxdogdg shglgd

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Dojxqrv gh orv huuruhv päv iuhfxhqwhv txh vh frphwhq do dfrwdu h{suhvlrqhv uhdohv sru rwudv frqvlvwhq hq xwlol}du ghvljxdogdghv txh qr vh fxpsohq hq orv frqmxqwrv gdgrv1 Sru hmhpsor/ od sursrvlflöq s m{|m m{|m m{m.m|m  m{m.m|m ;{> | 5 U  i3j qr hv yhugdghud/ hv ghflu/

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m{|m m{m.m|m

s

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D{lrpd gho Vxsuhpr

Ilqdophqwh suhvhqwduhprv ho ýowlpr d{lrpd gh orv qýphurv uhdohv/ d{lrpd txh qrv shuplwh ghflu txh orv qýphurv uhdohv frpsohwdq wrgrv orv hvsdflrv hq xqd uhfwd vl vrq uhsuhvhqwdgrv hq wdo irupd1 Hq sulphu oxjdu yhdprv txì hqwhqghprv sru xq frqmxqwr dfrwdgr1 Ghqlflöq 415: Vhd D  U / D 9@ >/ u 5 U 1 Gluhprv txh l, u hv xqd Frwd Vxshulru gh D vl | vöor vl=

du

> ;d 5 D

du

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hv ghflu/ u hv frwd vxshulru gh D vl u hv pd|ru r ljxdo txh fxdotxlhu hohphqwr gh D1

ll, u hv xqd Frwd Lqihulru gh D vl vöor vl=

hv ghflu/ u hv phqru r ljxdo txh fxdotxlhu hohphqwr gh D1

4171 D[LRPD GHO VXSUHPR

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Hmhpsor 9; D @`  4> 8` hv dfrwdgr vxshulruphqwh/ sxhv> H{lvwh u @ 8 5 U wdo txh d  8 sdud wrgr d 5 D/ shur qr hv dfrwdgr lqihulruphqwh/ sxhv qr h{lvwh u 5 U wdo txh/ sdud wrgr d 5 D> u  d Revhuyhprv txh fxdotxlhu u 5 ^8> 4^ hv xqd frwd vxshulru sdud D1 Oxhjr ho frqmxqwr gh frwdv vxshulruhv gh D hv ^8> 4^1 Hq fdpelr E @`  5> 4^ hv dfrwdgr lqihulruphqwh shur qr dfrwdgr vxshulruphqwh1 Ho frqmxqwr gh frwdv lqihulruhv gh E hv ho frqmxqwr `  4> 5`1 Gh or dqwhulru/ hv fodur txh ho frqmxqwr F @ ^5> 8` hv dfrwdgr Dvlplvpr/ ho frqmxqwr G @ i{ 5 U = {5 ? 45j hv dfrwdgr sxhvwr txh

hv ghflu/

{5 ? 45 /m { m?

s s s 45 /  45 ? { ? 45>

G @` 

s s 45> 45^

s s grqgh `  4>  45` hv ho frqmxqwr gh frwdv lqihulruhv | ho frqmxqwr ^ 45> 4^ hv ho frqmxqwr gh frwdv vxshulruhv1 Vlq hpedujr ho frqmxqwr> H @ i{ 5 ] = {5 ? 45j

qr wlhqh orv plvprv frqmxqwrv gh frwdv vxshulruhv h lqihulruhv txh G/ hq hihfwr/ s s H @ ]_`  45> 45^@ i6> 5> 4> 3> 4> 5> 6j

oxhjr H wdpelìq hv dfrwdgr shur ho frqmxqwr gh frwdv vxshulruhv gh H hv `  4> 6` | ho gh frwdv vxshulruhv gh H hv ^6> 4^1 Hmhuflflr 9< Vhdq

q D@ {5U=

4.m5{6m m{.8m

r s  ? 4 > E @ { 5 U =m {  6 m {5  4

Ghwhuplqh vl orv frqmxqwrv D/ E / D _ E/ D ^ E vrq dfrwdgrv | hqfxhqwuh/ vl h{lvwhq/ ho frqmxqwr gh frwdv vxshulruhv h lqihulruhv hq fdgd fdvr1 Vroxflöq= Hq ho hmhuflflr 95 rewxylprv txh

4 > :^ 6 Hvwh frqmxqwr hv dfrwdgr lqihulru | vxshulruphqwh> dvð/ D hv xq frqmxqwr dfrwdgr1 Ho frqmxqwr gh frwdv lqihulruhv gh D hv `  4> 64 ` | ho frqmxqwr gh frwdv vxshulruhv hv ^:> 4^1 Dghpäv hq ho hmhuflflr 96 ghwhuplqdprv txh D @`

8 E @ ^ > 4^> 6

ho fxdo hv xq frqmxqwr dfrwdgr lqihulruphqwh/ shur qr hv dfrwdgr vxshulruphqwh/ oxhjr E qr hv dfrwdgr1 Ho frqmxqwr gh frwdv lqihulruhv gh E hv `  4> 86 ` Frpr D @` 64 > :^ | E @ ^ 86 > 4^/ hqwrqfhv 8 D _ E @` > :` 6

FKDSWHU 41 ORV QÝPHURV UHDOHV

;7 |

D ^ E @` 

4 > 4^ 6

Oxhjr D _ E hv dfrwdgr1 Ho frqmxqwr gh frwdv vxshulruhv gh D _ E hv ^:> 4^ | ho gh frwdv lqihulruhv hv `  4> 86 `1 Vlq hpedujr E qr hv dfrwdgr sxhv qr hv dfrwdgr vxshulruphqwh1 Ho frqmxqwr gh frwdv lqihulruhv gh D ^ E hv `  4>  46 ^1

Ghqlflöq 415; Vhd D  U / D 9@ >

l, Vl D hv dfrwdgr vxshulruphqwh/ gluhprv txh O 5 U hv ho Vxsuhpr gh D vl | vöor vl O hv xqd frwd vxshulru gh D/ |/ sdud fxdotxlhu frwd vxshulru { gh D vh wlhqh txh O  {=

Hv ghflu/ O hv od phqru gh odv frwdv vxshulruhv gh D1 Vh dqrwduä vxs D @ O ll, Vl D hv dfrwdgr lqihulruphqwh/ gluhprv txh O 5 U hv ho Lqpr gh D vl | vöor vl O hv xqd frwd lqihulru gh D/ |/ sdud fxdotxlhu frwd lqihulru { gh D vh wlhqh txh O  {=

Hv ghflu/ O hv od pd|ru gh odv frwdv lqihulruhv gh D1 Vh dqrwduä

lqi D @ O

Hmhpsor :3 Frpr D @` 64 > :^ hv xq frqmxqwr dfrwdgr/ hqwrqfhv h{lvwhq vxs D @ :> lqi D @

4 6

Vlq hpedujr> E @`  4> 86 ` qr hv dfrwdgr lqihulruphqwh/ oxhjr qr h{lvwh lqi E/ shur h{lvwh vxs E sxhv E hv dfrwdgr vxshulruphqwh/ vxs E @ 86 1

D{lrpd gho Vxsuhpr=

Wrgr vxefrqmxqwr D gh U qr ydfðr | dfrwdgr vxshulruphqwh wlhqh vxsuhpr ho fxdo hv xq hohphqwr gh U1 Xvdqgr ho d{lrpd gho vxsuhpr/ vh sxhgh ghprvwudu txh= Wrgr vxefrqmxqwr D gh U qr ydfðr | dfrwdgr lqihulruphqwh wlhqh ðqpr/ ho fxdo hv xq hohphqwr gh U1 Dxqtxh hq pxfkrv fdvrv qr vh yh fodudphqwh fxäo hv ho vxsuhpr gh xq frqmxqwr/ ho d{lrpd qrv dvhjxud vx h{lvwhqfld1 Hmhpsor :4 Frqvlghuhprv D ho frqmxqwr gh wrgdv odv dsur{lpdflrqhv ghflpdohv vxfhvlydv h lqihulruhv d s 5/ hv ghflu/ D @ i4> 7> 4> 74> 4> 747> 4> 7475> = = =j

frpr D  U / D 9@ ! | D hv dfrwdgr vxshulruphqwh +sru hmhpsor 4> 8 5 U hv frwd vxshulru,/ hqwrqfhv sru d{lrpd gho vxsuhpr/ h{lvwh vxs D ho fxdo hv xq hohphqwr gh U1

4171 D[LRPD GHO VXSUHPR

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Qrwd= Hq ho hmhpsor dqwhulru

vxs D @

Hmhpsor :5 Vhd

 D@ {5U={@

Hv fodur txh D  U | D 9@ >/ dghpäv

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q

q

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hv ghflu/ { ? 4 sdud wrgr { 5 D1 Oxhjr D hv dfrwdgr vxshulruphqwh | sru d{lrpd gho vxsuhpr/ h{lvwh vxs D1 Frqvlghudqgr glvwlqwrv ydoruhv gh q 5 Q srghprv frqmhwxudu txh= vxs D @ 4

+4198,

Ho khfkr txh U vdwlvidjd orv d{lrpdv gh fxhusr/ gh rughq | gho vxsuhpr vh h{suhvd glflhqgr txh hv xq fxhusr rughqdgr | frpsohwr1 Hmhuflflr :6 Vhd D  U / D 9@ ! | dfrwdgr vxshulruphqwh1 Suxheh txh vxs D hv ýqlfr

Vroxflöq= Sru d{lrpd gho vxsuhpr hv fodur txh h{lvwh vxs D1 Vhd O @ vxs D1 Vxsrqjdprv txh vxs D qr hv ýqlfr/ hv ghflu/ vxsrqjdprv txh h{lvwh O3 5 U wdo txh O 9@ O3 | O3 @ vxs D

hqwrqfhv O | O3 vrq frwdv vxshulruhv gh D1 Sru ghqlflöq gh vxsuhpr whqhprv txh= Vl O @ vxs D/ hqwrqfhv O  O3 / shur vl O3 @ vxs D/ hqwrqfhv O3  O/ hqwrqfhv O @ O3 txh hv xqd frqwudglfflöq1 Gh irupd dqäorjd vh suxhed txh ho ðqpr gh D hv ýqlfr vl D hv dfrwdgr lqihulruphqwh1 Ho vljxlhqwh whruhpd qrv shuplwh/ hqwuh rwudv frvdv/ ghprvwudu od frqmhwxud khfkd hq 4198 gho hmhpsor :5 Whruhpd 415< Vhd D  U> D 9@ !/

l, Vhd O frwd vxshulru gh D1 Hqwrqfhv O @ vxs D / ;% A 3> <{ 5 D = { A O  %

ll, Vhd O frwd lqihulru gh D1 Hqwrqfhv

O @ lqi D / ;% A 3> <{ 5 D = { ? O . %

Ghprvwudflöq= l, +,, Sru ghprvwudu txh

O @ vxs D , ;% A 3> <{ 5 D = { A O  %

Sru ho pìwrgr gho devxugr= Vxsrqjdprv txh

+O @ vxs D, a +<% A 3,+;{ 5 D,+{  O  %,

+4199,

{  O  %> ;{ 5 D

+419:,

| rewhqjdprv xqd frqwudglfflöq1 Sru 4199/ h{lvwh % A 3 wdo txh

Hqwrqfhv sru ghqlflöq gh frwd O  % hv frwd vxshulru gh D shur O @ vxs D +od pd|ru gh odv frwdv vxshulruhv,/ oxhjr O  O  % gh grqgh %  3/ txh frqwudglfh % A 31

FKDSWHU 41 ORV QÝPHURV UHDOHV

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l, ++, Qxhydphqwh xvdprv ho pìwrgr gho devxugr1 Vhd O frwd vxshulru gh D1 Sru ghprvwudu Vxsrqjdprv txh

+; % A 3,+<{ 5 D,+{ A O  %, , O @ vxs D +; % A 3,+<{ 5 D,+{ A O  %, a O 9@ vxs D

Vl O 9@ vxs D/ vlhqgr O frwd vxshulru gh D/ hqwrqfhv O qr hv od phqru gh odv frwdv vxshulruhv/ hv ghflu/ h{lvwh O3 frwd vxshulru gh D wdo txh O3 ? O

oxhjr h{lvwh % A 3 wdo txh

Shur/ sru klsöwhvlv/ h{lvwh { 5 D wdo txh

Sru 419; | 419< vh wlhqh txh

O3 . % @ O

+419;,

{AO %

+419<,

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frq { 5 D

or txh hv xqd frqwudglfflöq/ sxhv O3 hv frwd vxshulru gh D1 Dqäorjdphqwh vh ghpxhvwud txh vl O hv frwd lqihulru gh D



O @ lqi D / ; % A 3> <{ 5 D = { ? O . %

Hq edvh do whruhpd dqwhulru/ ghprvwuduhprv txh ho frqmxqwr gh orv qýphurv qdwxudohv qr hv dfrwdgr1 Hvwr qrv shuplwluä d vx yh} suredu ho lpsruwdqwh whruhpd 4164 Sursrvlflöq 4163 Q qr hv dfrwdgr1

Ghprvwudflöq= Vxsrqjdprv or frqwudulr/ hv ghflu/ txh Q hv dfrwdgr vxshulruphqwh1 Hqwrqfhv/ sru d{lrpd gho vxsuhpr h{lvwh vxs Q @ O frq O 5 U 1 Oxhjr sru ho whruhpd 415< whqhprv= ; % A 3> q A O  %

Hq sduwlfxodu/ frqvlghuhprv % @ 4 A 3> hqwrqfhv h{lvwh q 5 Q wdo txh hv ghflu>

q A O  4> q.4AO

shur q . 4 5 Q / or txh hv xqd frqwudglfflöq |d txh O @ vxs Q 1



Whruhpd 4164 +Surslhgdg Dutxlphgldqd1, Wrgr qýphur uhdo sxhgh vhu vreuhsdvdgr sru xq qýphur qdwxudo1 Hv ghflu ;{ 5 U> ;q 5 Q = {  q

Hv fodur txh { hv frwd vxshulru gh Q +sru ghqlflöq gh frwd vxshulru,> oxhjr Q hv dfrwdgr vxshulruphqwh/ or txh hv xqd frqwudglfflöq 

4171 D[LRPD GHO VXSUHPR

;:

Revhuydflöq= Xqd yhuvlöq htxlydohqwh gh od surslhgdg Dutxlphgldqd hv od vljxlhqwh sursrvlflöq Hq hihfwr Vhd % A 3 | frqvlghuhprv ho uhdo

; % A 3>
4 / hqwrqfhv sru %

qA

gh grqgh hv fodur txh

4 ? % q

+41:4,

41:3/ h{lvwh q 5 Q wdo txh

4 %

4 ? % q

+Ho dujxphqwr sxhgh uhyhuwluvh, Ho vljxlhqwh hmhpsor pxhvwud xqd dsolfdflöq gh od surslhgdg Dutxlphgldqd1 Hmhpsor :7 Surehprv txh vl

D@ hqwrqfhv

 {q 5 U = {q @

q >q 5 Q q.4



vxs D @ 4

Vroxflöq= Sulphur sureduhprv txh O @ 4 hv frwd vxshulru gh D/ | vhjxqgr txh O @ 4 hv od phqru gh wrgdv/ hv ghflu/ ; % A 3> <{q 5 D = {q A O  % 41 Hv fodur txh

q ? q . 4 ;q 5 Q

oxhjr

.4 ? 4

q

hv ghflu/

Sru or wdqwr O @ 4 hv frwd vxshulru gh D1

q

;q 5 Q>

{q ? 4 ;q 5 Q

51 Revhuyhprv txh suredu od h{lvwhqfld gh {q 5 D wdo txh

{q ? 4  % ; % A 3

htxlydoh d suredu od h{lvwhqfld gh q 5 Q wdo txh

.4 ? 4 

q

Vhd % A 31 Frqvlghuhprv ho uhdo

4 % 1 %

q

Sru od surslhgdg Dutxlphgldqd/ h{lvwh q 5 Q wdo txh qA

shur qA

% ;%A3

4 % %

4 % %

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/ qAqq %.4 % / q A +q . 4,+4  %, q / A4 % q.4

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;; Hq frqvhfxhqfld Oxhjr Surehprv txh vl hqwrqfhv

+; % A 3,+<{q @

vxs D @ 4 D @ i{q 5 U = {q @

Vroxflöq= Sru ghprvwudu txh Hmhpsor :8

q 5 D,+{q A 4  %, q.4

lqi D @ 3

4 > q 5 Qj q

41 O @ 3 hv frwd lqihulru gh D1

51 O @ 3 hv od pd|ru gh wrgdv odv frwdv lqihulruhv/ hv ghflu 41 Hv fodur txh oxhjr dvð/ O @ 3 hv frwd lqihulru gh D1

; % A 3> <{q 5 D = {q ? O . % 3?

4

q

;q 5 Q

3 ? {q > ;{q 5 D>

51 Sru od surslhgdg Dutxlphgldqd 41:4 vh wlhqh txh

or txh htxlydoh d= grqgh O @ 31

Hq frqvhfxhqfld lqi D @ 31

; % A 3>
4 ? % q

; % A 3> <{q 5 D = {q ? O . %

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FKDSWHU 51 IXQFLRQHV

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5, Od yhorflgdg gh fuhflplhqwr gh od yhjhwdflöq ghshqgh gho wlhpsr 6, Ho suhflr gh xq duwðfxor ghshqgh gh od rihuwd | gh od ghpdqgd

7, Ho iudqtxhr srvwdo kdvwd xq ghvwlqr gdgr ghshqgh gho shvr +| gho yroxphq, gho sdtxhwh

Pxfkdv yhfhv orv sdshohv gh odv yduldeohv lqghshqglhqwhv frq odv yduldeohv ghshqglhqwhv vrq lqwhufdp0 eldeohv/ sru hmhpsor/ F @ 5u

txh qrv gd od orqjlwxg F gh xqd flufxqihuhqfld fxdqgr frqrfhprv ho udglr u/ shur wdpelìq sxhgh vhu ylvwr frpr F u@ 5 sdud ho fdvr hq txh vh qrv gì od orqjlwxg F gh od flufxqihuhqfld | txlvlìudprv fdofxodu vx udglr1 Hq orv vljxlhqwhv hmhuflflrv oohjduhprv d xqd ixqflöq/ txh vh rewlhqh gho hqxqfldgr gho sureohpd/ fru0 uhvsrqglhqgr hvwh wlsr gh hmhuflflr d or txh vh hqwlhqgh sru prghodflöq pdwhpäwlfd1 Ho grplqlr gh od ixqflöq rewhqlgd vh gheh uhvwulqjlu d odv frqglflrqhv gho sureohpd frqvlghudqgr grqgh wlhqh vhqwlgr od ydul0 deoh xwlol}dgd/ rewhqlhqgr xq qxhyr grplqlr txh oodpduhprv grplqlr gho sureohpd | or ghqrwduhprv GrpS +i ,1 Sru hmhpsor= Vxsrqjdprv txh { lqglfd od orqjlwxg gh xq odgr gh xqd fdmd | od ixqflöq D+{, @ {+43{, uhsuhvhqwd ho äuhd gh od fdmd1 Hqwrqfhv ho grplqlr gh od ixqflöq +pludgd qr hq ho frqwh{wr gho sureohpd, hv U/ shur frqvlghudqgr D+{, frpr äuhd | whqlhqgr suhvhqwh txh { hv xqd orqjlwxg hqwrqfhv ghehprv dvhjxuduqrv gh txh {A3

| 43  { A 3

gh grqgh 3 ? { ? 43/ oxhjr ho grplqlr gho sureohpd hv GrpS +D, @`3> 43^1

Hmhuflflr 433 Od frpsdôðd gh duulhqgr gh dxwrv Mlpìqh}%/ duulhqgd dxwrpöylohv Fkhyhwwh d ' 431833 sru gðd | 73 shvrv sru nloöphwur uhfruulgr/ plhqwudv txh od frpsdôðd Dudqfleld% duulhqgd ho plvpr wlsr gh dxwr d ' ;1<33 sru gðd | d 93 shvrv sru nloöphwur1 d, Vl { uhsuhvhqwd ho qýphur gh nloöphwurv uhfruulgr hq xq gðd/ hqfxhqwuh odv ixqflrqhv FD +{,/ FM +{, txh uhsuhvhqwdq ho frvwr sru duuhqgdu xq Fkhyhwwh hq od frpsdôðd Dudqfleld% | Mlpìqh}% uhvshfwlyd0 phqwh1

e, BcFxäqwrv nloöphwurv vh gheh uhfruuhu hq ho gðd sdud txh frqyhqjd duuhqgdu ho dxwr gh od frpsdôðd Mlpìqh}%B Hqwuhjxh vl hv srvleoh xqd vroxflöq juäfd1

Vroxflöq= d, Vl { uhsuhvhqwd ho qýphur gh nloöphwurv uhfruulgr hq xq gðd/ hqwrqfhv ho frvwr sru duuhqgdu xq Fkhyhwwh hq fdgd frpsdôðd hv> FM +{, @ 43833 . 73{>

FD +{, @ <;33 . 93{ frq {  3

e, Frqylhqh duuhqgdu ho dxwr gh od frpsdôðd Mlpìqh}% fxdqgr ho frvwr vhd phqru txh ho gh od frpsdôðd Dudqfleld/ oxhjr ghehprv uhvroyhu FD +{, A FM +{, or txh htxlydoh d

<;33 . 93{ A 43833 . 73{

FKDSWHU 51 IXQFLRQHV

457

gh grqgh 68 ? {1 Hv ghflu/ frqylhqh duuhqgdu ho dxwr gh od frpsdôðd Mlpìqh}% vl vh uhfruuhq päv gh 68 nloöphwurv1 Dkrud yhdprv xqd vroxflöq juäfd sdud ho sureohpd1 Frqvlghuhprv odv uhfwdv | @ 43833 . 73{ h | @ <;33 . 93{ txh uhsuhvhqwdq d odv ixqflrqhv FM > FD uhvshfwlydphqwh Revhuydqgr jxud txh vljxh/ yhprv txh orv ydoruhv gh od ixqflöq FM +{, vrq phqruhv txh orv gh od ixqflöq FD +{, vl { A 681

Hmhuflflr 434 Ho lpsxhvwr W d od yhqwd gh flhuwrv duwðfxorv +wd{deoh, hv gh 4; ( vreuh ho ydoru { gho duwðfxor1 Vl W | { hvwäq gdgrv hq shvrv1 d, H{suhvh W hq ixqflöq gh {1 e, Fdofxoh W +533, | W +8=98,1

4; { frq { A 31 Vroxflöq= d, W +{, @ 433 4; e, W +533, @ 433 533 @ 69 | W +8=98, @

4; 433 8=98

Hmhuflflr 435 Vl ho iudqtxhr hq ho vhuylflr gh fruuhrv gh X1V1D1 hv gh ; fhqwdyrv sru rq}d r iudfflöq gh rq}d/ { hv ho shvr gh od fduwd +hq rq}dv, h | uhsuhvhqwd ho iudqtxhr uhtxhulgr sdud wdo fduwd/ h{suhvh | hq ixqflöq gh {1

Vroxflöq= { sxhgh vhu xq qýphur hqwhur gh rq}dv/ r elhq sxhgh vhu dojxqdv rq}dv päv iudfflöq gh rq}d1 Hq ho sulphu fdvr ho iudqtxhr vhuä vlpsohphqwh shur hq ho vhjxqgr fdvr ho iudqtxhr vhuä sru or wdqwr/ r frq { 5 U. wdpelìq

|+{, @ 3=3;{ @ 3=3;^{` |+{, @ 3=3;+^{` . 4,

|+{, @



3=3;+^{` . 4, { 5 U.  ]. 3=3;{ { 5 ].

|+{, @ 3=3;+^{`  ^^{`  {`, |+{, @ 3=3; m ^{` m

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

458

Hmhuflflr 436 Vh ghvhd frqvwuxlu xq vlor hq irupd floðqgulfd uhpdwdgr sru xqd eöyhgd vhplhviìulfd frpr vh lqglfd hq od jxud

d, H{suhvh ho yroxphq gho vlor hq ixqflöq gh u | k1 e, H{suhvh ho äuhd gho vlor hq ixqflöq gh u | k1

f, Vl ho frvwr gh frqvwuxfflöq sru phwur fxdgudgr hv ho greoh hq od eöyhgd | ho vxhor frpsdudgr frq od sduwh floðqgulfd | ho yroxphq gho vlor hv Y3 ^p6 `/ h{suhvh ho frvwr gh frqvwuxfflöq F frpr ixqflöq gh u | hqfxhqwuh GrpS +F,1

Vroxflöq= d, Frpr ho yroxphq gh od vhplhvihud hv 56 u6 | ho yroxphq gho flolqgur hv u5 k vh wlhqh txh ho yroxphq gho vlor hv ho yroxphq gh od eöyhgd päv ho yroxphq gho flolqgur 5 Y @ u6 . u5 k 6

e, Frpr ho äuhd gho whfkr +vhplhvihud, hv 5u5 / ho äuhd gh odv sduhghv +flolqgur, hv 5uk | ho äuhd gho vxhor +fðufxor, hv u5 / vh wlhqh txh ho äuhd wrwdo gho vlor hv D @ 5u5 . 5uk . u5

f, Ho frvwr wrwdo gh frqvwuxfflöq gho vlor hvwä gdgr sru od vxpd gh orv frvwrv gh frqvwuxfflöq gh od eöyhgd/ odv sduhghv | ho vxhor1 Vl o hv ho frvwr sru phwur fxdgudgr gh frqvwuxfflöq gh od sduwh floðqgulfd +sduhghv,/ hqwrqfhv ho frvwr wrwdo gh frqvwuxfflöq hvwä gdgr sru xqd ixqflöq txh ghshqgh gh k | u F @ +5u5 ,+5o, . +5uk,o . +u5 ,+5o,=

Shur ho yroxphq gho vlor hv mr/ Y3 / sru or wdqwr

dvð/ ghvshmdqgr k

5 Y3 @ u6 . u5 k 6

k@



Y3  56 u6 u5



+516,

FKDSWHU 51 IXQFLRQHV

459 | sru or wdqwr ho frvwr gdgr ghshqglhqgr vrodphqwh gh u hv   Y3 5  u F @ 9u5 o . 5uo u5 6

grqgh u A 3 | k A 3 shur sru 516 k vhuä srvlwlyr vl Y3  56 u6 A 3 / frq u A 3 gh grqgh 3 ? u ? Oxhjr/ ho grplqlr gho sureohpd hv u 6 6Y3 ^ GrpS +i , @`3> 5

t 6

6Y3 1 5

Hmhuflflr 437 Frqvlghuh U xq uhfwäqjxor +DEFG, frq vxv odgrv sdudohorv d orv hmhv/ lqvfulwr hq od holsvh <{5 . 7| 5 @ 691 Vhd } od glvwdqfld hqwuh xq odgr yhuwlfdo gh U | ho hmh | +yhu jxud ,

d, Ghwhuplqh ho äuhd gh U frpr xqd ixqflöq gh }/ D+},1

e, ÁSdud txì ydoruhv gh }/ wlhqh D+}, vhqwlgr hq hvwh sureohpdB

Vroxflöq= d, Vhd F ho yìuwlfh gho uhfwäqjxor U xelfdgr hq ho sulphu fxdgudqwh F @ +}> |,/ hqwrqfhv orv odgrv gho uhfwäqjxor vrq m DE m@ 5} | m EF m@ 5|1 Sru or wdqwr/ ho äuhd gh U hv D @ 5}5| @ 7}|1 Shur F shuwhqhfh d od holsvh/ dvð txh vxv frrughqdgdv vdwlvidfhq od hfxdflöq <{5 . 7| 5 @ 69/ gh grqgh/ ghvshmdqgr |/ vh wlhqh s s | @ 45 69  <} 5 b | @  45 69  <} 5 s frpr | hv xqd glvwdqfld/ hqwrqfhv | @ 65 7  } 5 | sru or wdqwr ho äuhd gh U qrv txhgd D @ 7}

s 6s 7  } 5 @ 9} 7  } 5 5

s e, Od ixqflöq D+}, @ 9} 7  } 5 wlhqh xq grplqlr pdwhpäwlfr +Grp+D, @ i} = 7  } 5  3j, shur ho grplqlr gho sureohpd hv GrpS +D, @`3> 5^/ |d txh } hv xqd glvwdqfld+| sru or wdqwr gheh vhu srvlwlyd, Hmhuflflr 438 Od frqfxuuhqfld phgld hq xq flqh hq ho txh od hqwudgd ydoh ' 4533 hv gh 433 shuvrqdv1 Ho hpsuhvdulr wlhqh exhqdv ud}rqhv sdud fuhhu txh fdgd yh} txh vh uhgxfh ho suhflr hq ' ;3/ ho qýphur gh hvshfwdgruhv dxphqwd hq 531

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

45:

d, Ghwhuplqh od uhfdxgdflöq U hq ixqflöq gho suhflr s1

e, BcTxì suhflr | qýphur gh hvshfwdgruhv surgxfluäq pd|ru wdtxloodB f, BcFxäo hv od uhfdxgdflöq pä{lpd sru vhvlöqB

Vroxflöq= d, Vhd q hv ho qýphur gh hvshfwdgruhv | s ho suhflr gh od hqwudgd/ hqwrqfhv od uhfdxgdflöq vhuä U @ qs

Vhd o ho qýphur gh yhfhv txh vh uhgxfh ho suhflr hq ' ;3/ oxhjr ho suhflr s hvwduä gdgd sru s @ 4533  ;3o grqgh o 5 Q

Dghpäv/ sru fdgd yh} txh vh uhgxfh ho suhflr/ ho qýphur gh hvshfwdgruhv dxphqwd hq 53/ oxhjr shur sru 517/ o @ Sru or wdqwr

4533s ;3

+517,

q @ 433 . 53o    4533  s U+s, @ 433 . 53 s ;3

grqgh s 5 i4533  ;3o = o 5 Qj_`3> 4533` Od ixqflöq U+s, hv htxlydohqwh d>

   4533  s s U+s, @ 433 . 53 ;3

e, Frpr U+s, hv od sduäerod gh od jxud

od pä{lpd uhfdxgdflöq vh hqfxhqwud fxdqgr ho suhflr hv s @ ;33 | frq xqd wdtxlood q @ 733  733  533 @ 5331 f, Od uhfdxgdflöq pä{lpd sru vhvlöq hv ' 493=3331

;33 7

@

Hmhuflflr 439 Xq fdpshvlqr wlhqh glqhur sdud frpsudu 4933 phwurv gh dodpeuh gh sýdv/ | ghvhd fhufdu frq fxdwur kheudv gh dodpeuh xq whuuhqr uhfwdqjxodu hq vx suhglr1 Vl { uhsuhvhqwd xq odgr gho uhfwäqjxor d, Ghwhuplqh od ixqflöq D+{, äuhd gho whuuhqr fhufdgr frq hvwh dodpeuh hq wìuplqrv gh {1 e, Hqfxhqwuh odv glphqvlrqhv gho whuuhqr fhufdgr gh pd|ru äuhd1

FKDSWHU 51 IXQFLRQHV

45;

Vroxflöq= d, Vl { h | vrq orv odgrv gho uhfwäqjxor/ vh wlhqh txh ho äuhd gho whuuhqr fhufdgr hv frq { A 3 > | A 3

D+{, @ {| | ho dodpeuh hpsohdgr sdud hvwh uhfwäqjxor vhuä

7+5{ . 5|, txh gheh vhu ljxdo d od fdqwlgdg gh dodpeuh glvsrqleoh1 Dvð= 7+5{ . 5|, @ 4933 { . | @ 533 | @ 533  {

Sru or wdqwr ho äuhd txhgd vöor frpr ixqflöq gh {

grqgh GrpS +D, @`3> 533^1

D+{, @ +533  {,{

e, \d txh D+{, @ +533 {,{ hv xqd sduäerod/ frpr od gh od jxud vljxlhqwh vh wlhqh txh ho äuhd pä{lpd vh rewlhqh fxdqgr { @ 433/ sru or wdqwr odv glphqvlrqhv gho whuuhqr fhufdgr vrq gh 433 sru 433 pwv1

Hmhuflflr 43: Od rulood gh xqd slvflqd irupd xq uhfwäqjxor gh 93 slhv gh odujr | 63 slhv gh dqfkr1 Vx surixqglgdg dxphqwd xqliruphphqwh gh 7 d < slhv hq xq wudpr krul}rqwdo gh 73 slhv | ghvsxìv frqwlqýd do plvpr qlyho orv uhvwdqwhv 53 slhv/ frpr vh loxvwud hq od jxud txh pxhvwud xqd vhfflöq wudqvyhuvdo1 Vl od slvflqd vh kd oohqdgr ghvgh ixhud dofdq}dqgr xq qlyho k hq ho odgr päv surixqgr/ ghwhuplqh ho yroxphq Y +k, gho djxd hq ixqflöq gh od dowxud k1

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

Vroxflöq= Sulphu fdvr1 Vxsrqjdprv sulphur txh 3 ? k ? 81 Fodudphqwh whqhprv txh Y +k, @ Y4 +k, . Y5 +k, grqgh Y4 +k, @ 53  k  63

Sdud Y5 +k, frqvlghuhprv od jxud grqgh

Y5 +k, @ shur sru ho whruhpd gh Wkdohv whqhprv txh

63k| 5

8 k @ 73 |

or txh lpsolfd txh | @ ;k oxhjr/ Y5 +k, @ Sru or wdqwr/ vxpdqgr dperv yroýphqhv

573k5 63k+;k, @ = 5 5

Y +k, @ 933k . 453k5 Vhjxqgr fdvr1 Vxsrqjdprv dkrud txh <  k  8 | frqvlghuhprv od jxud

45<

FKDSWHU 51 IXQFLRQHV

463

Whqhprv txh Y +k, @ Y4 +k, . Y5 +k, . Y6 +k, grqgh Y4 +k, @ 53  63  k @ 933k Y5 +k, @ 738563 @ 9333 5 @ 6333 Y6 +k, @ +k  8,  73  63 @ 4533+k  8, sru or wdqwr/ Y +k, @ 933k . 6333 . 4533+k  8, | qdophqwh Y +k, @



933k . 453k5 933k . 6333 . 4533+k  8,

vl 3 ? k ? 8 vl 8  k  <

Hmhuflflr 43; Ghwhuplqh xqd ixqflöq txh uhsuhvhqwh ho yroxphq gh xq flolqgur lqvfulwr hq xqd hvihud gh udglr d +frpr hq od jxud , hq ixqflöq gh vx dowxud1

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

464

Vroxflöq= Vhdq u | k udglr | dowxud gho flolqgur uhvshfwlydphqwh1 Vdehprv txh ho yroxphq gho flolqgur hvwä gdgr sru od iöupxod Y @ u5 k +txh ghshqgh gh odv yduldeohv u | k,/ shur/ sru whruhpd gh Slwäjrudv/ whqhprv  5 k . u5 = d5 @ 5 Oxhjr/ ghvshmdqgr u5 | uhhpsod}dqgr hq od iöupxod gho yroxphq/ whqhprv  5 k 5 ,k Y +k, @ +d  5

5 grqgh GrpS +Y , @`3> 5d^/ sxhvwr txh k A 3 | d5  k5 A 3> gh grqgh 3 ? k ? 5d1

Hmhuflflr 43< Xq flfolvwd wlhqh txh kdfhu xq yldmh gh g npv/ hq xq wlhpsr ghwhuplqdgrv1 Fdofxod txh gheh yldmdu d Y npv@kuv1 Shur vdoh 6 plqxwrv wdugh | yldmd d 4 npv@kuv päv ghsulvd/ | oohjd d wlhpsr1 Ghwhuplqdu od yhorflgdg suhylvwd Y hq ixqflöq gh od glvwdqfld g1

Vroxflöq= Vxsrqjdprv txh ho flfolvwd yldmded d xqd yhorflgdg gh y np2ku> hqwrqfhv od yhorflgdg exvfdgd g vhuä Y @ +y  4, np2ku1 Ho flfolvwd yldmö uhdophqwh gy krudv plhqwudv txh ho wlhpsr suhylvwr hud gh y 4 krudv1 Sru klsöwhvlv g g 6 y4  y @ 93 yg+y4,g 4 +y4,y @ 53 53yg  53+y  4,g @ +y  4,y 53yg  53gy . 53g @ y5  y y5  y  53g @ 3 Frpr g A 3 | y A 3 whqguhprv txh

s 4 . ;3g 5 shur od yhorflgdg exvfdgd hv Y +g, @ y+g,  4/ sru or wdqwr s 4 . ;3g  4 Y +g, @ 5 y+g, @

4.

FKDSWHU 51 IXQFLRQHV

465

Hmhuflflr 443 Vh ghvhd frqvwuxlu xqd fdmd frq wdsd gh edvh fxdgudgd frq äuhd qr pd|ru gh 433fp5 frpr od gh od jxud / gh yroxphq 585fp6 1 Vl ho frvwr gh od wdsd hv ' 5 sru fp5 / od edvh ' 8 sru fp5 | ho gh orv odgrv hv gh ' 6 sru fp5 / h{suhvh ho frvwr F frpr ixqflöq gh { | GrpS +F,1

Vroxflöq= Sru klsöwhvlv vh vdeh txh ho yroxphq gh od fdmd hv gh 585fp6 > oxhjr {5 | @ 585

+518,

frq { A 3/ | A 3 | {5  4331 Ho frvwr gh od wdsd hv ' 5 sru fp5 1 Od wdsd wlhqh {5 fp5 / oxhjr ho frvwr gh od wdsd hv ' 5{5 1 Ho frvwr gh od edvh hv ' 8 sru fp5 | od edvh wlhqh {5 fp5 / oxhjr ho frvwr gh od edvh hv ' 8{5 1 Ilqdophqwh/ ho frvwr gh orv odgrv hv ' 6 sru fp5 | fdgd odgr wlhqh {|fp5 / oxhjr ho frvwr gh od edvh hv ' 6+7{|, @ ' 45{|1 Dvð/ ho frvwr wrwdo gh od fdmd hv= +519, F @ 5{5 . 8{5 . 45{|

shur sru 518/

|@

uhhpsod}dqgr hq 519 rewhqhprv od ixqflöq shglgd F+{, @ :{5 . 45{

585 {5

585 grqgh GrpS +F, @`3> 43` {5

Hmhuflflr 444 Vl ho udglr edvdo u gh xq frqr flufxodu uhfwr dxphqwd hq xq {(/ plhqwudv txh vx dowxud k glvplqx|h hq xq 53(/ irupxoh xqd ixqflöq hq wìuplqrv gh { txh shuplwd rewhqhu hq txì srufhqwdmh yduðdq= d, ho äuhd edvdo gho frqr1 e, ho yroxphq gho frqr1

Hq sduwlfxodu/ vl ho udglr dxphqwd hq 48(/ Bchq txì srufhqwdmh yduðdq ho äuhd edvdo | ho yroxphq gho frqrB +Vöor ghvduurooduhprv or uhodwlyr do äuhd,1

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

466

Vroxflöq= d, Vl ho udglr edvdo dxphqwd hq {( hqwrqfhv ho qxhyr udglr vhuðd  { {  u @ 4. u uQ @ u . 433 433 oxhjr ho äuhd edvdo vhuðd

5 De @ uQ

Dvð/ ho äuhd edvdo dxphqwduðd hv ghflu/ dxphqwduðd







 { 5 5 @  4. u 433   { 5  { @  4. . u5 83 433   { 5  { @ u5 .  . u5 83 433

 { 5  { . u5 xqlgdghv/ 83 433

 { 5  { . 433 sru flhqwr 83 433

Oxhjr od ixqflöq txh uhsuhvhqwd ho srufhqwdmh gh dxphqwr gho äuhd edvdo hv   { 5  { 433 frq { A 3 D+{, @ . 83 433

Hq sduwlfxodu/ vl ho udglr dxphqwd hq xq 48( hqwrqfhv ho srufhqwdmh gh dxphqwr gho äuhd hv D+48,1

51714

Surjudpdflöq Olqhdo

Hq orv vljxlhqwhv sureohpdv gh prghodflöq pdwhpäwlfd/ vh frqvlghudq ixqflrqhv gh grv yduldeohv8 gho wlsr } @ i+{> |, @ d{ . e|

vxmhwdv d uhvwulfflrqhv gh ghvljxdogdghv1

Hmhpsor 445 Xqd lqgxvwuld pdqxidfwxuhud surgxfh elflfohwdv | prwrqhwdv/ fdgd xqd gh odv fxdohv gheh surfhvduvh d wudyìv gh grv fhqwudohv gh pätxlqdv D | E1 Od fhqwudo D wlhqh xq pä{lpr gh 453 krudv glvsrqleohv | od fhqwudo E srvhh xq pä{lpr gh 4;3 krudv glvsrqleohv1 Od pdqxidfwxud gh xqd elflfohwd uhtxlhuh 9 krudv hq od fhqwudo D | 6 krudv hq od fhqwudo E> od pdqxidfwxud gh xqd prwrqhwd uhtxlhuh gh 7 krudv hq od fhqwudo D | 43 krudv hq od fhqwudo E1 Vdelhqgr txh ho ehqhflr sru elflfohwd hv gh ' 78 | sru prwrqhwd hv gh ' 88 irupxohprv xq prghor pdwhpäwlfr txh shuplwd rewhqhu ho qýphur gh elflfohwdv | gh prwrqhwdv txh vh ghehuäq ideulfdu sdud rewhqhu xq ehqhflr pä{lpr1 Vroxflöq= Vhdq { h | odv yduldeohv txh uhsuhvhqwdq odv fdqwlgdghv gh elflfohwdv | prwrqhwdv uhvshfwl0 ydphqwh | vhd } od jdqdqfld txh uhvxowd do yhqghu { elflfohwdv h | prwrqhwdv1 Ho remhwlyr hv hvfrjhu orv ydoruhv gh { h | gh pdqhud txh vh pd{lplfh od fdqwlgdg

+51:,

} @ 78{ . 88|

vxmhwd d od uhvwulfflöq fruuhvsrqglhqwh d od fhqwudo D

9{ . 7|  453

8 Xqd ixqflöq uhdo gh grv yduldeohv frq grplqlr D i G D

hohphqwrv E{> |  hq

D

h{lvwh xq ýqlfr hohphqwr

}

M U wdo txh

\ U2 3< U hv xqd fruuhvsrqghqfld wdo txh sdud fdgd sdu gh }

'

i E{> | 

+Yhu Dsìqglfh F,1

FKDSWHU 51 IXQFLRQHV

467 | od uhvwulfflöq fruuhvsrqglhqwh d od fhqwudo E 6{ . 43|  4;3

Frpr { h | vrq fdqwlgdghv gh yhkðfxorv hqwrqfhv ghehprv h{ljlu wdpelìq {3 a |3

Oxhjr ho prghor pdwhpäwlfr fruuhvsrqglhqwh do sureohpd hv

Pd{ } @ 78{ . 88|

vxmhwd d

9{ . 7| 6{ . 43| { |

   

453 4;3 3 3

+51;,

Qrwd= Hvwh hv xq sureohpd gh surjudpdflöq olqhdo1 Ho sureohpd jhqhudo gh surjudpdflöq olqhdo wudwd gh pd{lpl}du r plqlpl}du +hv ghflu/ rswlpl}du, xqd ixqflöq olqhdo gh yduldv yduldeohv oodpdgd ixqflöq remhwlyr +hfxdflöq 51: hq ho hmhpsor dqwhulru, vxmhwd d xq frqmxqwr gh hfxdflrqhv olqhdohv r ghvljxdogdghv olqhdohv oodpdgdv uhvwulfflrqhv +hfxdflrqhv h lqhfxdflrqhv hq 51; gho hmhpsor dqwhulru,1 Sdud xqd ixqflöq olqhdo gh grv yduldeohv i+{> |, @ d{ . e|

ho sureohpd srghprv uhvroyhuor jhrpìwulfdphqwh1 Hq hihfwr/ hq hvwh fdvr ho sureohpd vöor wlhqh grv yduldeohv gh ghflvlöq | sru or wdqwr vöor grv glphqvlrqhv/ sru or txh srghprv xvdu xq surfhglplhqwr juäfr sdud uhvroyhuor1 Hvwh surfhglplhqwr lqfox|h od frqvwuxfflöq gh xqd juäfd gh grv glphqvlrqhv frq { h | hq orv hmhv1 Ho sulphu sdvr hv lghqwlfdu orv ydoruhv gh +{> |, shuplwlgrv sru odv uhvwulfflrqhv> dvð/ sdud ho sureohpd dqwhulru/ vh judfdq odv uhfwdv 9{ . 7| 6{ . 43| { |

rewhqlhqgr od uhjlöq idfwleoh U hq od jxud

@ @ @ @

453 4;3 3 3

Ho sdvr qdo hv vhohfflrqdu/ ghqwur gh hvwd uhjlöq/ ho sxqwr +{> |, txh pd{lpl}d ho ydoru gh } @ 78{.88|1 Hvwh sdvr vh yxhoyh dxwrpäwlfr ghvsxìv gh xq srfr gh suäfwlfd/ shur sdud ghvfxeulu hq txì vh ixqgd0 phqwd ydoh od shqd lqwhqwdu dojxqrv ydoruhv sru suxhed | huuru1 Sru hmhpsor/ frqvlghuhprv } @ ;431 Vl vh glexmd od uhfwd 78{ . 88| @ ;43 vh sxhgh yhu +hq od jxud , txh h{lvwhq pxfkrv sxqwrv vreuh hvwd uhfwd txh hvwäq ghqwur gh od uhjlöq U1 Sru or wdqwr ghehprv lqwhqwdu xq ydoru pd|ru sdud }/ sru hmhpsor/ } @ <331

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

468

Judfdqgr od uhfwd 78{ . 88| @ <33 yhprv qxhydphqwh txh h{lvwhq lqqlwrv sxqwrv vreuh hvwd uhfwd txh hvwäq ghqwur gh od uhjlöq U/ gh pdqhud txh ho pä{lpr ydoru shuplvleoh gh } gheh vhu sru or phqrv <331 Qrwhprv txh vl vh gd xq ydoru päv judqgh d }/ od uhfwd txhgduä päv duuled r päv dohmdgd gho ruljhq txh od sulphud | txh wrgdv hvwdv uhfwdv vrq sdudohodv1 Hvwh hv ho surfhglplhqwr gh suxhed | huuru txh vöor qhfhvlwd txh vh judtxhq xqd idplold gh uhfwdv sdudohodv txh frqwhqjdq do phqrv xq sxqwr gh od uhjlöq idfwleoh | txh vh holmd od txh hvwì päv dohmdgd gho ruljhq +hq od gluhfflöq hq txh fuhfhq orv ydoruhv gh },1 Glfkd uhfwd sdvd sru ho sxqwr +43> 48, frpr vh lqglfd hq od jxud | od hfxdflöq txhgd 78{ . 88| @ 6  43 . 88  48 @ 45:8

Sru or wdqwr od vroxflöq txh exvfäedprv hv=

{ @ 43> | @ 48> } @ 45:8

hv ghflu/ ideulfdu 43 elflfohwdv | 48 prwrqhwdv rewhqlhqgr xq ehqhflr gh ' 45:81 Ghvsxìv gh xq srfr gh suäfwlfd/ vh sxhgh dsolfdu xq pìwrgr päv vhqfloor txh frqvlvwh hq glexmdu vöor xqd gh hvwdv uhfwdv sdud hvwdeohfhu od shqglhqwh | oxhjr pryhu ylvxdophqwh hvwd uhfwd sdudohod d vðplvpd1

Hmhuflflr 446 Xq kruwlfxowru ghvhd ph}fodu ihuwlol}dqwh txh sursruflrqh xq pðqlpr gh 48 xqlgdghv gh srwdvd/ 53 xqlgdghv gh qlwudwrv | 57 xqlgdghv gh irvidwrv1 Glvsrqh gh grv pdufdv gh ihuwlol}dqwhv1 Od pdufd 4 sursruflrqd 6 xqlgdghv gh srwdvd 4 gh qlwudwrv | 6 gh irvidwrv> vx frvwr hv gh 453 göoduhv +od xqlgdg,1 Od pdufd 5 sursruflrqd 4 xqlgdg gh srwdvd 8 gh qlwudwrv | 5 gh irvidwrv> vx frvwr hv gh 93 göoduhv +od xqlgdg,1 Irupxoh xq prghor pdwhpäwlfr txh shuplwd rewhqhu od frpelqdflöq gh ihuwlol}dqwhv gh phqru frvwr txh vdwlvidjd odv hvshflfdflrqhv ghvhdgdv1 Vroxflöq= Vhdq

{ +xqlgdghv, od fdqwlgdg gh ihuwlol}dqwh +Pdufd 4, | +xqlgdghv, od fdqwlgdg gh ihuwlol}dqwh +Pdufd 5, Hqwrqfhv ho remhwlyr hv plqlpl}du od ixqflöq=

vxmhwd d odv uhvwulfflrqhv=

} @ 453{ . 93|

41 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh srwdvd1 6{ . |  48

51 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh qlwudwrv1

{ . 8|  53

61 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh irvidwrv1 Fodudphqwh/ dghpäv Oxhjr ho prghor pdwhpäwlfr shglgr hv

6{ . 5|  57 {3 |3 Plq } @ 453{ . 93|

FKDSWHU 51 IXQFLRQHV

469 vxmhwr d

Judfdqgr odv uhfwdv

rewhqhprv od uhjlöq U gh od jxud

6{ . | { . 8| 6{ . 5| { | 6{ . | { . 8| 6{ . 5| { |

    

@ @ @ @ @

48 53 57 3 3 48 53 57 3 3

Sru rwud sduwh d sduwlu gh od ixqflöq remhwlyr/ frq f frqvwdqwh/ vh wlhqh txh |@

f  5{ 93

gh grqgh od shqglhqwh gh od uhfwd f @ 453{ . 93| hv p @ 51 Frqvlghudqgr od uhfwd +frq shqglhqwh p @ 5, päv fhufdqd do ruljhq/ txh sdvh sru do phqrv xq sxqwr gho frqmxqwr U/ whqhprv txh ho sxqwr öswlpr hv { @ 5> | @ <

hv ghflu/ 5 xqlgdghv gh ihuwlol}dqwh Pdufd 4 | < xqlgdghv gh ihuwlol}dqwh Pdufd 51

Hmhuflflr 447 Xqd hglwruldo sodqhd xvdu xqd vhfflöq gh vx sodqwd sdud surgxflu grv wh{wrv D | E1 Od jdqdqfld hv gh ' ; sru ho wh{wr D | ' 8 sru ho wh{wr E1 Ho wh{wr D uhtxlhuh gh 7 krudv sdud lpsulplu | gh 9 krudv sdud hqfxdghuqdu | hpsdvwdu1 Ho wh{wr E uhtxlhuh gh 8 krudv sdud lpsulplu | gh 6 krudv sdud hqfxdghuqdu | hpsdvwdu1 Vl vh glvsrqh gh 533 krudv sdud lpsulplu | gh 543 krudv sdud hqfxdghuqdu | hpsdvwdu | vl sru or phqrv vh ghehq surgxflu 53 wh{wrv1 41 Irupxoh xq prghor pdwhpäwlfr txh shuplwd rewhqhu ho ehqhflr pä{lpr1

51 Ghwhuplqh ho qýphur öswlpr gh fdgd wh{wr txh gheh surgxflu od hglwruldo | vx fruuhvsrqglhqwh jdqdq0 fld1 61 BcFxäqwdv krudv vh xvdq sdud lpsulpluB

5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ

46:

71 BcFxäqwdv krudv vh xvdq sdud hqfxdghuqdu | hpsdvwduB

81 Vl vh djuhjd od uhvwulfflöq= Od surgxfflöq gho wh{wr E gheh vhu sru or phqrv ho 83( gh od surgxfflöq gho wh{wr D1BcTxì sxhgh ghflu gho öswlprB

91 Vl vh glvsrqh gh 9 krudv päv sdud lpsulplu | od jdqdqfld sru fdgd wh{wr vh uhgxfh hq ' 3/35 Bctxì sxhgh ghflu gho öswlprB +QR frqvlghuh od uhvwulfflöq lpsxhvwd hq 8,1

Vroxflöq= 4,= Vhdq

{= qýphurv gh wh{wrv D1

|= qýphurv gh wh{wrv E1

Irupxodflöq gho sureohpd=

vxmhwr d

5, Judfdqgr odv uhfwdv

Pd{ } @ ;{ . 8| 7{ . 8| 9{ . 6| { | 7{ . 8| 9{ . 6| { |

rewhqhprv od uhjlöq U gh od jxud l}txlhugd

   

@ @ @ @

533 543 53 3 533 543 53 3

Frqvlghudqgr od uhfwd +frq shqglhqwh p @  ;8 , päv ohmdqd do ruljhq/ txh sdvh sru do phqrv xq sxqwr gho frqmxqwr U/ whqhprv txh ho sxqwr öswlpr hv { @ 58> | @ 53

frq } @ 633 6, | 7, Uhhpsod}dqgr od vroxflöq öswlpd hq od uhvwulfflöq gh lpsuhvlöq whqhprv txh vh xvdurq 7  58 . 8  53 @ 533 krudv sdud lpsulplu/ hv ghflu/ wrgdv odv krudv glvsrqleohv sdud wdo hihfwr1 Frqvlghudqgr od uhvwulfflöq gh hqfxdghuqdflöq | hpsdvwh/ rewhqhprv txh vh xvdurq odv 543 krudv glvsrqleohv sdud hvwd wduhd1 8, Do djuhjdu od uhvwulfflöq 5|  {  3

FKDSWHU 51 IXQFLRQHV

46;

rewhqhprv od uhjlöq U4 txh pxhvwud od jxud dqwhulru ghuhfkd1 gh grqgh hv fodur txh ho öswlpr qr fdpelö/ hv ghflu/ od vroxflöq öswlpd vljxh vlhqgr { @ 58> | @ 53

frq } @ 6331 9, Vl vh glvsrqhq gh 9 krudv päv sdud lpsulplu | od jdqdqfld sru fdgd wh{wr vh uhgxfh hq ' 3/35 od irupxodflöq pdwhpäwlfd vhuðd Pd{ } @ :> <;{ . 7> <;| vxmhwr d

7{ . 8| 9{ . 6| { |

   

539 543 53 3

od juäfd fruuhvsrqglhqwh d hvwh prghor vh hqfxhqwud hq od jxud

/ gh grqgh od vroxflöq öswlpd hv { @ 57> | @ 55> } @ :> <;  57 . 7> <;  55 @ 634> 3;

Oxhjr od jdqdqfld dxphqwd hq ' 4> 3;

518

Dojheud gh Ixqflrqhv

Xwlol}dqgr ixqflrqhv hohphqwdohv s |d frqrflgdv/ vh sxhgh ideulfdu qxhydv ixqflrqhv> sru hmhpsor xvdqgr odv ixqflrqhv i+{, @ vhq{/ j+{, @ { vh sxhgh frqvwuxlu od ixqflöq k+{, @ vhq{ .

s {

Xqd gh odv frvdv txh gheh hvshflfduvh dtxðhv ho grplqlr gh k+{,/ | hv qdwxudo shqvdu hq xq frqmxqwr wdo txh fdgd vxpdqgr hvwì ghqlgr hq U> dvðho grplqlr gh k+{, gheh vhu d or päv Grpi _ Grpj1 Hq ho fdvr dqwhulru vh xvö od rshudflöq . +vxpd,/ wdpelìq vh srguðd kdehu xvdgr od rshudflöq  +pxo0 wlsolfdflöq,/ glylvlöq/ hwf1

Ghqlflöq 516 Hq jhqhudo vxsrqjdprv txh i | j vrq grv ixqflrqhv | vhd G @ Grpi _Grpj/ hqwrqfhv vh ghqhq qxhydv ixqflrqhv +frq grplqlr G, frqvwuxlgdv hq edvh d rshudflrqhv hqwuh vxv lpäjhqhv uhvshfwlydv=

5181 DOJHEUD GH IXQFLRQHV

46<

d, Od vxpd gh i | j ghqrwdgd sru i . j | ghqlgd sru=

i . j = G $ U { $ i+{, . j+{,

e, Od glihuhqfld gh i | j ghqrwdgd sru i  j | ghqlgd sru=

i  j = G $ U { $ i+{,  j+{,

f, Ho surgxfwr gh i | j ghqrwdgr sru i  j | ghqlgr sru=

i j = G $ U { $ i+{,j+{,

g, Ho fxrflhqwh i | j ghqrwdgr sru ij | ghqlgr sru=

grqgh

i @j = G3 $ U { $ i +{,@j+{, G3 @ i{ 5 G = j+{, 9@ 3j

s Hmhpsor 448 Vhdq i +{, @ +5  {,6 / j+{, @m {  5 m1 Ghwhuplqhprv od vxpd/ od glihuhqfld/ ho surgxfwr s | ho fxrflhqwh gh i | j1 Vroxflöq= Qrwhprv txh i +{, @ 5  { m 5  { m1 Ho grplqlr gh i hv ho lqwhuydor `  4> 5` | ho grplqlr gh j hv U/ sru or wdqwr od lqwhuvhfflöq gh vxv grplqlrv hv `  4> 5` | odv ixqflrqhv txh vh uhtxlhuhq hvwäq gdgdv sru s +i . j,+{, @ s5  { m 5  { m . m {  5 m {  5 +i  j,+{, @ s5  { m 5  { m  m {  5 m {  5 +ij,+{, @ s 5  {+5  {,5 {5 5{m5{m +i @j,+{, @ s m{5m @ 5{ {?5 Qrwh txh vl j hv xqd ixqflöq frqvwdqwh wdo txh j+{, @ f sdud wrgr { | vl i hv fxdotxlhu ixqflöq/ hqwrqfhv fi ghqrwd ho surgxfwr gh j | i / +fi,+{, @ fi +{,> ;{ 5 Grpi

Hmhpsor 449 Vhdq i | j odv ixqflrqhv txh vh ghqhq d frqwlqxdflöq ;  {  4 ? 4> { . 4> 5 ? { ? 3 s 5> 4 ? {  8 i+{, @ j+{, @ {> 3{?7 = s {  4> { A 8

+yhu jxud , Ghwhuplqh i . j/ i  j/ i j/ i@j | dghpäv judiðtxhodv1

FKDSWHU 51 IXQFLRQHV

473 Vroxflöq= Hv fodur sru od ghqlflöq txh

  i sxhv j+{, 9@ 3 j @ Grpi _ Grpj @ @`  5> 7^_U @`  5> 7^

Grp+i . j, @ Grp+i  j, @ Grp+ij, @ Grp

Sxhvwr txh i | j vrq ixqflrqhv ghqlgdv sru sduwhv ghehprv dqdol}du orv srvleohv fdvrv gh dfxhugr d odv lqwhuvhfflrqhv srvleohv gh grplqlrv1 41 Vhd { 5`  5> 3^_`  4> 4`/ hv ghflu/ { 5`  5> 4`/ hqwrqfhv oxhjr/

i +{, @ { . 4> j+{, @ 4

+i . j,+{, +i  j,+{,  +i j,+{, i +{, j

@ i +{, . j+{, @ { . 5 @ i +{,  j+{, @ { @ i +{,j+{, @ { . 4 @ ij++{{,, @ { . 4

51 Vhd { 5`  5> 3^_`  4> 8`/ hv ghflu/ { 5`  4> 3^/ hqwrqfhv oxhjr/

i +{, @ { . 4> j+{, @ 5

+i . j,+{, +i  j,+{,  +i j,+{, i +{, j

@ i +{, . j+{, @ { . 6 @ i +{,  j+{, @ {  4 @ i +{,j+{, @ 5{ . 5 @ ij++{{,, @ {.4 5

61 Vhd { 5 ^3> 7^_`  4> 8`/ hv ghflu/ { 5 ^3> 7^/ hqwrqfhv s i +{, @ {> j+{, @ 5 oxhjr/

Hq frqvhfxhqfld=

+i . j,+{, +i  j,+{,  +i j,+{, i +{, j

s @ i+{, . j+{, @ s{ . 5 @ i+{,  j+{, @s {  5 @ i+{,j+{, @ 5 { s @ ij++{{,, @ 5{

; 5 ? {  4 ? {.5 { . 6 4 ? { ? 3 +i . j,+{, @ = s {.5 3{?7 ; 5 ? {  4 ? { {  4 4 ? { ? 3 +i  j,+{, @ = s {5 3{?7 ; ? { . 4 5 ? {  4 5{ . 5 4 ? { ? 3 +i j,+{, @ = s 5 { 3{?7 ;   ? { . 4 5 ? {  4 i {.4 4 ? { ? 3 +{, @ = s5{ j 3{?7 5

5181 DOJHEUD GH IXQFLRQHV

474

+juäfdv hq jxudv vljxlhqwhv,

51814

Frpsrvlflöq gh ixqflrqhv

Ghqluhprv xqd qxhyd ixqflöq txh vh frqvwux|h d sduwlu gh rwudv grv ixqflrqhv i | j shur/ hq hvwh fdvr/ qrv edvduhprv hq odv surslhgdghv dojheudlfdv gho frqmxqwr gh oohjdgd1 Ghqlflöq 517 Vhdq D/ E | F frqmxqwrv gh qýphurv uhdohv1 Vhd i xqd ixqflöq gh D hq E | vhd j xqd ixqflöq gh F hq G/ hv ghflu/

i

D $ E

j

a F $ G

Vl Uhfi _ F 9@ >/ hv ghflu/ ho uhfruulgr gh i | ho grplqlr gh j wlhqhq hohphqwrv hq frpýq/ hqwrqfhv hv srvleoh xvdu i | j sdud ghqlu xqd ixqflöq gh D4  D hq G Wdo ixqflöq vh ghqrplqd frpsrvlflöq gh j frq i/ | vh ghqrwd sru j  i | vh ghqh sru= j  i = D4 $ G { $ j+i +{,,

grqgh Grp+j  i, @ D4 @ i{ 5 D = i +{, 5 Fj/ vl Uhfi _ Grpj 9@ >/ hv ghflu/

Grp+j  i , @ D4 @ i{ 5 Grpi = i +{, 5 Grpj _ Uhfij

FKDSWHU 51 IXQFLRQHV

475

+yhu hq jxud ,1 9

Hq sduwlfxodu vl Uhfi  Grpj/ hqwrqfhv Grp+j  i , @ Grpi1

Hmhuflflr 44: Vhdq |

Ghwhuplqhprv j  i 1

i = ^4> 4^$ U s {4.5 { $ j = U $ U { $ +{  5,5

Vroxflöq= Sxhvwr txh Uhfi  U @ Grpj/ hqwrqfhv Grp+j  i, @ ^4> 4^ | s s +j  i,+{, @ j+i +{,, @ j+ {  4 . 5, @ + {  4 . 5  5,5 @ {  4

oxhjr/

j  i = ^4> 4^$ U { $ {  4

Dkrud ghwhuplqhprv i  j1 Hq hvwh fdvr ghehprv ghwhuplqdu Uhfj1 Fodudphqwh Uhfj @ ^3> 4^ | Uhfj 9 Grpi / vlq hpedujr Uhfj _ Grpi @ ^4> 4^9@ > oxhjr1 Shur

oxhjr/

Dghpäv/

Grp+i  j, @ i{ 5 Grpj = j+{, 5 Grpi j @ i{ 5 U = +{  5,5 5 ^4> 4^j

+{  5,5 5 ^4> 4^ +, +{  5,5  4 +, m {  5 m  4 +, {6 b {4 Grp+i  j, @ i{ 5 Grpj = j+{, 5 Grpi j @ `  4> 4` ^ ^6> 4^ @ U`4> 6^

+i  j,+{, @ i +j+{,, @ i++{  5,5 , @

s +{  5,5  4 . 5

9 Dojxqrv wh{wrv srqhq frpr frqglflöq qhfhvduld Uhfi \ Grp j sdud ghqlu od frpsxhvwd j  i shur frpr vh yh hvwh hv xq fdvr sduwlfxodu gh od ghqlflöq gdgd dtxð1

5181 DOJHEUD GH IXQFLRQHV oxhjr/

476

i  j = U`4> 6^$ s U +{  5,5  4 . 5 { $

Qrwh txh i  j 9@ j  i/ hv ghflu/ qr vh fxpsoh od surslhgdg gh frqpxwdwlylgdg sdud od frpsrvlflöq gh ixqflrqhv1 Vlq hpedujr vh yhulfd od dvrfldwlylgdg +i  j,  k @ i  +j  k,

Hq ho hmhpsor vljxlhqwh vh yhulfd hvwd ljxdogdg hq xq fdvr s s Hmhuflflr 44; Vhdq i+{, @ { . 4  5/ j+{, @ {5  4 | k+{, @ {5 ixqflrqhv ghqlgdv hq vxv grplqlrv pä{lprv1 Ghwhuplqh +i  j,  k | i  +j  k,1

. F Vroxflöq= Fodudphqwh vh yh txh Grpk @ U/ Uhfk @ U. 3 / Grpj @`4> 4^ / Uhfj @ U3 / Grpi @ ^4> 4^1 Sdud kdfhu od frpsxhvwd hv dsursldgr kdfhu ho gldjudpd gh od jxud vljxlhqwh

Grp+j  k, @ i{ 5 U = {5 5`  4> 4^F j @ i{ 5 U = {5  4j @ `  4> 4^F s Grp+i  j, @ i{ 5`  4> 4^F = s{5  4 5 ^4> 4^j @ i{ 5`  4> 4^F = {5  4  4j @ `  4> 4^F

Grp++i  j,  k, @ @ @ @

Grp+i  +j  k,, @ @ @ @

i{ 5 Grpk = {5 5 Grp+i  j,j i{ 5 U = {5 5`  4> 4^F j i{ 5 U = {5  4j `  4> 4^F

i{ 5 Grp+j  k, s = +j  k,+{, 5 Grpij i{ 5`  4>s4^F = {7  4 5 ^4> 4^j i{ 5 U = {7  4  4j `  4> 4^F

Dvð/ vh wlhqh txh wdqwr +i  j,  k frpr i  +j  k, vrq ixqflrqhv frq ljxdo grplqlr hq hvwh fdvr1 Dkrud d qlyho gh iöupxodv fdofxohprv ++i  j,  k,+{,1 Frpr s 5 +i  j,+{, @ i+ ss{  4, {5  4 . 4  5 @

FKDSWHU 51 IXQFLRQHV

477 whqhprv txh

+i  j,  k,,+{, @ +i  j,+{5 , ss {7  4 . 4  5 @ Sru rwud sduwh hq ho fäofxor gh +i  +j  k,,+{, whqhprv

+i  +j  k,,+{, @ is++j  k,+{,, @ t+j  k,+{, . 4  5 s +k+{,,5  4 . 4  5 @ ss {7  4 . 4  5 @

Dvð/

i  +j  k, @ +i  j,  k |d txh wlhqhq ljxdo grplqlr/ uhfruulgr | uhodflöq +iöupxod, txh odv ghqh1 Hq orv gldjudpdv vljxlhqwhv vh ylvxdol}dq orv sdvrv khfkrv hq ho ghvduuroor1

`  4> 4^F $ U. ^4> 4^$ ^5> 4^ U $ U. 3 > 3 >

j

k

Sdud i  +j  k,

`  4> 4^F $ U. ^4> 4^$ ^5> 4^ 3 >

jk

i

i +jk,

| sdud +i  j,  k

`  4> 4^F $ ^5> 4^

`  4> 4^F $ ^5> 4^ U $ U. 3 >

k

i j

i j,k,

`  4> 4^F $ +

Hmhuflflr 44< Vhdq

i

i+{, @

Ghwhuplqh i  j | j  i 1

^5> 4^

 5  s { > {A3 {  4> { A 4 j+{, @ 5{ . 4> {  3 {4 {6 >

Vroxflöq= Hq sulphu oxjdu ghwhuplqhprv +j  i,+{,1  s j+ {  4,> { A 4 +j  i ,+{, @ j+i+{,, @ j+{6 ,> {4 s 6 Shur od ixqflöq j wlhqh glvwlqwrv ydoruhv ghshqglhqgr vl {  4 | { vrq pd|ruhv r phqruhv txh fhur1 Frpr s {  4 A 3/ sdud wrgr { A 4 vh whqguä txh s s j+ {  4, @ + {  4,5 > vl { A 4 Shur {6 wlhqh grv frpsruwdplhqwrv hq ho lqwhuydor `  4> 4`/ d vdehu/ {6 A 3 vl { 5`3> 4` | {6  3 vl { 5`  4> 3`1 Dvð/ ; s ? + {  4,5 > { A 4 { 5`3> 4` +j  i,+{, @ +{6 ,5 > = 5+{6 , . 4> {  3 Ilqdophqwh fdofxohprv +i  j,+{,  {A4 i +{5 ,> +i  j,+{, @ i +j+{,, @ i +5{ . 4,> {  4

Shur od ixqflöq i wlhqh wdpelìq glvwlqwrv ydoruhv ghshqglhqgr vl {5 | 5{ . 4 vrq pd|ruhv r phqruhv txh xqr1 Fodudphqwh {5 A 4 vl { A 4 | 3 ? {5  4 vl 3 ? {  41 Sru rwur odgr 5{ . 4  4 vl {  31 Frq wrgd hvwd lqirupdflöq whqhprv txh ; 5 6 3?{4 ? +{ s , > +i  j,+{, @ {5  4> { A 4 = +5{ . 4,6 > {  3

5181 DOJHEUD GH IXQFLRQHV

51815

478

Ixqflrqhv Fuhflhqwhv | Ghfuhflhqwhv

D frqwlqxdflöq gduhprv xqdv ghqlflrqhv txh qrv vhuäq ýwlohv sdud uhvroyhu lqhfxdflrqhv/ |d txh hoodv wlhqhq txh yhu frq or txh vxfhgh do dsolfdu xqd ixqflöq d dperv odgrv gh xqd lqhfxdflöq/ sru hmhpsor hohydu do fxdgudgr1 Ghqlflöq 518 Xqd ixqflöq i = D $ E vh glfh fuhflhqwh hq D/ vl hv yhugdghud od vljxlhqwh lpsolfdflöq= +;x> y 5 D,+x ? y , i+x,  i +y,,>

ghfuhflhqwh hq D=

+;x> y 5 D,+x ? y , i+x,  i +y,,>

hvwulfwdphqwh fuhflhqwh hq D= hvwulfwdphqwh ghfuhflhqwh hq D=

+;x> y 5 D,+x ? y , i+x, ? i +y,,> +;x> y 5 D,+x ? y , i+x, A i +y,,=

Hv ghflu/ xqd ixqflöq i vhuä hvwulfwdphqwh ghfuhflhqwh vl lqylhuwh ho rughq/ hvwulfwdphqwh fuhflhqwh vl frqvhuyd ho rughq1

Hmhpsor 453 Od ixqflöq i +{, @ p{ . e hv hvwulfwdphqwh ghfuhflhqwh vl p hv qhjdwlyr1 Hq hihfwr/ vhdq x> y 5 U/ hqwrqfhv x ? y , px A py , px . e A py . e , i +x, A i+y,

Fodudphqwh/ i hv fuhflhqwh vl p A 31 Vl p @ 3 vh wlhqh od ixqflöq frqvwdqwh i +{, @ e txh hv fuhflhqwh | ghfuhflhqwh d od yh}1 Juäfdphqwh vh lqwhusuhwd frpr vljxh= Xqd ixqflöq vhuä hvwulfwdphqwh fuhflhqwh vl hvwdqgr sdudgr hq fxdotxlhu sxqwr gh od juäfd gh od ixqflöq vh frplhq}d d fdplqdu vreuh od juäfd hq od gluhfflöq srvlwlyd gho hmh R[/ qhfhvduldphqwh vh frplhq}d d vxelu hq ho juäfr1 Hq ho fdvr gh fuhflhqwh vöor vh sxhgh dvhjxudu txh qr edmd1 1

Hmhpsor 454 Od ixqflöq i+{, @ Hq jhqhudo=

s { hv hvwulfwdphqwh fuhflhqwh sxhv sdud wrgr {> | 5 U. s s { ? | , { ? |=

s Hmhpsor 455 Od ixqflöq i+{, @ q { frq q 5 Q hv hvwulfwdphqwh fuhflhqwh1 Hq hihfwr hq od vhfflöq gh qýphurv uhdohv vh sureö sru ho frurodulr 414; txh s s +;{> | 5 U. ,+{ ? | , q { ? q |, Hmhpsor 456 Od ixqflöq i+{, @ ^{` hv fuhflhqwh1 Hq hihfwr +;{> | 5 U,+{ ? | , ^{`  ^|`,

shur qr hv hvwulfwdphqwh fuhflhqwh sxhv vl { @ 4> | @ 4> 8

{ ? | shur ^{` @ ^|` @ 4

FKDSWHU 51 IXQFLRQHV

479

Hmhpsor 457 Od ixqflöq i+{, @ {5 hv hvwulfwdphqwh fuhflhqwh hq U. / sxhv vl {> | 5 U. wdo txh { ? | hqwrqfhv {5 ? |5 > shur qr hv hvwulfwdphqwh fuhflhqwh hq U/ sxhvwr txh h{lvwhq { @ 5 5 U/ | @ 4 5 U wdohv txh { ? | shur {5 9? | 5 = Sru rwud sduwh/ qrwhprv txh od ixqflöq i +{, @ {5 hv hvwulfwdphqwh ghfuhflhqwh hq U. / sxhv vl {> | 5 . U wdo txh { ? | hqwrqfhv {5 ? |5 oxhjr {5 A | 5 / hv ghflu/ +;{> | 5 U. ,+{ ? | , i +{, A i+|,,

Sursrvlflöq 519 Vl i | j vrq ixqflrqhv hvwulfwdphqwh fuhflhqwhv wdohv txh Uhfi  Grpj hqwrqfhv j  i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh ghqlgd hq Grpi 1 Ghprvwudflöq= Hv fodur txh vl Uhfi  Grpj hqwrqfhv Grpj  i @ Grpi 1 Vhd {> | 5 Grpj  i wdo txh { ? | hqwrqfhv/ gdgr txh i hv hvwulfwdphqwh fuhflhqwh whqhprv i +{, ? i+|,

| sxhvwr txh j wdpelìq hv hvwulfwdphqwh fuhflhqwh hqwrqfhv hq frqvhfxhqfld

j+i +{,, ? j+i +|,, { ? | , +j  i,+{, ? +j  i ,+|,



Hmhuflflr 458 Suxheh txh vl i hv xqd ixqflöq fuhflhqwh hq G hqwrqfhv i hv xqd ixqflöq ghfuhflhqwh hq G1

Vroxflöq= Vhd {> | 5 G wdo txh { ? | hqwrqfhv i+{,  i+|, sxhv i hv fuhflhqwh/ pxowlsolfdqgr sru 4 vh wlhqh i +{,  i +|,

hq frqvhfxhqfld

Sru or wdqwr i hv ghfuhflhqwh hq G1

{ ? | , i+{,  i+|,

Hmhuflflr 459 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh1 41 Vhdq i | j ixqflrqhv wdohv txh Grpi _ Grpj 9@ >/ hqwrqfhv

Uhf+i . j, @ Uhfi ^ Uhfj=

51 Vhdq i / j | k ixqflrqhv wdohv txh G @ Grpi @ Grpj @ Grpk 9@ >/ hqwrqfhv +i . j,k @ i k . jk hq G k+i . j, @ ki . kj hq G

61 Vl Uhfk _ Grp+i . j, 9@ > | G @ Grp+i . j,  k/ hqwrqfhv

+i . j,  k @ i  k . j  k hq G

71 Vhd G @ Grpj _ Grpk 9@ >1 Vl Uhf+j . k,  Grpi / hqwrqfhv

+i  +j . k,,+{, @ +i  j,+{, . +i  k,+{,> ;{ 5 G

81 Vl i | j vrq ixqflrqhv ghfuhflhqwhv hq U hqwrqfhv i  j hv ghfuhflhqwh hq U1

5181 DOJHEUD GH IXQFLRQHV

47:

91 Vl i hv fuhflhqwh hq `  4> d` | hq `d> 4^ hqwrqfhv i hv fuhflhqwh hq U

:1 Vhd i xqd ixqflöq frq grplqlr G h Lg od ixqflöq lghqwlgdg: / hqwrqfhv i  Lg @ Lg  i @ i

;1 Vhd i ixqflöq/ hqwrqfhv h{lvwh j ixqflöq wdo txh

j  i @ Lg> ;{ 5 Grpi

Vroxflöq=

41 Idovr1 Vhdq/ sru hmhpsor/

j = ^4> 4^$ U i = U $ U s { $ 4 { $ {4

ixqflrqhv1 Uhfi @ i4j/ Uhfj @ ^3> 4^1 Oxhjr Sru rwud sduwh/ fx|r uhfruulgr hv ^4> 4^1

Uhfi ^ Uhfj @ ^3> 4^^i4j

i . j = ^4> 4^$ s U {44 { $

51 Yhugdghur1 Vhd { 5 G/ hqwrqfhv/

oxhjr/ gh grqgh Dqäorjdphqwh vl { 5 G>

oxhjr/ gh grqgh

++i . j,k,+{, @ @ @ @ @

+i . j,+{,k+{, +i+{, . j+{,,k+{, i+{,k+{, . j+{,k+{, +ik,+{, . +jk,+{, +ik . jk,+{,

++i . j,k,+{, @ +i k . jk,+{, ;{ 5 G +i . j,k @ i k . jk

+k+i . j,,+{, @ @ @ @ @

k+{,+i . j,+{, k+{,+i+{, . j+{,, k+{,i +{, . k+{,j+{, +ki,+{, . +kj,+{, +ki . kj,+{,

+k+i . j,,+{, @ +ki . kj,+{, ;{ 5 G k+i . j, @ ki . kj hq G

: Hq uhdolgdg ho vðperor Lg hvwä ghqrwdqgr ho frqfhswr gh ixqflöq lghqwlgdg / shur jhqhudophqwh vhuäq ixqflrqhv glvwlqwdv/ ghshqglhqgr gho Grplqlr gh ghqlflöq1 Dvðkd| LgU / Lgdf>o hwf1

FKDSWHU 51 IXQFLRQHV

47; 61 Yhugdghur1 Vhd { 5 G/ hqwrqfhv>

oxhjr/ gh grqgh

++i . j,  k,+{, @ @ @ @

+i . j,+k+{,, i +k+{,, . j+k+{,, +i  k,+{, . +j  k,+{, +i  k . j  k,+{,

++i . j,  k,+{, @ +i  k . j  k,+{, ;{ 5 G +i . j,  k @ i  k . j  k hq G

71 Idovr1 Frqvlghuh odv vljxlhqwhv ixqflrqhv

i +{, @ { . 4 j+{, @ {5

k+{, @ 5{

hqwrqfhv Grp+j . k, @ Grpj _ Grpk @ U 9@ > | Uhf+j . k,  U @ Grpi / dghpäv +i  +j . k,,+{, @ @ @ @

i ++j . k,+{,, i +j+{, . k+{,, i +{5 . 5{, {5 . 5{ . 4

Shur +i  j,+{, @ i +j+{,, @ {5 . 4 | +i  k,+{, @ i+k+{,, @ 5{ . 4/ oxhjr/ sru or wdqwr

+i  j,+{, . +i  k,+{, @ {5 . 5{ . 5>

81 Idovr1 Odv ixqflrqhv i | j wdo txh

i  +j . k, 9@ i  j . i  k i +{, @ j+{, @ {

vrq ghfuhflhqwhv hq U/ vlq hpedujr ixqflöq fuhflhqwh1

91 Idovr1 Vhd

i  j+{, @ i+j+{,, @ i +{, @ {

i+{, @



{ {6

vl { 5`  4> 4` vl { 5`4> 4^

i+{, @ { hv fuhflhqwh hq `  4> 4` | i +{, @ {  6 hv fuhflhqwh hq `4> 4^1 Vlq hpedujr i qr hv fuhflhqwh `  4> 4`^`4> 4^@ U sxhv +sru hmhpsor,/ h{lvwh { @ 3 5 U/ | @ 5 5 U wdohv txh { ? | shur 3 @ i +{, 9? i +|, @ 41

:1 Yhugdghur1 Vhd { 5 G |

oxhjr/ ö

+i  Lg,+{, @ i +Lg+{,, @ i +{, ;{ 5 G

+Lg  i ,+{, @ Lg+i+{,, @ i +{, ;{ 5 G i  LgG @ LgUhf i  i @ i i  Lg @ Lg  i @ i

5181 DOJHEUD GH IXQFLRQHV

47<

;1 Idovr1 Vhd i xqd ixqflöq ghqlgd sru

i = U $ U { $ 4

| vxsrqjdprv txh h{lvwh xqd ixqflöq j wdo txh Uhfi  Grpj | j  i @ Lg/ hqwrqfhv Vl { @ 4 vh wlhqh txh Vl { @ 5/ vh wlhqh txh

+j  i,+{, @ {> ;{ 5 Grpi @ U +j  i,+4, @ j+i +4,, @ j+4, @ 4

+j  i ,+5, @ j+i+5,, @ j+4, @ 5>

hv ghflu/ j+4, wlhqh grv ydoruhv 4 | 5 gh grqgh j qr hv ixqflöq/ or txh frqwudglfh ho vxsxhvwr1

Dqäorjdphqwh/ vl i hv ixqflöq gh D hq E qr qhfhvduldphqwh h{lvwh xqd ixqflöq j wdo txh i  j @ LgE > ;{ 5 Grpj=

Vlq hpedujr/ edmr flhuwdv frqglflrqhv hv srvleoh txh hvwr rfxuud/ hv ghflu/ txh sdud xqd ixqflöq i gh D hq E h{lvwd j gh E hq D +oodpdgd od ixqflöq lqyhuvd gh i, wdo txh i  j @ LgE

|

j  i @ LgD

Yhuhprv päv dghodqwh txh xqd ixqflöq wlhqh lqyhuvd vl | vöor vl hv lq|hfwlyd | hsl|hfwlyd1 Ho frqfhswr gh lq|hfwlylgdg ; lqglfd txh d wrgr hohphqwr gho uhfruulgr gh i oh fruuhvsrqgh xqd ýqlfd suhlpdjhq hq ho grplqlr gh i 1 Ho gh hsl|hfwlylgdg lqglfd txh d wrgr hohphqwr gho frqmxqwr gh oohjdgd oh fruuhvsrqgh sru or phqrv xqd suhlpdjhq/ hq rwudv sdodeudv/ ho frqmxqwr gh lpäjhqhv gh i +uhfruulgr gh i , gheh vhu ljxdo do frqmxqwr gh oohjdgd1 Ghqlflöq 51: Xqd ixqflöq i = D $ E vh glfh lq|hfwlyd vl | vöor vl +;x> y 5 D,+i+x, @ i +y, , x @ y,

Hmhpsor 45: Vhd i od ixqflöq ghqlgd sru

i = U  i4j $ U { $ 5{{.4 4

Surehprv txh i hv xqd ixqflöq lq|hfwlyd1

Vroxflöq= Hq hihfwr= Vhdq x> y 5 U  i4j wdohv txh i +x, @ i+y,1 Vh wlhqh=

oxhjr/ ; Rwur

i +x, 5x.4 , x4 , +5x . 4,+y  4, , 5xy  5x . y  4 , x

@ @ @ @ @

i +y, 5y .4 y4 +5y . 4,+x  4, 5yx  5y . x  4 y

i +x, @ i +y, , x @ y> ;x> y 5 U  i4j

qrpeuh xwlol}dgr hq od olwhudwxud frpr vlqöqlpr gh lq|hfwlyd hv 404 +vh ohh xqr d xqr ,1

FKDSWHU 51 IXQFLRQHV

483 Hmhuflflr 45; Vhd i od ixqflöq ghqlgd sru

Ghwhuplqh vl i hv xqd ixqflöq lq|hfwlyd

i = U $ U { $ {5 . { . 4

Vroxflöq= i qr hv lq|hfwlyd/ hq hihfwr i+4, @ i +5, @ 6/ shur 4 9@ 51 Hv ghflu d | @ 6 5 Uhfi oh fruuhvsrqghq grv suhlpäjhqhv= 4/051 Revhuyhprv od juäfd gh i hq ho sulphu juäfr vljxlhqwh

Qrwhprv txh do wud}du xqd uhfwd krul}rqwdo/ ìvwd sxhgh fruwdu d od fxuyd hq grv sxqwrv +h{fhswr hq | @ 6@7 d txlhq vöor oh fruuhvsrqgh od suhlpdjhq {@0425, Fodudphqwh vh revhuyd txh vl vh uhvwulqjh od ixqflöq do lqwhuydor ^4@5> 4^ ö do lqwhuydor `  4> 4@5`/ hqwrqfhv vh rewlhqhq ixqflrqhv lq|hfwlydv/ hv ghflu/ i = ^4@5> 4^$ U { $ {5 . { . 4

|

i = `  4> 4@5^$ U { $ {5 . { . 4

vrq dpedv ixqflrqhv lq|hfwlydv +juäfrv fhqwudo | ghuhfkr gh jxud dqwhulru uhvshfwlydphqwh,1 Ghwhuplqhprv dqdoðwlfdphqwh hq txì lqwhuydor i hv lq|hfwlyd Vhdq x> y 5 U wdo txh i+x, @ i+y,1 vh wlhqh= i +x, @ i+y, +, x5 . x . 4 @ y 5 . y . 4 +, x5  y 5 @ y  x +, +x  y,+x . y, . +x  y, @ 3 +, +x  y,+x . y . 4, @ 3 +, x@y b y.x.4@3 Qrwhprv txh x @ y a x . y . 4 @ 3/ vl x @ y @ 4@51 Exvfduhprv xq vxefrqmxqwr G gho grplqlr gh i hq ho fxdo vh fxpsod=

x @ y | x . y . 4 9@ 3 ;x> y 5 G  i 45 j

Qrwhprv txh , , , ,

x @ y a x . y . 4 9@ 3 x @ y a +x . y . 4 A 3 b x . y . 4 ? 3, +x @ y a x . y . 4 A 3, b +x @ y a x . y . 4 ? 3, 5x . 4 A 3 b 5x . 4 ? 3 x A 4@5 b x ? 4@5

5181 DOJHEUD GH IXQFLRQHV

484

shur 4@5 shuwhqhfh do grplqlr gh lq|hfwlylgdg1 Oxhjr/ srghprv frqvlghudu frpr grplqlrv gh lq|hfwlylgdg= G @`  4> 4@5`

r elhq

G @ ^4@5> 4^

Dvð/

i mG = `  4> 4@5` $ U i mG = ^4@5> 4^$ U { $ {5 . { . 4 { $ {5 . { . 4

vrq dpedv ixqflrqhv lq|hfwlydv frpr |d vh ylr juäfdphqwh< Hmhuflflr 45< Ghpxhvwuh txh i = U $ U { $ {6 hv xqd ixqflöq lq|hfwlyd1

Vroxflöq= Vhdq x> y 5 Grpi @ U wdohv txh i+x, @ i +y, vh wlhqh i +x, @ +, x6 @ 6 +, x  y6 @ 5 +, +x  y,+x . xy . y5 , @ +, x@y b

i +y, y6 3 3 x5 . xy . y5 @ 3

Shur ho glvfulplqdqwh gh od hfxdflöq gh vhjxqgr judgr x5 . xy . y5 hq od yduldeoh x/ hv  @ y5  7y5 @ 6y 5 ? 3> ;y 5 U  i3j

oxhjr/

x5 . xy . y5 A 3> ;x> y 5 U  i3j

Sru or wdqwr x @ y1 Rwud irupd gh yhu txh x5 . xy . y5 9@ 3 hv

4 6 x5 . xy . y5 @ +x . y,5 . y 5 A 3 ;x> y 5 U  i3j 5 7

Hmhuflflr 463 Ghwhuplqh vl

j = U $ U { $ j+{, @ {5{.8 hv lq|hfwlyd1 Hq fdvr gh qr vhuor kdfhu odv uhvwulfflrqhv qhfhvduldv sdud txh or vhd1

Vroxflöq= Vhdq x> y 5 Grpj @ U wdohv txh j+x, @ j+y, vh wlhqh vxfhvlydphqwh= j+x, x +, x5 .8 5 +, x+y . 8, +, xy5 . 8x +, xy5  yx5 +, xy+y  x, +, xy+y  x,  8+y  x, +, +y  x,+xy  8, +, x@y

@ @ @ @ @ @ @ @ b

j+y, y y5 .8 y+x5 . 8, yx5 . 8y 8y  8x 8+y  x, 3 3 xy  8 @ 3

Revhuyhprv txh j qr hv 404 +lq|hfwlyd,/ sxhv j+4, @ j+8, @ 4@9 | 4 9@ 81 <

iG ƒ

lqglfd

i

uhvwulqjlgd do grplqlr

G

1

FKDSWHU 51 IXQFLRQHV

485

Dkrud ghwhuplqhprv xq vxefrqmxqwr G gho grplqlr grqgh j vhd lq|hfwlyd1 Qrwhprv txh x@y

vl

x@y@

a xy @ 8

s s 8 b x@y@ 8

Sdud ghwhuplqdu ho grplqlr gh lq|hfwlylgdg vh gheh yhulfdu x @ y a xy 9@ 8=

x @ y a xy 9@ 8 +, +, +, +, +, +,

Dkrud elhq=

x @ y a +xy A 8 b xy ? 8, +x @ y a xy A 8, b +x @ y a xy ? 8, x5 A 8sb x5 ? 8 s m x mAs 8 b m x m?s 8 s s +x A 8 b s x ? s 8, b + 8 ? s x asx ? 8, x 5`  4>  8^^` 8> 4^ b x 5`  8> 8^

s s Shur  8 | 8 shuwhqhfhq do grplqlr gh lq|hfwlylgdg/ oxhjr/ s s j4 = ^ 8> 8` $ U { $ j+{, @ {5{.8 | s s j5 =`  4>  8` ^ ^ 8> 4^ $ U { $ j+{, @ {5{.8 vrq dpedv ixqflrqhv lq|hfwlydv1

Revhuyhprv ho juäfr gh j hq

Hmhuflflr 464 Vhdq i = D $ E | j = E $ F ixqflrqhv | k @ j  i1 Ghpxhvwuh txh vl i | j vrq lq|hfwlydv/ hqwrqfhv k wdpelìq hv lq|hfwlyd1 Vroxflöq= Vhdq x> y 5 Grpk wdohv txh k+x, @ k+y,1 Ghehprv ghprvwudu txh x @ y k+x, @ k+y, +, +j  i,+x, @ +j  i,+y, +, j+i+x,, @ j+i +y,,

Frpr j hv 404 vh wlhqh hqwrqfhv txh i+x, @ i +y,/ | vlhqgr i lq|hfwlyd/ uhvxowd txh x @ y or txh ghpxhvwud od sursrvlflöq

Hmhuflflr 465 Vhdq i = D $ E | j = E $ F ixqflrqhv | k @ j  i 1 Ghpxhvwuh txh vl k hv lq|hfwlyd/ hqwrqfhv i wdpelìq or hv1 Vroxflöq= Vhdq x> y 5 Grpi @ D wdohv txh i +x, @ i +y,1 Ghehprv qxhydphqwh ghprvwudu txh x @ y1 Frpr i+x, @ i +y, | j hv ixqflöq whqguhprv txh j+i+x,, @ j+i +y,, gh grqgh k+x, @ k+y,/ shur frpr k hv 404 vh wlhqh txh x @ y1

Sursrvlflöq 51; Vl i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh r elhq hvwulfwdphqwh ghfuhflhqwh/ hqwrqfhv i hv lq|hfwlyd1

5181 DOJHEUD GH IXQFLRQHV

486

Ghprvwudflöq= Vlq sìuglgd gh jhqhudolgdg +v1s1j1,/ vxsrqjdprv txh i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh/ hv ghflu/ +;x> y 5 Grpi ,+x ? y , i +x, ? i +y,, +51<, | vxsrqjdprv dghpäv txh i qr hv lq|hfwlyd/ sru or wdqwr h{lvwluäq x> y 5 Grpi wdohv txh i +x, @ i+y, a x 9@ y

gh grqgh

xAybx?y

or txh frqwudglfh 51<1 Od ghprvwudflöq hv dqäorjd vl i hv hvwulfwdphqwh ghfuhflhqwh/ txhgd sdud ho ohfwru uhsurgxfluod1  Qrwh txh ho uhfðsurfr gh od sursrvlflöq dqwhulru qr hv flhuwr1 Xqd ixqflöq lq|hfwlyd qr hv qhfhvduldphqwh hvwulfwdphqwh fuhflhqwh r hvwulfwdphqwh ghfuhflhqwh1 Sru hmhpsor/ frqvlghuh od ixqflöq | @ 4@{ uhsuhvhqwdgd sru od juäfd gh od klsìuerod htxloäwhud> hvwd ixqflöq hv lq|hfwlyd shur qr hv hvwulfwdphqwh fuhflhqwh ql hvwulfwdphqwh ghfuhflhqwh Ghqlflöq 51< Xqd ixqflöq i = D $ E vh glfh hsl|hfwlyd r vreuh|hfwlyd vl | vöor vl Uhfi @ E1 Hmhpsor 466 Frqvlghuhprv od ixqflöq i ghqlgd sru

$ U i = U. 3 s { $ i +{, @ {

Hvwd ixqflöq qr hv hsl|hfwlyd/ sxhv Uhfi @ U. 9@ U1 Vlq hpedujr/ vl vh uhvwulqjh ho frqmxqwr gh oohjdgd d 3 / hqwrqfhv U. 3 $ U. i = U. 3 3 s { $ i +{, @ { vðhv xqd ixqflöq hsl|hfwlyd1

Revhuyh txh xqd ixqflöq i = D $ Uhfi hv vlhpsuh hsl|hfwlyd1

Hmhpsor 467 Frqvlghuhprv od ixqflöq i ghqlgd sru

i = `  4> 4^$ U { $ i+{, @

s {5 4

4

Uhvwulqmdprv grplqlr | uhfruulgr gh prgr txh i vhd lq|hfwlyd | hsl|hfwlyd1

Vroxflöq= Vhdq x> y 5 Grpi @`  4> 4^ wdohv txh i +x, @ i +y, vh wlhqh= i +x, @ i +y, s44 x5 @ s44y5 , s s +, 4  y5 @ 4  x5 +, +y  x,+x . y, @ 3 +, x@y b x.y @3

Qrwhprv txh vl x @ 3 hqwrqfhv vh yhulfd txh x @ y a y . x @ 31 Hv fodur txh orv lqwhuydorv gh lq|hfwlylgdg gh i vrq ^3> 4^ | `  4> 3`/ hv ghflu/ odv ixqflrqhv i4 = `  4> 3` $ U { $ i+{, @

vrq dpedv ixqflrqhv lq|hfwlydv1

s {5 4

4

i5 = ^3> 4^$ U { $ i+{, @

Hmhpsor 468 Frqvlghuhprv i5 | ghwhuplqhprv vx uhfruulgr1

s {5 4

4

FKDSWHU 51 IXQFLRQHV

487 Vhd { 5 ^3> 4^ h | 5 U wdo txh | @ i +{,/ vh wlhqh=

dkrud elhq/ frpr { 5 ^3> 4^/ hqwrqfhv

shur | A 3/ oxhjr/ |  41 Dvð/

+, +, +, +, +, +,

| +, | +, | 5 +, {5

@ i +{, 4 @ s4 {5 4 @ 4{5 5 4 @ | | 5

|A3

t 5 | 4 ? 4 |5 5 |  4 3  |5 ? 4 3  | 5  4 ? |5 3  | 5  4 ? |5 3  | 5  4 a |5  4 ? | 5 4  | 5 a 4 ? 3 |5  4

frq | 5  4  3

3

i = ^3> 4^$ ^4> 4^ { $ i +{, @

hv xqd ixqflöq lq|hfwlyd | vreuh|hfwlyd

s {5 4

4

Ghqlflöq 5143 Xqd ixqflöq i = D $ E vh glfh el|hfwlyd vl | vöor vl hv lq|hfwlyd | vreuh|hfwlyd1

Hmhpsor 469 Odv ixqflrqhv vljxlhqwhv vrq el|hfwlydv

i = U. $ U 3 3 s { $ i +{, @  { $ U. j = U 3 3 { $ j+{, @ {5 k = U $ U { $ k+{, @ {6

Qrwh txh/ sdud i | j/ vh fxpsoh=

|

5 +i  j,+{, @ i+j+{,, @ i +{ s, @  {5 @ m{m @ +{, @ {

hv ghflu/ Dqäorjdphqwh sdud k | o gh grqgh

o = U $ U s { $ o+{, @ 6 { ;{ 5 U 3

s s +j  i,+{, @ j+i +{,, @ j+ {, @ + {,5 @ { ;{ 5 U. 3 i  j @ LgU 3

a j  i @ LgU. 3

+k  o,+{, @ { > +o  k,+{, @ { ;{ 5 U k  o @ o  k @ LgU

Odv ixqflrqhv txh fxpsohq frq hvwd surslhgdg vh glfhq lqyhuwleohv/ vhjýq vh h{solfd d frqwlqxdflöq1

5181 DOJHEUD GH IXQFLRQHV

488

Surslhgdg 5144 Vl i = D $ E hv el|hfwlyd/ hqwrqfhv h{lvwh xqd ýqlfd i 4 gh E hq D wdo txh d, i  i 4 @ LgE | i 4  i @ LgD

e, +i 4 ,4 @ i

Od ixqflöq i 4 gh od sursrvlflöq dqwhulru hv ýqlfd | sxhgh ghqluvh gluhfwdphqwh frpr vljxh=

Ghqlflöq 5145 Vl i = D $ E hv xqd ixqflöq el|hfwlyd/ hqwrqfhv od ixqflöq lqyhuvd gh i vh ghqh frpr i 4 = E $ D | $ { @ i 4 +|,

grqgh

| @ i +{,

r

{ @ i 4 +|,

Hq ho hmhpsor dqwhulru od lqyhuvd gh i hv j | od lqyhuvd gh k hv o1 Uhfðsurfdphqwh od lqyhuvd gh j hv i | od

lqyhuvd gh o hv k1 Revhuyhprv odv juäfdv gh i | j 43 hq od jxud Sdud rewhqhu i 4 gh xqd ixqflöq i el|hfwlyd ghehprv ghvshmdu { gh od hfxdflöq | @ i +{,/ frpr vh h{solfd hq orv vljxlhqwhv hmhpsorv1 Hmhpsor 46: Hq ho hmhpsor 467 ylprv txh od ixqflöq

hv el|hfwlyd | rewxylprv txh> oxhjr/

i = ^3> 4^$ ^4> 4^ { $ i+{, @

4

4

| @ it+{, 5 4 +, { @ | | 5

{ @ i 4 +|, @ Sru or wdqwr

s {5

v

|5  4 |5

i 4 = ^4> 4^$ ^3> 4^ | $

i 4 +|, @

t 5 | 4 |5

43 Gdgd od juäfd gh i hq xqd krmd wudqvsdudqwh/ sdud frqrfhu od juäfd gh vx lqyhuvd edvwd urwdu orv hmhv hq eD frqwud uhorm | gdu yxhowd od krmd | pludu sru ghwuäv od jxud> hvwd ýowlpd fruuhvsrqghuä d od juäfd gh od lqyhuvd gh i 1

FKDSWHU 51 IXQFLRQHV

489

Hv frvwxpeuh uhvhuydu od ohwud { sdud orv hohphqwrv gho grplqlr gh xqd ixqflöq/ oxhjr/ or dqwhulru vh hvfuleh i 4 = ^4> 4^$ ^3> 4^

Hmhuflflr 46; Vhdq

ixqflrqhv1

{ $ i 4 +{, @

$ ^4> 4^s i = U. 3 { $ i +{, @ {  4

t

{5 4 {5

j = U $ U { $ j+{, @ { . 8

d, Suxheh txh i hv el|hfwlyd | ghwhuplqh i 4 1

e, Hqfxhqwuh xqd ixqflöq/ k wdo txh k  i @ j1

Vroxflöq= d1l, Sureduhprv txh i hv lq|hfwlyd1 wdo txh i+x, @ i +y, hqwrqfhv= Vhd x> y 5 U. 3 s s x ys 4 s 4 @ x @ y +, +, x @ y

Oxhjr i+x, @ i +y, , x @ y> ;x> y 5 U. 3 ll, Vhd { 5 U. h | 5 ^4> 4^ wdo txh | @ i+{, hqwrqfhv 3 s | @ s{  4 +, |.4 @ { > |.43 +, +| . 4,5 @ {

od h{suhvlöq +| . 4,5 hvwä ghqlgd sdud wrgr | hq U/ shur | . 4  3/ Sru or wdqwr i hv hsl|hfwlyd1 Dvðsru +l, | +ll, i hv el|hfwlyd1

Uhfi @ ^4> 4^

i 4 = ^4> 4^$ U. 3 { $ i 4 +{, @ +{ . 4,5

e, Sxhvwr txh Uhfi 4 @ U.  Grpj hqwrqfhv h{lvwh 3 od ljxdogdg k  i @ j/ hv ghflu/ ki +, k  +i  i 4 , +, k  Lg +, k

j  i 4 1 Oxhjr/ sdud ghvshmdu k dsolfdprv i 4 d @ @ @ @

Sru or wdqwr Grpk @ Grpj  i 4 @ Grpi 4 @ ^4> 4^ |

j j  i 4 j  i 4 j  i 4 =

k+{, @ j+i 4 +{,, @ j++{ . 4,5 , @ +{ . 4,5 . 8 k = ^4> 4^$ U { $ +{ . 4,5 . 8

Hmhuflflr 46< Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq Yhugdghudv r Idovdv

41 Vl j hv lqyhuwleoh/ hqwrqfhv j 4 @ 4j +grqgh 4 lqglfd od ixqflöq frqvwdqwh 4,1

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV 51 Vl j = D $ E hv lqyhuwleoh/ hqwrqfhv

48:

j  j4 @ j 4  j

61 Vl i = D $ E | j = E $ F vrq ixqflrqhv lqyhuwleohv/ hqwrqfhv j  i hv wdpelìq lqyhuwleoh | Vroxflöq=

+j  i ,4 @ i 4  j4

41 Idovr1 Od ixqflöq j ghqlgd sru hv lqyhuwleoh | vlq hpedujr/ oxhjr 51 Idovr1 Frqvlghuhprv od el|hfflöq

| vx lqyhuvd

hq fdpelr

. $ U j = U. 3 s3 { $ {

$ U. j 4 = U. 3 3 { $ {5

  4 4 j +{, @ j+{, @

s{ >

j 4 9@ j=

4

4 j

;{ 5 U.

^4> 4^$ s U. 3 { $ {4

$ ^4> 4^ j 4 = U. 3 { $ {5 . 4

+j  j4 ,+{, @ { ;{ 5 U. 3

+j 4  j,+{, @ { ;{ 5 ^4> 4^

oxhjr/ j  j 4 9@ j4  j srutxh wlhqhq glvwlqwrv grplqlrv1

61 Yhugdghud1 Vx frpsuredflöq txhgd gh hmhuflflr sdud ho ohfwru1

519

Odv ixqflrqhv H{srqhqfldohv | Orjduðwplfdv

Frqvlghuhprv ho vljxlhqwh sureohpd= Vl xqd edfwhuld hq flhuwr fxowlyr vh gxsolfd hq fdgd krud | ho qýphur gh edfwhuldv txh kdeuðd do lqlfldu ho h{shulphqwr hv xqd/ hqwrqfhv ho qýphur i+q, gh edfwhuldv txh kd| hq ho fxowlyr ghvsxìv gh q krudv hv i +q, @ 5q

Qrwh txh od h{suhvlöq dqwhulru hv xqd ixqflöq fx|r grplqlr hv Q/ hv ghflu i = Q $ U q $ 5q

Txhuhprv h{whqghu od ixqflöq i d xqd ixqflöq päv jhqhudo/ frq grplqlr U1 Shur/ Bctxì vhqwlgr wlhqh h{whqghu d od h{suhvlöq 5{ frq { 5 UB

FKDSWHU 51 IXQFLRQHV

48; Vdehprv txh vl { 5 Q/ hqwrqfhv Vl { 5 ] / { 5 ]. / hqwrqfhv Vl { @ p q 5 T/ hqwrqfhv

5{ @ 5  5    5 +{ yhfhv, 4 5{ @ { 5

s q 5{ @ 5p

s s Shur vl { 95 T/ hv ghflu/ { hv xq luudflrqdo/ sru hmhpsor/ { @ 5/ Bctxì vljqlfdgr wlhqh 5 5 B s Frqvlghuhprv odv dsur{lpdflrqhv ghflpdohv vxfhvlydv h lqihulruhv d 5 | iruphprv ho frqmxqwr D @ i54>7 > 54>74 > 54>747 > 54>7475 > = = =j

Sxhvwr txh D hv xq s5frqmxqwr dfrwdgr/ hqwrqfhv sru d{lrpd gho vxsuhpr h{lvwh VxsD1 Vhd VxsD @= 5 1 Hq jhqhudo vl e A 3 | { 5 U/ vh ghqh |

e{ =@ Vxsie{3 >{4 > e{3 >{4 {5 > e{3 >{4 {5 {6 > = = =j

e{ =@ Lqiie{3 >{4 > e{3 >{4 {5 > e{3 >{4 {5 {6 > = = =j

vl e A 4

vl 3  e  4

grqgh {3 > {4 > {3 > {4 {5 > {3 > {4 {5 {6 >    vrq odv dsur{lpdflrqhv ghflpdohv vxfhvlydv h lqihulruhv d { 5 U1 Oxhjr srghprv ghqlu od ixqflöq= i = U $ U { $ 5{

Od uhsuhvhqwdflöq juäfd gh hvwd ixqflöq hv xqd fxuyd gh wud}dgr frqwlqxr/ frpr vh yh hq ho judfr ghuhfkr vljxlhqwh

Dqäorjdphqwh/ j ghqlgd frpr

j = U $ U { $ + 45 ,{

vh uhsuhvhqwd juäfdphqwh sru od fxuyd gh wud}dgr frqwlqxr/ frpr vh sxhgh yhu hq ho juäfr ghuhfkr dqwhulru/ fx|r uhfruulgr hv U. 1 Hq jhqhudo srghprv ghqlu od ixqflöq h{srqhqfldo hq edvh e frq e A 3 frpr

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

48<

h{se = U $ U { $ e{

fx|d uhsuhvhqwdflöq juäfd hvwä hq od jxud

Qrwh txh vl e A 4 od ixqflöq h{se hv hvwulfwdphqwh fuhflhqwh | vl 3 ? e ? 4/ od ixqflöq h{se hv hvwulfwdphqwh ghfuhflhqwh1 Hq dperv fdvrv hv lq|hfwlyd | vx uhfruulgr hv U. 1 Revhuyhprv txh od fxuyd qr fruwd ho hmh {/ hv ghflu/ qr fruwd d od uhfwd | @ 31 Gluhprv txh od uhfwd | @ 3 hv xqd dvðqwrwd krul}rqwdo gh od ixqflöq | @ e{ 1 Hvwr vljqlfd txh vl e A 4 | { ghfuhfh lqghqlgdphqwh/ odv lpäjhqhv | @ e{ wlhqghq d fhur1 Hq hvwh fdvr hvfuleluhprv>

Vl { $ 4/ hqwrqfhv e{ $ 3

Hq fdpelr vl 3 ? e ? 4 | { fuhfh lqghqlgdphqwh/ hqwrqfhv odv lpäjhqhv | @ e{ wlhqghq d fhur1 Hq hvwh fdvr hvfuleluhprv

Hmhuflflr 473 Judtxh od ixqflöq

Vl { $ 4/ hqwrqfhv e{ $ 3 | @ 4  h{

Vroxflöq= Revhuyhprv od juäfd gh | @ h{ | gh | @ h{

FKDSWHU 51 IXQFLRQHV

493

Hq hvwh fdvr e @ 4h ? 41

Dvð/ | @ 4h{ hv xq wudvodflöq gh | @ h{ frpr vh pxhvwud hq od jxud

Qrwh txh od dvðqwrwd krul}rqwdo gh | @ 4  h{ hv dkrud | @ 4/ sxhv vl { $ 4 hqwrqfhv 4  h{ $ 4

1

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

494

Hmhpsor 474 Ghwhuplqh od juäfd gh od ixqflöq | @ de{ . f sdud orv glihuhqwhv ydoruhv srvleohv gh d/ e | f hq U

Vroxflöq= Frqvlghuduhprv glvwlqwrv fdvrv txh vh pxhvwudq hq odv juäfdv 10 -4

10 2 x

-2

4

-4

2 x

-2

0

0

-10

-10

-20

-20

-30

-30

d ? 3> e A 4> f A 3

d ? 3> e A 4> f ? 3

4

FKDSWHU 51 IXQFLRQHV

495

Surslhgdg 5146 Vl d 5 U. / e 5 U. | {> | 5 U/ hqwrqfhv l, d{ d| @ d{.| ll, d{ e{ @ +de,{ lll, +d{ ,| @ d{|  { { ly, de @ de{

Hmhpsor 475 Uhvroydprv odv vljxlhqwhv hfxdflrqhv h{srqhqfldohv 41 8{  6{4 @ 6{.4  8{4 51 5{.4 . 7{ @ ;3

Vroxflöq= 41

8{  6{4 @ 6{.4  8{4

oxhjr/ V @ i5j1

+, +, +, +,

{4 8{ . 8{4 @ 6{.4  . 64  4 { { 8 4. 8 @6 6. 6 8{ 9 @ 6{ 43  8{8  865  @ 65 6{  8 {  8 5 @ 6 +, 6 +, { @ 5

51 Frqvlghuhprv ho fdpelr gh yduldeohv x @ 5{ / hqwrqfhv 5{.4 . 7{ @ ;3 +, 5{.4 . 7{  ;3 @ 3 +, 5  5{ . +5{ ,5  ;3 @ 3 +, x5 . 5x  ;3 @ 3 +, +x . 43,+x  ;, @ 3 +, x @ 43 b x @ ;

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

496

Iljxuh 514= | @ vlqk+{, / | @ frvk+{, oxhjr/ V4 @ i43> ;j/ shur x A 3/ oxhjr/ V @ i;j/ oxhjr/ gh grqgh { @ 6

5{ @ ;

Od edvh h{srqhqfldo gh xvr päv iuhfxhqwh hv h1 Sru vx xvr wdq h{whqvr/ d od ixqflöq i +{, @ h{ vh oh oodpduä ixqflöq h{srqhqfldo1 Hq edvh d hvwd ixqflöq srghprv ghqlu od vljxlhqwhv ixqflrqhv= Ghqlflöq 5147 Vh ghqh od ixqflöq vhqr klshueöolfr frpr>

| frvhqr klshueöolfr frpr>

vhqk = U $ U { { { $ h 5h frvk = U $ U { { { $ h .5h

Odv juäfdv fruuhvsrqglhqwhv vrq odv txh vh pxhvwudq hq od jxud Khprv ylvwr txh vl e 9@ 4 od ixqflöq h{se hv xqd ixqflöq lq|hfwlyd frq uhfruulgr U. / oxhjr/ od ixqflöq h{se = U $ U. { $ e{

4 hv el|hfwlyd1 Sru or wdqwr h{lvwh vx lqyhuvd h{s e txh ghqrplqduhprv od ixqflöq orjdulwpr hq edvh e | hvfuleluhprv 4 h{s e @ orje hv ghflu/ od ixqflöq orjdulwpr hq edvh e vh ghqh frpr

grqgh

orje = U. $ U { $ orje +{, @ | | @ orje +{, +, { @ e|

FKDSWHU 51 IXQFLRQHV

497 Hmhpsor 476

41 orj5 +7, @ 5 sxhv 55 @ 71

51 orj5 + 47 , @ 5 sxhv 55 @ 47 1

61 orj 45 +7, @ 5 sxhv + 45 ,5 @ 71

Od uhsuhvhqwdflöq juäfd gh od ixqflöq orjdulwpr hv od txh pxhvwud od jxud Revhuydflöq= 41 Vl e A 4/ hqwrqfhv orje hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh1

51 Vl 3 ? e ? 4/ hqwrqfhv orje hv xqd ixqflöq hvwulfwdphqwh ghfuhflhqwh1 Dghpäv gh od juäfd hv fodur txh 61 Vl e A 4 | { A 4/ hqwrqfhv orje +{, A 31

71 Vl e A 4 | 3 ? { ? 4/ hqwrqfhv orje +{, ? 31

81 Vl 3 ? e ? 4 | 3 ? { ? 4/ hqwrqfhv orje +{, A 31 91 Vl 3 ? e ? 4 | { A 4/ hqwrqfhv orje +{, ? 3

Hvwd hv xqd exhqd lqirupdflöq sdud uhvroyhu dojxqdv lqhfxdflrqhv | judfdu ixqflrqhv uhodflrqdgdv frq ixqflrqhv orjduðwplfdv1 Sru hmhpsor/ ho frqmxqwr vroxflöq gh od lqhfxdflöq s orj 45 +{,  orj5 +{, . {  4 +5143, hv ho lqwhuydor ^4> 4^1 Hq hihfwr/ orv wìuplqrv hq + 5143, wlhqhq vljqlfdgr hq U vl { A 3 | {  4/ hv ghflu/ { 5 ^4> 4^ | frpr> s orj5 +{,  3 | {  4  3

sdud {  4/ hqwrqfhv Sru rwud sduwh

orj5 +{, .

s {  4  3 ;{ 5 ^4> 4^ orj 45 +{,  3

sdud {  4/ hq frqvhfxhqfld/ od ghvljxdogdg 5143 vh fxpsoh sdud wrgr { 5 ^4> 4^ oxhjr V @ ^4> 4^

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

498

Revhuyhprv txh hq { @ 3 od ixqflöq orje +{, qr hvwä ghqlgd1 Gluhprv txh od uhfwd { @ 3 hv xqd dvðqwrwd yhuwlfdo sdud od ixqflöq1 Xqd dvðqwrwd yhuwlfdo { @ {3 qrv lqglfd txh sdud { fhufdqrv d {3 odv lpäjhqhv i+{, fuhfhq r ghfuhfhq lqghqlgdphqwh1 Hq dojxqrv fdvrv/ sdud judfdu dsur{lpdgdphqwh od fxuyd txh uhsuhvhqwd d xqd ixqflöq/ hv vxflhqwh ghwhuplqdu ho grplqlr/ odv dvðqwrwdv/ odv lqwhuvhfflrqhv frq orv hmhv gh frrughqdgdv/ | hq txì lqwhuydorv od ixqflöq hv srvlwlyd +qhjdwlyd,1 Sru hmhpsor Judtxhprv od ixqflöq | @ i+{,/ grqgh Vroxflöq= Ghwhuplqhprv ho grplqlr

Grpi

Oxhjr

| @ orj 56 mm { . 5 m 8 m @ i{ 5 U =mm { . 5 m 8 mA 3j @ i{ 5 U =mm { . 5 m 8 m9@ 3j @ i{ 5 U = { 9@ 6 a { 9@ :j Grpi @ U  i:> 6j

Odv dvðqwrwdv yhuwlfdohv gh od ixqflöq sdvdq sru dtxhoorv sxqwrv grqgh ho dujxphqwr gho orjdulwpr hv fhur/ oxhjr/ odv dvðqwrwdv vrq odv uhfwdv { @ 6> { @ :

Dkrud ghwhuplqhprv odv lqwhuvhfflrqhv frq orv hmhv frrughqdgrv Vl | @ 3/ hqwrqfhv orj 56 mm { . 5 m 8 m@ 3

gh grqgh

or txh htxlydoh d

mm { . 5 m 8 m@ 4

m { . 5 m 8 @ 4 +, m { . 5 m@ 9 +, { . 5 @ 9 b { . 5 @ 9 +, { @ 7 b { @ ;

b m { . 5 m 8 @ 4 b m { . 5 m@ 7 b { . 5 @ 7 b { . 5 @ 7 b { @ 5 b { @ 9

oxhjr/ od fxuyd sdvd sru orv sxqwrv S4 +7> 3,/ S5 +;> 3,/ S6 +5> 3,/ S7 +9> 3,1 Vl { @ 3/ hqwrqfhv | @ orj 56 m 6 m@ orj 56 6 oxhjr/ od fxuyd sdvd sru S8 +3> orj 56 6,1 Qrv idowd ghwhuplqdu orv lqwhuydorv grqgh od ixqflöq hv srvlwlyd | grqgh hv qhjdwlyd1 Odv lpäjhqhv gh od ixqflöq vhuäq srvlwlydv vl or txh hv htxlydohqwh d

orj 56 mm { . 5 m 8 mA 3 3 ?mm { . 5 m 8 m? 4 +, 7 ?m { . 5 m a m { . 5 m? 9 a { 9@ 6 a { 9@ : +, { 5 +`  ;> 9^^`5> 7^,  i:> 6j

Odv lpäjhqhv gh od ixqflöq vhuäq qhjdwlydv vl

orj 56 mm { . 5 m 8 m? 3 +, mm { . 5 m 8 mA 4

+, { 5`  9> 5^^`7> 4^^`  4> ;^

Hq edvh d od lqirupdflöq rewhqlgd vh wlhqh od vljxlhqwh juäfd gh od jxud

FKDSWHU 51 IXQFLRQHV

499

| @ orj 56 mm { . 5 m 8 m Gdgr txh orje hv od ixqflöq lqyhuvd gh h{se hqwrqfhv od frpsrvlflöq uhvxowd vhu od ixqflöq lghqwlgdg= hv ghflu/

orje  h{se @ LgU

a h{se  orje @ LgU.

{ @ orje +h{se +{,, @ orje +e{ , ;{ 5 U { @ h{se +orje +{,, @ eorje { ;{ 5 U.

oxhjr vh wlhqhq odv vljxlhqwhv surslhgdghv ixqgdphqwdohv=

Surslhgdg 5148 Vhd e xq qýphur uhdo srvlwlyr glvwlqwr gh 4 hqwrqfhv orje e{ @ { eorje { @ {

;{ 5 U

;{ 5 U.

Hmhpsor 477 Ghwhuplqh sdud txì ydoruhv gh d 5 U od lqhfxdflöq wlhqh frpr frqmxqwr vroxflöq d U1 Vroxflöq=

 8{.5.{5 5 4 6 
 8{.5.{5 5 6 46 
Frpr orj6 hv xq ixqflöq hvwulfwdphqwh fuhflhqwh/ sxhv e @ 6 A 4/ hqwrqfhv

5 5 orj6 6{ 8{4  orj6 65d{ 7{ +, {5  8{  4  5d{5  7{ +, +5d . 4,{5 . { . 4  3

Od lqhfxdflöq wlhqh frpr vroxflöq d U vl

+4  7+5d . 4, ? 3 a 5d . 4 9@ 3, b +4  7+5d . 4, @ 3 a 5d . 4 A 3,

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV hv ghflu

49:

 6;  d a d A  45

Hmhuflflr 478 Frpsohwh odv vljxlhqwhv iudvhv

6 d 5 ^ > 4^ ;

ixqflöq/ hqwrqfhv ho grplqlr pä{lpr G gh i hv 1111111111 41 Vhd i = G $ ^6> 4^ { $ h{  4

51 Ho uhfruulgr gh

i =`  4> 9` $ U { $ orj4@5 +{ . 5,  orj5 +4@49,

hv 11111111111111 s 61 Vl i+{, @ {  5 | j+{, @ orj5 +<  m{m, . 4 vrq ixqflrqhv hqwrqfhv h{lvwh i  j frq grplqlr 1111111111 | +i  j,+{, @ 1111111111

hv 111111111111 71 Od ixqflöq lqyhuvd gh od ixqflöq el|hfwlyd i = U $ `  5> 4^ { $ +4@5,{4  5

Vroxflöq=

41 Grpi @ i{ 5 U = h{  4  6j shur

oxhjr { 5`  4>  oq 7`1 Dvð/ Grpi @`  4>  oq 7`

h{  4  6 +, h{  7 +, {  oq 7 +, {   oq 7

51 Vhd { 5`  4> 9` h | 5 U wdo txh

shur { 5`  4> 9` oxhjr

| @ orj4@5 +{ . 5,  orj5 +4@49, +, | @ orj4@5 +{ . 5, . 7 +, |  7 @ orj4@5 +{ . 5, 4 +, + ,+|7, @ { . 5 5 4 +, { @ + ,+|7,  5 5

hq frqvhfxhqfld Uhfi @ ^4> 7^1

4 4 ? + ,+|7,  5  9 @ . 5 5 4 +, 4 ? + ,+|7,  ; @ orj4@5 5 +, 3 A |  7  6 +, 7 A |  4

FKDSWHU 51 IXQFLRQHV

49; 61 Grpj

@ @ @ Grpi @ Grp+i  j, @ @

shur

i{ 5 U = <  m{m A 3j i{ 5 U = m{m ? <j `  <> <^= i{ 5 U = {  5  3j @ ^5> 4^= i{ 5 Grpj = j+{, 5 Grpi j i{ 5`  <> <^ = orj5 +<  m{m, . 4  5j=

orj5 +<  m{m, . 4  5 +, +, +, +,

DvðGrp+i  j, @`  <=<^_^:> :` @ ^:> :`1

orj5 +<  m{m,  4 <  m{m  5 m{m  : :  {  :

+i  j,+{, @ i+j+{,, @ i +orj5 +<  m{m, . 4, @

71 Vhd { 5 U h | 5`  5> 4^ wdo txh | @ i +{, hqwrqfhv

s orj5 +<  m{m,  4

4 | @ + ,{4  5 5 4 +, | . 5 @ + ,{4 5 +, orj4@5 +| . 5, . 4 @ {

Dvð i 4 =`  5> 4^ $ U { $ orj4@5 +{ . 5, . 4

Uhvxhoyd od lqhfxdflöq

Vroxflöq= Uhvwulqjlhqgr whqhprv

orj 46 +orj8 +{5  7,, A 5

{5  7 A 3 a orj8 +{5  7, A 3 +, {5 A 7 a {5  7sA 4 +, m{m A 5 a m{m A 8

s s Oxhjr { 5`  4>  8^^` 8> 4^/ dvð/

s s U @`  4>  8^^` 8> 4^

Dkrud uhvroyhprv od lqhfxdflöq frq { 5 U

shur { 5 U1

+, +, +, +,

orj 46 +orj8 +{5  7,, A 5  5 orj8 +{5  7, ? 46 {5  7s ? 8< m {sm? 8< . 7 s  8< . 7 ? { ? 8< . 7

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

49<

Hmhuflflr 479 Oxhjr ho frqmxqwr vroxflöq hv s s s s s s V @`  8< . 7> 8< . 7^_U @`  8< . 7>  8^^` 8> 8< . 7^ Hmhuflflr 47: Uhvxhoyd od hfxdflöq

Vroxflöq= Od lqhfxdflöq htxlydoh d

 {  +{4, 5: orj 7 7 @ < ; orj ;  5 5{  5 6+{4,  65 5{  65 6+4{,

@

+, @ 6 6 +, 5{ . 6+4  {, @ 4 +, { @ 5

5 orj 5 6 orj 5

5 6

Surslhgdg 5149 Vhdq e 5 U.  i4j/ {> | 5 U. / u 5 U/ hqwrqfhv= l, orje +{|, @ orje { . orje |   l, orje {| @ orje {  orje | l, orje {u @ u orje {

Ghprvwudflöq= Vhd x @ orje {/ y @ orje |/ hqwrqfhv

ex @ {> ey @ |

dvð/ oxhjr/ Sru rwud sduwh/ oxhjr/ Ilqdophqwh/ sru or wdqwr/

orje +{|, @ orje +ex ey , @ orje ex.y @ x . y orje +{|, @ orje { . orje |

    { @ orje ex ey @ orje exy @ x  y orje |   { orje @ orje {  orje | |

orje {u @ orje +ex ,u @ orje eux @ ux orje {u @ u orje {

or txh suxhed qdophqwh od surslhgdg1 Uhvxhoyd od hfxdflöq orj 5 . orj+7{5 . <, @ 4 . orj+5{5 . 4,



Vroxflöq= Qrwh txh od uhvwulfflöq hv U4 @ U sxhv orv dujxphqwrv gh orv orjdulwprv vrq dperv vxpdv gh qýphurv srvlwlyrv/ oxhjr od hfxdflöq vh uhgxfh d orj 5 . orj+7{5 . <, @ 4 . orj+5{5 . 4, +, orj 5+7{5 . <, @ orj 43+5{5 . 4, +, 5+7{5 . <, @ 43+5{5 . 4, +, 55{  535{ . 97 @ 3

FKDSWHU 51 IXQFLRQHV

4:3 Vhd x @ 5{ / uhhpsod}dqgr rewhqhprv od hfxdflöq fxdguäwlfd

x5  53x . 97 @ 3 +, x @ 49 b x @ 7 +, 5{ @ 49 b 5{ @ 7 +, { @ 7 b { @ 5

qdophqwh/

V @ i5> 7j

Hmhuflflr 47; Uhvxhoyd od lqhfxdflöq

orj< +orj 45 +{5  4,  orj 45 +{ . 4,, ? 3

Vroxflöq= Uhvwulqjlhqgr whqhprv txh=

U4 {5  4 A 3 sru or wdqwr { 5`  4> 4^^`4> 4^ U5 { . 4 A 3> sru or wdqwr { 5`  4> 4^

U6 orj 45 +{5  4,  orj 45 +{ . 4, A 3> oxhjr ghehprg uhvroyhu od vljxlhqwh lqhfxdflöq

Ilqdophqwh/ od uhvwulfflöq wrwdo hv

+, +, +, +,

orj 45 +{5  4,  orj 45 +{ . 4, A 3 orj 45 +{5  4, A orj 45 +{ . 4, {5  4 ? { . 4 +{  5,+{ . 4, ? 3 { 5`  4> 5^=

U @ +`  4> 4^^`4> 4^,_`  4> 4^_`  4> 5^@`4> 5^

Dkrud od uhvroxflöq1 Vl { 5 U/ hqwrqfhv

Ilqdophqwh/ od vroxflöq hv

+, +, +, +, +,

Uhvxhoyd od lqhfxdflöq

orj< +orj 45 +{5  4,  orj 45 +{ . 4,, ? 3 orj< +orj 45 +{  4,, ? 3 orj 45 +{  4, ? 4 orj 45 +{  4, ? orj 45 45 {  4 A 45 { 5` 65 > 4^ 6 6 V @`4> 5^_` > 4^@` > 5^ 5 5 orj5 +{  45 , ?5 orj5 {

Vroxflöq= Uhvwulqjlhqgr/ whqhprv

oxhjr { 5` 45 > 4^i4j/ dvð/

Dqdolfhprv orv vljxlhqwhv fdvrv=

{

4 A 3 a { A 3 a { 9@ 4 5 4 U @` > 4^i4j 5

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV

4:4

Fdvr 4= Vxsrqjdprv txh { A 4/ hqwrqfhv orj5 { A 3 |

5 { 45 , ? 5 5{ orj5 +{  45 , ? 5 orj5 { orj5 +{  45 , ? orj5 {5 {  45 ? {5 {5  { . 45 A 3 orj +

+, +, +, +,

orj

Frpr ho glvfulplqdqwh gh od hfxdflöq gh vhjxqgr judgr

hv  @ 4 ? 3/ hqwrqfhv Oxhjr/

{5  { . {5  { .

4 5

4 @3 5

A 3> ;{ 5 U

V4 @`4> 4^_U _ U4 @`4> 4^

Fdvr 5=Vxsrqjdprv txh 3 ? { ? 4frq { 5 U4 / hv ghflu/ vxsrqjdprv txh | 45 orj5 +{ , ?5 orj5 { 4 +, orj5 +{  5 , A 5 orj5 { +, orj5 +{  45 , A orj5 {5 +, {  45 A {5 +, {5  { . 45 ? 3

4 5

? { ? 4/ hqwrqfhv orj5 { ? 3

Sru orv plvprv dujxphqwrv gho Fdvr 4 whqhprv txh hq hvwh fdvr V5 @ >

Ilqdophqwh/ od vroxflöq wrwdo gh od hfxdflöq hv

VW @ V4 ^ V5 @`4> 4^

Ho vljxlhqwh whruhpd qrv shuplwh fdpeldu gh edvh fxdqgr vhd qhfhvdulr= Whruhpd 514: +Fdpelr gh Edvh, Vhdq d> e 5 U.  i4j/ { 5 U. / hqwrqfhv orje { @

orjd { orjd e

Ghprvwudflöq= Vhd x @ orjd {/ y @ orje {/ hqwrqfhv dx @ { | ey @ {/ dvð/ whqhprv=

or txh suxhed qdophqwh ho whruhpd1 Hmhuflflr 47< Uhvroyhu

+, +, +, +,

dx @ ey orjd dx @ orjd ey x @ y orjd e orjd { @ orje { orjd e d{ orje { @ orj orjd e

orj{ +8{5 ,+orj8 {,5 @ 4



FKDSWHU 51 IXQFLRQHV

4:5 Vroxflöq= Uhvwulqjlhqgr whqhprv txh { A 3 | { 9@ 4/ sru or wdqwr/ U4 @ U.  i4j

Uhvroylhqgr od hfxdflöq whqhprv

Frpr hqwrqfhv/ vh wlhqh=

orj{ +8{5 ,+orj8 {,5 @ 4 +, +orj{ +8, . 5 orj{ +{,,+orj8 +{,,5 @ 4 4 orj{ 8 @ orj8 {



 4 . 5 orj58 { @ 4 orj8 {

Vhd x @ orj8 {/ hqwrqfhv od hfxdflöq dqwhulru vh wudqvirupd vxfhvlydphqwh hq 4  5 x .5 x @4 +, +4 . 5x,x @ 4 +, 5x5 . x  4 @ 3 +, x @ 45 b x @ 4 4 +, orj8 { s@ 5 b orj4 8 { @ 4 +, { @ 8 b { @ 8

Oxhjr/

Hmhuflflr 483 Uhvxhoyd

4 s 4 s V @ i > 8j _ +U.  i4j, @ i > 8j 8 8 orj6{ +<{, . orj6 +{6 ,  5

Vroxflöq= Ho frqmxqwr uhvwulfflöq gh od lqhfxdflöq hv U @ U.  i4@6j1 Vh wlhqh= orj6{ +<{, . orj6 +{6 ,  5 orj6 +<{, +, . orj6 +{6 ,  5 orj6 +6{, orj6 < . orj6 { +, . 6 orj6 +{,  5 orj6 6 . orj6 {

Vhd x @ orj6 {/ hqwrqfhv od lqhfxdflöq htxlydoh

+,

+,

5.x . 6x  5 4.x 5x . 6x5 3 4.x x+5 . 6x, 3 4.x

Uhvroylhqgr od lqhfxdflöq + 5144,/ rewhqhprv txh

x  4 b  56  x  3 5 +, orj6 {  4 b  6  orj6 {  3 5 b 6 6  {  4 +, {  64

+5144,

5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV shur { A 3 | { 9@

4 6

hqwrqfhv

sru or wdqwr ho frqmxqwr vroxflöq hv

4 3 ? { ? 4@6 b s 6 5 {4 6 4 4 V @`3> ^^^ s 6 5 > 4` 6 6

Hmhuflflr 484 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv +mxvwltxh,1 41 Vhdq d> e 5 U.  i4j/ hqwrqfhv

orje d  orje d5

51 i{ 5 U = orj6 {5 A 4j @ i{ 5 U = 5 orj6 { A 4j 61 ;{> | 5 U.  i4j/

orj {4 | @ orj{

71 ;d> e 5 U.  i4j/

  4 |

orjd +e4 , @ +orjd e,4

81 Ho frqmxqwr vroxflöq gh

s

8 {4 ? orj 48 {

hv >1

Vroxflöq=

41 Idovr1 Frqvlghuhprv d @ 47 / e @ 5 shur 5 9 71

orj5

4 7

@ 5> orj5

 4 5 7

@ 7

51 Idovr1 Gdgr txh orj6 +6,5 @ 5 A 4/ hqwrqfhv vlq hpedujr/

6 5 i{ 5 U = orj6 {5 A 4j

6 95 i{ 5 U = 5 orj6 { A 4j=

61 Yhugdghur1 Vhdq {> | 5 U.  i4j/ hqwrqfhv

orj {4 |

71 Idovr1 Vhdq d @ 5 | e @ 5 shur

@

{| 4 {+ { , orj{ | orj

orj

@ 4 @  orj{ | @ orj{ |4 @ orj{ |4

orjd e4 @ orj5 54 @ 4

+orjd e,4 @ +orj5 5,4 @ 4

4:6

FKDSWHU 51 IXQFLRQHV

4:7

s 81 Yhugdghur1 Odv h{suhvlrqhv {  4 | orj4@8 { wlhqhq vljqlfdgr hq U vl { A 4/ shur orj4@8 { ? 3 | s{4 8 A 3 sdud { A 4 gh grqgh hv fodur txh ho frqmxqwr vroxflöq gh od lqhfxdflöq hv ! Hq edvh d od ixqflöq h{srqhqfldo | orjdulwpr qdwxudo srghprv ghqlu rwud ixqflöq gh judq lqwhuìv

Ghqlflöq 514; Vhdq i | j ixqflrqhv wdohv txh Grpi _ Grpj 9@ !1 Vl i+{, A 3 hq dojýq vxefrqmxqwr D gh Grpi hqwrqfhv vh ghqh od ixqflöq k frpr k+{, @ i +{,j+{, @= hj+{, oq i +{,

frq grplqlr G @ D _ Grpj/ hv ghflu/

G @ Grpi j @ i{ 5 Grpi _ Grpj = i +{, A 3j

Hmhpsor 485 Ho grplqlr gh od ixqflöq k+{, @ {{ hv

Grpk @ i{ 5 U = { A 3j

Qrwhprv txh od h{suhvlöq {{ wdpelìq hvwä ghqlgd hq U vl { shuwhqhfh d orv hqwhurv qhjdwlyrv/ sru hmhpsor vl { @ 5 hqwrqfhv 55 @ +45,5 @ 47 5 U/ shur hq hvwh fdvr {{ qr vh sxhgh lqwhusuhwdu frpr h{ oq { sxhv od ghqlflöq dqwhulru h{ljh txh i+{, vhd srvlwlyr1

Hmhuflflr 486 Vhjýq od ghqlflöq dqwhulru/ ghwhuplqh ho grplqlr gh odv vljxlhqwhv ixqflrqhv {  41 k+{, @ 4 . {4

s

51 k+{, @ +{ . 4, {

Vroxflöq=

41 Grpk @ i{ 5 U = { 9@ 3 a 4 . {4 A 3j 4.

51

51:

4 {.4 A 3 +, A3 { { +, +{ . 4 A 3 a { A 3, b +{ . 4 ? 3 a { ? 3, +, +{ A 4 a { A 3, b +{ ? 4 a { ? 3, +, +{ A 3 b { ? 4,

Sru or wdqwr Grpk @ U  ^4> 3`

51:14

Grpk @ i{ 5 U _ U. = { . 4 A 3j @ i{ 5 U. = { A 4j @ U.

Hmhuflflrv Sursxhvwrv

Hmhuflflrv gh Grplqlr | Uhfruulgr

41 Ghwhuplqh Grpi > Uhfi sdud fdgd xqd gh odv vljxlhqwhv ixqflrqhv uhdohv d, f, h, j,

s

i +{, @ {{4 s i +{, @ s+{ . 4,5  +5{ . 6,5 s i +{, @ s 5  5{ . 4 {4 i +{, @ s {

e, g, i, k,

{5 .5{6 i +{, @ s {4 s i +{, @ {.4.5 s i +{, @ m{ . 4m  4 { i +{, @ { 6

51:1 HMHUFLFLRV SURSXHVWRV

4:8

51 Hq fdgd fdvr/ ghwhuplqh D  U gh prgr txh od uhodflöq vhd xqd ixqflöq s d, i = D  U $ ^4> 6^ > i+{, @ 4  { 4 e, i = D  U $ `4> 4` > i+{, @ { s5 f, i = D  U $ ^4> <^ > i+{, @ 5s {  : s g, i = D  U $ ^4> :^ > i+{, @ s {  5 h, i = D  U $ ^43> 4^ > i+{, @ 4  m{m {8,+{.6, i , i = D  U $ U. > i+{, @ +{5+. {.4,+{5 7, s j, i = D  U $ ^3> 6` > i+{, @ 6  7 4  {5

61 Ghwhuplqh Uhfi / sdud fdgd xqd gh odv vljxlhqwhv ixqflrqhv d, e, f, g, h,

51:15

i i i i i

= `  4> 6` = `4> 4` = `<> 4` = `3> 4` = `  4> 6`

Hmhuflflrv gh Juäfd

41 Judfdu odv vljxlhqwhv ixqflrqhv s d, i+{, @ s{  5 f, i+{, @ {  4 . 8 h, i+{, @ m{  4mm{ . 6m  4 j, i+{, @ ^ 56{ . 4` l, i+{, @ { . ^{`

51 Judfdu odv vljxlhqwhv ixqflrqhv +d, i +{, @

; 5 ? { =

+e, i +{, @ +f, i +{, @

= s { {A3 ; ? ^{` {  4 =

+g, i +{, @

=

U U U U U

> > > > >

s i +{, @ 4  { 4 i +{, @ {.5 s i +{, @ 5 {  : 4 i +{, @ s { i +{, @ m{m  6

+frqvlghudqgr vx grplqlr pä{lpr, | ghwhuplqdu ho uhfruulgr1 s e, i+{, @ m{  4m f, i +{, @ {  4 . 8 g, i+{, @ m{  4m . m5{  6m h, i +{, @ m{  4mm{ . 6m  4 j, i +{, @ ^ 56{ . 4` i, i+{, @ ^ 56{ ` 5{ k, i+{, @ ^ 6 ` . 4 l, i +{, @ { . ^{` m, i+{, @ 5m{m . ^{`

{3

{ . 4 { ? 3 ; 6 {3 ? {

$ $ $ $ $

{5 { ? 4 ; s ? 4  m{m m{m  4

5 m{m A 4 ; 6{  4 6 ? { ? 4 A A A A ? 4?{?; h{ +h, i +{, @ A A A A = s { . 4 ; ? { ? 48 ; ? oq +4  {, { ? 4 +i, i +{, @ = {.4 {4 h

FKDSWHU 51 IXQFLRQHV

4:9

51:16

Hmhuflflrv gh Prghodflöq Pdwhpäwlfd

41 Hq xqd sdufhod/ vh ghvhd hqfhuudu grv sruflrqhv gh whuuhqr gh ljxdo äuhd +frpr hq od jxud, frq xqd pdood gh orqjlwxg O1 H{suhvdu ho äuhd gh od sdufhod hq ixqflöq gh {1 {

{ |

51 Ho äuhd gh xqd slvflqd uhfwdqjxodu frq erughv hv gh 4; p5 1 Vl ho erugh vxshulru h lqihulru plghq 46 p | orv erughv odwhudohv plghq 47 p1 H{suhvdu ho äuhd frpsuhqglgd hqwuh orv erughv hq ixqflöq gh xqr gh orv odgrv gh od slvflqd1

61 Xq krpeuh vh hqfxhqwud hq xqd odqfkd/ d fxdwur ploodv gho sxqwr S päv fhufdqr gh od frvwd/ txh hv uhfwd | ghvhd oohjdu d xq sxqwr T/ txh hvwä d 43 ploodv gho sxqwr S d or odujr gh od frvwd1 Od odqfkd sxhgh ghvsod}duvh d 4; ploodv2kuv | hq fxdotxlhu sxqwr U gh od frvwd srguä derugdu xq dxwrpöylo txh sxhgh yldmdu d 63 ploodv2kuv hq gluhfflöq gh T1 Vhd w ho wlhpsr uhtxhulgr sdud uhfruuì od glvwdqfld kdvwd T1 H{suhvdu w hq ixqflöq gh {/ grqgh { hv od glvwdqfld ghvgh S d U1 71 Hq od ulehud gh xq uðr gh 6 np gh dqfkr kd| xqd sodqwd hoìfwulfd/ hq od rwud ulehud 7 np fruulhqwh duuled kd| xqd iäeulfd1 Ho frvwr gh whqghu xq fdeoh sru wlhuud +oðqhd dìuhd, hv gh ' 63 sru phwur | gh ' 83 sru phwur/ vl vh wlhqgh edmr ho djxd +fdeoh vxepdulqr,1 Ghwhuplqh od ixqflöq frvwr gho whqglgr gh fdeoh gh od sodqwd hoìfwulfd d od iäeulfd +hq ixqflöq gh { Yhu jxud,1 81 Xq pd|rulvwd hq deduurwhv yhqgh odwdv gh wdpdôr qýphur 5 gh xq flhuwr yhjhwdo hq orwhv gh fdmdv gh dfxhugr frq od vljxlhqwh olvwd gh suhflrv1 ' 5183 sru fdmdv hq od frpsud gh 53 fdmdv r phqrv

' 5133 sru fdmdv hq od frpsud gh päv gh 53 fdmdv shur phqrv gh 83 +r ljxdo,

' 41:8 sru fdmdv sdud öughqhv txh lqfox|hq päv gh 83 fdmdv shur phqrv gh 433 fdmdv +r ljxdo,1 ' 4183 sru fdmdv sdud öughqhv gh päv gh 433 fdmdv1

Vl | hv ho suhflr wrwdo | { hv od fdqwlgdg gh fdmdv1 H{suhvdu | hq ixqflöq gh {1

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Ho txlqwr wìuplqr hv 89 1 Vl q hv vxshu judqgh% BcFhufd gh txì ydoru vh hqfxhqwud {q B

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4>

s s 5> 6> 5>   

s ho txlqwr wìuplqr hv 8 Bc {q vh dfhufd d xq qýphur uhdo fxdqgr q hv judqgrwhB

Hmhpsor 489 Pxfkdv yhfhv odv vxfhvlrqhv vrq ghqlgdv hq irupd uhfxuvlyd/ sru hmhpsor Vhd {3 5`3> 4^ vh ghqh {4 @ {3 {q @ {5q4 vl q A 4 Vl txhuhprv hvwd vxfhvlöq frpr xq olvwdgr/ ghehprv fdofxodu {5 > {6 > {7 >    hwf1 Sru hmhpsor ho fxduwr wìuplqr hv Dvð od vxfhvlöq hv Vl {3 @

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BcD txì ydoru wlhqgh% {q vl {3 @ 45 B Od uhvsxhvwd wdpelìq ixh gdgd hq od jxud I} ghuhfkd1 Hmhpsor 48: Vhd +{q ,4 q@4 ghqlgd uhfxuvlydphqwh sru {4 {q

@ 4 @ vhq+{q4 , vl q A 4

Phgldqwh od wìfqlfd juäfd gdgd do lqlflr yhd d txh ydoru vh dfhufd {q fxdqgr q hv judqgh +hv ghflu/ judtxh | @ { h | @ vhq+{,1 Uhfxhugh txh 4 ? 5 ,1 Bc Txì rfxuuh vl {4 @ 4 vh fdpeld sru {4 @ 43B

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Ghqlflöq 614 Xqd vxfhvlöq gh qýphurv uhdohv4 hv xqd ixqflöq fx|r grplqlr hv ho frqmxqwr gh orv qýphurv qdwxudohv r vxefrqmxqwr G gh Q gh od irupd G @ ^d> 4^_Q | vx uhfruulgr hv xq vxefrqmxqwr gh orv uhdohv1 r { = G $ U Vh hvfuleh> { = Q $ U q $ {+q, @ {q q $ {+q, @ {q Uhfrughprv txh xqd ixqflöq hv hq sduwlfxodu xqd uhodflöq> dvð xqd vxfhvlöq { hv xq vxefrqmxqwr gh Q  U/ hv ghflu/ od vxfhvlöq { hv ho frqmxqwr i+4> {+4,,> +5> {+5,,> +6> {+6,,>    +q> {+q,,>    j

shur d fdpelr gh hvwr vh dqrwd vlpsohphqwh

{4 > {5 > {6 > {7 >   

grqgh {q lqglfd ho ydoru gh {+q, oodpdgr q0ìvlpr wìuplqr Qrwdflöq= Rwudv irupdv uhvxplgdv gh dqrwdu vxfhvlrqhv vrq= +{q ,4 q@4 >

i{q j4 q@4 >

+{q ,q4 >

i{q jq4 >

r h{wuhpdqgr od vlpsolfdflöq vh dqrwd vlpsohphqwh +{q ,> i{q j  4 4 Hmhpsor 48; +{q ,4 q@4 @ 5q q@4 /

Revhuydflöq= Qr vh gheh frqixqglu xqd vxfhvlöq frq vx uhfruulgr1 Hq ho fdvr gh od vxfhvlöq +{q , @ +4,q / hv ghflu/ od vxfhvlöq 4> 4> 4> 4> 4> 4>    wlhqh frpr uhfruulgr ho frqmxqwr i4> 4j @ Uhf+{,1 Xqd vxfhvlöq vlhpsuh wlhqh lqqlwrv wìuplqrv/ ho txlqwr wìuplqr gh hvwd hv 04/ ho ghflpr wìuplqr hv 41 Ghqlflöq 615 Gluhprv txh xqd vxfhvlöq +{q , hv dfrwdgd vl vx uhfruulgr hv xq frqmxqwr dfrwdgr/ l1h1/ m{q m ? N=

Dqäorjdphqwh vh ghqh vxfhvlöq dfrwdgd vxshulruphqwh | dfrwdgd lqihulruphqwh1

Ghqlflöq 616 Gluhprv txh xqd vxfhvlöq +{q , hv fuhflhqwh vl hood frpr ixqflöq hv fuhflhqwh/ hv ghflu/ {4  {5  {6  {7    

Dqäorjdphqwh vh ghqh hvwulfwdphqwh fuhflhqwh/ ghfuhflhqwh/ hvwulfwdphqwh ghfuhflhqwh1 Gh xqd pdqhud päv irupdo

41 +{q , hv xqd vxfhvlöq fuhflhqwh vl | vöor vl

+;q 5 G,+{q  {q.4 ,

51 +{q , hv xqd vxfhvlöq hvwulfwdphqwh fuhflhqwh vl | vöor vl +;q 5 G,+{q ? {q.4 ,

61 +{q , hv xqd vxfhvlöq ghfuhflhqwh vl | vöor vl

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Gluhprv txh xqd vxfhvlöq +{q , hv prqöwrqd vl | vöor vl hv fuhflhqwh r elhq ghfuhflhqwh

4 Ghqlflöq 617 Vhd +nq ,4 q@4 xqd vxfhvlöq hvwulfwdphqwh fuhflhqwh gh qýphurv qdwxudohv | vhd +{q ,q@4 xqd 4 vxfhvlöq duelwuduld gh qýphurv uhdohv/ od frpsxhvwd {  n/ hv ghflu/ +{nq ,q@4 hv oodpdgd vxevxfhvlöq gh +{q ,4 q@4

Hmhpsor 48< Od vxfhvlöq + q4 , hv dfrwdgd sxhv h{lvwh n @ 4 5 U wdo txh sdud wrgr q 5 Q vh wlhqh> m q4 m  41 Sru rwud sduwh + q4 , hv hvwulfwdphqwh ghfuhflhqwh+prqöwrqd,1 Hq hihfwr= Hv fodur txh sdud wrgr q 5 Q> q ? q . 4 hqwrqfhv sru surslhgdg gh orv qýphurv uhdohv vh wlhqh 4 4 A q q.4

oxhjr ;q 5 Q > {q A {q.4

Hmhpsor 493 Od vxfhvlöq +q,4 q@4 qr hv dfrwdgd sxhv qr h{lvwh n hq U wdo txh mqm  N +Uhfrugdu txh ho frqmxqwr gh orv qýphurv qdwxudohv Q qr hv dfrwdgr,1 Vlq hpedujr hv fodur txh +q, hv hvwulfwdphqwh fuhflhqwh/ sxhv ;q 5 Q > q ? q . 4

q 4 Hmhpsor 494 +{q ,4 q@4 @ + q.4 ,q@4 hv xqd vxfhvlöq dfrwdgd +4 hv xqd frwd vxshulru, | prqöwrqd/ +hvwulf0 wdphqwh fuhflhqwh,1

q5 4 Hmhpsor 495 +{q ,4 q@4 @ + q.6 ,q@4 hv xqd vxfhvlöq qr dfrwdgd vxshulruphqwh vöor lqihulruphqwh/ {q hv hvwulfwdphqwh fuhflhqwh1

qr hv prqöwrqd | wlhqh d + 494q5 ,4 Hmhpsor 496 +{q , @ + +54, q@4 frpr vxe0 q , hv xqd vxfhvlöq dfrwdgd/ 5 4 vxfhvlöq |d txh od vxfhvlöq gh qýphurv qdwxudohv +;q ,q@4 hv hvwulfwdphqwh fuhflhqwh1 Frqvlghudqgr od 6q +4, 4 vxfhvlöq +6q,4 q@4 vh wlhqh txh + 9q ,q@4 wdpelìq hv xqd vxevxfhvlöq gh {q 1 q

Hmhuflflr 497 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh1 41 Wrgd vxfhvlöq dfrwdgd hv pöqrwrqd1

51 Wrgd vxfhvlöq pöqrwrqd hv dfrwdgd1 61 Od vxfhvlöq + +q4, , hv prqöwrqd1 q

s 71 Od vxfhvlöq +q, hv xqd vxevxfhvlöq gh + q,1

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41 Idovr1 Od vxfhvlöq ++4,q , hv dfrwdgd shur qr hv fuhflhqwh ql ghfuhflhqwh1

51 Idovr1 Od vxfhvlöq +q, hv fuhflhqwh oxhjr hv prqöwrqd shur qr hv dfrwdgd1 61 Idovr1 Vhd {q @



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71 Oxhjr {4 ? {5 shur {5 A {6 1 Dvð{q qr hv fuhflhqwh ql ghfuhflhqwh sru or wdqwr qr hv prqöwrqd1 s 81 Yhugdghur1 Vhd {q @ q | vhd nq @ q5 1 Fodudphqwh +nq , hv xqd vxfhvlöq gh qdwxudohv fuhflhqwh | s +{  n,+q, @ {+nq , @ {+q5 , @ q5 @ q oxhjr ++{  n,q , @ +q,q hv xqd vxevxfhvlöq gh +{q ,1

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Revhuydflöq= Revhuydu txh xqd vxfhvlöq +|q , hv xqd vxevxfhvlöq gh xqd vxfhvlöq +{q , vl +|q , hvwäq lqfoxlgrv5 hq orv wìuplqrv gh od vxfhvlöq +{q , | dghpäv h{lvwh xqd vxfhvlöq +nq , gh qdwxudohv fuhflhqwh wdo txh +{  n,+q, @ |q 1 Dojxqdv vxfhvlrqhv gh lqwhuìv hq fxuvrv srvwhulruhv vrq odv txh hvwxglduhprv hq orv vljxlhqwhv hmhpsorv1 Hmhpsor 498 Vl u 5 `3> 4^ hqwrqfhv od vxfhvlöq +{q , @ +uq , hv dfrwdgd | hvwulfwdphqwh ghfuhflhqwh1

Ghprvwudflöq= Vhd u 5`3> 4^ | q 5 Q1 Frpr u A 3 hqwrqfhv uq A 3/ oxhjr +uq , hvwä dfrwdgd lqihulruphqwh sru 31 Rfxsdqgr ho khfkr txh od ixqflöq h{srqhqfldo h{su hv hvwulfwdphqwh ghfuhflhqwh vh wlhqh txh= q A 3 , uq ? u3

hv ghflu uq ? 41 Hq frqvhfxhqfld +uq , hvwä dfrwdgd vxshulruphqwh sru 41 Sru ho plvpr dujxphqwr vh wlhqh txh= q ? q . 4 , uq.4 ? uq

Sru or wdqwr +uq , hv hvwulfwdphqwh ghfuhflhqwh1

Hmhpsor 499 Vl u A 4 hqwrqfhv +uq , hv hvwulfwdphqwh fuhflhqwh/ shur qr hv dfrwdgd1 Ghprvwudflöq= Vhd q 5 Q> u A 4

q ? q . 4 , h{su +q, ? h{su +q . 4, , uq ? uq.4

oxhjr +uq , hv hvfwulfwdphqwh fuhflhqwh1 Dkrud sureduhprv txh +uq , qr hv dfrwdgd vl u A 4 +Sru pìwrgr gho devxugr,1 Vxsrqjdprv txh +uq , hv dfrwdgd oxhjr/ h{lvwh n 5 U. wdo txh uq  n

shur

uq  n

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;q 5 Q

+, uq  uorju +n, +, q  orju +n,

q  orju n

;q 5 Q

gh grqgh orju +n, hv frwd vxshulru gho frqmxqwr gh orv qýphurv qdwxudohv/ or fxdo hv xqd frqwudglfflöq +sxhv Q qr hv dfrwdgr,1 Revhuydflöq= Rwud irupd gh suredu txh +uq , qr hv dfrwdgr +vl u A 4, hv od vljxlhqwh= Frpr d vh sxhgh hvfulelu d @ 4 . k frq k A 3/ whqhprv sru ho ghvduuroor gho elqrplr txh= dq @ +4 . k,q @ 4 . qk .    . qkq4 . kq

| frpr wrgrv orv vxpdqgrv vrq srvlwlyrv rewhqhprv od ghvljxdogdg gh Ehuqrxool dq A 4 . qk >

;q

q frpr od vxfhvlöq +4 . qk,4 q@4 qr hv dfrwdgd/ wdpsrfr or hv +{q , @ +d , sxhv +{q , hv hvwulfwdphqwh fuhflhqwh Hq edvh d od vxfhvlöq dqwhulru ghqdprv xqd qxhyd vxfhvlöq= 5 shqvdgr

frpr uhfruulgr gh od vxfhvlöq

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dvð od vxfhvlöq hv

6

+{q , uhfxuvlydphqwh frpr vljxh

@ 4 @ {q4 . uq > q  4

4> 4 . u> 4 . u . u5 > 4 . u . u5 . u6 >    S Ho q0ìvlpr wìuplqr hv {q @ 4 . u . u5 .    . uq @ ql@3 ul txh fruuhvsrqgh d od vxpd gh q . 4 wìuplqrv gh xqd surjuhvlöq jhrpìwulfd gh ud}öq u/ dvð txh 4  uq.4 > ;q 5 Q {q @ 4u

Revhuydflöq= Od ljxdogdg dqwhulru vh sxhgh suredu sru lqgxfflöq1 Fodudphqwh vh yh txh +{q , hv 4 hvwulfwdphqwh fuhflhqwh | hv dfrwdgd vxshulruphqwh sru 4 u1 Frpr fdvr sduwlfxodu gh od vxfhvlöq dqwhulru wrphprv u @ 45 > ho wìuplqr q0ìvlpr qrv txhgd {q @ 4 .

4  5q4.4 4 4 4 4 4 4 @ 5+4  q.4 , ? 5 . . . .. q @ 4 5 7 ; 49 5 5 4 5

txlìq hv dfrwdgd sru 5/ hv ghflu/ m{q m ? 5 sdud wrgr q1

Hmhpsor 49; Xqd vxfhvlöq px| xwlol}dgd hq fäofxor hv od vxfhvlöq txh wlhqh frpr q0ìvlpr wìuplqr d {q @ 4 .

4 4 4 4 . . .. 4$ 5$ 6$ q$

+{q ,4 q@4 hv fodudphqwh fuhflhqwh1 Hood hv dfrwdgd frpr vh sxhgh yhu hq od ghvljxdogdg {q @ 4 .

4 4 4 4 4 4 4 4 . . . . ?4.4. . . .    . q4 ? 4 . 5 @ 6 4$ 5$ 6$ q$ 5 55 555 5

|d txh hvwd ýowlpd vxfhvlöq hv dfrwdgd +frpr vh ylö dqwhulruphqwh,1

Hmhpsor 49< Vhd +|q ,4 q@4 od vxfhvlöq txh wlhqh frpr wìuplqr jhqhudo +hv ghflu/ q0ìvlpr wìuplqr, |q @ +4 . q4 ,q 1 Sru ho ghvduuroor gh Elqrplr gh Qhzwrq vh wlhqh txh 4 q+q  4, 4 5 @ 4 . q+ , . + , . q 5$ q q+q  4,+q  5,+q  6,    +6,+5,+4, 4 q + , . q$ q 4 4 4 4 5 @ 4 . 4 . +4  , . +4  ,+4  , .    5$ q 6$ q q 4 4 5 6 q4 . +4  ,+4  ,+4  ,    +4  , q$ q q q q Revhuyhprv txh wrgrv orv idfwruhv gh orv vxpdqgrv gho wìuplqr |q vrq srvlwlyrv/ dghpäv vh wlhqh/ hq fdgd vxpdqgr/ txh ho idfwru txh dfrpsdôd n4$ hv phqru txh 4/ dvð frpsdudqgr vxpdqgr sru vxpdqgr frq od vxfhvlöq {q gho hmhpsor dqwhulru/ rewhqhprv txh |q

|q ? {q

;q 5 Q

4 | frpr +{q ,4 q@4 hv dfrwdgr/ vh wlhqh txh +|q ,q@4 hv wdpelìq dfrwdgd1 4 Bc Hv +|q ,q@4 xqd vxfhvlöq fuhflhqwh B s q Hmhpsor 4:3 Od vxfhvlöq +}q ,4 q@4 @ q hv ghfuhflhqwh d sduwlu gh xq q3 dsursldgr | fodudphqwh hv dfrwdgd lqihulruphqwh sru fhur/ wdpelìq vh sxhgh yhu txh hv dfrwdgd vxshulruphqwh frpsdudqgrod frq +|q ,4 q@4 6 Qr

fdpeld od ghqlflöq vl vh wrpd

Q ö Qf hq od ghqlflöq gh vxfhvlöq

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Hq hvwd vhfflöq fodulfduhprv ho vljqlfdgr gh {q vh dfhufd d O%> O 5 U1 Yhdprv orv vljxlhqwhv hmhpsorv  dq @ q4 vh dfhufd d 3

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Sdud hvwr qhfhvlwdprv ho frqfhswr gh yhflqgdg Ghqlflöq 618 Vhd s 5 U1 Xqd yhflqgdg r yhflqgdg delhuwd gho sxqwr s hv vlpsohphqwh xq frqmxqwr/ txh frqwlhqh d s/ gho wlsr Ys>% @ i{ 5 Um m{  sm ? %j > grqgh % 5 U. =

Ghvduuroodqgr od lqhfxdflöq txh ghqh d Ys>% whqhprv txh od yhqflgdg Ys>% hv ho lqwhuydor delhuwr txh wlhqh frpr fhqwur7 d s/ hv ghflu/ Ys>% @`s  %> s . %^

Hmhpsor 4:4 Vl s @ 5/ vrq yhflqgdghv gh 5/ orv lqwhuydorv `4/ 6^> `41 `41<<<< 5 . %^ uhsuhvhqwd wrgdv odv yhflqgdghv gh 5/ edvwd wrpdu % 5 U. 1 Hq ho sulphu hmhpsor % @ 4/ ghvsxhv % @ 3=4 | hq ho ýowlpr % @ 3=333341 Hvwd ýowlpd yhflqgdg fruuhvsrqgh do wlsr gh yhqflgdghv txh xvduhprv1 Od lghd hv wrpdu % shtxhôr +vxshu shtxhôr, Ghqlflöq 619 Vhd O 5 U | +{q ,4 q@4 xqd vxfhvlöq gh qýphurv uhdohv/ gluhprv txh {q frqyhujh d O fxdqgr q wlhqgh d lqqlwr r txh ho oðplwh gh {q hv O/ fxdqgr q wlhqgh d lqqlwr r vlpsohphqwh {q frqyhujh d O vl vh wlhqh txh sdud fxdotxlhu yhflqgdg gh O> YO>% / txh vh wrph +hv ghflu/ dýq txh % vhd vxshu shtxhôr%, vlhpsuh txhgduä vöor xqd fdqwlgdg qlwd gh wìuplqrv gh od vxfhvlöq ixhud gh od yhflqgdg YO>% +| sru or wdqwr wrgrv orv rwurv lqqlwrv wìuplqrv txhgdq dghqwur,1 Qrwdflöq=

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Dvð/

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5q  8 olp @5 q$4 q . 4

Ghprvwudflöq= Whqhprv txh ghprvwudu txh

    5q  8  5 ? %, +;% A 3,+
Dqwhv gh frphq}du frq od ghprvwudflöq ghwhuplqhprv xq uhdo dsursldgr do fxdo dsolfduoh od surslhgdg dutxlphgldqd1 Qrwhprv txh         5q  8  5q  5   :   5q  8 : : @ @    5 +614,    q . 4 @ q . 4 ? q  q.4 q.4

Gdgr % A 3/ txhuhprv suredu txh h{lvwh xq qdwxudo Q wdo txh sdud fxdotxlhu q 5 Q frq q A Q / vh whqjd txh    5q  8   ?%  4 +615,  q.4  8 Qrwh

txh

%

hv duelwudulr/ s hur vh slhqvd vlhpsuh txh hv s htxhôr/

%

'

vhuðd px| judqgh

FKDSWHU 61 VXFHVLÖQ

4<7

Hv fodur sru +614, txh= vl h{lvwh Q wdo txh q: ? % sdud q A Q / whqguhprv od ghvljxdogdg +615,1 Sru or wdqwr exvtxhprv wdo Q1 Revhuyhprv txh q: ? % htxlydoh d :% ? q1 Hqwrqfhv gdgr % A 3/ frqvlghuhprv ho uhdo :% | xvdqgr od surslhgdg Dutxlphgldqd ghprvwuduhprv +BB,1 Ghprvwudflöq= Vhd % A 3/ hqwrqfhv sru surslhgdg Dutxlphgldqd h{lvwh Q 5 Q wdo txh Q A :% 1 Vhd q 5 Q wdo txh q A Q / shur Q A :% / oxhjr q A :% / or txh htxlydoh d q: ? % shur    5q  8  : : :  ? |  5 @  q.4 q q.4 q.4    8   5 ? q: ? %= Dvð/ khprv suredgr +BB,1 sru or wdqwr  5qq.4 Hmhpsor 4:9 Ghpxhvwuh gdgr s> t 5 ]. 1 Sru ghprvwudu

4 olp @3 q$4 qs@t

4 +;% A 3,+
Ghprvwudflöq= Vhd % A 3/ hqwrqfhv frqvlghuhprv ho uhdo + 4% ,t@s Sru surslhgdg Dutxlphgldqd h{lvwh Q 5 Q wdo txh 4 Q A + ,t@s %

Vhd q 5 Q wdo txh q A Q hqwrqfhv sru +616, gh grqgh

+616,

4 q A + ,t@s %

4 % A + ,s@t sdud q A Q q Dvð sdud % A 3 gdgr hqfrqwudprv Q wdo txh vl q A Q vh wlhqh

4 @ 31 Sru frqvljxlhqwh olpq$4 qs@t

4 + ,s@t ? % q

q4 Hmhpsor 4:: Od vxfhvlöq +{q ,4 q@4 @ ++4, ,q@4 glyhujh |d txh vl wrpdprv od yhflqgdg Y4> 47 /hv ghflu/ 4 O @ 4 | % @ 7 / ghmd lqqlwrv wìuplqrv gh od vxfhvlöq ixhud gh hood +orv wìuplqrv frq ðqglfh q lpsdu,> or plvpr rfxuuh frq Y4> 47 yhu jxud QQ

Iljxud QQ

6161 OÐPLWH GH XQD VXFHVLÖQ

4<8

Frq hvwh plvpr hmhpsor yhprv od qhfhvlgdg gh ghprvwudu txh ho oðplwh gh xqd vxfhvlöq hv ýqlfr sru hvr xvdprv ho% | qr orv%1 Whruhpd 61; +Xqlflgdg gho Oðplwh, Ho oðplwh gh xqd vxfhvlöq +{q ,4 q@4 hv ýqlfr/ hv ghflu/ olp {q @ d a olp {q @ e , d @ e

Ghprvwudflöq= Surfhghuhprv sru ho devxugr/ hvwr vljqlfd txh vxsrqguhprv txh d 9@ e | oohjduhprv d xqd frqwudglfflöq1 Hvhqfldophqwh kduhprv or plvpr txh hq ho hmhpsor dqwhulru1 Frpr d 9@ e vh wlhqh txh ho qýphur mdem 9@ 3 | srvlwlyr1 \d txh olp { @ d/ sdud % @ mdem h{lvwh xq Q 5 Q wdo txh vl q A Q> { 5 Y > q q d> % 7 7 sru rwud sduwh vh wlhqh txh olp {q @ e/ dvð sdud ho plvpr % h{lvwh xq P 5 Q wdo txh q A P> {q 5 Ye> % 1 Qrwh txh Yd> % _ Ye> % @ ! +frpr hq od jxudQQ frq d @ 4 | e @ 4,/ shur vl wrpdprv q wdo txh q A Q a q A P vh wlhqh txh {q 5 Yd> % a {q 5 Ye> % / hv ghflu/ {q 5 Yd> % _ Ye> % / dvð Yd> % _ Ye> % 9@ !/ or fxdo qrv oohyd d xqd frqwudglfflöq1  Hvwh whruhpd hv px| ýwlo sdud prvwudu txh xqd vxfhvlöq hv glyhujhqwh/ sru hmhpsor 4 4 q  +{q ,4 q@4 @ +4, +6 . q ,q@4 glyhujh  4  +{q ,4 q@4 @ +vhq+ ; q,,q@4 glyhujh

Whruhpd 61< Vl od vxfhvlöq +{q , frqyhujh d O hqwrqfhv wrgd vxevxfhvlöq gh hood wdpelìq frqyhujh d O1

Qrwhprv txh hv xq whruhpd gho wlsr s , t Ghprvwudflöq= Vhd +{nq , xqd vxevxfhvlöq gh +{q ,1 Frpr nq hv xqd ixqflöq fuhflhqwh hq Q vh wlhqh

4  n4  n5  n6      nq frq q  nq

Sru or wdqwr/ frpr olpq$4 {q @ O vh wlhqh txh oxhjr Dvð

; % A 3 %

; % A 3 % olp { @ O n$4 nq

Frurodulr 6143 Vl olpq$4 {q @ d hqwrqfhv sdud wrgr n 5 Q olpq$4 {q.n @ d



Or txh glfh ho frurodulr hv txh sdud hvwxgldu od frqyhujhqfld gh xqd vxfhvlöq qr lpsruwd or txh rfxuud frq orv sulphurv wìuplqrv +orv n sulphurv wìuplqrv,

Hmhpsor 4:; Od vxfhvlöq

{q @ frqyhujh d fhur +sru Frurodulr 6143,1

 q 5 4

q

q  4333 q A 4333

Dsolfduhprv ho whruhpd dqwhulru/ hq irupd gluhfwd/ sdud rewhqhu ho oð plwh gh vxfhvlrqhv hq orv hmhpsorv txh vljxhq 4 4 Hmhpsor 4:< + 5q4 5 ,4 q@4 frqyhujh d fhur |d txh hood hv xqd vxevxfhvlöq gh + 5q ,q@4 1

FKDSWHU 61 VXFHVLÖQ

4<9 Hmhuflflr 4;3 Fdofxodu ho

olp q$4



q5 . 4 q5

q5

vl vh vdeh txh olpq$4 +4 . q4 ,q @ O1  5 q5 q  q5 q5 r  Vroxflöq= \d txh qq.4 @ 4 . q45 | 4 . q45 irupd xqd vxevxfhvlöq gh +4 . q4 ,q 5 +wrpdqgr q5 frpr od ixqflöq fuhflhqwh, hqwrqfhv olp q$4



q5 . 4 q5

q5

@O

Revhuydflöq= Xqd frqvhfxhqfld gho whruhpd hv txh vl xqd vxfhvlöq wlhqh grv vxevxfhvlrqhv frqyhujhqwhv d oðplwhv glvwlqwrv/ hqwrqfhv od vxfhvlöq lqlfldo qr wlhqh oðplwh |d txh vh frqwudglfh od xqlflgdg gho oðplwh Hmhpsor 4;4 Od vxfhvlöq



3 vl q hv lpsdu q vl q hv sdu q.5 qr hv frqyhujhqwh/ sxhv od vxevxfhvlöq {5q irupdgd sru orv wìuplqrv gh oxjdu sdu frqyhujh d 4/ vlq hpedujr od vxevxfhvlöq {5q.4 irupdgd sru orv wìuplqrv gh oxjdu lpsdu frqyhujh d 31 {q @

Dqäorjdphqwh vh wlhqh

Hmhpsor 4;5 Od vxfhvlöq ++4,q , qr frqyhujh sxhv h{lvwhq odv vxevxfhvlrqhv +{5q , @ +4, | +{5q.4 , @ +4, txh frqyhujhq d 4 | 4 uhvshfwlydphqwh1

Revhuyh txh vl +{nq , $ O qr vh wlhqh qhfhvduldphqwh txh +{q , $ O1 Hvwr vh sxhgh ylvxdol}du frq q 4 4 4 +{q ,4 q@4 @ +vhq+ 5 ,,q@4 1 Od vxevxfhvlöq +{5q ,q@4 frqyhujh d 3 shur +{q ,q@4 glyhujh1 Dvð srghprv qrwdu txh ho whruhpd dqwhulru hv gh od irupd s , t | qr s +, t1 Xq whruhpd gho plvpr wlsr hv ho vljxlhqwh1 Whruhpd 6144 Wrgd vxfhvlöq frqyhujhqwh hv dfrwdgd

Ghprvwudflöq= Vhd +{q ,4 q@4 frqyhujhqwh d O/ sdud % @ 4 h{lvwh Q 5 Q wdo txh vl q A Q vh wlhqh txh {q 5 YO>4 +hv ghflu/ O  4 ? {q ? O . 4,1 Vhdq  @ pd{i{4 > {5 > {6 >    > {Q 4 > {Q > O . 4j |

 @ plqi{4 > {5 > {6 >    > {Q 4 > {Q > O  4j

dvð sdud wrgr q  4 vh wlhqh {q 5 ^> ` sru or wdqwr ho frqmxqwr i{q = q 5 Qj hv dfrwdgr vxshulruphqwh sru  h lqihulruphqwh sru  sru or wdqwr +{q ,4 q@4 hv dfrwdgd1  Ho vljxlhqwh hmhpsor pxhvwud/ txh ho whruhpd 6144 hv vöor gh od irupd s , t1

q +4, q 4 4 Hmhpsor 4;6 Vhd +{q ,4 q@4 @ +frv + ; , . 7 ,q@4 od vxfhvlöq +{q ,q@4 hv dfrwdgd +dfrwdgd vxshulru0 8 8 phqwh sru 7 h lqihulrphqwh sru  7 , shur qr hv frqyhujhqwh1

Ho vljxlhqwh whruhpd hv lpsruwdqwh | vx ghprvwudflöq vh dsr|d ixhuwhphqwh hq ho d{lrpd gh frpsohwlwxg gh orv qýphurv uhdohv1 Ho whruhpd päv elhq hv gh h{lvwhqfld gho oðplwh gh xqd vxfhvlöq gdgd/ d glihuhqfld gh orv whruhpdv txh shuplwhq fdofxodu xq oðplwh1 Shur wlhqh lpsruwdqfld frqrfhu hvwh whruhpd |d txh sxhgh ydolgdu dojýq oðplwh rewhqlgr frpsxwdflrqdophqwh/ |d txh ho oðplwh rewhqlgr hv ýqlfr | fxdotxlhu vxevxfhvlöq wlhqh ho plvpr oðplwh1

6171 FÄOFXOR GH OÐPLWHV

4<:

Whruhpd 6145 Whruhpd frqyhujhqfld prqöwrqd= Wrgd vxfhvlöq prqöwrqd dfrwdgd hv frqyhujhqwh

Ghprvwudflöq= Uhfrughprv txh prqöwrqd sxhgh vljqlfdu fuhflhqwh/ ghfuhflhqwh/ hwf1 Vhd +{q ,4 q@4 xqd vxfhvlöq fuhflhqwh | dfrwdgd +dqäorjr sdud orv rwurv fdvrv/ frpr ghfuhflhqwh/ hwf,/ dvð {4  {5  {6      {q  {q.4    

Vhd  @ vxs+D, grqgh D @ i{q = q 5 Qj1 Sru fdudfwhul}dflöq gh vxsuhpr gh D vh wlhqh txh sdud wrgr % A 3 h{lvwh xq hohphqwr gh D> {Q wdo txh   % ? {Q  

shur vl q A Q vh wlhqh {Q  {q +fuhflhqwh, | {q   + @ vxs D,1 Dvð ; % A 3

hv ghflu/

+; % A 3,+ % , or txh vljqlfd txh olpq$4 {q @   Revhuydflöq= Vl +{q , hv ghfuhflhqwh | dfrwdgd lqihulruphqwh hqwrqfhv olp {q @  grqgh  @ lqii{q = q 5 Qj Vl +{q , hv fuhflhqwh | dfrwdgd vxshulruphqwh hqwrqfhv olp {q @  grqgh  @ vxsi{q = q 5 Qj1 Hmhpsor 4;7 Od vxfhvlöq +{q ,4 q@4 frq

{q @ 4 .

4 4 4 . . . 4$ 5$ q$

hv fodudphqwh fuhflhqwh | dfrwdgd vxshulruphqwh sru 6 +dqdol}dgd hq ghwdooh dqwhulruphqwh,/ sru or wdqwr +{q , frqyhujh d dojýq ydoru O1 Xvh od fdofxodgrud sdud yhu fxdo vhuðd ho ydoru gh O1 Hmhpsor 4;8 Od vxfhvlöq +|q ,4 q@4 frq

4 q , q hv fuhflhqwh | dfrwdgd sru 6 sru or wdqwr olpq$4 |q @ V+h{lvwh,1 Xvh od fdofxodgrud sdud rewhqhu V1 Bc Txh uhodflöq h{lvwh hqwuh V | O +O gho hmhpsor dqwhulru,1

617

|q @ +4 .

Fäofxor gh oðplwhv

Orv whruhpdv txh yhuhprv d frqwlqxdflöq vrq sulqflsdophqwh xwlol}dgrv sdud fdofxodu oðplwhv

Whruhpd 6146 +Fhur Dqltxlod, Vhd olpq$4 {q @ 3 h +|q ,4 q@4 xqd vxfhvlöq dfrwdgd hqwrqfhv olpq$4 {q |q @ 3 Sru ghprvwudu txh

; % A 3
. Ghprvwudflöq= Frpr +|q ,4 q@4 hv dfrwdgd/ h{lvwh n 5 U wdo txh m|q m ? n> ;q 5 Q1 Dghpäv frpr +{q ,4 frqyhujh d 3 vh wlhqh txh q@4

 5 Q wdo txh vl q A Q+  %,> m{q m ? % ; % A 3 gdgr % A 3 edvwd wrpdu % @ n% | sdud hvwh  wdo txh q A Q  , m{q m ? % / dvð m{q |q m ? nm{q m ? %1 ydoru whqhprv txh h{lvwh Q n Sru or wdqwr/  + % , wdo txh q A Q , m{q |q m ? % ; % A 3
FKDSWHU 61 VXFHVLÖQ

4<;

Hmhpsor 4;9 Xq hmhpsor grqgh vh sxhgh xwlol}du hvwh ýowlpr whruhpd hv hq od vxfhvlöq 4 q{ ,,j i}q j @ i +frv + 5 q q . 53

grqgh { hv mr1 ,j txh vdwlvidfhq Hvwd vxfhvlöq wlhqh frpr idfwruhv d odv vxfhvlrqhv i{q j @ i q4 j | i|q j @ ifrv + q5q{ .53 odv klsöwhvlv gho whruhpd/ dvð txh }q $ 31

Hmhpsor 4;: Ghprvwudu

olp q$4

vhq+q, @3 q

Hq hihfwr olpq$4 q4 @ 3 | ivhq+q,j hv dfrwdgd/ oxhjr sru whruhpd gho fhur dqltxlod whqhprv od gh0 prvwudflöq1

Dsduhqwhphqwh hvwh ýowlpr whruhpd hv vöor dsolfdeoh sdud fäofxor gh oðplwhv txh vhdq fhur/ shur qr hv dvð sru od vljxlhqwh sursrvlflöq1 Sursrvlflöq 6147

olp { @ O +, olp +{q  O, @ 3 q$4 q q$4

Hmhpsor 4;; Vl d 5 `3> 4^ hqwrqfhv olpq$4 +4 . d .    . dq , @ 44 d q.4 Ghprvwudflöq= Frpr 4 . d .    . dq @ 44dd q.4 sru or wdqwr olpq$4 +4 . d .    . dq , @ olpq$4 44dd Xvdqgr od sursrvlflöq dqwhulru/ od ghprvwudflöq frqvlvwh hq suredu txh

Shur | frpr vh wlhqh txh hv htxlydohqwh d

4  dq.4 4  ,@3 olp + q$4 4  d 4d

4  dq.4 4 4 dq.4  @ @ dq.4 + , 4d 4d 4d 4d

4 olp dq.4 @ 3 | + , hv xqd vxfhvlöq dfrwdgd/ q$4 4d 4  dq.4 4 olp +  ,@3 q$4 4  d 4d

4 olp +4 . d . d5 .    . dq , @ q$4 4d

Whruhpd 6148 +Dojheud gh Oðplwhv, Vhdq d> e 5 U1 Vl olpq {q @ d | olpq |q @ e1 Hqwrqfhv 4, olpq +{q . |q , @ olpq {q . olpq |q @ d . e

5, olpq +{q  |q , @ olpq {q  olpq |q @ d  e q {q @ d vl e 9@ 3 a | 9@ 3> ;q 6, olpq + {|qq , @ olp q olpq |q e

5*, olpq +{q , @  olpq {q @ d>  5 U +Edvwd wrpdu od vxfhvlöq frqvwdqwh +,4 q@4 hq 5 | vh rewlhqh od ljxdogdg,

4*, olpq +{q  |q , @ olpq {q  olpq |q @ d  e +Edvwd wrpdu  @ 4 hq 5* | oxhjr xvdu 4 sdud rewhqhu od ljxdogdg,

6171 FÄOFXOR GH OÐPLWHV

4<<

Vl elhq hv flhuwr txh ho whruhpd vh uhhuh d ljxdogdghv/ gheh whqhuvh hq phqwh txh ho whruhpd wdpelìq 4 kdeod gh od frqyhujhqfld gh odv vxfhvlrqhv +{q . |q ,4 q@4 > +{q |q ,q@4 / hwf/ hv ghflu/ ho whruhpd vh uhhuh d od h{lvwhqfld gh orv oðplwhv sxhvwrv d od l}txlhugd gh odv ljxdogdghv | dghpäv vh uhhuh do ydoru txh frqyhujhq1 Ghprvwudflöq= +4, Oðplwh gh od vxpd1 Frpr {q $ d hv ghflu/ ; %d A 3
Frpr |q $ e> ; %e A 3 Qe j wdo txh

q A Q , m{q  dm ? %d a m|q  em ? %e =

Sru rwud sduwh m{q . |q  +d . e,m @ m+{q  d, . +|q  e,m  m{q  dm . m|q  em1 Dvð vh wlhqh/ sru or dqwhulru/ txh vl q A Q m{q . |q  +d . e,m ? %d . %e @ % Sru or wdqwr/ ; % A 3
4 Frpr +{q ,4 q@4 hv frqyhujhqwh/ hood hv dfrwdgd1 Od vxfhvlöq frqvwdqwh +e,q@4 wdpelìq hv dfrwdgd1 Sru sursrvlflöq dqwhulru olpq +|q  e, @ 3 | olpq +{q  d, @ 3= Sru whruhpd fhur dqltxlod | oðplwhv gh od vxpd olpq$4 +{q |q  de, @ olpq$4 {q +|q  e, . olpq$4 +{q  d,e @ 3.3 @ 3 Dvð

olp +{ | , @ de q$4 q q

+6, Oðplwh gho fxrflhqwh1 Frqvlghuhprv od ljxdogdg

4 e{q  d|q {q d  @ @ +e|q  d{q , |q e e|q e|q

| ho khfkr txh + |q4 e ,4 q@4 hv xqd vxfhvlöq dfrwdgd1 Sru +5*, olpq$4 +e{q  d|q , @ ed  de @ 3 vh wlhqh txh

{q d olp +  ,@3 q$4 |q e

q5 .5 Hmhpsor 4;< Fdofxohprv ho oðplwh gh od vxfhvlöq +{q ,4 q@4 grqgh {q @ 8q5 .q 1 Frpr q5 . 5 4 . q55 q5 . 5 5 5 @ 8qq5 qq @ {q @ 5 8q . q 8 . q4 q5 . q5 | |d txh olpq$4 +4 . q55 , @ 4 | olpq$4 +8 . q4 , @ 8 vh wlhqh txh olp+4 . q55 , 4 olp {q @ @ 4 q 8 olp+8 . q ,



FKDSWHU 61 VXFHVLÖQ

533

Revhuyhprv txh sdud fdofxodu hvwh oðplwh/ qr srghprv xvdu gluhfwdphqwh od sduwh +6, gho whruhpd dqwhulru sxhv qr h{lvwhq orv oðplwhv olpq$4 8q5 . q | olpq$4 q5 . 51 +Hq hvwh fdvr whqhprv txh olpq$4 8q5 . q @ 4 | olpq$4 q5 . 5 @ 4 shur 4 95 U1 Hq jhqhudo/ vh sxhgh ghprvwudu xvdqgr ho dojheud gh oðplwhv txh= 4,

dn dn qn . dn4 qn4 .    . d3 olp @ q$4 en qn . en4 qn4 .    . e3 en

n +hq hvwh fdvr pxowlsolfdqgr sru 44@q @qn ,1  dn qn . dn4 qn4 .    . d3 3 5, olp @ qr h{lvwh q$4 eo qo . eo4 qo4 .    . e3 q 5 r frqyhujh d 31 Hmhpsor 4<3 Od vxfhvlöq qq6 .4 .5

Vroxflöq=

q5 . 4 olp 6 q$4 q . 4

5 oxhjr olpq$4 qq6 .4 @ 31 .5

vl o A n vl o ? n

q5 . 4 4@q6 olp 6  q$4 q . 5 4@q6 4@q . 4@q6 @ olp q$4 4 . 5@q6 3 olpq$4 +4@q . 4@q6 , @ @3 @ 6 olpq$4 +4 . 5@q , 4 @

6 qr h{lvwh Hmhuflflr 4<4 Ghpxhvwuh txh olpq$4 qq5 .5 .4

Vroxflöq= +Sru ho pìwrgr gho devxugr, 6 h{lvwh | vx ydoru hv O 5 U Vxsrqjdprv txh ho olpq$4 qq5 .5 .4 6 .5 q Gdgr txh olpq$4 q5 .4 h{lvwh +sru hmhpsor dqwhulru, hqwrqfhv sru dojheud gh oðplwhv/

| Dvð

q6 . 5 q5 . 4  h{lvwh olp 5 q$4 q . 4 q6 . 5

q6 . 5 q5 . 4 q6 . 5 q5 . 4  6 @ olp 5 olp 6 @O3@3 olp 5 q$4 q . 4 q . 5 q$4 q . 4 q$4 q . 5

Sru rwud sduwh/ hv fodur txh/

q6 . 5 q5 . 4 olp 5  q$4 q . 4 q6 . 5

+617,

q6 . 5 q5 . 4 olp 5 +618,  @ olp 4 @ 4 q$4 q . 4 q6 . 5 q$4 Dvð sru +617, | +618, | sru od xqlflgdg gho oð plwh vh wlhqh txh 3 @ 4 or txh hv xqd frqwudglfflöq1 Od 6 qr h{lvwh1 frqwudglfflöq vh surgxmr do vxsrqhu txh h{lvwðd ho oðplwh/ sru or wdqwr olpq$4 qq5 .5 .4 Fdofxodu +7,q . 8q O @ olp q$4 +7,q.4 . 5+8,q

Vroxflöq=

+ 87 ,q . 4 O @ olp q$4 +7,+ 87 ,q . 5

6171 FÄOFXOR GH OÐPLWHV

534

Shur olpq$4 + 87 ,q @ 3 sxhv m 87 m ? 4 Oxhjr sru äojheud gh oðplwhv

+ 7 ,q . 4 olpq$4 ++ 87 ,q . 4, 4 @ @ O @ olp 87 q q$4 + 8 , . 5 olpq$4 +7,+ 87 ,q . 5 5

Sursrvlflöq 6149 +Pdqwhqflöq gho vljqr gho oðplwh hq xqd vxfhvlöq , Vl olpq {q @ O/ frq O 5 U. vh wlhqh txh h{lvwh Q 5 Q wdo txh vl q A Q hqwrqfhv {q 5 U. Ghprvwudflöq= Frpr {q $ O vh wlhqh txh

; % A 3
wdo txh

q A Q , {q 5 YO> % = O Dvð wrpdqgr xq % elhq sduwlfxodu % @ 7 A 3 vh wlhqh txh O . %, sru or wdqwr vl q A Q vh wlhqh O 6O @ O  @ O  % ? {q 3? 7 7 

Frurodulr 614: Vl {q $ d/ |q $ e | vl vh wlhqh od ghvljxdogdg {q  |q d sduwlu gh xq ðqglfh Q3 / hqwrqfhv olp {q  olp |q +r htxlydohqwhphqwh d  e, q q

Frurodulr 614; +Whruhpd gho Dfrwdplhqwr, Vhdq {q  }q  |q d sduwlu gh xq ðqglfh Q hq dghodqwh1 Vl olp {q @ olp |q @ O hqwrqfhv olp }q @ O

Ghprvwudflöq= Frpr {q $ O/ vh wlhqh txh sdud %4 A 3 h{lvwh Q4 5 Q wdo txh m{q  Om ? %4 / sdud wrgr q A Q4 1 Frpr |q $ O +, ; %5 A 3 Q5 j vh wlhqh vlpxowdqhdphqwh txh vl q A Q O  %4 ? {q ? O . %4 a O  %5 ? |q ? O . %5 a {q  }q  |q

dvð/ xvdqgr od wudqvlwlylgdg gh od ghvljxdogdg vh rewlhqh

O  %4 ? {q  }q  |q ? O . %5 vl q A Q

Wrpdqgr %4 @ %5 @ % +lqlfldophqwh, vh wlhqh txh
Hmhpsor 4<5 Suredu txh olpq$4

q

Vroxflöq= Sdud wrgr q 5 Q/ vh wlhqh> gh grqgh

Frpr olpq$4  q4 @ olpq$4 Hq jhqhudo/ vh wlhqh>

q, @ 3 xvdqgr ho whruhpd gh dfrwdplhqwr

vhq +

4  vhqq  4

4 vhqq 4   ;q 5 Q q q q 4 q @ 3/ hqwrqfhv sru whruhpd gho dfrwdplhqwr vhqq @3 olp q$4 q 



FKDSWHU 61 VXFHVLÖQ

535 Hmhpsor 4<6 Vl i{q j hv xqd vxfhvlöq dfrwdgd hqwrqfhv olpq$4 {qq @ 3

Vroxflöq= Vl i{q j hv xqd vxfhvlöq dfrwdgd hqwrqfhv h{lvwh N 5 U. wdo txh/ m{q m  n> ;q 5 Q oxhjr> n  {q  n ;q 5 Q or txh lpsolfd

3

{q n n   ;q 5 Q q q q {q n q @ 3/ sru whruhpd gho dfrwdplhqwr vh wlhqh hqwrqfhv txh olpq$4 q @ 

Frpr olpq$4  qn @ 3 | olpq$4

Fdofxodu

q . vhq +q, O @ olp q$4 q  +4,q

Vroxflöq=

4 . vhqq+q, O @ olp q$4 4  +4,q q

shur sdud wrgr q 5 Q

vhq +q, 4 4   q q q Oxhjr sru ho whruhpd gho dfrwdplhqwr

Dvð/ sru äojheud gh oðplwh/

olp q$4

4 +4,q 4   q q q

a

vhq +q, @3 q

olp q$4

a

olp

+4,q @3 q

4 . vhqq q 4 O @ olp @ @4 q$4 4  +4,q 4 q

Hmhpsor 4<7 Fdofxodu

olp q$4

Vroxflöq=

5 5q + ,q @ q  6 6 | olpq$4 + 56 ,q @ 3/ hqwrqfhv sru ho whruhpd gho

q . frv +q, 6q

q 5q 5  @ + ,q q q 6 6 6 dfrwdplhqwr

;q 5 Q

q olp @ 3= q$4 6q

Sru rwud sduwh dghpäv olpq$4 + 46 ,q @ 3 Oxhjr sru whruhpd gho dfrwdplhqwr Hq frqvhfxhqfld/ sru äojheud gh oðplwhv olp q$4

4 frv q 4  q  q 6 6 6 frv q olp @3 q$4 6q

q . frv q q frv q @ olp q . olp @3 q q $4 q $4 6 6 6q

4 @ 3= q

6171 FÄOFXOR GH OÐPLWHV

536

Hmhpsor 4<8 Fdofxodu olpq$4 {q grqgh {q @

Vroxflöq= oxhjr shur

s s q  q . 41

s s s s 4 q. q.4 s s {q @ + q  q . 4, s @s q. q.4 q. q.4

4 4 s s @  olp s olp { @ olp s q$4 q q$4 q . q . 4 q$4 q . q . 4 4 4 s s 3 s q q. q.4

| olpq$4 s4q @ 3/ oxhjr sru whruhpd gho dfrwdplhqwr

| sru frqvljxlhqwh olpq$4

4 s @3 olp s q$4 q . q . 4

s s q  q . 4 @ 31

Hmhpsor 4<9 Ghprvwudu = Vl mum ? 4 hqwrqfhv olpq$4 quq @ 3 Vroxflöq= Frqvlghuhprv 3 ? u ? 4 Vhd k @ 4u  4 Sdud q  5/ vh wlhqh Oxhjr

+4 . k,q @ 4 . qk .

3 ? quq @

shur Sru whruhpd gho dfrwdplhqwr Hmhpsor 4<:

q+q . 4, 5 q+q . 4, 5 k .    . kq  k 5 5

q q 5q  @ +4 . k,q q+q  4,@5 q+q  4,

5q 5 @ olp @3 olp q$4 q+q  4, q$4 q  4 olp quq @ 3 q$4

5, Vhd 4 ? u  3 Sdud wrgr q 5 Q/ vh wlhqh

mumq  uq  mumq

Frpr olpq$4 qmumq @ 3 sxhv 3  mum ? 4 hqwrqfhv sru whruhpd gho dfrwdplhqwr vh wlhqh olpq$4 quq @ 31 Hmhpsor 4<; Uhwrphprv orv hmhpsorv 49; | 49<

4 .    q4$ {q @ 4 . 4$4 . 5$ q  4 |q @ 4 . q4 @ 4 . 4 . 45 +4  q4 , .    q4$ +4  q4 ,+   ,+4  q q ,

Frpr |q  {q ;q | dpedv frqyhujhq vh wlhqh O5 @ olpq |q  olpq {q @ O4 / sru or wdqwr O5  O4 1 Qxhvwur remhwlyr hv suredu txh O4 @ O5 sru or wdqwr vöor qrv idowd yhu txh O4  O5 Sdud s mr od vxfhvlöq +sq ,4 q@4 frq sq @ 4 . 4 .

4 4 s4 4 .    +4  ,+   ,+4  , 5$ s$ q q

FKDSWHU 61 VXFHVLÖQ

537

frqyhujh +sru rshudwruld gh oðplwhv, do wìuplqr {s fxdqgr q wlhqgh d 41 Shur sdud fdgd s mr vh wlhqh |q  sq +sxhvwr txh sq frqwlhqh d orv sulphurv s . 4 sulphurv gh |q ,/ wrpdqgr oðplwh fxdqgr q $ 4 vh rewlhqh/ phgldqwh ho xvr gho frurodulr 614:/ txh O5  {s Dkrud srghprv kdfhu whqghu s $ 4/ dvð O5  olp {s @ O4 s$4

sru or wdqwr Dvð/ odv vxfhvlrqhv +{q ,4 q

O5  O4

| +|q ,4 q

frqyhujhq dpedv do plvpr ydoru/ oodpdgr h +Xvdqgr od fäofxodgrud @4 @4 | fxdotxlhud gh odv vxfhvlrqhv dqwhulruhv ylvxdolfh txh h @ 5=:4;5;4;===, Revhuydflöq=

n olp +4 . ,q @ hn q$4 q Mxvwlfduhprv vöor hq sduwh9 hvwd revhuydflöq1 Vl n @ 4s > s 5 Q/ od vxfhvlöq vh sxhgh yhu frpr xqd 4 p vxevxfhvlöq gh {p @ +4 . p , |d txh +4 .

n q 4 q 4 sq n , @ +4 . , @ ++4 . , , > q sq sq

grqgh p @ sq | frq d|xgd gho whruhpd 61< srghprv rewhqhu od revhuydflöq Hmhpsor 4<< Fdofxodu

Vroxflöq= d,

d,

 q 4 olpq$4 qq .4

e,

olpq$4

 q 4 olpq$4 qq .4



 q6 q 8q.7

5

  q+4 4 , q @ olpq$4 q+4. q4 , q @



4 q q$4 +4 q , 4 q +4. q$4 q,

olp olp

4 @ hh4 e,

olpq$4



@ h5    6 , q 5q+4 q6 q @ olp q$4 8q+4. 857qq , 8q.7 5

6 q +4 olp 5q , @ olpq$4 + 58 ,q olpqq$4 $4 +4. 7 ,q 8q

6

@ 3 h 75 h8 @ 3=

Hmhpsor 533 Vxsrqlhqgr txh od vxfhvlöq +{q ,4 q@4 frqyhujh1 Fdofxodu vx oðplwh/ grqgh od vxfhvlöq {q hv ghqlgd uhfxuvlydphqwh frpr vljxh {4 A 3 9 Hq

4 d {q.4 @ +{q . , 5 {q

d 5 U. >

ehqhflr gho fäofxor/ khprv ghvduuroodgr hvwd revhuydflöq frpr wìfqlfd gh fäofxor gh oðplwhv gh vxfhvlrqhv1

6171 FÄOFXOR GH OÐPLWHV

538

Vroxflöq= Vl olpq$4 {q @ O 9@ 3 wrpdqgr oðplwh hq dperv odgrv gh od ghvljxdogdg dqwhulru vh wlhqh olpq {q.4 @ 45 +olpq {q . O @ 45 +O . Od ,

d

q {q

olp

,

s s 5 @ 45 +O5 . d,1 Sru or wdqwr O5 @ d dvð dvð Os s O @ d ö O @  d/ shur frpr od vxfhvlöq hv srvlwlyd O @ d1 Qrwh txh= dýq qr vdehprv s vl {q $ d/ wrgr vh edvö hq od frqyhujhqfld gh {q 1 Sru rwud sduwh vl {4 kxelhvh vlgr qhjdwlyr {q $  d s Hmhpsor 534 Vhd {q @ q d> d 5 U. 1 Srghprv yhu txh {q frqyhujh/ |d txh vl= d, d A 4 hqwrqfhv {q hv ghfuhflhqwh | dfrwdgd lqihulruphqwh r elhq vl e, d 5 `3> 4^ hqwrqfhv {q hv fuhflhqwh | dfrwdgd s s vxshulruphqwh1 hv ghflu/ olpq$4 q d @ O h{lvwh Sru or wdqwr q d frqyhujh/ s s Vl d A 4 O @ lqii q d = q 5 Qj | vl 3 ? d ? 4/ O @ vxsi q d = q 5 Qj= Sru yhu txh hq dperv fdvrv O @ 4/ fodudphqwh vh yh txh O 9@ 31 4 s 4 Frpr {q @ q d @ d q wlhqh frpr vxevxfhvlöq d q+q.4, sru whruhpd 61< whqhprv txh

shur

O @ olp q$4

s 4 q d @ olp d q+q.4, q$4

d q+q.4,

dq

4

oxhjr Dvð/ O @ 4 Sru or wdqwr

O @ olp q$4

@

d q+q.4,

olp q$4

4  d q.4

4

4

@

dq 4

d q.4 4

olpq d q O @ @ @4 4 O q .4 olpq d 4

s q d@4

;d 5 U.

s Hmhpsor 535 Fdofxodu olpq$4 q q1 s Vroxflöq= Vdehprv txh |q @ q q irupd xqd vxfhvlöq ghfuhflhqwh +d sduwlu gho whufhu wìuplqr, | dfrwdgd lqihulruphqwh> sru or wdqwr frqyhujh1 s Vhd O @ olpq$4 |q / fodudphqwh O  4/ frpr 5q 5q hv xqd vxevxfhvlöq gh |q vh wlhqh txh vx oðplwh wdpelìq hv O Sru or wdqwr whqhprv s s 4 4 s 5q q s O5 @ olp + 5q,5 @ olp +5q, q @ olp 5 q q q @ olp 5 q q @ 4  O q$4 q$4 q$4 oxhjr O5 @ O/ dvð O @ 4/ sru or wdqwr

olp q$4

s q q@4

Sursrvlflöq 614< Vhd s 5 ]. s A 41 Vl olpq$4 {q A 3 hqwrqfhv t s olp s {q @ s olp {q q$4 q$4

Hmhpsor 536 Fdofxodu

olp q$4

u 6

;.

5 q6

FKDSWHU 61 VXFHVLÖQ

539 Vroxflöq=

olp q$4

Hmhpsor 537 Fdofxodu

Vroxflöq= Frpr

u

u

6

6

5 ;. 6 @ q

O @ olp q$4 4

u q

u q

4.

s 5 6 olp +; . 6 , @ ; @ 5 q$4 q

q 4 .4. s 6 q +5q,6 . 5

4 4 4. q q

q5

Sxhvwr txh olpq$4 4 @ olpq$4 +4 . q4 , @ 4/ hqwrqfhv sru whruhpd gh1 dfrwdplhqwr u 4 q 4. @4 olp q$4 q Sru rwud sduwh>

q q @ olp t olp s q$4 6 +5q,6 . 5 q$4 6 q ;.

Hq frqvhfxhqfld/ sru äojheud gh oðplwhv= O @ olp q$4 Hmhpsor 538 Fdofxodu Vroxflöq=

shur/

dghpäv

u q

4.

4 4 @ olp t @ q $4 5 5 6 ; . q56 q6

q 6 4 4 .s @4. @ 6 q 5 5 +5q,6 . 5

s q7 . 6q . 5  q5 O @ olp frv +q$, q$4 q . q$

s q7 . 6q . 5  q5 6q . 5 olp @ olp s 7 q$4 q$4 + q . 6q . 5 . q5 ,+q . q$, q . q$ 6q . 5 6q . 5 3? s ?s 7 5 7 + q . 6q . 5 . q ,+q . q$, q . 6q . 5 . q5 6 5 6q . 5 q . q5 t @ olp olp s 7 @3 q$4 q . 6q . 5 . q5 q$4 4 . q66 . q57 . 4

Oxhjr sru whruhpd gho dfrwdplhqwr

s q7 . 6q . 5  q5 @3 olp q$4 q . q$

| frpr od ixqflöq frvhqr hv dfrwdgd vh wlhqh sru whruhpd 6146 txh O @ 3 Hv lpsruwdqwh kdfhu qrwdu dojxqrv oðplwhv ixqgdphqwdohv+|d phqflrqdgrv, txh ghehprv whqhu suhvhqwh sdud ho fäofxor gh rwurv oð plwhv1

6171 FÄOFXOR GH OÐPLWHV

53:

Oðplwhv Ixqgdphqwdohv 41 olpq$4 f @ f>

4 51 olpq$4 qk @ 3>

f5U

k5U

4 61 olpq$4 qs@t @ 3> s 5 ]> t 5 ] vl mum ? 4 71 olpq$4 uq @ 3> k5U 81 olpq$4 vhqqkq @ 3> s 91 olpq$4 q d @ 4> d 5 U. s :1 olpq$4 q q @ 4>  q ;1 olpq$4 4 . q4 @ h>  q <1 olpq$4 4 . qn @ hn > n 5 U

431 olpq$4 +4 . 4 .

.  . q4$ , @ h 4 441 olpq$4 +4 . u . u5 .  . uq , @ 4 u > vl mum ? 4 ; 3 > vl t A s ? s ds d sq vl t @ s 451 olpq$4 dt qt @ dt = glyhujh vl t ? s 4

5$

.

4

6$

Rwur oðplwh ixqgdphqwdo txh ghprvwuduhprv päv dghodqwh xvdqgr ho whruhpd gh hqodfh hv

Hmhuflflr 539 Fdofxodu

n olp q vhq+ , @ n q$4 q

olp q$4

n5U

s 6 q frv+q, 5 vhq+6@q, . q.7 4@q

s6 q, / e @ 5 vhq+6@q, Vroxflöq= Vhd dq @ qqfrv+ q .7 4@q Fdofxohprv olp dq | olp eq 1 +Gh h{lvwlu/ srghprv dsolfdu dojheud oðplwhv, Sdud wrgr q 5 Q/ vh wlhqh u s s 6 6 4 q q 6 4 ? s ? ? 3? 6 5 q.7 q q q5 frpr olpq$4

s6 q5 4

@ 3/ hqwrqfhv sru whruhpd gh dfrwdplhqwr s 6 q olp @3 q$4 q . 7

| frpr od vxfhvlöq +frv q, hv dfrwdgd hqwrqfhv sru ho whruhpd gho fhur dqltxlod olp d @ 3 q$4 q

Sru rwud sduwh/ dsolfdqgr ho oðplwh ixqgdphqwdo

olp q vhq+n@q, @ n q$4

@q, @ 5 olp rewhqhprv txh olpq$4 5 vhq+6 q$4 q vhq+6@q, @ 5  6 @ 9 oxhjr olpq$4 eq @ 9 | dsolfdqgr ho 4@q dojheud gh oðplwhv/ vh wlhqh olp d . eq @ 3 . 9 @ 9 q$4 q

FKDSWHU 61 VXFHVLÖQ

53;

Hmhuflflr 53: Ghpxhvwuh txh vl olp {q h{lvwh | olp |q qr h{lvwh hqwrqfhv olp {q . |q qr h{lvwh Vroxflöq= +Sru ho pìwrgr gho devxugr, Vxsrqjdprv txh olp {q . |q h{lvwh1 Gdgr txh olp {q h{lvwh hqwrqfhv sru dojheud gh oðplwhv> olp+{q . |q ,  {q

hv ghflu

olp |q

or txh frqwudglfh ho khfkr txh olp |q qr h{lvwh1

h{lvwh

h{lvwh

Hmhuflflr 53; Ghwhuplqh vl ho vljxlhqwh oðplwh h{lvwh>

Vroxflöq= Vhd {q @

s7 q7q 6

.8

6q q6 . 5 olp s . q$4 7 q7 . 8 q5 . 4

6 | |q @ qq5 .5 .4

6q 6q 6 @ olp s olp { @ olp s @ olp s @6 q$4 q q$4 7 q7 . 8 q$4 q 7 4 . 8@q7 q$4 7 4 . 8@q7

oxhjr h{lvwh olpq$4 {q 1 Vlq hpedujr sru hmhpsor 4;< olpq$4 |q qr h{lvwh sru frqvljxlhqwh dsolfdqgr ho hmhuflflr dqwhulru whqhprv txh ho oðplwh gdgr qr h{lvwh1   q   Hmhuflflr 53< Ghwhuplqh { 5 U  i3j wdo txh olpq$4  dqdq.4  ? 4 grqgh/ dq @ qq${q 1 Vroxflöq=

     dq.4   +q . 4,${q.4 qq  m{mqq @     dq   +q . 4,q.4 q${q  @ +q . 4,q

hqwrqfhv

  q  dq.4    @ olp m{mq olp   q$4 dq q$4 +q . 4,q q  q @ m{m olp q$4 q . 4 q  4 @ m{m olp q$4 4 . 4@q 4 4 @ m{m olp @ m{m q$4 +4 . 4@q,q h

gh grqgh

   dq.4   ? 4 +, olp  q$4 dq  +, +, +,

4 ?4 h m{m ? h a { 9@ 3 h ? { ? h a { 9@ 3 { 5`  h> h^i3j m{m

Ghwhuplqh sdud txì ydoruhv gh { hq U  i3j ho oðplwh    dq.4   olp  q$4 dq  hv phqru txh 4/ vlhqgr dq @ q45 +{ . 4,q

Hmhuflflr 543

: Hvwd

:

wìfqlfd hv xvdgd hq or txh vxhoh oodpduvh lqwhuydorv | udglrv gh frqyhujhqfld gh vhulhv gh s rwhqfldv1

6171 FÄOFXOR GH OÐPLWHV

53<

Vroxflöq= Frpr whqhprv      dq.4   q5 q5  4 .4 q @  @ m{ . 4m   dq   +q . 4,5 +{ . 4, +{ . 4,q +q . 4,5 oxhjr     q5 m{ . 4m olpq$4  dqdq.4  @ olpq$4 +q.4, 5 q5 @ m{ . 4m olpq$4 q5 .5 q.4 @ m{ . 4m  4 Sru or wdqwr

Hmhuflflr 544

    olpq$4  dqdq.4  ? 4 +, m{ . 4m ? 4 a { 9@ 4 +, 4 ? { . 4 ? 4 a { 9@ 4 +, 5 ? { ? 3 a { 9@ 4 +, { 5`  5> 3^i4j

Hmhuflflr 545 Ghwhuplqh vl odv vljxlhqwh dupdflrqhv vrq yhugdghudv r idovdv1 41 Vl idq j hv dfrwdgd hqwrqfhv idq j hv frqyhujhqwh

51 Vl olpq$4 mdq m h{lvwh hqwrqfhv idq j hv frqyhujhqwh 61 Vl idq j hv dfrwdgd hqwrqfhv olpq$4 dqq5 @ 3 71 Od vxfhvlöq

hv dfrwdgd



q 5, q .7q

+

5

81 Wrgd vxfhvlöq frqyhujhqwh hv prqöwrqd1

Vroxflöq=

41 Idovr1 Od vxfhvlöq dq @ vhq+ q , hv dfrwdgd shur qr frqyhujhqwh 5 51 Idovr1 Vhd dq @ +4,q > mdq m @ 4 olpq$4 mdq m @ 4 shur idq j qr hv frqyhujhqwh1

61 Yhugdghur1 Vl idq j hv dfrwdgd/ hqwrqfhv h{lvwh n 5 U. wdo txh mdq m ? n ;q 5 Q1 Oxhjr dq n n  5  5 > ;q 5 Q 5 q q q n Sru whruhpd gho dfrwdplhqwr/ gdgr txh olpq$4 q5 @ 3/ vh wlhqh dq @3 olp q$4 q5

71 Yhugdghur1 Od vxfhvlöq dq @ Hq hihfwr>



q 5, q .7q

+ 5

hv frqyhujhqwh1

+5,q +4,q @ olp @3 olp q q q$4 5 . 7 q$4 4 . 5q Sru whruhpd/ wrgd vxfhvlöq frqyhujhqwh hv dfrwdgd

81 Idovr1 Vhd dq @

4, q / od vxfhvlöq idq j hv frqyhujhqwh shur qr hv prqöwrqd1



+

Hmhuflflr 546 Fdofxodu

q

olp q$4

+4@q,5 . +5@q,5 .    . +q@q,5 q

FKDSWHU 61 VXFHVLÖQ

543 Vroxflöq=

+4@q,5 . +5@q,5 . +6@q,5 .    . +q@q,5

shur

4 . 55 . 65 .    . q5 @

+hvwd ýowlpd ljxdogdg vh sxhgh ghprvwudu sru lqgxfflöq, Oxhjr olpq$4

@q,5 .+5@q,5 ..+q@q,5 q

5 5 5 @ q45 . q5 5 . q6 5 .    . qq5

@

5

5

 q5

4.5 .6 .

q5

.

q+q . 4,+5q . 4, 9

q.4, @ olpq$4 q+q.4,+5 9q6

+4

6 q5 .q @ olpq$4 5q .6 9q6

@

Hmhuflflr 547 Fdofxodu O @ olp q$4

4 6

4 . 8 . 85 . 86 .    . 8q4 8q . q

Vroxflöq= Od vxpd 4.8.85 .86 .  .8q4 fruuhvsrqgh d xqd vxpd gh xqd surjuhvlöq jhrpìwulfd gh q wìuplqrv | ud}öq u @ 8/ oxhjr 4  8q 4 . 8 . 85 . 86 .    . 8q4 @ 48 Sru or wdqwr q 48 O @ olpq$4 7+8 q .q, @8q ,4 @ olpq$4 ++4 7,+4.q@8q , shur olpq$4 84q @ olpq$4 + 48 ,q @ 3 +sru hmhpsor oðplwhv ixqgdphqwdohv 7, Sru rwud sduwh frpr q ? 5q / sdud wrgr q 5q q 5q 5 q 5 + q  q  q , +, ++ ,q  q  + ,q , 8 8 8 8 8 8 Frpr olpq$4 + 58 ,q @ 3/ sru whruhpd gh dfrwdplhqwr olpq$4 8qq @ 3 Hq frqvhfxhqfld sru äojheud gh oðplwhv whqhprv O @ 4@71

Hmhuflflr 548 Ghwhuplqh vl od vljxlhqwh vxfhvlöq idq j frqyhujh grqgh q [ 4 dq @ 5 q m @4 . m Vroxflöq=

dq @

Sxhvwr txh q5 . n  q5 . 4> Sru or wdqwr/

4 4 4 4 . . .. 5 q5 . 4 q5 . 5 q5 . 6 q .q

;n 5 Q/ hqwrqfhv

4 4  5 > ;n 5 Q q5 . n q .4 3  dq 

q q .4 5

6171 FÄOFXOR GH OÐPLWHV Frpr olpq$4 q5q.4 @ olpq$4 vh wlhqh

544

@q @ 3 @ q5 ,

4

4.4 +

olp d @ 3 q$4 q

dvð od vxfhvlöq idq j frqyhujh1

Hmhuflflr 549 Vh ghqh od vxfhvlöq gh qýphurv uhdohv srvlwlyrv +{q , frpr vljxh s {4 @ 4 {q.4 @ 6{q > ;q 5 Q Hvwxglh od frqyhujhqfld gh od vxfhvlöq +{q ,1

s s s Vroxflöq= Edvwd suredu txh hv dfrwdgd | fuhflhqwh1 {4 @ 4 {5 @ 6 {6 @ 6 6 l, +{q , hv xqd vxfhvlöq dfrwdgd vxshulruphqwh sru 6 Hq hihfwr= d, q @ 4 {4 @ 4  6 e, Vxsrqjdprv {n  6 +fodudphqwh {n A 3,/ sru ghprvwudu {n.4  6 s s {n.4 @ 6  {n  6  6 @ 6 Oxhjr sru lqgxfflöq {q  6 > ;q 5 Q ll, Á +{q , hv fuhflhqwhB Frpr 6  {n  3

{q +6  {q , @ {5q.4  {5q @ +{q.4 . {q ,+{q.4  {q ,  3

Frpr {q.4 . {q A 3 > ;q 5 Q hqwrqfhv {q.4  {q  3 > ;q 5 Q Sru or wdqwr {q.4  {q > ;q 5 Q Oxhjr +{q , hv fuhflhqwh1 Sru Whruhpd 6145 h{lvwh olpq$4 {q @ O Sru or wdqwr O5 @ olpq$4 {5q.4 @ olpq$4 6  {q @ 6  O , O5  6O @ 3 , O+O  6, @ 3 ,O@3öO@6

shur O A 3/ oxhjr olpq$4 {q @ 6

Hmhuflflr 54: Vhd

Ghwhuplqh od ixqflöq | judtxh1

{5q  4 i+{, @ olp 5q q$4 { . 4

Vroxflöq= Sdud fdofxodu hvwh oðplwh/ ghehprv dqdol}du sdud orv glvwlqwrv ydoruhv gh {1 Fdvr 41 Vl m{m ? 4 hqwrqfhv olpq$4 {5q @ 3 oxhjr/ sru äojheud gh oðplwhv1 olpq$4 +{5q  4, 4 {5q  4 olp 5q @ @ @ 4 q$4 { . 4 olpq$4 +{5q . 4, 4

Fdvr 51 Vl m{m A 4 hqwrqfhv>

4  {45q {5q  4 @ olp olp 5q q$4 { . 4 q$4 4 . {45q

shur olpq$4 {45q @ olpq$4 + {4 ,5q @ 3 sxhv m {4 m ? 4/ oxhjr {5q  4 olp 5q @4 q$4 { . 4

FKDSWHU 61 VXFHVLÖQ

545 Fdvr 61 Vl m{m @ 4 hqwrqfhv {5q @ 4 Hq frqvhfxhqfld

; ? 4 vl m{m ? 4 i+{, @ 4 vl m{m A 4 = 3 vl m{m @ 4

juäfd gh i+{, hq od jxud

Hmhuflflr 54;

{5q  4 olp 5q @3 q$4 { . 4

5q 4 i +{, @ olpq$4 {{5q  .4

;

Ghwhuplqdu d 5 U gh prgr txh

+q . 4,d h{lvwh O @ olp q$4 +5q5 . d, 67

Vroxflöq=

qd +4. q4 ,d d 6 +q5 +5. 5 ,, 7 q4 d d q +4. q , @ olpq$4 6 6 q 5 +5. qd5 , 7 4 d +4. q, @ olpq$4 qd6@5 d 6

O @ olpq$4

+5.

q5 ,

7

41 Vl d  6@5 A 3 ho oðplwh qr h{lvwh

51 Vl d  6@5 ? 3 ho oðplwh h{lvwh | ydoh fhur 61 Vl d  6@5 @ 3 sru äojheud gh oðplwh

Hmhuflflr 54<

; Hvwh

+4 . q4 , 5 6 4 @ 6 @ 5 7 O @ olp q$4 +5 . 6 5 , 67 7 5 5q 6

sureohpd | ho dqwhulru fruuhvs rqgh d or txh vh oodpd fäofxor gho oðplwh sxqwxdo gh ixqflrqhv/ hq hvwh fdvr

i Ed

6171 FÄOFXOR GH OÐPLWHV

546

Fdofxodu

olp q$4

Vroxflöq=

q [ n@4

4 n+n . 4,

4 4 4 @  n+n . 4, n n.4

oxhjr dsolfdqgr od surslhgdg whohvfrslfd vh rewlhqh>

q [

Dvð/

Hmhuflflr 553 Fdofxodu Vroxflöq=

q [ 4 4 4 4 @  @4 n+n . 4, n n.4 q.4 n@4 n@4

olp q$4

q [ n.4

4 4 @ olp 4  @43@4 n+n . 4, q$4 q.4 olp q$4

q [

q [ n@f

dun > vl mum ? 4

dun @ duf . duf.4 .    . duq

n@f hvwd vxpdwruld fruuhvsrqgh d od vxpd gh xqd surjuhvlöq jhrpìwulfd/ oxhjr q [ n@f

dun

duf +4  uqf.4 , 4u duq.4 duf  4u 4u

@ @

oxhjr sru dojheud gh oðplwhv | gdgr txh olpq$4 uq @ 3/ vh wlhqh q [ duf duq.4 dun @ olp olp  olp q$4 q$4 4  u q$4 4  u n@f duf @ 3 4u sru or wdqwr

olp q$4

q [ n@f

duq @

duf = 4u

Ghqlflöq 6153 +Oðplwh lqqlwr, Gluhprv txh xqd vxfhvlöq +{q ,4 q@4 wlhqgh d lqqlwr vl sdud wrgr uhdo srvlwlyr N judqgh/ h{lvwh +qxhydphqwh, Q 5 Q wdo txh vl q A Q vh wlhqhq txh {q A N1

Qrwdflöq= Dqrwduhprv hvwr gh od vljxlhqwh irupd

{q $ .4 ö wdpelìq olp {q @ .4 q$4 q$4

Dqäorjdphqwh whqhprv txh {q $ 4 hv htxlydohqwh d ghflu q$4 ;N 5 U.
FKDSWHU 61 VXFHVLÖQ

547

Hmhpsor 554 Vhd {q @ q hqwrqfhv olp {q @ .4 hq fdpelr vl {q @ q hqwrqfhv olpq$4 {q @ 41 Hmhpsor 555 Vhd {q @ dq vl d A 4 d hqwrqfhv {q $ .4

Vroxflöq= Frpr d A 4 vh sxhgh hvfulelu frpr d @ 4 . k frq k A 3/

Hmhpsor 556 Sru ghvduuroor gho elqrplr d od q vh wlhqh dq @ +4 . k,q A 4 . qk Hmhpsor 557 Vl e A 4 hqwrqfhv olpq$4 orje q @ .4 Vl 3 ? e ? 4 hqwrqfhv orje q @ 41 Srghprv revhuydu gh od juäfd gh | @ orje { +yhu jxud ,

txh d phglgd txh q fuhfh lqghqlgdphqwh odv lpdjhqhv orje q wdpelìq fuhfh lqghqlgdphqwh vlq hp0 edujr/ vl 3 ? e ? 4/ odv lpdjhqhv orje q ghfuhfhq lqghqlgdphqwh1

618

Hmhuflflrv Sursxhvwrv

Hmhuflflr 558 Ghpxhvwuh txh

6 6q  4 l, olp @ q$4 7q . 5 7

4 q5 . q . 4 ll, olp @ q$4 5q5 . q 5

Hmhuflflr 559 Ghpxhvwuh txh orv vljxlhqwhv oðplwhv qr h{lvwhq 5q5  4 l, olp q$4 q . 4

6q5 . 4 lll, olp 5 @6 q$4 q . 4

q6 ll, olp 5 q$4 q . 4

Hmhuflflr 55: Fdofxodu orv vljxlhqwhv oðplwhv s 6 .6q.4 5= olpq$4 q5 . 4  q 4= olpq$4 65qq6 .8 q5 .6 q 6= olpq$4 s 7= olpq$4 6qq 6 q6 .43 q 5q 8= olpq$4 vhqq+q, 9= olpq$4 55q  .5q q q 5q.vhq +4@q, := olpq$4 +s5,6q.frv ;= olp q $4 8.q q q 6 vhq.7qm +7, .8 <= olpq$4 qqm.5 43= olpq$4 + 7,q.4 .8q s s s q 44= olpq$4 q . 5+ q . 6  q  5, 45= olpq$4 frv 5q

6181 HMHUFLFLRV SURSXHVWRV s s 46= olpq$4 6 q . 4  6 q 48= olpq$4 qvhq+ q , s s 4:= olpq$4 q5 . 6q  q5 . 5q . 5 5 4<= olpq$4 5qq.4 vhq+ q , .+5q,5, 54= olpq$4 +7.49.69. q6 q.6, 56= olpq$4 +q.4,+qq.5,+ 6 q q.4 q.4 .+4, 58= olpq$4 qq . 5 5q .6 +4,q .6q

548

q

q , 5 .vhq + 47= olpq$4 6q .5q5 7 qfrv q s 5 49= olpq$4 + 78qq7.9 ,+ q . ;q  q, .8+4,q s 5 + q7 .6q.5q , frv +q$, 4;= olpq$4 q.q$ .+5q4, 53= olpq$4 4.6.8.:. 5q5 55= olpq$4 + q45 . q55 . q65 .    qq54 , +5q.4,  5q.4 57= olpq$4 ^ 4.6.8.: q.46 5 5 q ..8 59= olpq$4 4.8.8 8.8 q .q

Hmhuflflr 55; Fdofxoh oðplwh fxdqgr q $ 4 gh dq +fxdqgr fruuhvsrqgd, +q, 4= dq @ + qqvhq 5 4 ,

5q5,+q.4, 6= dq @ +5q.6,+6 q7 .6q.4 s s 8= dq @ q . 5  q  4 wdq+  , 5 q := dq @ 8 q q <= dq @ q$

5=

dq @

7=

dq @ dq @

9=

s

q s q q.sq q5s 43.q q vhq +q5 , sq 5 q +vhq q$, q.5q5 q

t

;= dq @ 43= dq @ 5q

Hmhuflflr 55< Fdofxoh oðplwh fxdqgr q $ 4 gh dq +fxdqgr fruuhvsrqgd,

41 Vhd dq @ Sq / grqgh Sq hv ho shuðphwur gh xq sroð jrqr uhjxodu gh q odgrv lqvfulwr hq xqd flufxqihu0 hqfld gh udglr u

51 Vhd dq @ Dq / grqgh Dq hv ho äuhd hqfhuudgd sru ho sroð jrqr gh q odgrv gho sureohpd dqwhulru

61 Vhd dq @  q / grqgh  q hv ho äqjxor irupdgr hqwuh orv odgrv dg|dfhqwh gho sroðjrqr uhjxodu gho sureohpd dqwhulru

71 Vhd dq @ {4 +q, . {5 +q, grqgh {4 +q, | {5 +q, vrq odv vroxflrqhv gh od hfxdflöq gh vhjxqgr judgr q{5 . e{ . f @ 3 frq e> f frqvwdqwhv | {4 A {5

Hmhuflflr 563 Hvwdeoh}fd ho ydoru gh yhugdg gh odv vljxlhqwhv dupdflrqhv

41 Vl +{q , hv xqd vxfhvlöq frq xqd vxevxfhvlöq {qn dfrwdgd/hqwrqfhv hood wdpelìq or hv1 51 Vl +{qn , hv xqd vxevxfhvlöq prqöwrqd fuhflhqwh hqwrqfhv +{q , hv prqrwrqd fuhflhqwh

61 Vl +{qn , hv dfrwdgd vxshulruphqwh/ +{q , wdpelìq or hv 71 Vl +m{q m, hv dfrwdgr hqwrqfhv +{q , wdpelìq or hv1

81 Vl +m{q m, hv prqöwrqd hqwrqfhv +{q , wdpelìq or hv1

91 Vl wrgd +{qn , hv dfrwdgd hqwrqfhv +{q , wdpelìq or hv1

:1 Vhd {q  |q ;q hqwrqfhv

Vl +{q , hv fuhflhqwh +|q , wdpelìq or hv1

Vl +|q , hv ghfuhflhqwh +{q , wdpelìq or hv1

Vl +|q , hv dfrwdgd/ +{q , wdpelìq or hv1

Vl +{q , hv qr dfrwdgd +|q , wdpelìq qr or hv1

FKDSWHU 61 VXFHVLÖQ

549 Hmhuflflr 564 Vhd +{q ,4 q@4 ghqlgd uhfxuvlydphqwh sru {4 @ 3

{5 @ 4

4 {q.5 @ +{q . {q.4 , 5

hv ghflu/ {q hv ho surphglr gh orv grv dqwhulruhv d, Fdofxodu {5q | {5q.4 | hvfuleluorv frpr xqd vxpd gh wìuplqrv gh xqd surjuhvlöq jhrpìwulfd1 e, Suredu txh {q hv dfrwdgd +xvdu d,1 f, Fdofxodu olpq$4 {q | mxvwlfdu vx h{lvwhqfld +D|xgd= {5q $ 5@6/ {5q.4 $ 5@6, Hmhuflflr 565 Vhdq +{q , | +|q , vxfhvlrqhv frqyhujhqwhv wdo txh

< Q 5 Q a {q  |q > ;q A Q Suredu txh olp {q  olp |q

+D|xgd= Xvh ho khfkr txh vl olp }q @ O A 3 ,

Hmhuflflr 566 Vhd +{q , frqyhujhqwh | q A Q hqwrqfhv olpq$4 {q  d

Hmhuflflr 567 Vh ghqh oðplwh lqihulru +olp lqi {q , | oðplwh vxshulru +olp vxs {q , gh xqd vxfhvlöq +{q , dfrwdgd/ gh od vljxlhqwh irupd Vhd Dq ho vljxlhqwh frqmxqwr Dq @ i{q > {q.4 > {q.5 >    j =@ i{n = n  qj

Frpr +{q , hv xqd vxfhvlöq dfrwdgd/ ho frqmxqwr Dq wdpelìq or hv | sru d{lrpd gh frpsohwlwxg hq U h{lvwh d, Xvdqgr ho khfkr txh

vq @ vxs Dq

> lq @ lqi Dq

D4  D5  D6    

lqgltxh sru txì whruhpd vh wlhqh od frqyhujhqfld gh +vq , | +lq , e, Vh ghqh olp vxs {q @ olp vq a olp lqi @ olp lq pxhvwuh txh olp lqi {q  olp vxs {q

f, Vhd {q @ | |q @ vhq+ q , 5 ; olp lqi {q > olp vxs |q Fdofxodu olp vxs {q > g, Hq txh fdvr vh wlhqh od ljxdogdg hq +e, q +4.+4, ,

>

olp lqi |q

Hmhuflflr 568 d, Wrpdu +|q , gho sureohpd dqwhulru/ hqfrqwudu xqd vxevxfhvlöq gh |q frqyhujhqwh +qrwh txh |q qr frqyhujh, e, Wrgd vxfhvlöq dfrwdgd +{q , srvhh xqd vxevxfhvlöq frqyhujhqwh +Pxhvwuh hvwd dupdflöq, D|xgd= Sxhgh wrpdu xqd {qn $ olp vxs {q +r olp lqi {q , Hmhuflflr 569 Vhd +{q , vxfhvlöq dfrwdgd= Pxhvwuh od htxlydohqfld

+{q , frqyhujh +, olp vxs {q @ olp lqi {q

Fkdswhu 7

Oðplwhv gh Ixqflrqhv 714

Oðplwhv gh Ixqflrqhv

71414

Sxqwr gh Dfxpxodflöq

Hq hvwd vhfflöq yhuhprv ho frqfhswr gh duelwuduldphqwh sur{lpr%/ lghd txh |d ixh ghvduuroodgd hq oðplwh gh vxfhvlrqhv/ hv ghflu/ hq ixqflrqhv gh Q $ U/ od glihuhqfld hq hvwh hvwxglr vhuä sulqflsdophqwh hq ho grplqlr/ txh hq hvwd rsruwxqlgdg hv xq vxefrqmxqwr gh U1 Uhfrughprv txh xqd yhflqgdg gh s hv gh od irupd Ys> % @ `s  %> s . %^/ grqgh % A 3 +hq jhqhudo xqd yhflqgdg gh s hv xq delhuwr txh frqwlhqh d s,1 Ghqlflöq 714 Vhd s 5 U | D  U +s qr qhfhvduldphqwh hq D,1 Gluhprv txh s hv xq sxqwr gh dfxpxodflöq gh D vl wrgd yhflqgdg gh s/ Ys / frqwlhqh dojýq sxqwr d 5 D glihuhqwh gh s1 Hq qrwdflöq päv irupdo ö

; % A 3 / +Ys> %  isj, _ D 9@ !=

; % A 3
Ho frqmxqwr ghulydgr gh D/ ghqrwdgr sru D3 / hv ho frqmxqwr gh wrgrv orv sxqwrv gh dfxpxodflöq gh D/ hv ghflu/ D3 @ is 5 U = s hv xq sxqwr gh dfxpxodflöq gh Dj=

Hmhpsor 56: Vhd D @ `6> :`1 Frpr wrgd yhflqgdg gh 6 lqwhuvhfwd d D vh wlhqh txh 6 hv xq sxqwr gh dfxpxodflöq gh D +hv ghflu/ 6 5 D3 ,1 Hq irupd iäflo vh sxhgh yhu txh D3 @ ^6> :`1

Hmhpsor 56; Vhd D @`  4> 5^^i7j ^ ^8> :^1 7 qr hv sxqwr gh dfxpxodflöq gh D sxhv h{lvwh Y @`6> 8^ yhflqgdg gh 7 wdo txh +Y  i7j, _ D @ !1 Xqd irupd gh ylvxdol}du orv sxqwrv gh dfxpxodflöq/ hvwä gdgd hq od vljxlhqwh sursrvlflöq

Sursrvlflöq 715 Vhd D  U qr ydfðr s 5 D3 vl | vror vl wrgd yhflqgdg gh s frqwlhqh lqqlwrv sxqwrv gh D

Ghprvwudflöq= + + , Wulyldo + , , Vl wrpdprv yhflqgdghv gh od irupd Ys> q4 @ Yq + hv ghflu/ frq % @ q4 ,1 Sdud q @ 4 h{lvwh {4 5 D _ +Y4  isj, | h{lvwh q5 5 Q wdo txh m{4  sm A q45 1 Sdud q5 h{lvwh {5 5 D _ +Yq5  isj, | h{lvwh q6 5 Q wdo txh m{5  sm A q46 / hwf1 54:

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

54;

Sru xq surfhvr lqgxfwlyr vh sxhgh jdudqwl}du od h{lvwhqfld gh {4 > {5 > {6 >   

wrgrv sxqwrv gh D1

Frurodulr 716 Vhd D  U qr ydfðr s 5 D3 +, <+{q ,4 q@4 > {q 5 D  isj | {q $ s

Hvwr vh sxhgh yhu hq od ghprvwudflöq gh od sursrvlflöq dqwhulru

Hmhpsor 56< D @ i q4 = q 5 Qj Fodudphqwh vh yh txh 3 5 D3 / päv dýq D3 @ i3j 4, ,q Hmhpsor 573 E @ i +4q+5.9 q = q 5 Qj 3 E @ i5j

q

5

Hmhpsor 574 F @ ivhq +{, = 3 ? { ? 7 j @ vhq +`3> @7^, s F 3 @ ^3> 55 `

Hmhpsor 575 G @ `6> 7^^`7> :` G3 @ ^6> :`

Hmhpsor 576 Vhd H @ T hqwrqfhv H 3 @ U/ or plvpr rfxuuh frq I @ L +luudflrqdohv,/ I 3 @ U1 Hvwrv uhvxowdgrv vh ghehq do khfkr txh wrgr lqwhuydor wlhqh lqqlwrv udflrqdohv/ hq ho sulphu fdvr/ r lqqlwrv luudflrqdohv/ hq ho vhjxqgr fdvr

Hmhpsor 577 Vhd J @ ]1 Fodudphqwh vh yh txh qlqjýq sxqwr gh ] hv sxqwr gh dfxpxodflöq |d txh wrpdqgr/ s 5 ]/ yhflqgdg Ys> 47 / hood vöor wlhqh xq sxqwr hq frpýq frq ]/ | vl s 5 U  ] vlhpsuh h{lvwh xqd yhflqgdg txh qr frqwlhqh sxqwrv gh ]1 Oxhjr whqhprv J3 @ !1

Ghqlflöq 717 Xq sxqwr s hq D vh glfh dlvodgr/ vl h{lvwh xqd yhflqgdg gh s/ Ys / wdo txh +Ys isj,_D @ !

Hq ho hmhpsor dqwhulru ] hvwä irupdgr vöor sru sxqwrv dlvodgrv/ or plvpr rfxuuh frq Q | frq i q4 = q 5 Qj Revhuydflöq= Vl s 5 D3 / whqhprv od fhuwh}d txh duelwuduldphqwh fhufd gh s vlhpsuh kd| hohphqwrv gh D/ hvwr qr rfxuuh vl s hv xq sxqwr dlvodgr gh D1

71415

Ghqlflöq gh oðplwh gh ixqflrqhv

Ghqlflöq 718 Vhdq s 5 D3 > i = D $ U ixqflöq | O 5 U Gluhprv txh ho oðplwh gh i+{, hv O fxdqgr { wlhqgh d s vl sdud wrgd yhflqgdg gh O> YO > h{lvwh xqd yhflqgdg gh s> Ys / wdo txh4 i +Ys  isj, @ i ++Ys  isj, _ D,  YO >

hv ghflu/ orv hohphqwrv duelwuduldphqwh fhufdqrv d s +orv sxqwrv gh od yhqflqgdg Ys ,/ vdoyr ho plvpr s/ txh vrq wrpdgrv sru i+{, +|d txh ghehq hvwäu hq D @ Grpi , wlhqh vxv lpdjhqhv duelwuduldphqwh fhufdqdv d O +|d txh ghehq hvwdu hq YO , Qrwdflöq=

olp i+{, @ O ö olp i +{, @ O ö i +{, $ O {$s s {$s

Od ghqlflöq dqwhulru hv qhfhvdulr hvfuleluod hq xq ohqjxdmh txh qrv shuplwd rshudu frq hood1 Frpr odv yhflqgdghv gh O vrq gh od irupd YO> % / od iudvh sdud wrgd yhflqgdg gh O htxlydoh d wrpdu fxdotxlhu % A 3 ö +; % A 3,> or plvpr rfxuuh frq odv yhflqgdghv gh s/ Ys> / hq hvwh fdvr ho fxdqwlfdgru hq od ghqlflöq hv h{lvwh/ < A 3 4 Vh

hqwlhqgh txh i ED

'

i ED

K Grp i /|d txh d orv hohphqwrv gh D txh qr hvwdq hq ho Grp i> i

ohv kdfh qdgd1

7141 OÐPLWHV GH IXQFLRQHV

54<

Sru or dqwhulru whqhprv txh vl | vöor vl

olp i +{, @ O {$s

; % A 3 < A 3 / i++Ys>  isj, _ D,  YO> % +yhu jxud QQ,

Iljxud QQ

Dkrud uhfrugdqgr txh vh sxhgh hvfulelu=

Ys> @ i{ 5 U = m{  sm ? j | YO> % @ i| 5 U = m|  sm ? %j>

Ghqlflöq Wudglflrqdo= vl | vöor vl

olp i +{, @ O {$s

+; % A 3,+ < A 3,++3 ? m{  sm ?  a { 5 Grpi , , mi +{,  Om ? %,=

Dvð/ ghprvwudu txh olps i+{, @ O frqvlvwh hq= hqfrqwudu xq  A 3 sdud xq % A 3 gdgr +duelwudulr, | txh yhultxh od lpsolfdflöq

71416

+3 ? m{  sm ?  a { 5 D, , mi +{,  Om ? %

Ghprvwudflöq gh Oðplwh %  1

Hmhuflflr 578 Vhd i+{, @ 6{ . 51 Ghprvwudu txh olp{$4 i+{, @ 8 Vroxflöq= \d txh sdud wrgr % A 3 h{lvwh < +@ 6% , wdo txh 3 ? m{  4m ? 

Dvð/

; % A 3 < @

+, +, +, +, +,

3 ? m{  4m ? 6% 3 ? 6m{  4m ? % 3 ? m6{  6m ? % 3 ? m6{ . 5  8m ? % 3 ? mi+{,  8m a mi+{,  8m ? %

% / +3 ? m{  4m ?  , mi +{,  8m ? %,= 6

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

553

BcSru txì wrpdprv  @ 6% B Sdud txh mxvwdphqwh vh gì od ghvljxdogdg mi+{,  8m ? % | qr rwud ghvljxdogdg1 Qrwhprv txh= Ho  exvfdgr gheh vdwlvidfhu= i +Y4>  i4j,  Y8> % ö i +`4  > 4 . ^, `8  %> 8 . %^ +vdoyr i +4, ,

| frpr i+{, hv fuhflhqwh/  gheh vdwlvidfhu r htxlydohqwhphqwh gh or fxdo whqhprv

`6+4  , . 5> 6+4 . , . 5^`8  %> 8 . %^ +vdoyr i +4,, 8  %  6+4  , . 5 a 6+4 . , . 5  8 . %

 %  6 a 6  %= Dvð/ sdud % gdgr/  exvfdgr gheh fxpsolu   6% +fxdotxlhu  srvlwlyr phqru r ljxdo txh 6% ,

Revhuydflöq= Hq jhqhudo vh wlhqh txh vl i +{, @ d{ . e frq d 5 U  i3j | e 5 U | s 5 U/ hqwrqfhv olp{$s d{ . e @ ds . e1 Hq hvwh fdvr edvwd wrpdu  @ md%m |d txh d 9@ 31 Vl d @ 3 vh wlhqh txh olp{$s e @ e1 5 Hmhuflflr 579 Vhd j+{, @ {{6< / +Yhu jxud päv dedmr,1 Ghprvwudu txh

olp j+{, @ 9= {$6

Vroxflöq= Qrwdu txh Grpj @ U  i6j1 Ho oðplwh hv fruuhfwr |d txh or vljxlhqwh hv yhugdghur= Sdud fdgd % A 3 h{lvwh >  @ % wdo txh= 3 ? m{  6m ?  , mj+{,  9m ? %1 Od lpsolfdflöq dqwhulru hv yhugdghud sru od vljxlhqwh fdghqd= 3 ? m{  6m ? 

+, 3 ? m{ . 6  9m ? % 6 +, 3 ? m+{ . 6, {{ 6  9m ? % = +, 3 ? mj+{,  9m ? %

5 | @ { . 6 /j+{, @ {{6<

Hmhuflflr 57: Vhd k+{, @ {5 . 51 Ghprvwudu olp{$4 k+{, @ 6 Vroxflöq= Gdgr % A 3 < @ plqi4> 6% j 3 ? m{  4m ? 

, m{  4m ? 6% a m{  4m ? 4 , m{  4m ? 6% a m{ . 4m ? 6 , m{  4mm{ . 4m ? 6 6% @ %

7141 OÐPLWHV GH IXQFLRQHV hv ghflu/

554

3 ? m{  4m ? 

, 3 ? m+{ . 4,+{  4,m ? % , 3 ? m{5  4m ? % , 3 ? m{5 . 5  6m ? % , 3 ? mk+{,  6m ? %

Dvð/ ; % A 3 < @ plqi4> 6% j+3 ? m{  4m ?  , mk+{,  6m ? %,1

Od suhjxqwd qdwxudo hv= Sdud % A 3 BcFxäo  wrpdu +föpr vh rewlhqh xq , dsursldgr sdud txh wrgr txhgh frpr od ghqlflöq wudglflrqdoB Od wìfqlfd frqvlvwh hq exvfdu xqd uhodflöq hqwuh %> / hvwd vh frqvljxh exvfdgr xqd uhodflöq hqwuh m{  sm | mi +{,  Om

Hq ho hmhpsor 578 vh frqvlghud m{  4m | m6{ . 5  8m> Gdgr txh m6{ . 5  8m @ m6{  6m @ 6m{  4m/ vl txhuhprv m6{ . 5  8m ? % +hvwr htxlydoh d 6m{  4m ? %,/ qhfhvlwdprv hqwrqfhv wrpdu m{  4m ? 6% hv ghflu/ qhfhvlwdprv wrpdu m{  4m ?  frq   6% 1 5 Hq ho hmhpsor 579 vh frqvlghud m{  6m | m {{6<  9m1 BcFöpr hvwä uhodflrqdgr m{  6m frq od vhjxqgd h{suhvlöqB Sdud frqwhvwdu hvwr qhfhvlwdprv wudedmdu frq od vhjxqgd h{suhvlöq Vl { 9@ 6> m

{5  < +{  6,+{ . 6,  9m @ m  9m @ m{ . 6  9m @ m{  6m {6 {6

Sru or wdqwr/ od vhjxqgd h{suhvlöq hv phqru txh % vl od sulphud wdpelìq or hv/ hv ghflu/  gheh vhu ljxdo d %1 Hq ho hmhpsor 57: orv wìuplqrv d frpsdudu vrq m{  4m | m{5 . 5  6m1 Hq hvwh fdvr txhuhprv txh m{5 . 5  6m @ m{5  4m @ m{  4mm{ . 4m ? %=

% / hvwh ýowlpr sdvr vxjlhuh txh  gheh vhu wrpdqgr frpr % / shur hvwr qr hv Sru or wdqwr/ m{  4m ? m{.4 m m{.4m fruuhfwr |d txh  hv xq qýphur srvlwlyr | qr xqd ixqflöq gh {1 Hq hvwh fdvr vh dfrwd vxshulruphqwh m{5 . 5  6m | hvd frwd vh kdfh phqru txh % frq dojýq  dsursldgr1 Qrwh txh vl  @ 4 | m{  4m ? 

, 4 ? {  4 ? 4 , 4?{.4?6 , m{ . 4m ? 6 Sru or wdqwr m{  4m ? 4 , m{  4mm{ . 4m ? 6m{  4m | vl   6% | d od yh}   4 + @ plqi4> 6% j, vh wlhqh Sru or wdqwr

m{  4m ?

% a m{ . 4m ? 6 6

% 6@ % 6 Hvwd wìfqlfd vh xvd jhqhudophqwh fxdqgr m{  sm dsduhfh vöor frpr xq idfwru hq mi+{,  Om | wdo surfhglplhqwr hv oodpdgr dfrwdplhqwr1 Yhdprv dkrud xq fdvr päv frpsolfdgr sdud xq hvwxgldqwh txh whqjd sureohpdv frq odv ghvljxdogdghv hqwuh lqyhuvrv1 Ho hmhpsor vh yhuä glihuhqwh do dqwhulru shur vöor xvduhprv od wìfqlfd gh dfrwdplhqwr1 s Hmhuflflr 57; Ghprvwudu s olp{$3 { @ 31 Qrwhprv txh vl i+{, @ { hqwrqfhv Grpi @ U. 3 m{  4mm{ . 4m ?

Vroxflöq= Gdgr % A 3/ h{lvwh  @ %5 A 3 wdo txh

3 ? m{m ?  a { 5 U. 3

+, +, +, $

3?{? 3 ? { ? %5 s 3? {?% s {?%

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

555 Sru or wdqwr

s +;% A 3,+< A 3,+{ 5 Grpi a 3 ? m{m ?  $ m {m ? %,= s s {@ d Revhuydflöq= Hq jhqhudo/ vh sxhgh ghprvwudu txh vl d 5 U. 3 > olp{$d 6 @ 5 Hmhuflflr 57< L, Ghprvwudu txh olp{$4 {{ 5 Ghwhuplqh vl od vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv 4 wdo txh LL, Gdgr % @ 47 / h{lvwh  @ 433 LLL, Gdgr % @ 47 / h{lvwh  @

4 6

3 ? m{  4m ?  , m

wdo txh

3 ? m{  4m ?  , m

{6  5m ? % {5 {6  5m ? % {5

6 Vroxflöq= L, Yhdprv frpr vh hqfxhqwud m{  4m hq m {{ 5  5m1 Frpr {6 {  6  5+{  5, m  { . 4m m{  4m m  5m @ m m@ @ {5 {5 m{  5m m{  5m 4 sru xq qýphur hqwrqfhv ghehprv dfrwdu m{ 5m 4 Vl m{  4m ? 6 vh wlhqh txh=

 46 ? {  4 ?

4 6

+,  76 ? {  5 ?  56 ,

5 6

,

Sru or wdqwr/ gdgr % A 3 < @ plqi 46 > 56% j wdo txh

d, 3 ? m{  4m ?  ? 46 e, 3 ? m{  4m ?  ? 56%

Sru or wdqwr sru +d, | +e, whqhprv

? m{  5m ? 4

m{5m

?

6 5

4 6 , m{5m ? 55% , m{  4m ? 6

mi+{,  5m ? %> vl 3 ? m{  4m ? 

LL, Yhugdghur1 Sru or dqwhulru= Gdgr % @ 47 / h{lvwh  @ plqi 46 > 56% j @ plqi 46 > 49 j @ | frpr  @

4 433

?

4 9

3 ? m{  4m ?  , m

hqwrqfhv

3 ? m{  4m ?

LLL, Idovr1 Vl  @ 46 whqhprv

{6  5m ? % {5

8 7

4 9

wdo txh

4 {6 4 ? ,m  5m ? % 433 9 {5

3 ? m{  4m ?  +, 3 ? m{  4m ?

dkrud vl % @ 47 / h{lvwh { @ sxhv

7 6

4 5 7 +, { 5` > ^i4j 6 6 6

6 5` 56 > 76 ^i4j shur m {{ 9 % 5  5m ?    8@7  6  4   @ 9? 4  5  8@7  5  6 7

7141 OÐPLWHV GH IXQFLRQHV

556

.4 @ Hmhpsor 583 Ghprvwudu txh olp{$4@6 57{{ 4

4 8

Vroxflöq= Qhfhvlwdprv exvfdu ho wìuplqr +idfwru, m{ . 4@6m +r htxlydohqwhphqwh ho wìuplqr    7{ . 4 4     5{  4  8  = \d txh

m6{.4m , hq 6

       7{ . 4 4   53{ . 8  5{ . 4   4;{ . 9  9m6{ . 4m        @  8+5{  4,  @ 8m5{  4m  5{  4  8  @  8+5{  4,

edvwduä dfrwdu vxshulruphqwh

m5{4m / sru xq qýphur srvlwlyr n/ sxhvwr txh vl 4

4 n m5{  4m

vh whqguðd

   7{ . 4 4  9m7{ . 4m 9n 4;n m6{ . 4m    5{  4  8  @ 8m5{  4m  8 m6{ . 4m @ 8 6 8% dvð/ edvwduðd wrpdu  @ 4; n | vh rewhqguðd     m{ . 4@6m ? 986%n a  5{44   n , m{ . 4@6m  m5{44m ? ,

m{ m m { m

9 6 .4 85 4

?%

n6  n @ 96

9

%

8

%

8

   .4 4  ,  57{{  8 ? % 4

Yhdprv vl hv srvleoh dfrwdu vxshulruphqwh

4

m5{4m

sdud { hq xqd yhflqgdg gh 4@61

m{ . 4@6m ? 4 +, 4 ? { . 4@6 ? 4 +, 7@6 ? { ? 5@6 +, 5+7@6,  4 ? 5{  4 ? 5+5@6,  4 +, 44@6 ? 5{  4 ? 4@6 ? 44@6 , m5{  4m ? 44@6

Dvð/ vl m5{  4m 9@ 3 vh wlhqh m5{44m A 6@44 qr hv dfrwdgr vxshulruphqwh1 Ho sureohpd hq hvwh fdvr hv txh od yhflqgdg wrpdgd hq wruqr do 4@6 lqfox|h do ydoru { @ 4@5 txlìq dqxod d m5{  4m1 Od yhflqgdg gheh h{foxlu d 4@5 | fhufdqr d ìo1 Sru hmhpsor od yhflqgdg gdgd sru   4@43 m{ . 4@6m ? 4@43 , 4@43 ? { . 4@6 ? 4@43 , 5;@48 ? 5{  4 ? 55@48 , 55@48 ? m5{  4m ? 5;@48 , 48@5; ? m5{44m ? 48@55

oxhjr vl m{ . 4@6m ? 4@43 , m5{44m ? 48 55 4 Gh hvwd irupd/ gdgr % A 3 edvwd wrpdu  @ plqi 4;+8%48 , > 43 j |d txh vl  ? 55

m{ . 4@6m ?  ?

4 4 48 , ? 43 m5{  4m 55

4 43

vh wlhqh

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

557 | dghpäv m{ . 4@6m ? 

, m{ . 4@6m ? ,

m{.4@6m m5{4m

?

%

8 4;+ 48 55 ,

% @

8 48 55 4;+48 55,

   4 4 ,  75{{  4 8  ? %

Ho dfrwdplhqwr uhdol}dgr hq hvwh sureohpd frqvlvwlö uhdophqwh hq ghwhuplqdu hq txì frqmxqwr vh hq0 fxhqwud ho uhfruulgr gh od ixqflöq j+{, @ m5{44m sdud ydoruhv fhufdqrv d 4@61 Dqdol}dqgr ho juäfr gh j+{,/ sdud xqd yhflqgdg dsursldgd gho 4@6 vh wlhqh qdwxudophqwh txh j+{, vhuä dfrwdgd> vl od yhflqgdg hv judqgh frpr sdud frqwhqhu d 4@5 qr vhuä dfrwdgd vxshulruphqwh j+{, |d txh hv lqqlwdphqwh judqgh fhufd gh 4@5 +yhu jxud ,

j+{, @

4

m5{4m

Hmhpsor 584 Ghpxhvwuh txh olp 8{+{  6, @ 43 {$5 Vroxflöq= Qxhvwur remhwlyr hv exvfdu m{  5m hq m8{+{  6, . 43m1 Frpr m8{+{  6, . 43m @ 8m{  4mm{  5m>

whqguhprv txh dfrwdu m{  4m hq xqd yhflqgdg gho sxqwr 51 Yhdprv hvwr juäfdphqwh hq od jxud txh vljxh

7141 OÐPLWHV GH IXQFLRQHV

558

j+{, @ m{  4m

1 grqgh j+{, @ m{  4m Qdwxudophqwh txh j+{, hv dfrwdgr/ wrpdqgr xqd yhflqgdg gh 5 gh udglr 4 hv ghflu/ 4 ? { ? 6 vh wlhqh sru ho juäfr txh j+4, ? j+{, ? j+6, +, 3 ? j+{, ? 5

hv ghflu/ vl Y @ i{ 5 U = m{5m ? 4j/ hqwrqfhv vh wlhqh txh= ;{ 5 Y> m{4m ? 5 | m8{+{6,.43m ? 85m{4m1 Gh hvwd irupd/ gdgr % A 3/ h{lvwh  ? 43% |  ? 4 wdo txh m{  5m ? 

, m{  4m ? 5 +|d txh  ? 4 , { 5 Y ,

m{  5m ? 

, m{  4m ? 5 a m{  5m ? 43%

m{  5m ?  Dvð/

Sru or wdqwr

71417

%, , m{  4m ? 43% +|d txh  ? 43

m{  4mm{  5m ? 5 43% @ 8% , 8m{  4mm{  5m ? % , m8{+{  6, . 43m ? %

; % A 3> < A 3> +m{  5m ?  , m8{+{  6, . 43m ? %,

Whruhpdv Eävlfrv vreuh Oðplwhv

D frqwlqxdflöq yhuhprv xq whruhpd txh hv xwlol}dgr sdud suredu od qr h{lvwhqfld gh oðplwhv/ ho whruhpd gh xqlflgdg Whruhpd 719 +gh xqlflgdg, Ho oðplwh gh xqd ixqflöq +hq xq sxqwr, hv ýqlfr Vhd D  U> i = D $ U ixqflöq > d 5 D3 1 Vl h{lvwh ho oðplwh gh i fxdqgr { $ d hqwrqfhv olp{$d i+{, hv ýqlfr1 Hv ghflu/ vl olp i+{, @ O4 | olp i +{, @ O5 hqwrqfhv O4 @ O5 {$d {$d

Ghprvwudflöq= Sru ho devxugr1 Vxsrqjdprv O4 9@ O5 / hqwrqfhv srghprv wrpdu yhflqgdghv glvmxqwdv YO4 | YO5 / hv ghflu YO4 _ YO5 @ !1 Frpr olp{$d i+{, @ O4 / h{lvwh xqd yhflqgdg gh d/ Y4 / wdo txh i +Y4  idj _ D,  YO4 / sru rwud sduwh vh wlhqh olp{$d i +{, @ O5 / hqwrqfhv h{lvwh xqd yhflqgdg gh d/ Y5 wdo txh i +Y5  idj _ D,  YO5 1

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

559

Hv fodur txh E @ D _ +Y5  idj, _ +Y4  idj, 9@ ! +hv lpsrvleoh txh vhd ydfðd od lqwhuvhfflöq |d txh dpedv vrq yhflqgdghv gh d | d hv xq sxqwr gh dfxpxodflöq gh D i+E,  YO4

sru or wdqwr

shur i +E, 9@ ! frqwudglflöq1 Dvð O4 @ O5

a

i +E,  YO5

i+E,  YO4 _ YO5 @ !

Whruhpd 71: 5 Vhd E  D  U> s 5 E 3 +sru or wdqwr s 5 D3 ,1 Vhd i = D $ U | j @ imE 1 Vl olp{$s i +{, @ O hqwrqfhv olp{$s j+{, @ O1 Uhflsurfdphqwh1 Vl E frqwlhqh xq lqwhuydor delhuwr txh frqwlhqh d s vh wlhqh txh vl olp{$s j+{, @ O hqwrqfhv olp{$s i+{, @ O1

Ghprvwudflöq= Od ghprvwudflöq gh od sulphud sduwh hv edvdgd hq whruðd gh frqmxqwr1 Vl olp{$s i +{, @ O vh wlhqh txh sdud wrgd yhflqgdg YO gh O h{lvwh xqd yhflqgdg Ys gh s wdo txh i++Ys  isj, _ D,  YO

sru or wdqwr wdpelìq vh wlhqh |d txh

j++Ys  isj, _ E,  YO

j++Ys  isj, _ E,  i ++Ys  isj, _ D,  YO

oxhjr

olp j+{, @ O {$s

Od rwud sduwh gho whruhpd uhdo}d ho frqfhswr gh oðplwh hq ho vhqwlgr txh edvwd dqdol}du orv sxqwrv fhufdqrv d s1

Hmhpsor 585 Fodudphqwh vh yh txh olp{$6 {5  5 @ :1 Dtxð vh hvwä shqvdqgr hq od ixqflöq i = U $ U> i+{, @ {5  51 Ho whruhpd dqwhulru qrv glfh txh i m`6>4^ @ j =`6> 4^$ U> j+{, @ {5  5 fxpsoh txh olp j+{, @ : {$6

5m Hmhpsor 586 Vhd i +{, @ m{{ 5 > { 9@ 5/ Surehprv olp{$5 i+{, qr h{lvwh +Sru ho devxugr,1 Vxsrqjdprv txh olp{$5 i+{, h{lvwh/ hv ghflu/ olp{$5 i +{, @ O1 frq O 5 U oxhjr/ vl frqvlghudprv

hqwrqfhv vhjýq ho whruhpd dqwhulru1

j @ i m`5>4^

olp j+{, @ O {$5

| k @ im`lqiw|>5^ |

olp k+{, @ O {$5

shur fodudphqwh j+{, @ 4 sdud wrgr { 5 Grpj | k+{, @ 4 sdud wrgr { 5 Grpk1 Sru or wdqwr olp j+{, @ 4 | olp k+{, @ 4 {$5 {$5

or txh frqwudglfh 714 Sru frqvljxlhqwh olp{$5 i +{, qr h{lvwh1 5 Vl i G D


| E

a

D> i

ƒE < U E

hv od ixqflöq i uhvwulqjlgd d xq grplqlr E | hv oodpdgd uhvwulfflöq gh i d E 1

+714,

7151 OÐPLWHV ODWHUDOHV

55:

Hmhpsor 587 Hq irupd dqdorjd vh suxhed od qr h{lvwhqfld gh olp{$3 i +{,/ grqgh i +{, @ vhq+ {4 ,/ { 5 U  i3j Vhd 4 4 = q 5 Qj> E @ i = q 5 Qj D@i 5q 5q . @5 dperv wlhqhq d 3 frpr sxqwr gh dfxpxodflöq Vxsrqjdprv txh olp{$3 i +{, @ O/ sru or wdqwr

olp i m @ O+sru whruhpd 71:, {$3 E   @ vhq +5q, @ 3> ;q 5 Q Shur imD +d, @ vhq + d4 ,> d 5 D/ | vhq+ d4 , @ vhq 44 5q sru or wdqwr olp i m @ olp 3 @ 3> dvð O @ 3 {$3 D {$3 4 Sru rwud sduwh/ imE +e, @ vhq+ 4e ,> e 5 E | frpr e @ 5q. @5 /hqwrqfhv olp i m @ O {$3 D

vhq

a

+715,

4 @ vhq +5q . @5, @ vhq@5 @ 4= e

Dvð i mE +e, @ 4> ;e 5 E gh grqgh O @ 4/ or txh frqwudglfh 715

715

Oðplwhv Odwhudohv

Ghqlflöq 71; Ho oðplwh odwhudo ghuhfkr gh i hq d vh hqwlhqgh frpr ho oðplwh vljxlhqwh olp{$d im`d>4^ | vh dqrwd olp{$d. i+{,1+vl d hv sxqwr gh dfxpxodflöq gh `d> 4^_Grpi 9@ !,6 Dqäorjdphqwh vh ghqh oðplwh odwhudo l}txlhugr frpr olp{$d im`4>d^ | vh ghqrwd frpr olp{$d i +{,1+vl d hv sxqwr gh dfxpxodflöq gh `  4> d^_Grpi 9@ ! Sru whruhpd dqwhulru vh wlhqh txh= Vl i hvwä ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d +qr qhfhvduldphqwh ghqlgd hq d, hqwrqfhv l, Vl olp{$d i +{, h{lvwh | hv ljxdo d O/ hqwrqfhv dperv oðplwhv odwhudohv h{lvwhq | vrq ljxdohv1 Sru frqvljxlhqwh xvdqgr od htxlydohqfld s , t +, t , s whqhprv=

ll, Vl orv oðplwh odwhudohv qr h{lvwhq r vrq glihuhqwh hqwrqfhv olp{$d i +{, qr h{lvwh1 +lqvlvwlpr hq txh vlhpsuh txh i hvwä ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d,

Hmhpsor 588 Vhd

+

{5 .9{.; { A 7 {.7 {5 49 {5 .:{.45 { ? 7 Fdofxohprv orv oðplwhv odwhudohv hq { @ 71 Qrwhprv txh i hvwä ghqlgd hq U  i7j1 i+{, @

{5  49 olp  5 d, olp  i+{, @ {$7 {$7 { . :{ . 45 +{ . 7,+{  7, @ olp {$7 +{ . 6,+{ . 7, {7 @ olp  {$7 { . 6 ; @ 4 @ ;

6 Hq fdvr frqwudulr/ qr h{lvwh ho oðplwh +r qr vh sxhgh fdofxodu,1

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

55;

{5 . 9{ . ; +{ . 5,+{ . 7, e, olp i +{, @ olp @ olp . @ olp . { . 5 @ 5 {.7 {.7 {$7. {$7. {$7 {$7 Sru d, | e, vh wlhqh olp{$7 i+{, 9@ olp{$7. i +{, oxhjr olp{$7 i+{, qr h{lvwh1 s Hmhpsor 589 Frqvlghuhprv od ixqflöq i +{, @ { s Khprv ylvwr hq xq hmhpsor dqwhulru txh olp{$3 { h{lvwh | vx ydoru hv 31 Vlq hpedujr qr srghprv dupdu txh dperv oðplwhv odwhudohv h{lvwdq1 Hq hvwh fdvr/ gho Grpi @ U. 3 / oxhjr i qr hvwd ghqlgd hq wrgr xq lqwhuydor delhuwr txh frqwhqjd d 31 Revhuydprv od ghprvwudflöq gho hmhpsor 57; hv fodur txh s s olp { @ olp. { @ 3 {$3 {$3 Vreuh ho oðplwh odwhudo l}txlhugr vh wlhqh txh od sursrvlflöq

s +; % A 3,+< A 3,+{ 5 Grpi a  ? { ? 3 , m {  Om ? %,

hv yhugdghud sdud wrgr O +O 5 U, |d txh qr s h{lvwh { hq U txh fxpsoh { 5 Grpi a  ? { ? 31 Gh hvwd irupd dsduhqwdphqwh olp{$3 { @ O sdud wrgr O 5 U Shur 3 qr hv xq sxqwr gh dfxpxodflöq gh `  4> 3^_Grpi @ !1 Sru or wdqwr ho oðplwh odwhudo l}txlhugr hq { @ 3 qr vh sxhgh fdofxodu +qr wlhqh vhqwlgr od h{suhvlöq,1

Revhuydflöq= Hq wìuplqrv gh > % od ghqlflöq gh oðplwh odwhudo ghuhfkr | odwhudo l}txlhugr hv= olp i+{, @ O +, {$d. +; % A 3,+< 4 A 3,++3 ? {  d ?  4 a { 5 Grpi, , mi+{,  Om ? %, olp i+{, @ O +, {$d +; % A 3,+< 5 A 3,++ 5 ? {  d ? 3 a { 5 Grpi, , mi+{,  Om ? %,

+716, +717,

Dkrud/ vxsrqjdprv txh h{lvwhq | vrq ljxdohv orv oðplwhv odwhudohv gh xqd ixqflöq i hq { @ d/ hv ghflu/ olp i +{, @ olp i+{, @ O {$d {$d.

Hqwrqfhv srghprv dupdu txh olp{$d i +{, h{lvwh | olp{$d i +{, @ O1 Hq hihfwr= Gdgr txh h{lvwhq orv oðplwhv odwhudohv hqwrqfhv vh wlhqh 716 | 7171 Vhd % A 3 | frqvlghuhprv  @ plqi 4 >  5 j hqwrqfhv vh wlhqh

gh grqgh=

{ 5 Grpi a 3 ? {  d ?  ?  4 , mi +{,  Om ? % { 5 Grpi a  5 A  A {  d A 3 , mi +{,  Om ? %

+; % A 3,+< A 3,+{ 5 Grpi a  ? {  d ?  a { 9@ d, , mi+{,  Om ? %

Khprv ghprvwudgr gh hvwd irupd ho vljxlhqwh whruhpd1

Whruhpd 71< Vl orv oðplwhv odwhudohv hq { @ d h{lvwhq +vh sxhghq fdofxodu, | dperv vrq ljxdohv d O hqwrqfhv vh wlhqh txh olp{$d i+{, h{lvwh | olp{$d i+{, @ O1

Hmhpsor 58: Vhd

i +{, @

+

{6 4 {4 s <{

{?4 {4

Ghwhuplqhprv vl h{lvwh olp{$4 i+{,1 6 Frpr olp{$4 i +{, @ olp{$4 {{44 @ olp{$4 {5 . { . 4 @ 6 s | olp{$4. i+{, @ olp{$4. <{ @ 6 hqwrqfhv olp{$4 i+{, @ olp{$4. i +{, @ 61 Oxhjr/ sru whruhpd dqwhulru olp{$4 i+{, @ 61

7151 OÐPLWHV ODWHUDOHV

55<

5m Hmhpsor 58; Vhd i +{, @ { . m{{ 5 / yhultxhprv txh i+{, qr wlhqh oð plwh hq 51 Frpr olp{$5. i +{, @ olp{$5. +{ . 4, @ 6 | dghpäv olp{$5 i+{, @ olp{$5. +{  4, @ 4/ vh wlhqh txh olp{$5 i +{, qr h{lvwh1

Hmhpsor 58< Ghwhuplqhprv ho ydoru gh yhugdg gh odv vljxlhqwhv sursrvlflrqhv l, olp{$3.

4

ll, olp{$3

4

m4{m

@ 31

m4{m

{ @ olp{$3 5 {

{

{

lll, olp{$4 ^{` qr h{lvwh1  ^{` {4 ly, Vl i +{, @ 5{  4 { ? 4

hqwrqfhv ho olp{$4 i+{, @ 4

s7 5 y, olp{$3 { {.{ @ 4

Vroxflöq=

Hmhpsor 593 l, Idovr1 wdqwr



Vl { $ 3 hqwrqfhv 4  { A 3 hq xqd yhflqgdg Y gho fhur/ oxhjr m4  {m @ 4  { hq Y sru or olp {$3

4  m4  {m 4  +4  {, { @ olp @ olp @ 4 {$3 {$3 { { {

ll, Idovr1 Sru or dqwhulru/ vh wlhqh=

oxhjr

olp {$3

4  m4  {m 4  m4  {m @ olp. @4 { { {$3

olp {$3

4  m4  {m 5{ 9@ olp { { {$3

Qrwh txh 4  { fdpeld gh vljqr fhufd gho sxqwr 4 | do wrpdu { whqglhqwh d fhur hvwdprv wrpdqgr sxqwr fhufd gh fhur grqgh 4  { wlhqh ho ydoru srvlyr1

lll, Yhugdghur1 olp{$4. ^{` @ 4> olp{$4 ^{` @ 3/ oxhjr olp{$4. ^{` 9@ olp{$4 ^{`1

ly, Yhugdghur1 olp{$4. i +{, @ olp{$4. ^{` @ 4 | olp{$4 i+{, @ olp{$4 +5{  4, @ 4 oxhjr orv oðplwhv odwhudohv vrq ljxdohv sru or wdqwr olp i +{, @ 4 {$4

s5 s7 5 s ly, Idovr1 olp{$3 { {.{ @ olp{$3 m{m {{ .4 @ olp{$3 m{{m {5 . 41 Fdofxodqgr orv oðplwhv odwhudohv s s whqhprv olp{$3. {{ {5 . 4 @ olp{$3. {5 . 4 @ 4 | sru od l}txlhugd s s olp{$3 {{ {5 . 4 @  olp{$3 {5 . 4 @ 4

Hmhuflflr 594 Ghwhuplqh d 5 U gh prgr txh O @ olp {$d

{5  m{  dm  d5 m{  dm

h{lvwd

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

563

Vroxflöq= Qrwhprv txh m{  dm @ {  d vl { A d | m{  dm @ { . d vl { ? d/ oxhjr fdofxohprv orv oðplwhv odwhudohv O. @ olp. {$d

{5  m{  dm  d5 m{  dm

@ @ @

@

@

Sru rwud sduwh O @ olp {$d

{5  m{  dm  d5 m{  dm

@ @ @ @

@

{5  d5  +{  d, {d {5  d5  +{  d, olp {d {$d. +{  d,+{ . d  4, olp {d {$d. olp. { . d  4 {$d 5d  4 olp {$d.

{5  d5 . +{  d, +{  d, 5 {  d5 . +{  d, olp +{  d, {$d +{  d,+{ . d . 4, olp +{  d, {$d olp +{ . d . 4, {$d +5d . 4, olp {$d

Oxhjr O h{lvwh vl | vöor vl olp{$d. i+{, @ olp{$d i +{,/ hv ghflu/ 5d  4 @ 5d  4

gh grqgh d @ 31 Sru or wdqwr/ ho oðplwh h{lvwh vl | vöor vl d @ 31

Hmhuflflr 595 Fdofxohprv

olp {$3

4  m4  6{m {

Hq hvwh fdvr/ qrwhprv vl { wlhqgh d fhur ho idfwru 4  6{ hv srvlwlyr/ oxhjr m4  6{m @ 4  6{ sdud wrgr { fhufd gh 3 | olp {$3.

Oxhjr olp{$3

71514

m46{m

4

{

4  m4  6{m {

4  m4  6{m olp { {$3 4  +4  6{, @ olp {$3 { 6{ @ olp {$3 { @ 6 @

@ 61

Dojheud gh Oðplwhv

Whruhpd 7143 Vhd [  Grpi _ Grpj  U> s 5 [ 3 | vhdq i | j wdo txh h{lvwdq orv oðplwhv olp{$s i +{,/ olp{$s j+{, hqwrqfhv d, oðplwh gh od vxpd i . j wdpelìq h{lvwh hq s | vh wlhqh olp +i . j,+{, @ olp i +{, . olp j+{, {$s {$s {$s

7151 OÐPLWHV ODWHUDOHV

564

e, oðplwh gho surgxfwr i  j h{lvwh hq s | vh wlhqh

olp +i  j,+{, @ olp i+{,  olp j+{, {$s {$s {$s

f, oðplwh gho fxrflhqwh i @j h{lvwh hq s fxdqgr olp{$s j+{, 9@ 3 | vh wlhqh olp{$s i +{, olp +i @j,+{, @ {$s olp{$s j+{,

Ghprvwudflöq= Xvdu uhvxowdgr gh dojheud gh oðplwhv gh vxfhvlrqhv | whruhpd gho hqodfh/ txh yhuhprv srvwhulruphqwh1  Revhuydflöq= Hq sduwlfxodu gh +e, vh ghgxfh txh Vl  5 U hqwrqfhv olp{$s i +{, @  olp{$s i +{,1 Revhuydflöq= Hv lpsruwdqwh hqidwl}du txh sdud dsolfdu ho dojheud gh oðplwhv/ ghehprv hvwdu vhjxurv gh txh vh yhultxhq odv klsöwhvlv1 Vl qr hv dvð srghprv rewhqhu frqwudglfflrqhv frpr yhuhprv d frqwlqxdflöq1 5 Hv fodur txh olp{$5 {{ 5 @ olp{$5 4 @ 41 Vlq hpedujr/ vl hvfulelprv 4 {5 @ olp +{  5, olp {$5 {  5 {$5 {5

| dsolfdprv ho dojheud gh oðplwhv vlq dvhjxuduqrv gh txh dpedv oð plwhv h{lvwdq srghprv ghflu htxlyrfdgd0 phqwh txh1 4 4 4 @ olp +{  5, olp @ 3 olp @3 olp +{  5, {$5 {$5 {  5 {$5 {  5 {  5 {$5

5 4 gh grqgh olp {{ 5 @ 3 or txh hv xqd frqwudglfflöq +| hvwr vxfhgh srutxh olp{$5 {5 qr h{lvwh1 Revhuydflöq= Vl olp{$s i +{, @ O vh wlhqh txh

+; % A 3, +< A 3,+^{ 5 Grpi a 3 ? m{  sm ? ` , mi+{,  Om ? %,=

Wrpdqgr/ hq sduwlfxodu/ % @ 4 vh wlhqh txh

;{ 5 +Ys  isj, _ Grpi > O  4 ? i +{, ? O . 4=

Hv ghflu/ txh i hv dfrwdgd vxshulruphqwh sru O . 4 hq od yhflqgdg Ys  isj | wdpelìq dfrwdgd lqihul0 ruphqwh sru O  4 hq ho plvpr frqmxqwr1

Whruhpd 7144 +Dfrwdplhqwr, Vhd D  U> s 5 D3 > i> j> k = D $ U1 Vhd Y xqd yhflqgdg gh s wdo txh 4, ;{ 5 +Y  isj, _ D vh wlhqh= i+{,  j+{,  k+{, | dghpäv 5, olp{$s i+{, @ olp{$s k+{, @ O/ hqwrqfhv olp j+{, @ O {$s

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

565

Ghprvwudflöq= +yhu jxud dqwhulru, Gdgr % whqhprv sdud i | k or vljxlhqwh +< i A 3,+3 ? m{  sm ?  i a { 5 D, , mi +{,  Om ? %

+< k A 3,+3 ? m{  sm ?  k a { 5 D, , mk+{,  Om ? %

shur +mi+{,  Om ? % +, O  % ? i+{, ? O . %, | +mk+{,  Om ? % +, O  % ? k+{, ? O . %, dvð/

vh wlhqh

Sru or wdqwr/

Sru or wdqwr

;{ 5 Y _ i{ = m{  sm ?  i a m{  sm ?  k j O  % ? i +{,  j+{,  k+{, ? O . %= ; % A 3 < @ plqi i >  k j+{ 5 D a 3 ? m{  sm ? , , mj+{,  Om ? % olp j+{, @ O {$s 

Or vljxlhqwh hvwä uhodflrqdgr frq od jxud 1 Fxdqgr vh ghqh jhrpìwulfdphqwh od ixqflöq vhq +{, | frv +{, vh xwlol}d od flufxqihuhqfld gh udglr 4 frpr hq od jxud / hv dvð txh odv frrughqdgdv gho sxqwr S vrq S @ +frv+{,> vhq+{,, grqgh { hvwd phglgr hq udgldqhv1

7151 OÐPLWHV ODWHUDOHV

566

Iljxud 

Sru rwud sduwh/ shqvdqgr hq od ghqlflöq gh odv ixqflrqhv wuljrqrpìwulfdv xvdqgr xq wuläqjxor uhfwäq0 jxor/ vh wlhqh txh7 wdq { @

fdwhwr rsxhvwr GV @ fdwhwr dg|dfhqwh RV

Gh hvwd irupd/ odv frrughqdgdv gho sxqwr G vrq +4> wdq+{,, Sru ho khfkr txh od flufxqihuhqfld wlhqh udglr 4 | ho äqjxor { hvwä gdgr hq udgldqhv/ vh wlhqh txh ho dufr gh flufxqihuhqfld gh S d V> g S V hv od orqjlwxg {1 Qrwd= hq od jxud ho äqjxor { vdwlvidfh 3  { ? 5 / shur fodudphqwh vh sxhgh yhu txh vl { hv qhjdwlyr | pd|ru txh  5 / vöor kd| txh whqhu fxlgdgr frq ho vljqr hq odv uhodflrqhv txh d frqwlqxdflöq gdprv sdud {/ { 5`  5 > 5 ^1 g S V @ m{m SF @ m vhq+{,m

m vhq+{,m  m{m

Sru or wdqwr/ sdud { 5`  5 > 5 ^ vh wlhqh od ghvljxdogdg

|d txh S F  g SV

m{m  vhq+{,  m{m

Hmhpsor 596 Sru or dqwhulru vh wlhqh txh olp{$3 vhq { @ 31 Hq hihfwr= Frpr   m{m  vhq {  m{m > { 5`  > ^ 5 5 | olp{$3 m{m @ olp{$3 m{m @ 31 hqwrqfhv sru whruhpd gho dfrwdplhqwr vh wlhqh olp vhq { @ 3 {$3

+718,

Wdpelìq xvdqgr od jxud  vh sxhgh yhu txh fxdqgr { wlhqgh do äqjxor fhur/ od klsrwhqxvd gho wuläqjxor RFS / RS wlhqgh do fdwhwr dg|dfhqwh RF/ gho plvpr wuläqjxor/ shur frpr RS hv vlhpsuh ljxdo d 4 | RF @ frv+{, vh wlhqh txh olp frv+{, @ 4> {$3 oðplwh txh mxvwlfduhprv gh rwud irupd pdv dghodpwh1 7 Qrwdflöq

DE

lqglfd od glvwdqfld hqwuh

D E |

r orqjlwxg gho wud}r gh uhfwd gh

D E d

1

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

567 Hmhpsor 597 Surehprv txh

vhq { @4 olp {$3 { Sdud uhvroyhu hvwr xwlol}duhprv xqd uhodflöq h{lvwhqwh hqwuh grv äuhdv gh od jxud  äuhd gh vhfwru flufxodu RVS  äuhd gho wuläqjxor RVG

Fdofxodqgr fdgd äuhd8 vh wlhqh

4 {  wdq+{, > 5 5

vl { srvlwlyr

Pxowlsolfdqgr hvwd ýowlpd ghvljxdogdg d dperv odgrv sru ho qýphur srvlwlyr frv+{, 

vhq+{, {

{ rewhqhprv

5 frv+ ,

{

Xwlol}dqgr ho khfkr txh frv+{, hv xqd ixqflöq sdu +frv+{, @ frv+{,, | frpr vhq+{,  {/ srghprv ideulfdu od vljxlhqwh fdghqd dsursldgd sdud xwlol}du ho whruhpd gho dfrwdplhqwr

vhq+{, 4 { ghvljxdogdg yäolgd sdud { 9@ 3 | { 5`  5 > 5 ^ + |d txh od ixqflöq vhq+{, | { vrq ixqflrqhv lpsduhv | vx fxrflhqwh hv sdu,1 Gh hvwd irupd sru whruhpd 7144 | xwlol}dqgr txh frv+{, $ 4 fxdqgr { $ 3 rewhqhprv frv+{, 

olp {$3

Hmhuflflr 598 Fdofxodu Vroxflöq=

vhq {  5{ olp {$3 6{ . 7vhq{ vhq {  5{ O @ olp {$3 6{ . 7vhq{

Frpr

vhq+{, @4 {

vhq {  5{ 4@{ olp  {$3 6{ . 7 vhq { 4@{ vhq { 5 @ olp { vhq { {$3 6 . 7 { @

vhq { vhq { olp 6 . 7 @ olp 6 . 7 olp @ 6 . 7 @ : 9@ 3= {$3 {$3 {$3 { {

|

olp {$3

vhq { vhq {  5 @ olp  olp 5 @ 4  5 @ 4= {$3 { {$3 {

hqwrqfhv sru dojheud gh oðplwhv O @

4 1 :

Hmhpsor 599 Fdofxohprv olp{$3. { vhq {d grqgh d 5 U1 Vroxflöq= Vdehprv txh

m vhq

Pxowlsolfdqgr sru m{m od hfxdflöq 719 rewhqhprv 8 uhfxhugh

txh

{ hvwä

hq udgldqhv1

d m4 {

d m{ vhq m  m{m {

+719, +71:,

7151 OÐPLWHV ODWHUDOHV

568

Dvð/ 3

d m{m  { vhq + ,  m{m= { Frpr olp{$3 m{m @ 3 | olp{$3 m{m @ 3 hqwrqfhv sru whruhpd gho dfrwdplhqwr vh wlhqh olp{$3. m{ vhq {d m @

Hvwh ýowlpr fdvr vh sxhgh jhqhudol}du d ixqflrqhv dfrwdgdv1 Uhfrughprv txh xqd ixqflöq i vh glfh dfrwdgd hq D vl h{lvwh n 5 U. wdo txh= mi+{,m  n

Whruhpd 7145 +Fhur Dqltxlod, Vhd i xqd ixqflöq dfrwdgd hq xqd yhflqgdg gh s> Ys +qr qhfhvduld0 phqwh ghqlgd hq s,1 Vl olp{$s j+{, @ 3 hqwrqfhv olp{$s j+{,i+{, @ 3

Ghprvwudflöq= Frpr mi+{,m  n vh wlhqh hv ghflu/

mj+{,i+{,m  nmj+{,m

nmj+{,m  j+{,i+{,  nmj+{,m

| sru whruhpd gh dfrwdplhqwr vh rewlhqh or uhtxhulgr1 Hmhpsor 59: Ghprvwudu

5 @3 +71;, olp +{  4, frv {$4 {4 5 , hv dfrwdgd/ oxhjr sru whruhpd fhur dqltxlod/ Hq hihfwr olp{$4 {  4 @ 3 | od ixqflöq i +{, @ frv+ { 4 vh wlhqh 71;1 Hmhuflflr 59; Fdofxodu

olp {$6

4 vhq+ { , 6 4

{6   4 , vhq + { 6 @ olp 4 Vroxflöq= O @ olp{$6 4 { $ 6 +{  6, vhq {6 1 {6 4 Frpr olp{$6 {  6 @ 3 | i+{, @ vhq { hv dfrwdgd/ hqwrqfhv sru whruhpd fhur dqltxlod O @ 31 6 Hmhuflflr 59< Ghprvwudu xvdqgr ho whruhpd dqwhulru txh

olp vhq +{, @ vhq +d, {$d

Vroxflöq= Edvwd ghprvwudu txh olp{$d +vhq {  vhq d, @ 31 Xvdqgr od lghqwlgdg wuljrqrpìwulfd vhq {  vhq d @ 5 vhq+

{.d {d , frv+ , 5 5

d , hv dfrwdgr | olp {d {d | ho khfkr txh frv+ {. {$d vhq+ 5 , @ olpx$3 vhq x @ 3 grqgh 5 @ x1 Vh wlhqh sru 5 whruhpd fhur dqltxlodgru txh= olp +vhq {  vhq d, @ 3 {$d Hq irupd dqäorjd vh sxhgh suredu txh olp frv { @ frv d {$d

Hmhuflflr 5:3 Vhdq d A 3> e A 3/ Ghprvwudu txh

  e { e @ olp. d {$3 d {

D|xgd= Xvdu od vljxlhqwh surslhgdg gh od sduwh hqwhud

{  4  ^{`  { > ;{ 5 U=

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

569 Vroxflöq= Sru surslhgdg dqwhulru whqhprv

  e e e 4  { { {

pxowlsolfdqgr sru {d frq d A 3> { A 3 vh wlhqh     { e {e { e 4   d { d { d{

r htxlydohqwhphqwh

  e { { e e  4  d d d { d sxhvwr txh olp{$3. de  {d @ olp{$3. de @ de hqwrqfhv sru whruhpd gh dfrwdplhqwr   e { e @ olp. d {$3 d {

Hmhuflflr 5:4 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 41 olp{$3

@{, @{ @ 41

vhq +4 4

51 olp{$7 +{ . 7,^{` @ 31 61 olp{$3 vhq{ { @ 31 71 olp{$ vhq{ { @ 41 81 olp{$3 frv{ { @ 31 91 Qr h{lvwh olp{$3 frv{ { 1 :1 olp{$3 +{ vhq+4@{, .

{ { , qr h{lvwh1

frv+ ,

;1 Vl olp{$d ij h{lvwh | olp{$d i h{lvwh hqwrqfhv olp{$d j h{lvwh1

<1 Vl olp{$d +i . j, h{lvwh | olp{$d i qr h{lvwh hqwrqfhv olp{$d j h{lvwh1

Vroxflöq=

@{, 41 Idovr1 Frpr olp{$3 vhq4+4 @{ @ olp{$3 { vhq +4@{, | olp{$3 { @ 3 | i +{, @ vhq +4@{, hv dfrwdgd hqwrqfhv olp{$3 { vhq +4@{, @ 3 sru whruhpd fhur dqltxlod1 51 Yhugdghur1 Od ixqflöq i +{, @ ^{` hv dfrwdgd hq xqd yhflqgdg gho 7/ hv ghflu/ 8  ^{`  7 i rudoo{ 5`  8> 6^

| olp{$7 +{ . 7, @ 3 oxhjr sru whruhpd fhur dqltxlod

olp +{ . 7,^{` @ 3= {$7 61 Yhugdghur1 olp{$3 vhq{ { @ olp{$3 +vhq+4{,,@{ @ 71 Idovr1 olp{$ vhq{+{, @ vhq+, @ 3 3 81 Yhugdghur1 olp{$3 frv{ { @ frv+3, @3

4 4

@ 41

7151 OÐPLWHV ODWHUDOHV

56:

{, 91 Yhugdghur1 Vxsrqguhprv txh olp{$3 frv+ { h{lvwh | vx ydoru hv O1 Gdgr txh olp{$3 { h{lvwh sru {, dojheud gh oðplwhv olp{$3 frv+ { { gheh h{lvwlu shur olp {$3

| sru dojheud gh oðplwhv olp {$3

frv +{, { @ olp frv+{, @ 4 {$3 {

frv+{, frv+{, { @ olp olp { @ O  3 @ 3 {$3 { {$3 {

or fxdo hv xqd frqwudglfflöq sru od xqlflgdg gho oðplwh1 Sru or wdqwr ho ydoru O qr h{lvwh1

:1 Yhugdghur19 Vdehprv sru +4, txh> olp{$3 { vhq +4@{, h{lvwh +| vx ydoru hv fhur, | olp{$3 frv{ { qr h{lvwh sru +9, Oxhjr

frv { olp +{ vhq 4@{ . , qr h{lvwh= {$3 { |d txh gh h{lvwlu wdpelìq ghehuðd h{lvwlu +sru oðplwh gh od vxpd,

| hvwr ýowlpr hv idovr sru +9,1

olp +{ vhq 4@{ . {$3

frv {  { vhq+4@{,, {

;1 Idovr1 Vhdq i +{, @ vhq {> j+{, @ {4 > olp{$3 i+{,j+{, h{lvwh | vx ydoru hv 4/ dghpäv olp{$3 i +{, h{lvwh | vx ydoru hv fhur/ shur olp{$3 j+{, qr h{lvwh1

<1 Idovr1 Vhdq i +{, @ {4 | j+{, @  {4 oxhjr olp{$3 +i . j, h{lvwh | vx ydoru hv fhur/ shur olp{$3 i+{, @ olp{$3 {4 qr h{lvwh | olp{$3 j+{, @ olp{$3  {4 qr h{lvwh1

71515

Oðplwh gh ixqflrqhv 0 vxfhvlrqhv

Ho vljxlhqwh whruhpd qrv shuplwh xvdu orv uhvxowdgrv rewhqlgrv hq vxfhvlrqhv

Hmhpsor 5:5 Vhd i +{, @ {/ od ixqflöq lghqwlgdg hq U1 Vh vdeh txh olp{$s i +{, @ s1 Xvhprv ho whruhpd dqwhulru sdud suredu txh = d, olp{$s {5 @ s5 1 e, olp{$s   {n @   sn grqgh  5 U | f, olp{$s t+{, @ t+s,/ grqgh t+{, @ 3 . 4 { . 5 {5 .    . q {q Vroxflöq= d, Frpr {5 @ i +{,i+{, vh wlhqh

olp {5 @ olp i+{, olp i+{, @ s  s @ s5 = {$s {$s {$s

Dvð/ olp{$s {5 @ s5 e, Hvwd vh kduä xvdqgr lqgxfflöq hq n1 Sdud n @ 4 hv flhuwr |d txh olp{$s { @ olp{$s  olp{$s { @   s Vxsrqguhprv dkrud txh hv yäolgr sdud n | ghprvwuduhprv txh hv yäolgr sdud n . 4 olp {n.4 @ olp +{n ,{ @ olp +{n , olp { @ sn s @ sn.4 = {$s {$s {$s {$s

f, Od whufhud wdpelìq vh kduä sru lqgxfflöq hq n Sdud n @ 4/ vh wlhqh olp{$s 3 . 4 { @ 3 . 4 s1 Dvð sru dojheud vh fxpsoh sdud n @ 41 9 Vh

yhuä hq wìuplqrv päv jhqhudohv hq hmhuflflr 5::1

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

56;

Vxsrqjdprv txh ho oðplwhv hv yäolgr sdud q @ n ghprvwuduhprv txh hv yäolgr sdud n . 41 Xvdqgr dojheud gh oðplwhv vh wlhqh= olp  . 4 { .    . n {n . n.4 {n.4 @ olp 3 . 4 { .    . n {n . olp n.4 {n.4 {$s 3 {$s {$s

Od h{lvwhqfld gh hvwh oðplwh vh jdudqwl}d sru od klsöwhvlv lqgxfwlyd

olp  . 4 { .    . n {n @ 3 . 4 s .    . n sn {$s 3

dghpäv sru hmhuflflr dqwhulru oxhjr

olp  {n.4 @ n.4 sn.4 {$s n.4

olp  . 4 { .    . n {n . n.4 {n.4 @ 3 . 4 s .    . n sn . n.4 sn.4 {$s 3

Sru frqvljxlhqwh

olp t+{, @ t+s, {$s

1

Hmhuflflr 5:6 Fdofxodu

olp {$5

{5 . 6{ . 4 = {.4

Vroxflöq= Frpr olp{$5 { . 4 @ 6 9@ 3 | olp{$5 {5 . 6{ . 4 @ 44/ hqwrqfhv sru dojheud gh oðplwhv +f, olp {$5

{5 . 6{ . 4 {.4

@

@

olp{$5 {5 . 6{ . 4 olp{$5 { . 4 44 6

Revhuydflöq= Qrwdu txh qr vh wxhgh dsolfdu ho dojheud gh oðplwhv1 Vl { wlhqgh d 41 Hq orv vljxlhqwhv h mhuflflrv wdpsrfr srghprv xvdu ho dojheud gh oð plwhv gluhfwdphqwh/ shur hvr qr vljqlfd txh ho oðplwhv qr h{lvwh/ frpr vh yhud d frqwlqxdflöq1 s s Hmhuflflr 5:7 Vdelhqgr txh olp{$s t+{, @ t+s,/ vl t+s, A 31 Fdofxodu s 6  5{ . 4 olp {$7 {5  49

Vroxflöq=

O @ olp {$7

s 6  5{ . 4 {5  49

s s 6  5{ . 4 6 . 5{ . 4 s  {5  49 6 . 5{ . 4 <  +5{ . 4, s @ olp 5 {$7 +{  49,+6 . 5{ . 4, 5+7  {, s @ olp {$7 +{  7,+{ . 7,+6 . 5{ . 4, 5 s = @ olp {$7 +{ . 7,+6 . 5{ . 4, @

olp {$7

Gdgr txh olp{$7 5 @ 5 | olp{$7 +{ . 7, @ ;1 s s olp 6 . 5{ . 4 @ olp 6 . olp 5{ . 4 @ 6 . 6 @ 9> {$7 {$7 {$7

7151 OÐPLWHV ODWHUDOHV

56<

hqwrqfhv sru dojheud gh oðplwhv vh wlhqh O@

5 4 olp{$7 5 s @ @ ;9 57 olp{$7 +{ . 7,  olp{$7 +6 . 5{ . 4

Revhuydflöq= Vh sureduä hq od vhfflöq gh frqwlqxlgdg txh vl q 5 Q> t+s, A 3 hqwrqfhv s s olp q t+{, @ q t+s, {$s

Hmhuflflr 5:8 Fdofxodu

s 6 5 6 +{ . d,5  s d > vl d 9@ 3= olp {$3 {

Vroxflöq= Uhfrughprv txh d6  e6 @ +d  e,+d5 . de . e5 ,1 s 6 5 6 +{ . d,5  s d O @ olp {$3 { s 6 5 s 6 55 6 +{ . d,5  s 6 +{ . d,5 5 . s 6 +{ . d,5 d5 . s d d @ olp  s s s 5 6 {$3 6 +{ . d,5 . 6 +{ . d,5 d5 . d5 5 { +{ . d,5  d5 @ olp s 6 55 {$3 {+ 6 +{ . d,5 5 . s 6 +{ . d,5 d5 . s d , @

{5 . 5d{ s = s olp s {$3 {+ 6 +{ . d,7 . 6 +{ . d,5 d5 . 6 d7 ,

s s s s 5 6 5 6 Gdgr txh olp{$3 { . 5d @ 5d | olp{$3 @ olp{$3 6 +{ . d,5 . 6 +{ . d,5 d5 . d5 @ 6 d7 9@ 3/ hqwrqfhv sru dojheud gh oðplwhv vh wlhqh 5d O@ s 6 6 d7

Hmhuflflr 5:9 Ghwhuplqdu d 5 U gh prgr txh h{lvwd olp{$5 i +{, grqgh + s s {s {s ; { A 5 { 5 i +{, @ = d{ . 4 { ? 5

Vroxflöq= Gdgr txh Grpi @ U  i5j1 Ho oðplwh h{lvwlud vl vöor vl olp i+{, @ olp i +{, {$5. {$5

olp i +{, @ {$5. @ @ @ @ @ @

s s { { ; s olp s {$5. {  5 s s s s +{ {  ;,+ {  5 s s olp s s {$5. + {  5,,+ { . 5, s s s s {5 . { 5{  ; {  49 olp {5 {$5. s s 5 +{  7, . +{ 5{  5 5{, olp {5 {$5. s +{  5,+{ . 5, . +{  5,+ 5{, olp {5 {$5. s +{  5,+{ . 5 . 5{, olp {5 {$5. s olp { . 5 . 5{= {$5.

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

573 Frpr olp{$5. { @ 5> olp{$5 5 @ 5> olp{$5

s s 5{ @ 5  5 @ 5/ hqwrqfhv sru dojheud gh oðplwh vh wlhqh

olp i+{, @ 5 . 5 . 5 @ 9= {$5.

Sru rwud sduwh olp{$5 i+{, @ olp{$5 d{ . 4 @ 5d . 4/ oxhjr olp{$5 i+{, h{lvwh vl | vöor vl 5d . 4 @ 9/ gh grqgh d @ 85 1 Hmhuflflr 5:: Ghpxhvwuh txh> Vl olp{$d i+{, h{lvwh | olp{$d j+{, qr h{lvwh hqwrqfhv olp{$d +i . j,+{, qr h{lvwh1

Vroxflöq= Sru pìwrgr gho devxugr

++s , t, vl | vöor vl +s a t,

Vxsrqjdprv txh olp{$d +i . j, h{lvwh/ oxhjr sru dojheud gh oðplwhv/ gdgr txh olp{$d i h{lvwh hqwrqfhv olp{$d +i . j,  i h{lvwh/ shur i . j  i @ i or txh frqwudglfh od qr h{lvwhqfld gh olp{$d j1 Hmhuflflr 5:; Ghpxhvwuh txh orv vljxlhqwhv oðplwhv qr h{lvwh1 41 olp{$3 {4

51 olp{$3 +{ . {4 ,1

Vroxflöq=

Hmhuflflr 5:< 41 Vxsrqjdprv txh olp{$3 {4 h{lvwh | vx ydoru ydoru hv O1 Gdgr txh olp{$3 { h{lvwh/ hqwrqfhv sru dojheud gh oðplwhv/ h{lvwh ho oðplwh gho surgxfwr/ hv ghflu/ olp{$3 {4  { h{lvwh 4 4 olp  { @ olp  olp { @ O  3 @ 3= {$3 { {$3 { {$3

Sru rwud sduwh

4 olp { @ olp 4 @ 4> {$3 { {$3 oxhjr sru xqlflgdg gho oðplwhv vh wlhqh txh 3 @ 4 or txh hv xqd frqwudglfflöq1

51 Frpr olp{$3 { h{lvwh | olp{$3 {4 qr h{lvwh/ hqwrqfhv sru hmhuflflr 5:: vh wlhqh 4 olp { . {$3 {

qr h{lvwh

Revhuydflöq= Hq jhqhudo/ vh sxhgh ghprvwudu sru ho pìwrgr gho devxugr txh= Vl olp{$d i+{, @ f 9@ 3 | olp{$d j+{, @ 3 hqwrqfhv olp{$3 ij++{{,, qr h{lvwh1 Hq hvwh fdvr vh glfh txh ho oðplwh olp{$d ij++{{,, hv gh od irupd 3f  Hmhpsor 5;3 Orv vljxlhqwhv oðplwhv vrq gh od irupd 3f  m{ . 5m olp > {$4 {  4

oxhjr orv oðplwhv qr h{lvwh1

{.5 olp {$5 {5  7

Hmhpsor 5;4 Frqvlghuhprv od ixqflöq i +k, @

^{` olp {$6@5 6  5{

d5 . d . 6dk5  5 k5

olp {$4

Ghwhuplqhprv sdud txì ydoru gh d 5 U/ ho oðplwh olpk$3 i +k, h{lvwh1

4  frv5 { {4

7151 OÐPLWHV ODWHUDOHV

574

5 5 5 sxhgh h{lvwlu vl: olpk$3 d5 . d . 6dk5  5 @ 3 Gdgr txh olpk$3 k5 @ 3/ hqwrqfhv olpk$3 d .dk6dk 5 5 .d.6dk5 5 d 5 5 qr h{lvwh,1 +vl olpk$3 d . d . 6dk  5 9@ 3 hqwrqfhv olpk$3 k5 Dkrud>

olp d5 . d . 6dk5  5 @ 3 +, d5 . d  5 @ 3 k$3 +, +d . 5,+d  4, @ 3 +, d @ 5 b d @ 4 5 9k5 dk5 5 @ olp Vl d @ 5 hqwrqfhv olpk$3 d .d.6 5 k$3 k5 @ 91 k 5 dk5 5 @ olp 6k5 Vl d @ 4 hqwrqfhv olpk$3 d .d.6 k$3 k5 @ 61 k5 5 5 dk 5 h{lvwh vl d 5 i5> 4j1 Sru or wdqwr olpk$3 olp d .d.6 k5 s s Hmhpsor 5;5 Prvwudu txh olp{$d { @ d/ vl d  31 s s Vroxflöq= Vl d A 3 edvwd suredu txh olp{$d + {  d, @ 3 shur s s s s s s {. d s { d @ { d s {. d {d s @ s {. d 4 s +{  d, @ s {. d

|

4 4 s ?s 3? s {. d d

| |d txh olp{$d { @ d sru whruhpd fhur Dqltxlod s s olp {  d @ 3 {$d

71516

Whruhpd gh Vxvwlwxflöq

Whruhpd 7146 +Vxvwlwxflöq, Vhdq [> \  U | i = [ $ U/ j = \ $ U ixqflrqhv wdohv txh i +[,  \ 1 Vxsrqjdprv txh olp{$s i+{, @ e | txh h{lvwh xqd yhflqgdg gh s> Ys wdo txh i+{, 9@ e1 sdud wrgr { 5 Ys  isj  [1 Vl olpx$e j+x, @ O hqwrqfhv olp{$s j+i+{,, @ O1 hv ghflu> olp j+i +{,, @ olp j+x, @ O= {$s x$e

Hmhpsor 5;6 Fdofxohprv

olp {$

vhq+{  , {

Vroxflöq= Frqvlghuhprv od vxvwlwxflöq r fdpelr gh yduldeoh x @ {  1 Hv fodur txh olp{$ {   @ 3/ hv ghflu/ vl { $  hqwrqfhv x $ 3/ gh grqgh {   9@ 3 hq Y  ij/ oxhjr sru whruhpd gh vxvwlwxflöq +gdgr txh olpx$3 vhqx x @ 4,/ vh wlhqh vhq+{  , vhq x olp @ olp @4 {$ {   x$3 x

Hmhpsor 5;7 Fdofxohprv

: Frqglflöq

qhfhvduld shur qr vxflhqwh1

6{ olp {$3 vhq+7{,

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

575

Vroxflöq= Vhd x @ 7{1 Vl { $ 3 hqwrqfhv x $ 3/ +dghpäv 7{ 9@ 3 sdud { hq xqd yhflqgdg gho fhur vlq wrpdu ho fhur, hqwrqfhv/ 6 6{ 6 x @ olp @ olp olp {$3 vhq 7{ x$3 7 vhq x 7 x$3

sru whruhpd gh vxvwlwxflöq/ dvð

6 olpx$3 4 6 vhq x @ = x @ 7 olp 7 x$3 x x

4

vhq

6{ 6 x 6 olp @ olp @ x$3 vhq+7{, x$3 7 vhq x 7

Frurodulr 7147 Vhdq [> \  U | i = [ $ U/ j = \ $ U | i +[,  \ 1 Vhd s 5 [ 3 a e 5 \ _ \ 3 a olp{$s i +{, @ e a olp{$e j+{, @ j+e, hqwrqfhv olp{$s +j  i ,+{, @ j+e,=

Ghprvwudflöq= Vhd j+e, @ f

[ s

$ i +[,  \ :$ e

$ U : $ j+e,

Sdud xqd yhflqgdg gh f> Yf / sxhvwr txh olp{$e j+{, @ f/ h{lvwh xqd yhflqgdg gh e> Ye / wdo txh j+Ye  iej,  Yf 1 Sru rwud sduwh/ frpr olp{$s i +{, @ e sdud od yhflqgdg Ye h{lvwh xqd yhflqgdg gh s wdo txh i+Ys isj,  Ye 1 Sru or wdqwr j+i+Ys  isj,,  j+Ye ,1 Dvð/ sdud xqd yhflqgdg Yf / h{lvwh xqd yhflqgdg gh s> Ys wdo txh +j  i ,+Ys  isj,  Yj+e, / hv ghflu/ olp +j  i,+{, @ j+e,= {$s

Hvwh whruhpd vh xwlol}d hq or txh vh vxhoh oodpdu fdpelr gh yduldeoh r vxvwlwxflöq



Hmhpsor 5;8

s t s +71<, olp {5 . 7 @ olp {5 . 7 @ 7 @ 5> {$3 {$3 s s s sxhvwr txh= Vl wrpdprv j+{, @ { | i +{, @ {5 . 7 | gdgr txh olp{$7 j+{, @ olp{$7 { @ 7 @ 5 | olp{$3 i +{, @ olp{$3 {5 . 7 @ 7/ hqwrqfhv sru ho frurodulr dqwhulru vh wlhqh 71< Hmhpsor 5;9 Vdelhqgr txh olp{$d h{ @ hd 1 Surehprv txh vl h{lvwh olp{$e i+{, hqwrqfhv

olp hi +{, @ holp{$e i +{, {$e Vroxflöq= Vhd j+{, @ h{ frpr +j i,+{, @ hi +{, | sxhvwr txh olp{$d h{ @ hd / hqwrqfhv olp{$e +j i ,+{, @ j+olp{$e i +{,, @ holp{$e i +{, 1 Dvð whqhprv sru hmhpsor= s6 {4 olp h {4 @ h4@6 {$4 s 6 Hq hihfwr1 Vhd x @ {1 Vl { $ 4 hqwrqfhv x $ 4 | s 6{4 x4 @ olp 6 olp {$4 {  4 x$4 x  4 x4 @ olp x$4 +x  4,+x5 . x . 4, 4 @ olp 5 x$4 x . x . 4 4 @ = 6

7151 OÐPLWHV ODWHUDOHV Oxhjr

576

s6 {4 s6 {4 olp h {4 @ holp{$4 {4 @ h4@6 = {$4

Sdud dsolfdu ho whruhpd dqwhulru/ ghehprv dvhjxuduqrv txh vh fxpsodq od klsöwhvlv1 Gh qr kdfhuor srghprv rewhqhu frqwudglfflrqhv1 Hmhpsor 5;: Vhd d 5 ]

olp ^{` 9@ ^ olp {` {$d {$d

Hq hihfwr> Vdehprv txh olp{$d ^{` qr h{lvwh sxhv orv oðplwhv odwhudohv vrq glvwlqwrv olp ^{` @ d {$d.

>

olp ^{` @ d  4 {$d

Vlq hpedujr> gdgr txh olp{$d { @ d/ hqwrqfhv vh wlhqh ^olp{$d {` @ ^d` @ d1

Revhuydflöq= Qrwhprv txh1 Vl d 95 ] hqwrqfhv olp{$d ^{` @ ^olp{$d {`=

Hmhuflflr 5;; Suredu txh vl olp{$d mi +{,m @ 3 hqwrqfhv olp{$d i +{, @ 31

Vroxflöq= Vxsrqjdprv txh olp{$d mi+{,m @ 3 hqwrqfhv olp{$d mi+{,m @ 3 dghpdv mi +{,m  i+{,  mi +{,m | sru whruhpd gh dfrwdplhqwr whqhprv txh olp{$d i +{, @ 31

Hmhuflflr 5;< Fdofxodu orv vljxlhqwhv oðplwhv d, e, f, Vroxflöq= d, olp {$3

s 4 . {5  4 {

s 4 . {5  4 olp {$3 s{ 4  6 4  5{ olp {$3 { s s 4 . {5  6 4  5{ olp {$3 { s s 4 . {5  4 4 . {5 . 4 s { 4 . {5 . 4 5 4.{ 4 @ olp s {$3 {+ 4 . {5 . 4, { @ olp s {$3 4 . {5 . 4 @ 3

@

olp {$3

e, Vhd x6 @ 4  5{1 Vl { $ 3 hqwrqfhv x $ 4 oxhjr/ s 4  6 4  5{ 4x @ olp olp {$3 x$4 +4  x6 ,@5 { 4x @ 5 olp x$4 +4  x,+4 . x . x5 , 4 @ 5 olp x$4 4 . x . x5 5 @ 6

577

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV f, olp {$3

s

s 4 . {5  6 4  5{ {

@ @ @ @

s

s 4 . {5  4 . 4  6 4  5{ olp {$3 { s s 4 . {5  4 4  6 4  5{ olp{$3 . { { 5 3. 6 5 = 6

Hv qhfhvdulr uhfrugdu/ sdud ho fäofxor gh oðplwh wuljrqrpìwulfrv dojxqdv lghqwlgdghv1 41 frv5 { . vhq5 { @ 4

51 frv+5{, @ 4  5 vhq5 {

61 frv+5{, @ 5 frv5 {  4

71 vhq+{ . |, @ vhq { frv | . vhq | frv {

81 vhq+{  |, @ vhq { frv |  vhq | frv { 91 frv+{ . |, @ frv { frv |  vhq { vhq | :1 frv+{  |, @ frv { frv | . vhq { vhq |

| , vhq+ {| , ;1 vhq {  vhq | @ 5 frv+ {. 5 5

| , vhq+ {.| , <1 frv {  frv | @ 5 vhq+ { 5 5

Hmhpsor 5<3 Ghprvwudu xvdqgr ho whruhpd dqwhulru txh olp vhq +{, @ vhq +d, {$d

Vroxflöq= Edvwd ghprvwudu txh olp{$d +vhq {  vhq d, @ 31 Shur xvdqgr od lghqwlgdg wuljrqrpìwulfd vhq {  vhq d @ 5 vhq+

{.d {d , frv+ , 5 5

d , hv dfrwdgr | olp {d {d | ho khfkr txh frv+ {. {$d vhq+ 5 , @ olpx$3 vhq x @ 3 grqgh 5 @ x1 Vh wlhqh sru 5 whruhpd fhur dqltxlodgru txh= olp +vhq {  vhq d, @ 3 {$d Hq irupd dqäorjd vh sxhgh suredu txh

olp frv { @ frv d {$d

Hmhpsor 5<4 Xvdqgr dojheud gh oðplwh | lghqwlgdghv wuljrqrpìwulfdv fdofxodu ho oðplwh gh od vljxlhqwh ixqflöq hq d i+{, @ wdq { > d 9@ 5 . q > q 5 Q

7151 OÐPLWHV ODWHUDOHV Vroxflöq=

578

vhq+{, olp wdq+{, @ olp {$d {$d frv+{, olp{$d vhq+{, @ olp{$d frv+{, vhq+d, @ frv+{, @ wdq+d,

Hmhpsor 5<5 Fdofxodu/ xvdqgr ho whruhpd gh fdpelr gh yduldeoh/ txh olp frv+{, @ 4 {$3

Vroxflöq= Hq xqd yhflqgdg gh 3 vh wlhqh od lghqwlgdg s frv+{, @ 4  vhq5 {

Wrpdqgr x @ 4  vhq5 +{,/ fxdqgr { wlhqgh d 3> x wlhqgh d 4 s s olp frv+{, @ olp x @ 4 @ 4 {$3 x$4 s s |d txh olp{$d { @ d1 Hmhuflflr 5<6 Fdofxodu

wj+{  5, olp s {$5 8{ . 9  7

Vroxflöq= Vhd x @ {  51 Vl { $ 5 hqwrqfhv x $ 3 +grqgh {  5 9@ 3 hq Y5  i5j,1 Oxhjr sru whruhpd gh vxvwlwxflöq

gh grqgh

wj+{  5, wj+x, olp s @ olp s {$5 8{ . 9  7 x$3 8x . 49  7

wj+x, olp s x$3 8x . 49  7

s vhq x 8x . 49 . 7 olp  x$3 frv x +8x . 49,  49 s vhq x 8x . 49 . 7 @ olp x$3 8x frv x s 4 vhq x 8x . 49 . 7 @ olp olp 8 x$3 x x$3 frv x ; @ 4 = 8 @

Revhuydflöq= Hq ho fäofxor gh oðplwhv +olp{$d i +{,, txh lqyroxfuh ixqflrqhv wuljrqrpìwulfdv/ or kdelwxdo hv kdfhu ho fdpelr gh yduldeoh x @ {  d r x @ d  {1 Hmhpsor 5<7 Fdofxodu

Vroxflöq=

6  6{, wdq { olp + {$@5 5

6   6{, wdq { @ 6 olp +  {, wdq { olp + {$@5 5 {$@5 5

Vhd x @ 5  { vl { $ 5 hqwrqfhv x $ 3 oxhjr/

579

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

Hmhpsor 5<8 Fdofxodu

6  olp +  6{, wdq { @ 6 olp +  {, wdq { {$@5 5 {$@5 5 vhq +@5  x, @ 6 olp x+ , x$3 frv +@5  x, frv x , @ 6 olp x+ x$3 vhq x x @ 6 olp frv x+ , x$3 vhq x x , @ 6 olp frv x olp + x$3 x$3 vhq x @ 644@6 olp {$4

5  5 frv +{  4, +{  4,5

Vroxflöq= Vhd x @ {  4 vl { $ 4 hqwrqfhv x $ 3 oxhjr/ olp 5+ {$4

4  frv +{  4, 4  frv x , @ 5 olp + , x$3 +{  4,5 x5 vhq5 x @ 5 olp + 5 , x$3 x +4 . frv x,  vhq x 5 4 @ 5 olp olp x$3 4 . frvx x$3 x 4 @ 5 4@4 5

Hmhuflflr 5<9 Fdofxodu +vl h{lvwh, olp{$4 i +{,/ olp{$3 i +{,1 Vl + 4{5 4?{?5 vhq +{s , i +{, @ s dufvhq +{@ 5,  5 ? { ? 4 a { 9@ 3 { Vroxflöq=

d, Frpr i hvwä ghqlgd hq xqd yhflqgdg gh 4/ srghprv fdofxodu orv oðplwhv odwhudohv1 l,

4  {5 +4  {,+4 . {, O4 @ olp. i+{, @ olp. @ olp vhq+{, {$4 {$4 vhq +{, {$3

Vhd x @ 4  { vl { $ 4. hqwrqfhv x $ 3 +sxhv 4  { ? 3,

x+5  x, olp x$3 vhq++4  x,, x +5  x, @ olp x$3 vhq+x, x 4 olp +5  x, @  x$3 vhq+x, x 4 @ olp olp +5  x,  x$3 vhq+x, x$3 x @ olp  w @ 4 vh wlhqh txh oðplwh dqwhulru hv |d txh olpx$3 vhq+ w$3 vhq+w, x, O4

@

O4 @

4 5 45 @  

7151 OÐPLWHV ODWHUDOHV

57:

ll, Sru rwud sduwh/

s s  olp{$4 dufvhq +{@ 5, 4 dufvhq +{@ 5, @ @ dufvhq+ s , @ = O5 @ olp { olp{$4 { 7 {$4 5

Sru +l, | +ll, O4 9@ O5 oxhjr qr h{lvwh olp{$4 i+{,1

e, i hvwä ghqlgd hq xqd yhflqgdg shuirudgd gho fhur

s dufvhq +{@ 5, olp i +{, @ olp {$3 {$3 { s s Vhd x @ dufvhq+{@ 5, hqwrqfhv; vhq +x, @ s{5 oxhjr { @ 5 vhq+x,

Vl { $ 3 hqwrqfhv x $ 3 Dvð

Sru or wdqwr olp{$3 i+{, @

Hmhuflflr 5<: Suxheh txh

s45 1

x olp i+{, @ olp s {$3 x$3 5 vhq+x, x 4 @ s olp 5 x$3 vhq +x, 4 @ s  4= 5

olp {$d

Vroxflöq= olp {$d

wj {  wj d {d

@ @ @ @ @ @

@

wj {  wj d {d

{ vhq {  d {d frv d vhq { frv dvhq d frv { frv { frv d olp {$d {d vhq+{  d, olp {$d frv { frv d+{  d, vhq+{  d, 4 olp {$d +{  d, frv { frv d vhq+{  d, 4 olp olp {$d +{  d, {$d frv { frv d 4 4 frv5 d 5 vhf d olp {$d

vhq frv

Whruhpd 7148 +gho Hqodfh, Vhd [  U> i = [ $ U ixqflöq s 5 [ 3 Hqwrqfhv whqhprv od vljxlhqwh htxlydohqfld olp i+{, @ O +, {$s

; Dtxð

vh wrpd

x ' @hUti?E I{2  frpr xqd

Sdud wrgd vxfhvlöq +{q ,4 q@4 frq {q 9@ s | {q $ s

vh wlhqh txh olp i +{q , @ O q$4

ixqflöq txh wlhqh lqyhuvd hq xqd yhflqgdg gh fhur sdud

x frpr

sdud

{1

57;

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

Ghprvwudflöq= +,, Vhd +{q , vxfhvlöq frq {q 9@ s | {q $ s1 Sru ghprvwudu txh olpq$4 i +{q , @ O1 Edvwd wrpdu ho frqmxqwr E  [> E @ i{q = q 5 Qj1 Frpr {q $ s vh wlhqh txh s 5 E 3 | xvdqgr ho whruhpd 71: vh wlhqh olp{$s i mE @ O/ hv ghflu/ olp i +{q , @ O q$4

++, Ghprvwudflöq sru ho devxugr/ hv ghflu/ vxsrqguhprv txh sdud wrgr +{q ,4 q@4 txh vdwlvidfh od klsöwhvlv vh wlhqh txh olpq$4 i +{q , @ O | qr vh wlhqh olp{$s i +{, @ O1 Sru hvwr ýowlpr/ h{lvwh xqd yhflqgdg gh O> YO> % wdo txh sdud wrgd Ys> +hvwr hv or plvpr txh ghflu sdud wrgr , vh wlhqh txh  ^i +Ys>  isj,  YO> % `/ hv ghflu/ sdud fdgd  q @ q4 h{lvwh do phqrv xq sxqwr {q 5 Ys>q  isj wdo txh i+{q , 95 YO> % 1 Dvð/ khprv ideulfdgr xqd vxfhvlöq +{q ,4 q@4 1 Frpr  q $ 3/ {q $ s | dghpäv i +{q , 9$ O +|d txh i+{q , hvwä do phqrv % ohmrv gh O sxhvwr txh i+{q , 95 YO> % , or txh frqwudglfh ho khfkr wrgd vxfhvlöq% gh odv  shglgdv> oxhjr i+{q , $ O1

Hmhpsor 5<; Fdofxohprv

4 olp q  vhq q$4 q

Frqvlghuhprv i+{, @ vhq{ { | vhd {q @ q4 > {q 9@ 3> ;q 5 Q Vl q $ 4 hqwrqfhv {q $ 3 | sru whruhpd gh hqodfh vh wlhqh

Oxhjr olpq$4 vhq+ q4 , @ 41

Hmhuflflr 5<< Suredu Vroxflöq=

4 olp i+{q , @ olp q vhq+ , q$4 q$4 q 4 @ olp vhq { { vhq { @ olp { @ 4

q7 . 5q 4 4 olp vhq+ , @ q$4 8q6 . 6 q 8 4 4 q7 . 5q q6 . 5 olp vhq+ , @ olp q vhq+ , q$4 8q6 . 6 q$4 8q6 . 6 q q 4 4 @ 8 4 @ = 8

Hmhpsor 633 Xvdqgr ho khfkr txh olp{$d {5 @ d5 1 Fdofxodu olpq$4 +4 . q5 . q45 ,q Vroxflöq= Frpr +4 . q5 . q45 ,q @ ++4 . q4 ,5 ,q @ ++4 . q4 ,q ,5 | olpq$4 +4 . q4 ,q @ h/ xvdqgr whruhpd gho hqodfh vh wlhqh txh olp {5q @ h5 grqgh {q @ +4 . q4 ,q Hmhpsor 634 Vxsrqlhqgr txh olp{$d oq { @ oq d> d A 3 Fdofxoh

Vroxflöq= Vl |q @

4 4 4 olp oq+4 . . .    . , q$4 4$ 5$ q$

Sq 4 l@4 l$ / olpq$4 |q @ h xvdqgr whruhpd gho hqodfh olp oq |q @ oq h @ 4

7161 FRQWLQXLGDG

57<

Hmhpsor 635 Vxsrqlhqgr txh olp{$d h{ @ hd Fdofxohprv

s

s s h q.4 sq @ olp h q.4 q O @ olp q$4 h q$4

Vroxflöq= Frpr olpq$4

s s q . 4  q @ 3 rewhqhprv sru whruhpd gho hqodfh O @ h3 @ 4

Gho whruhpd dqwhulru/ vh rewlhqh xqd khuudplhqwd px| ýwlo sdud suredu od qr h{lvwhqfld gh dojxqrv oðplwhv1 Frurodulr 7149 Vhd [  U> i = [ $ U ixqflöq s 5 [ 3 1 Vl h{lvwhq +{q ,> +|q , grv vxfhvlrqhv wdohv {q 9@ s> |q 9@ s> {q $ s> |q $ s shur olp i +{q , @ O4 | olp i +|q , @ O5 frq O4 9@ O5 q$4 q$4

r dojxqrv gh hoorv qr h{lvwh/ hqwrqfhv olp{$s i+{, qr h{lvwh1 Ghprvwudflöq= +dsolfdu s , t +, t , s,1

Hmhpsor 636 Surehprv txh olp{$3 vhq+ {4 , qr h{lvwh1 4 Frqvlghuhprv {q @ .54 q > |q @  .5 q 1

Fodudphqwh {q $ 3/ |q $ 3 grqgh {q 9@ 3> |q 9@ 31 Vhd i +{, @ vhq+ {4 ,= Oxhjr olpq$4 i+{q , @ olpq$4 vhq  @ 3/ | olpq$4 i +|q , @ olpq$4 vhq 5 @ 41 Sru or wdqwr/ sru frurodulr dqwhulru olp{$3 vhq+ {4 , qr h{lvwh1 5

716

Frqwlqxlgdg

Ghqlflöq 714: Vhd D  U> s 5 D/ gluhprv txh i hv frqwlqxd hq ho sxqwr s vl sdud wrgd yhflqgdg gh i+s,> Yi +s,> % h{lvwh xqd yhflqgdg gh s> Ys> wdo txh i +Ys> ,  < Yi +s,> %

Gluhprv txh i hv frqwlqxd hq D vl i hv frqwlqxd hq fdgd sxqwr s 5 D1

Hmhpsor 637 i+{, @ 5{ hv frqwlqxd hq s @ 6 sxhvwr txh= Xqd yhflqgdg duelwuduld gh i+6, hv gho wlsr =

Yi +6,> % @`i +6,  %> i+6, . %^@`9  %> 9 . %^

< Qrwh

txh qr qhfhvduldphqwh

Ys> a D ' i Es

whqjdq lpdjhq hood hvwh fhufdqd d

Grp

i

/ or lps ruwdqwh hv txh orv

{

txh wrpd od ixqflöq

i

/ hv ghflu/ orv

{ M Ys>

583

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

Vl sdud od yhflqgdg dqwhulru gh i+6,/ h{lvwlhud xqd yhflqgdg gh 6> `6  %> 6 . %^ hvwd ghehuðd fxpsolu i +`6  > 6 . ^ Yi +6,> % >

hv ghflu/ shur frpr i hv fuhflhqwh/ whqhprv

i+`6  > 6 . ^`9  %> 9 . %^

i+`6  > 6 . ^@`5+6  ,> 5+6 . ,^@`9  5> 9 . 5^

Sru or wdqwr/  gheh fxpsolu

`9  5> 9 . 5^`9  %> 9 . %^

Sdud % gdgr/  gheh fxpsolu hv ghflu/

9  % ? 9  5

a

a

5 ? %

9 . 5 ? 9 . %

>

5 ? % >

Sru or wdqwr/ sdud od yhflqgdg Yi +6,> % h{lvwh do phqrv xqd yhflqgdg gh 6 |d txh edvwd wrpdu 5 ? % +hv ghflu/  ? %@5, wdo txh i +Y6> ,  Y9> % 1 Yhu jxud dqwhulru Hmhpsor 638 Vhd

Ghwhuplqdu vl hv frqwlqxd hq { @ 61

i +{, @



5 {6 5=34 { A 6

Vroxflöq= Gdgd xqd yhflqgdg gh i +6,> Y5> % / ghehuðd h{lvwlu xqd yhflqgdg gh 6> Y6> / + hv ghflu/ h{lvwh  A 3 , wdo txh i +Y6> ,  Y5> % Or dqwhulru hv htxlydohqwh d ; % A 3> < A 3> i +Y6> ,  Y5> %

7161 FRQWLQXLGDG

584

ö Dvð/ ghehuðd rfxuulu

i+`6  > 6 . ^, `5  %> 5 . %^ ; % A 3> < A 3 wdo txh i+Y6> , @ i5> 5=34j `5  %> 5 . %^>

shur sdud % @ 3=334 r phqru hv lpsrvleoh rewhqhu od lqfoxvlöq dqwhulru1 Frqfoxvlöq= Frpr sdud wrgr % A 3 qr hv srvleoh hqfrqwudu xqd yhflqgdg gh 6> Y6> wdo txh i +Y6> ,  Y5> % vh wlhqh txh i qr hv frqwlqxd hq 61 Vh sxhgh yhu fodudphqwh txh i hv frqwlqxd hq orv rwurv sxqwrv Hmhpsor 639 Vhd i +{, @ 8/ +ixqflöq frqvwdqwh,1 Hood hv frqwlqxd hq wrgrv orv uhdohv1

Vroxflöq= Yhdprv od frqwlqxlgdg hq xq sxqwr duelwudulr s1 Qrwhprv txh= ; A 3 i+Ys> , @ i8j

Dvð/ ; % A 3> < A 3> i+Ys> ,  Yi +s,> % |d txh i+Ys> , @ i8j `8  %> 8 . %^ > ; % A 3

71614

Htxlydohqfldv gh Frqwlqxlgdg

D frqwlqxdflöq gduhprv fxdwur fdudfwhul}dflrqhv gh frqwlqxlgdg 41 Vl s hv xq sxqwr gh dfxpxodflöq gh D | s 5 D1

51 Vl s hv xq sxqwr gh dfxpxodflöq gh D | D frqwlhqh xqd yhflqgdg gh s1

61 Vl s qr hv xq sxqwr gh dfxpxodflöq | s 5 D1 71 Fdudfwhul}dflöq sru vxfhvlrqhv1

41 Vl s hv sxqwr gh dfxpxodflöq gh D | s 5 D hqwrqfhv i hv frqwlqxd hq s vl | vöor vl olp{$s i+{, @ i +s, Hvwd ýowlpd ljxdogdg glfh txh

l, olp{$s i+{, h{lvwh ll, Ho oðplwh dqwhulru hv ljxdo d i+s, h lpsoðflwdphqwh hvwdprv glflhqgr txh i +s, h{lvwh1

51 Vl D frqwlhqh xqd yhflqgdg gh s +sru or wdqwr s hv xq sxqwr gh dfxpxodflöq, whqhprv or vljxlhqwh= i hv frqwlqxd hq s vl | vöor vl Hmhpsor 63: Vhd i +{, @

d, olp{$s. i +{, h{lvwh e, olp{$s i +{, h{lvwh f, orv oðplwhv dqwhulruhv vrq ljxdohv d i +s,

s {  4 BcHv frqwlqxd hq 4 B

Vroxflöq= Qrwhprv txh Grpi @ D @ ^4> 4^/ | dtxð ylhqh qdwxudophqwh od htxlyrfdgd frqfoxvlöq i qr hv frqwlqxd hq s @ 4 |d txh olp{$s i +{, qr h{lvwh%/ frpr vh h{ljh hq 5f1

Revhuydflöq= Qr srghprv xvdu od fdudfwhul}dflöq 51 sxhv qr vh fxpsoh od klsrwìvlv gh txh D frqwhqjd xqd yhflqgdg gh 41 Hq fdpelr/ vl srghprv xvdusod fdudfwhul}dflöq 4/ sxhv ssxqwr gh dfxpxodflöq gh D | 4 5 D/ s 4 hv xqd hqwrqfhv gdgr txh olp{$4 {  4 h{lvwh | olp{$4 {  4 @ 4  4 @ 3 @ i +4,1 vh wlhqh txh i hv frqwlqxd hq 41

585

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV Surehprv i hv frqwlqxd hq 4 hq wìuplqrv gh yhflqgdghv1

Wrphprv xqd yhflqgdg duelwuduld gh i +4, @ 3> Y3> % qhfhvlwdprv hqfrqwudu xqd yhflqgdg gh 4> Y4> / wdo txh rfxuud= +7143,

i+Y4> ,  Y3> %

Yhdprv txì frqglflöq gheh fxpsolu  sdud txh hvwd ýowlpd lqfoxvlöq vhd yäolgd1

Frpr Y4> @`4  > 4 . ^ | i hv fuhflhqwh +yhu jxud, y

2

1.5

1

0.5

0 0

1.25

i +{, @ vh wlhqh txh=

3.75

s {4

5 x

s i +`4  > 4 . ^, @ i +^4> 4 . ^, @ ^3> ^

Dvð/ od lqfoxvlöq hq 7143 qrv txhgd Sru or wdqwr  gheh fxpsolu

2.5

s ^3> ^`  %> %^

s  ? % r htxlydohqwhphqwh  ? %5 1

Uhvxplhqgr1 Sdud wrgd yhflqgdg Y3> % h{lvwh xqd yhflqgdg gh 4> Y4> +edvwd txh  vhd phqru txh %5 wdo txh i +Y4> ,  Y3> % =

Oxhjr i hv frqwlqxd hq s @ 41

61 Vl s 5 D qr hv xq sxqwr gh dfxpxodflöq +s hv xq sxqwr dlvodgr, vh wlhqh txh i hv frqwlqxd hq s1

Ghprvwudflöq= Frpr s qr hv xq sxqwr gh dfxpxodflöq/ h{lvwh xqd yhflqgdg gh s> Ys> +hv ghflu/ h{lvwh  A 3, wdo txh Ys> _ D @ isj r htxlydohqwhphqwh

`s  > s . ^_D @ isj>

sru or wdqwr/ i +Ys> , @ i+isj, @ ii +s,j `i +s,  %> i+s, . %^

Od lqfoxvlöq ýowlpd rfxuuh sdud wrgr % A 3> gh hvwd irupd whqhprv od vljxlhqwh dupdflöq1

Sdud wrgd yhflqgdg gh i +s,> Yi +s,> % / h{lvwh xqd yhflqgdg gh s> Ys> +od wrpdgd do lqlflr, wdo txh i +Ys> ,  Yi +s,> %

Or txh txlhuh ghflu txh i hv frqwlqxd hq s1

7161 FRQWLQXLGDG

586

Hmhpsor 63; Wrgdv odv vxfhvlrqhv {q = Q $ U vrq frqwlqxdv/ |d txh wrgr sxqwr gh Q hv sxqwr gh dlvodgr1 Hq od suäfwlfd/ qr vh hqfxhqwudq pxfkrv grplqlrv frq sxqwrv dlvodgrv/ shur vl dsduhfhq hq vlwxd0 flrqhv hvshfldohv | sru hvh prwlyr or khprv ghvduuroodgr hq ghwdooh dtxð/ dpìq txh ohv idflolwduä ho/ hqwhqglplhqwr gho frqfhswr gh frqwlqxlgdg hq yduldv yduldeohv1 71 Ghqlflöq wudglflrqdo1

i hv frqwlqxd hq s> s 5 Grpi @ D vl

; % A 3 < A 3> m{  sm ?  a { 5 Grpi , mi +{,  i +s,m ? %

Qrwh txh m{  sm ?  | mi+{,  i+s,m ? % kdeodq gh odv yhflqgdghv gh s | i +s, uhvshfwlydphqwh | od lpsolfdflrq +,, vh uhhuh d od lqfoxvlöq gh frqmxqwrv> od phqflöq h{soðflwd gh { 5 Grpi hv sdud txh whqjd vhqwlgr od qrwdflöq i +{,1

81 Vhd i = D $ U> s 5 D1 i hv frqwlqxd hq s vl | vöor vl

Sdud wrgd vxfhvlöq43 +{q ,/ wdo txh {q frqyhujh d s vh wlhqh txh od vxfhvlöq i+{q , frqyhujh d i +s,1 Hvwh fulwhulr hv pdv ýwlo sdud suredu od qr frqwlqxlgdg hq xq sxqwr txh od frq|lqxlgdg1 Or dqwhulru vh sxhgh hvfulelu frpr

i hv frqwlqxd hq { @ 3 +, ;+{q ,> {q $ s , i +{q , $ i+s,

Dvð/ od qhjdflöq hv= i qr hv frqwlqxd hq s vl | v*ror vl

Hmhpsor 63< Vhd

<+{q ,> {q $ s a i +{q , 9$ i +s, +i+{, qr frqyhujh d i+s,, i+{, @



vhq + {4 , { 9@ 3 4 {@3

BcHv frqwlqxd hq 3 B 4 Vroxflöq= Frqvlghuhprv od vxfhvlöq +{q , frq {q @ 5q Fodudphqwh vh yh txh {q $ 3/ shur

i +{q , @ vhq+

Dvð/ i +{q , qr frqyhujh d 4

4 , @ vhq+5q, @ 3 ;q 5 Q 4@5q

Frqfoxvlöq=

i qr hv frqwlqxd hq 3 |d txh h{lvwh xqd vxfhvlöq +{q , frq {q $ 3 a i+{q , 9$ i+3, @ 4

Wdpelìq srghprv ghflu txh i qr hv frqwlqxd hq { @ 3 sxhv olp{$3 i +{, qr h{lvwh +yhu hmhpsor 636,1 43 Qdwxudophqwh

od qrflöq ghe h vhu gh hohphqwrv gh

D

1

587

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

71615

Frphqwdulrv

41 Frphqwdulrv lqwxlwlyrv44

l, i hv frqwlqxd hq s vl odv lpdjhqhv gh sxqwrv fhufdqrv d s hvwäq fhufdqrv d i +s, | sru px| h{ljhqwh txh vh srqjd +; % A 3, sdud shglu hvwdu fhufd gh i+s,/ vlhpsuh kd| xq prqwöq gh sxqwrv fhufd gh s txh hvwäq vxshu fhufd gh i+s,1

ll, Xqd ixqflöq frqwlqxd hq s qr sxhgh whqhu xq vdowr hq s/ qr sxhgh hvwdu fruwdgd hq s1

51 Frphqwdulr Uhgxqgdqwh

l, Fodudphqwh i qr hv frqwlqxd hq xq sxqwr s vl s 95 Grpi

ll, Gh od iudvh Xqd ixqflöq hv glvfrqwlqxd hq xq sxqwr s% qr vh sxhgh frqfoxlu txh qr wlhqh oðplwh |d txh shuihfwdphqwh od glvfrqwlqxlgdg sxhgh ghehuvh d txh d, s 95 Grpi r d txh

e, i+s, 9@ olp{$s i+{,1

5 Hmhpsor 643 i +{, @ {{44 / i+4, qr h{lvwh shur olp{$4 i +{, @ 51 Hvwd ixqflöq frqvwlwx|h xq hmhpsor sdud ho fdvr +d, Hmhpsor 644

i+{, @



{6, {6 8

vhq+

{ 9@ 6 {@6

Hq hvwd ixqflöq vh wlhqh i +6, @ 8 9@ olp{$6 i +{,1Hvwd ixqflöq frqvwlwx|h xq hmhpsor sdud ho fdvr +e,1

lll, Vl olp{$s i +{, qr h{lvwh/ qr hv frqwlqxd i hq s1 Od qr h{lvwhqfld gho oðplwh vh sxhgh ghehu d yduldv fdvrv/ hq ho hmhpsor 645 whqhprv ho fdvr hq txh fdgd oðplwh odwhudo h{lvwh/ shur olp{$s. i +{, 9@ olp{$s i+{,

Hmhpsor 645

i +{, @ Frpr

; A A ?

{5 < {6 9

A A = {5 9{.< {6 5:

{A6 {@6 {?6

olp i+{, @ 9 @ i +6, 9@ olp i+{, @ 3 {$6. {$6 Sru or wdqwr qr hv frqwlqxd hq 6

Ho vljxlhqwh whruhpd hv ho txh vluyh sdud prvwudu frq fodulgdg odv revhuydflrqhv +frphqwdulrv, dqwhulruhv

Whruhpd 714; l, Wrgd uhvwulfflöq gh xqd ixqflöq frqwlqxd hv frqwlqxd1 Glfkr gh rwud irupd Vhd i = [ $ U frqwlqxd hq ho sxqwr s 5 U1 Vl s 5 \  [ | j @ 45 i m\ hqwrqfhv wdpelìq j = \ $ U hv frqwlqxd hq s ll, Hq ho fdvr hvshfldo gh txh \ @ L _ [/ frq L xqd yhflqgdg gh s hqwrqfhv od uhfðsurfd vh fxpsoh1 Glfkr gh rwud irupd1 Vhd j @ im\ frqwlqxd hq ho sxqwr s +frq \ @ Ys> _ [ hqwrqfhv i = [ $ U hv frqwlqxd hq s

Hvwd ýowlpr qrv glfh txh od frqwlqxlgdg hq ho sxqwr s ghshqgh vöor gh orv sxqwrv txh hvwäq fhufd/ | hvwr hv or txh vh vxhoh oodpdu frqfhswr orfdo1 Dvð frqwlqxlgdg hv xq frqfhswr orfdo% do ljxdo txh oðplwh1 Orv vljxlhqwhv whruhpdv vrq xwlohv sru ho khfkr txh qrv vluyhq sdud ideulfdu qxhydv ixqflrqhv frqwlqxdv r sdud ylvxdol}du txh xqd ixqflöq hv frqwlqxd1 4 4 Hvwr

qr vluyh sdud qdgd hq odv ghprvwudflrqhv/ s hur vl sdud ylvxdol}du1

4 5 Uhfxhugh

txh

i\ ƒ

hv od ixqflöq uhvwulfflöq gh

[ \ d

7161 FRQWLQXLGDG

588

Whruhpd 714< +Dojheud gh ixqflrqhv frqwlqxdv, Vhd i = [ $ U > j = \ $ U frqwlqxdv fdgd xqd hq s 5 [ _ \ hqwrqfhv l, i . j> i  j> i  j vrq frqwlqxdv hq s ll, Vl j+s, 9@ 3 hqwrqfhv i@j hv frqwlqxd hq s Hq ho fdvr txh s vhd xq sxqwr gh dfxpxodflöq gh [ _ \ / od ghprvwudflöq vh rewlhqh idflophqwh gho dojheud gh oðplwhv1 Hq fdvr frqwudulr hv gluhfwd1 Frq hvwh ýowlpr whruhpd whqhprv/ sru hmhpsor/ txh orv srolqrplrv s+{, @ d3 . d4 { . d5 {5 .    dq {q >

frq dl frqvwdqwhv uhdohv

vrq ixqflrqhv frqwlqxdv1 Uhfrugdqgr orv uhvxowdgr gh oðplwhv whqhprv txh odv vljxlhqwhv ixqflrqhv vrq frqwlqxdv+hq wrgr vx grplqlr, 41 vhq {1

51 frv {1

61 wdq { vl { 9@ 5 . n> 71 vhq { . 5 frv {1

n 5 ]1

81 6 frv { . fvf {1

91 frw { . : vhq +{, vhf {1

:1 vhq5 {1 ;1 frv5 {1

| pxfkdv rwudv frpelqdflrqhv gh ixqflrqhv wuljrqrpìwulfdv | srolqrplfdv txh vhd rshudflrqhv dojh0 eudlfdv hqwuh ixqflrqhv eävlfdv frpr vhq +{,> frv +{, | d{1

Whruhpd 7153 +Od frpsxhvwd gh ixqflrqhv frqwlqxdv hv frqwlqxd, Vhdq i = [ $ U frqwlqxd hq s/ j = \ $ U frqwlqxd hq t @ i+s, wdo txh i+[,  \ hqwrqfhv j  i = [ $ U hv frqwlqxd hq s Hvwh whruhpd qrv d|xgd d prvwudu txh odv vljxlhqwhv ixqflrqhv vrq frqwlqxdv+hq wrgr U,1

41 vhq {5 |d txh hv od frpsxhvwd gh vhq { | {5

51 frv +6{8 . :, |d txh hv od frpsxhvwd gh frv { | 6{8 . : 61 vhq +5 frv {, frpsxhvwd gh vhq { frq 5 frv {

71 6 vhq5 {6 . : frv {6 frpsxhvwd 6 vhq5 +{, . : frv +{,> {6

Hmhuflflr 646 Frqvlghu od ixqflöq i =`  4> 4^$ U ghqlgd sru ; 5 {?3 ? { frv+4@{, 3 {@3 i+{, @ = {6 3 ? {?4 5 vhq+{ , Hvwxglh od frqwlqxlgdg gh i1

Vroxflöq=

Hmhuflflr 647 l, Vhd { ? 3/ hqwrqfhv i +{, @ {5 frv+4@{,1 Sru frpsrvlflöq | surgxfwr gh ixqflrqhv frqwlqxd/ i hv frqwlqxd hq `  4> 3^1

589

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

ll, Vhd 3 ? { ? 4/ hqwrqfhv I +{, @ frqwlqxd hq `3> 4^1

{6 sru frpsrvlflöq | fxrflhqwh gh ixqflrqhv frqwlqxd/ i hv {5 ,

vhq+

lll, Yhdprv od frqwlqxlgdg gh i hq { @ 31 Fodudphqwh 3 5 Grpi | i +3, @ 31 Frpr 3 hv sxqwr gh dfxpxodflöq gho Grpi / ghehprv yhu vl h{lvwh olp{$3 i +{,1 Sdud hvwr/ fdofxohprv orv oðplwhv odwhudohv

{6 {5 @ olp. {  olp. i+{, @ olp. @34@3 5 vhq {5 {$3 {$3 vhq+{ , {$3 sru fhur dqltxlod1

olp i+{, @ olp {5 frv+4@{, @ 3 {$3 {$3

hv ghflu/ olp{$3 i +{, h{lvwh dghpäv

oxhjr i hv frqwlqxd hq { @ 31

olp i +{, @ olp i +{, @ 3 {$3. {$3 olp i +{, @ i+3, @ 3 {$3

Sru frqvljxlhqwh i hv frqwlqxd hq wrgr vx grplqlr

71616

Ixqflrqhv frqwlqxdv vreuh frpsdfwr

Vl i hv xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr +frpsdfwr, ^d> e` vh ghehuðd whqhu txh 41 Hq qlqjxq sxqwr gh ^d> e` kd| vdowrv/ |d txh vl orv kxelhud vdowr orv olplwhv odwhudohv vrq glvwlqwrv jxud gh pdv dedmr

r elhq ho oðplwh gh od ixqflöq hq ho sxqwr qr frlqflgh frq ho ydoru gh od ixqflöq hq ho sxqwr yhu od vljxhqwh jxud

7161 FRQWLQXLGDG

58:

51 Wdpsrfr od ixqflöq sxhgh lu d lqqlwr hq dojýq sxqwr gh ^d> e` / yhu jxud txh vljxh/ |d txh hq hvh fdvr oðplwh qr h{lvwh

61 Wdpsrfr sxhgh rvflodu hq xq sxqwr {3 +yhu jxud txh vljxh, |d txh ho oðplwh qr h{lvwluld1

58;

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

Dvð whqhprv +lqwxlwlydphqwh, txh od juäfd gh xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr ^d> e` gheh vhu gh wdo irupd txh xqr od sxhgh judfdu vlq ohydqwdu ho oäsl} hpsh}dqgr hq +d> i+d,, | whuplqdqgr hq +e> i+e,,1 Hq ho irqgr xqd ixqflöq frqwlqxd hq ^d> e` hv vöor xqd ghvirupdflöq gho lqwhuydor ^d> e`/ vlhqgr vx juäfd vrodphqwh xqd fxhugd qr lqqlwd txh xqh orv sxqwrv +d> i+d,, | +e> i+e,,1 Or dqwhulru qrv shuplwh fuhhu orv vljxlhqwhv whruhpdv1

Whruhpd 7154 gho Ydoru Lqwhuphglr Vhd i = ^d> e` $ U frqwlqxd1 Vl  hv xq qýphur uhdo txh fxpsoh i +d, ?  ? i +e, hqwrqfhv  wlhqh do phqrv xqd suhlpdjhq hq ho lqwhuydor `d> e^1 +hv ghflu/ < f 5`d=e^ wdo txh i+f, @ , Frpr päv vh xwlol}d hvwh whruhpd/ hv hq od h{lvwhqfld gh udðfhv gh xqd ixqflöq1

Frurodulr 7155 +H{lvwhqfld gh Udlfhv, Vhd d> e 5 U wdo txh 4, i+d,i+e, ? 3 +hv ghflu/ i +d,| i+e, wlhqh glvwlqwrv vljqrv, 5, i frqwlqxd hq xq frqmxqwr txh lqfox|d do lqwhuydor ^d> e` hqwrqfhv i wlhqh xqd udl} hq ho lqwhuydor `d> e^ +hv ghflu/ h{lvwh f 5`d=e^ wdo txh i +f, @ 3 ,

Hmhpsor 648 Od ixqflöq i +{, @ {6 . {5 . 6{ . 5 wlhqh xqd udð} hq ho lqwhuydor `  4> 3^ Hq hihfwr i+4, @ 4 . 4  6 . 5 @ 4 ? 3 | i+3, @ 5 A 3 oxhjr i +4,i+3, ? 31 Dghpäv i hv frqwlqxd hq ^4> 3` oxhjr sru frurodulr dqwhulru h{lvwh xqd udð} hq ho lqwhuydor `  4> 3^1 Hmhuflflr 649 Suxheh txh od hfxdflöq

h{  6{ @ 3

wlhqh do phqrv xqd udð} uhdo hq ho lqwhuydor `3> 4^1 Vroxflöq= Vhd i+{, @ h{  6{ i+3, @ h3  6  3 @ 4 A 3 | i+4, @ h  6 ? 3 oxhjr i+3,i +4, ? 3 Dghpäv i hv xqd ixqflöq frqwlqxd hq U hq sduwlfxodu hq ^3> 4`/ oxhjr sru ho frurodulr dqwhulru/ h{lvwh f 5`3> 4^ wdo txh i+f, @ 3/ hv ghflu f hv xqd udð} r vroxflöq gh od hfxdflöq1

Frurodulr 7156 Vhd i frqwlqxd hq xq lqwhuydor fhuudgr L/ hqwrqfhv i+L, wdpelìq hv xq lqwhuydor fhuudgr1

7161 FRQWLQXLGDG

58<

Frurodulr 7157 Vhd i frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr ^d> e`/ hqwrqfhv i +L, hv xq lqwhuydor fhuudgr | dfrwdgr/ sru or wdqwr i hv dfrwdgd1 Hmhuflflr 64:s Hvwxglh s od frqwlqxlgdg gh d, i +{, @ s {  5 . 5s { s e, i +{, @ {5  4 . 8 {+4  {, . {+4  {, D|xgd= Fxlgdgr frq orv grplqlrv1 f, i = ] $ U i+{, @ ^{` +sduwh hqwhud, g, i = U $ U i+{, @ ^{` +sduwh hqwhud,

Hmhuflflr 64; Hvwxglh od frqwlqxlgdg gh od ixqflöq hq orv sxqwrv s lqglfdgrv1 4 frv+ { , { 9@ 6 6 s @ 61 d, i +{, @ 3 {@6 6 > s @ 6 | s @ 31 e, i +{, @ m{{  6m 4 +{  6, frv+ { , { 9@ 6 6 s @ 6 | s @ 31 f, i +{, @ 3 {@6

Hmhuflflr 64< Sdud txì ydoruhv gh > i hv frqwlqxd hq ho sxqwr lqglfdgr1 +{  , vhq+ {4  , { @ 9  d, i +{, @ s@ s@3 3 {@ > s @ 3> s @ 5 e, i +{, @ {  f, i +{, @ {{57 s@3 s@5 Hmhpsor 653 Frqvlghuh od vljxlhqwh ixqflöq ; d oq+{ . h5 ,6 . e vhq5{7{ {?3 A A A A ? d{ . 6 {@3 i +{, @ A A A s4.e{5 4 A = 3?{? vhq {5

grqgh d | e vrq qýphurv uhdohv 4, Fdofxoh ho oðplwh odwhudo ghuhfkr gh i hq 3 5, Fdofxoh ho oðplwh odwhudo l}txlhgr gh i hq 3 6, Hqfxhwuh orv ydoruhv gh d | e txh kdfhq frqwlqxd d i hq 31 7, Ghwhuplqh sdud txì ydoru gh e h{lvwh ho oðplwh odwhudo l}txlhugr gh i hq  Vroxflöq= 4,

4 . e{5  4 vhq5 { s s 4 . e{5  4 4 . e{5 . 4 s olp. 5 vhq { {$3 4 . e{5 . 4 5 +4 . e{ ,  4 s olp {$3. vhq5 {+ 4 . e{5 . 4, e{5 s olp {$3. vhq5 {+ 4 . e{5 . 4, {5 4 olp. s e olp. 5 {$3 vhq { {$3 4 . e{5 . 4 4 e4 5 e 5

olp i+{, @ olp. {$3. {$3 @ @ @ @ @ olp i+{, @ {$3.

s

593

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV 5,

vhq 7{ , olp i+{, @ olp +d oq +{ . h5 ,6 . e 5{ {$3 {$3 vhq 7{ @ olp d oq +{ . h5 ,6 . olp 5{ {$3 {$3 5 vhq 7{ @ d oq +h5 ,6 . olp e {$3 5  5{ vhq 7{ @ 9d . 5e olp 7{ {$3 @ 9d . 5e  4 @ 9d . 5e olp i+{, @ 9d . 5e {$3.

6, i vhuä frqwlqxd hq 3 vl olp{$3 i +{, @ i +3, oxhjr olp{$3. i+{, @ olp{$3 @ i +3,/ hv ghflu/ e e @ 9d . 5e @6 5 5 Dvð e @ 9 | d @  65 s e{54 710 olp{$ i+{, @ olp{$ 4. vhq {5 Vl e @ 3 hqwrqfhv i+{, @ 3 | olp{$ i +{, @ 3 Vl e 9@ 3 hqwrqfhv ho olplwh qr h{lvwh1 s Hq hihfwr> Vxsrqjdprv txh h{lvwh/ hv ghflu/ olp{$ i+{, @ O> O 5 U hqwrqfhv do fdofxodu olp{$ + 4 . e{5  4, gh grv irupdv glvwlqwdv whqhprv s 4 . e5  4 @

@ @

oxhjr hvwr hv xqd frqwudglfflöq1

@ @

s olp + 4 . e{5  4, {$ s 4 . e{5  4 olp +vhq5 {, vhq5 { {$ s 4 . e{5  4 olp olp +vhq5 {, vhq5 { {$ {$ O3 3

Hmhpsor 654 Ghwhuplqh vl odv vljxlhqwh dupdflrqhv vrq yhugdghudv r idovdv1 41 Od ixqflöq i+{, @ ^5{` hv glvfrqwlqxd vöor hq ]  ^{` {  7 51 Od ixqflöq i+{, @ hv glvfrqwlqxd hq { {?7

G @ i{ 5 ] = {  7j

61 Vl i . j hv frqwlqxd hq d | i hv frqwlqxd hq d hqwrqfhv j hv frqwlqxd hq d1

71 Vhdq i> j ixqflrqhv ghqlgdv hq xqd yhflqgdg gh d/ wdo txh i hv glvfrqwlqxd d hqwrqfhv i . j hv glvfrqwlqxd hq d1

81 Vl i | j vrq glvfrqwlqxdv hq d hqwrqfhv i . j hv glvfrqwlqxd hq d1

7161 FRQWLQXLGDG

594

Vroxflöq= 41 Idovr= od ixqflöq hq h{lvwh1

4 5

hv glvfrqwlqxd |d txh olp{$ 4 . ^5{` @ 4 | olp{$ 4  ^5{` @ 3 oxhjr ho oðplwh qr 5 5

51 Idovr= i hv frqwlqxd hq 7 sxhv olp{$7. i +{, @ olp{$7. ^{` @ 7 dghpäv olp{$7 i+{, @ olp{$7 { @ 7 oxhjr olp{$7 i +{, @ 7 @ i+7,1 dghpäv hv fodur sru od judfd +yhu jxud vljxlhqwh,

txh i hv glvfrqwlqxd hq i{ 5 ] = { A 7j1

61 Yhugdghur= Sru dojheud gh oðplwhv whqhprv1 Vl i . j/ i vrq frqwlqxdv hq d hqwrqfhv +i . j,  i hv frqwlqxd hq d/ hv ghflu j hv frqwlqxd hq d1

71 Idovr= H{lwhq i +{, @ ^{` | j+{, @ {  ^{` ixqflrqhv ghqlgdv hq xqd yhflqgdg gh d @ 4 wdo txh i hv glvfrqwlqxd hq { @ 4 vlq hpedujr +i . j,+{, @ { hv frqwlqxd hq d @ 41 81 = Ho plvpr hmhpsor1

Hmhpsor 655 Frqvlghuh od ixqflöq i +{, @

; ? =

{5 .5{

 vhq 5{ 4 oq+6{5 . h,4

@5 ? { ? 3 {@3 {A3

ÁHv i frqwlqxd hq { @ 3B Hq fdvr gh qr vhuor uhghqlu i vl hv srvleoh/ gh prgr txh i vhd frqwlqxd1 Vroxflöq= { @ 3 hv xq sxqwr gh dfxpxodflöq gho Grpi @`  @5> 4^ oxhjr i vhuä frqwlqxd hq { @ 3 vl | vöor vl 4, 3 5 Grpi 5, olp{$3 i +{, h{lvwh1 6, olp{$3 i +{, @ i +3,1 Dvð= 4, Hv fodur txh 3 5 Grpi sxhv i +3, @ 4 5, Fdofxohprv olp{$3 i+{,

595

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

{5 . 5{ olp i+{, @ olp {$3 {$3  vhq +5{, {+{ . 5, @ olp {$3 5 frv { vhq +{, 4 {+{ . 5,, @ olp 5 {$3 vhq+{, frv +{, 4 { {.5 @ olp olp 5 {$3 vhq +{, {$3 frv { 5 @ 4  @ 4 5 olp i+{, @ olp. oq +6{5 . h,4 {$3. {$3

Sxhvwr txh od ixqflöq orjdulwpr hv frqwlqxd hq U. |

olp 6{5 . h @ h 5 U. {$3

olp i+{, @ olp. oq +6{5 . h,4 {$3. {$3 @  olp. oq +6{5 . h, {$3 @  oq olp. +6{5 . h, {$3 @  oq+h, @ 4

oxhjr h{lvwh olp{$3. i +{, @ 4 6, olp{$3 i +{, 9@ i+3, oxhjr i qr hv frqwlqxd hq { @ 31 Vlq hpedujr/ gdgr txh olp{$3 i +{, h{lvwh hqwrqfhv hv srvleoh uhghqlu i gh prgr txh i vhd frqwlqxd hq { @ 3/ gh od vljxlhqwh irupd= ; {5 .5{ ?  vhq 5{ @5 ? { ? 3 i+{, @ 4 {@3 = {A3 oq+6{5 . h,4 Hmhpsor 656 Vhd i =`3> ^$ U od ixqflöq ghqlgd sru ; {  ? frw {  5 frv { { ? @5 i +{, @ = d{  4 {  @5

BcTxì ydoru +vl h{lvwh, gheh dvljqduvh d od frqvwdqwh d sdud txh i vhd frqwlqxd hq 5 B Vroxflöq= i vhuä frqwlqxd hq @5 vl l,

 olp i +{, @ olp i+{, @ i + , 5 {$ 5  {$ 5 .

 {  i+{, @ olp olp     frw { 5 frv { {$ 5 {$ 5 { vhq {  @ olp  5 frv { {$ 5  frv { 5{ vhq {   @ olp   5 frv { {$ 5

+7144,

7161 FRQWLQXLGDG

596

Vhd x @ 5  { vl { $ 5  hqwrqfhv x $ 3. l, olp olp i +{, @ {$ 5  {$ 5 

5{ vhq {   5 frv {

5+@5  x, vhq +@5  x,   5 frv +@5  x, +  5x, frv x   @ olp. 5 vhq x x$3 +frv x  4,  5x frv x @ olp. 5 vhq x x$3 frv x  4 x frv { @  olp  olp. . x$3 5 vhq x {$3 vhq x @

sru rwud sduwh>

olp x$3.

frv x  4 olp x$3. 5 vhq x

dghpäv

frv5 x  4 olp. x$3 5 vhq x+frv x . 4,  vhq5 x @ olp. x$3 5 vhq x+frv x . 4,  vhq x @ olp. x$3 5+frv x . 4, @ 3 @

x frv x x olp @ olp. olp frv x @ 4  4 @ 4 x$3. vhq x x$3 vhq x x$3. Oxhjr sru dojheud gh olplwh olp {$ 5 

5{ vhq {   frv x  4 x frv { @  olp.  olp. @ 4 5 frv { x$3 5 vhq x {$3 vhq x

ll, olp{$ 5 . i +{, @ olp{$ 5 . d{  4 @ 5 d  4 Sru +l, | +ll, | sru 7144 vh wlhqh 4 @

sru or wdqwr whqhprv d @ 31

Hmhpsor 657 Vhd

d, BcHv i frqwlqxd hq { @ 3B e, BcHv i frqwlqxd hq UB i +3, f, H{lvwh G. @ olp{$3. i +{{, 3 i +3, g, H{lvwh G @ olp{$3 i +{{, 3

  d  4 @ i+ , 5 5

; 6 ? { frv + {45 , { ? 3 3 {@3 i +{, @ = s 5 { . 5{6 { A 3

Vroxflöq= d, Fdofxohprv orv oðplwh odwhudohv

4 olp i +{, @ olp {6 frv + 5 , @ 3 { {$3 {$3

sxhv olp{$3 {6 @ 3 | frv + {45 , hv dfrwdgd1

597

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

olp i+{, @ olp {$3. {$3.

s {5 . 5{6 @ 3=

Oxhjr olp{$3. i +{, @ olp{$3 i +{, @ i +3, @ 3 Sru or wdqwr i hv frqwlqxd hq { @ 3 e, Vhd i4 +{, @ {6 frv + {45 , sdud { ? 3 frpr i4 hv surgxfwr gh ixqflrqhv frqwlqxdv hq `  4> 3^ hqwrqfhv +sru dojheud gh ixqflrqhv frqwlqxdv, i4 hv frqwlqxd1 Sru vxpd | frpsrvlflöq gh ixqflrqhv/ vh wlhqh txh i5 hv frqwlqxd hq `3> 4^1 Dghpäv sru +d, i hv frqwlqxd hq { @ 3/ oxhjr i hv frqwlqxd hq U1 f, G.

@ @ @ @ @

@

Oxhjr h{lvwh G. 1 g, G

i +{,  i +3, {3 s {5 . 5{6  3 olp. > sxhv sdud { A 3 {3 {$3 s {5 +4 . 5{, olp. { {$3 s m{m 4 . 5{ olp > sxhv sdud { A 3 m{m @ { { {$3. s +4 . 5{, olp {$3. 4 olp {$3.

i +{,  i +3, {3 6 4 { frv+4@{5 ,  3 @ olp sxhv i +{, @ {6 frv+ 5 , sdud { ? 3 {3 { {$3 4 @ olp {5 frv+ 5 , { {$3 @ 3 @

olp {$3

sxhv j+{, @ frv+4@{5 , hv dfrwdgd | olp{$3 {5 @ 31 Oxhjr h{lvwh G 1

Hmhuflflr 658 Ghwhuplqh d> e hq U gh prgr txh i vhd frqwlqxd hq { @ 3 ; vhq d{vhq e{ {?3 ? { d{ . 6 { @3 i+{, @ = es7{5 .{6 { A 3 { Vroxflöq=

Hmhuflflr 659

41 3 5 Grpi> i 3, @ 61

51 olp{$3 i +{, h{lvwh vl | vöor vl olp{$3. i+{, @ olp{$3 i +{,1 s e 7{5 . {6 olp i +{, @ olp. { {$3. {$3 s em{m 7 . { @ olp. {A3 { {$3 s @ olp e 7 . { @ 5e {$3.

7161 FRQWLQXLGDG

598

vhq +d{,  vhq +e{, { vhq +d{, vhq +e{,  olp e @ olp. d d{ e{ {$3 {$3 vhq +d{, vhq +e{,  e olp @ d olp. d{ e{ {$3 {$3 @ de

olp i +{, @ olp {$3 {$3

sru or wdqwr d  e @ 5e/ dghpäv i +3, @ 6/ oxhjr i vhud frqwlqxd hq { @ 3 vl d  e @ 6 a 5e @ 6 +, d @ <@5 a e @ 6@5

Hmhuflflr 65: Ghwhuplqdu e hq U gh prgr txh i vhd frqwlqxd hq { @ 41  {5 5.e5 {A4 { i +{, @ +e  4,{ { ? 4

Vroxflöq= i vhuä frqwlqxd hq { @ 4 vl olp{$4 i +{, @ i+4, olp i+{, @ olp {$4. {$4.

{5  5 . e5 s {4

Hvwh oðplwh srguðd h{lvwlu vl olp{$4. {5  5 . e5 @ 3 + hq fdvr frqwudulr ho oðplwh vhuðd gh od irupd 3f | qr h{lvwluðd,1 Dkrud olp {5  5 . e5 @ 3 +, 4  5 . e5 @ 3 +, e @ 4 b e @ 4 {$4 l, Vl e @ 4 vh wlhqh txh

olp {$4.

sru rwud sduwh/

{5  5 . e5 s {4

{5  4 olp. s {$4 {4 s +{ . 4,+{  4, {  4 @ olp {4 {$4. s @ olp. +{ . 4, {  4 @ 3 {$4 @

olp i +{, @ olp +e  4,{ @ e  4 @ 4  4 {$4 {$4

Oxhjr olp{$4. i+{, @ olp{$4 i +{, | sru or wdqwr i hv frqwlqxd hq { @ 4 vl e @ 4/ sxhv olp{$4 i+{, @ i+4, @ 3 ll, Vl e @ 4/ vh wlhqh txh {5  4 . e5 {5  4 @ olp s @3 olp. s {$4 {$4. {  4 {4 shur

oxhjr

olp i +{, @ olp +e  4,{ @ e  4 @ 5 {$4 {$4 olp i+{, 9@ olp i +{, {$4. {$4

gh grqgh olp{$4 i +{, qr h{lvwh | sru or wdqwr i qr hv frqwlqxd hq { @ 4 vl e @ 41

599

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

717

Uhfwd wdqjhqwh d od Fxuyd

Od qrflöq gh oðplwh hv ixqgdphqwdo sdud ho hvwxglr gh pxfkrv frqfhswrv gh odv pdwhpäwlfdv/ gh od iðvlfd | rwudv flhqfldv1 Sdud loxvwudu hvwr hvwxglduhprv ho sureohpd gh hqfrqwudu od uhfwd wdqjhqwh d xqd fxuyd hq xq sxqwr S gdgr1

Hq od jhrphwuðd sodqd/ od uhfwd wdqjhqwh hq xq sxqwr S vreuh xqd flufxqihuhqfld vh sxhgh ghqlu frpr od uhfwd txh wlhqh vrodphqwh ho sxqwr S hq frpýq frq wdo flufxqihuhqffld/ frpr vh loxvwud hq od jxud

1 Hvwd ghqlflöq qr vh sxhgh dsolfdu d fxdotxlhu juäfd/ |d txh xqd uhfwd wdqjhqwh hq xq sxqwr S sxhgh fruwdu d xqd juäfd yduldv yhfhv frpr vh pxhvwuh hq od jxud/ juäfr l}txlhugr,1 ö hq ho fdvr gh od juäfd gh ixqflöq ydoru devroxwr i +{, @ m{m1 ÁFxäo hv od wdqjhqwh hq S @ +3> 3,B

Sdud lghqwlfdu od uhfwd wdqjhqwh O d od juäfd gh xqd ixqflöq hq xq sxqwr S / edvwd hvshflfdu od shqglhqwh p gh O/ |d txh ìvwd | ho sxqwr S ghwhuplqd frpsohwdphqwh d od uhfwd1 Sdud hqfrqwudu p/ frqvlghuduhprv odv uhfwdv OST / uhfwdv vhfdqwhv txh sdvd sru S 1

Vhd pST od shqglhqwh gh OS T 1 Fxdqgr T wlhqgh d S  +S mr, sru od ghuhfkd jxud

txh vljxh

7171 UHFWD WDQJHQWH D OD FXUYD

59:

| sru od l}txlhugd jxud

vh revhuyd txh pST wlhqgh d p +shqglhqwh gh od uhfwd O,1 Hvwr qrv vxjlhuh txh vl pST wlhqgh d xq ydoru mr1 fxdqgr T wlhqgh d S / hqwrqfhv hvh ydoru ghehuðdprv xvdu sdud ghqlu od shqglhqwh p gh od uhfwd wdqjhqwh O hq S 1 Hv fodur txh od shqglhqwh gh od uhfwd vhfdqwh OST hv pST @

i +{,  i +d, {d

|d txh sdvd sru orv sxqwrv +d> i+d,, | +{> i+{,,1 Frq hvwd qrwdflöq srghprv vxvwlwxlu od iudvh T wlhqgh d S % sru { wlhqgh d d%/ dvð oohjdprv d od vljxlhqwh ghqlflöq

Ghqlflöq 7158 Vhd i xqd ixqflöq ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d1 Od shqglhqwh p gh

59;

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> i +d,, hv p @ olp {$d

vlhpsuh | fxdqgr ho oðplwh h{lvwd1+Yhu jxud ,

i +{,  i+d, {d

| hq hvh fdvr vh gluä txh h{lvwh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> i+d,,1

Hmhpsor 65; Vhd i +{, @ {6 | d xq qýphur uhdo1 Hqfrqwuduhprv= d, Od shqglhqwh pd gh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> d6 ,1 e, Od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd hq ho sxqwr +4> 4,1 f, Od hfxdflöq gh od uhfwd qrupdo +uhfwd shushqglfxodu d od uhfwd wdqjhqwh, d od juäfd hq ho sxqwr +4> 4,1 Vroxflöq= d,

pd

Dvð/ od shqglhqwh hv

i +{,  i +d, {d {6  d6 @ olp {$d {  d +{  d,+{5 . d{ . d5 , @ olp {$d {d @ olp +{5 . d{ . d5 , @ 6d5 {$d

@

olp {$d

pd @ 6d5

e, \d txh od shqglhqwh gh od uhfwd wdqjhqwh O hq ho sxqwr +4> 4, od srghprv rewhqhu frpr ho fdvr hvshfldo hq txh d @ 4/ whqhprv txh p @ 61 Hq jhqhudo od hfxdflöq gh od uhfwd txh sdvd sru +d> i+d,, | frq shqglhqwh p hv Hq qxhvwur fdvr r

|  i +d, @ p+{  d, |  4 @ 6+{  4, | @ 6{  5

7171 UHFWD WDQJHQWH D OD FXUYD

59<

f, Vl p> p 9@ 3 hv od shqglhqwh gh od uhfwd wdqjhqwh | pQ od shqglhqwh gh od uhfwd qrupdo d od juäfd gh | @ i+{, hqwrqfhv vh gheh fxpsolu p  pQ @ 4 Dvð/ od hfxdflöq gh od uhfwd qrupdo d od juäfd gh i+{, @ {6 hq +4> 3, hv/ |  i +d, @ pQ +{  d, |4 @  46 +{  4, | @  46 { . 76

+, +,

Revhuyh txh vl p @ 3 sdud dojýq sxqwr hqwrqfhv od uhfwd wdqjhqwh hv krul}rqwdo/ fx|d hfxdflöq hv | @ i +d,/ | hq hvwh fdvr od hfxdflöq gh od uhfwd qrupdo hv { @ d1 Hmhpsor 65< d, Ghprvwuduhprv txh

olp {$d

vhq+{,  vhq+d, @ frv+d, {d

| e, Ghwhuplqhprv od hfxdflöq gh od uhfwd wdqjhqwh d | @ vhq+{, hq S @ + 5 > 4,1

Vroxflöq= Vhd x @ {  d/ vl { $ d hqwrqfhv x $ 3/ oxhjr/ xvdqgr vxvwlwxflöq/ O @ @ @ @ @

vhq+{,  vhq+d, {d vhq+x . d,  vhq+d, olp x$3 x vhq+x, frv+d, . vhq+d, frv+x,  vhq+d, olp x$3 x frv+x,  4 vhq+x, olp +frv+d, . vhq+d, , x$3 x x vhq+x, frv+x,  4 olp frv+d, . olp vhq+d, x$3 x$3 x x olp {$d

Fdofxodqgr sru vhsdudgr orv oðplwh whqhprv |

olp x$3 olp vhq+d, x$3

vhq+x, frv+d, vhq+x, @ frv+d, olp @ frv+d, x $ 3 x x

frv+x,  4 x

@

@ @ Oxhjr sru dojheud gh oðplwhv

frv+x,  4 x +frv+x,  4,+frv+x, . 4, vhq+d, olp x$3 x+frv+x, . 4, frv5 +x,  4 vhq+d, olp x$3 x+frv+x, . 4, vhq+x, vhq+x,  vhq+d, olp olp x$3 x x$3 frv+x, . 4 vhq+d,  4  3 @ 3=

@ vhq+d, olp x$3

@

vhq+x, frv+x,  4 . olp vhq+d, x$3 x x vhq+x, frv+x, . 4 @ frv+d, olp . vhq+d, olp x$3 x x$3 x @ frv+d,

O @

olp frv+d, x$3

5:3

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

e, Frpr od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd gh | @ vhq+{, hq fxdotxlhu sxqwr +d> vhq+d,, hvwä gdgd sru vhq+{,  vhq+d, p @ olp {$d {d | sru +d, whqhprv txh p @ frv+d,/ uhhpsod}dqgr d @ 5 rewhqhprv txh p @ frv+ 5 , @ 3 | dvð od uhfwd wdqjhqwh hv> |  4 @ p+{  5 , +, |4@3 +, |@4

Revhuyh txh od hfxdflöq gh od qrupdo hq S @ + 5 > 4, hv { @ 5 1

Hmhuflflr 663 Ghwhuplqh od shqglhqwh gh od uhfwd wdqjhqwh d od sduäerod | @ d{5 . e{ . f hq ho sxqwr +{3 > |3 , Vroxflöq=

p @ @ @ @ @ @ @

@

i+{,  i+{3 , {  {3 d{5 . e{ . f  +d{53 . e{3 . f, olp {${3 {  {3 d{5  d{53 . e{  e{3 olp {${3 {  {3 5 d+{  {53 , . e+{  {3 , olp {${3 {  {3 d+{  {3 ,+{ . {3 , . e+{  {3 , olp {${3 {  {3 +{  {3 ,+d+{ . {3 , . e, olp {${3 {  {3 olp d+{ . {3 , . e {${3 5d{3 . e olp {${3

Hmhuflflr 664 Xq htxlsr gh lqjhqlhurv gh fdplqrv glvhôd xq wudpr gh fduuhwhud txh gheh frqhfwdu xqd dxwrslvwd krul}rqwdo frq rwud gh shqglhqwh p @ 4@8 1 Ho hqodfh gheh/ uhdol}duvh vreuh xqd glvwdqfld krul}rqwdo gh ;33 slhv xvdqgr xqd fxuyd sdudeöolfd sdud xqlu orv sxqwrv D | E1 Rewhqjd xqd hfxdflöq gho wlsr | @ d{5 . e{ . f sdud od sduäerod uhvshfwlyd | ghwhuplqh odv frrughqdgdv gh E1

Vroxflöq= Odv frrughqdgdv gh D vrq +3> 3, | orv gh E vrq +;33> |3 , grqgh |3 @ d+;33,5 .e+;33,.f/ dghpäv frpr +3> 3, wdpelìq shuwhqhfh d od sduäerod/ gheh vdwlvidfhu od hfxdflöq gh hood/ dvð3 @ d  3 . e  3 . f/ gh grqgh f @ 31 sru hmhuflflr dqwhulru/ rewxylprv txh od shqglhqwh gh | @ d{5 . e{ . f hq +{3 > |3 , hv p @ 5d{3 . e/ oxhjr hq {3 @ ;33 hv p @ 5d  ;33 . e | hq {3 @ 3 hv p @ e1 Dghpäv sdud frqhfwdu od dxwrslvwd gh shqglhqwh 4@8 frq od sduäerod/ gheh rfxuulu txh 5d  ;33 . e @ 4@8

+7145,

e@3

+7146,

| sdud frqhfwdu od dxwrslvwd krul}rqwdo +gh shqglhqwh p @ 3, frq od sduäerod> Dvð/ gh 7145 | 7146 whqhprv

4 @d ;333

7171 UHFWD WDQJHQWH D OD FXUYD

5:4

4 > e @ 3> f @ 31 Sru rwud sduwh E @ +;333> |3 , shuwhqhfh d od sduäerod/ oxhjr Hq frqvhfxhqfld d @ ;333 vdwlvidfh vx hfxdflöq/ hv ghflu 4 +;33,5 @ ;3 |3 @ ;333

Sru or wdqwr odv frrughqdgdv gh E vrq +;33> ;3,1 Hmhpsor 665 Gdgd od ixqflöq

i +{, @

s {

; A A A A ?

{5 {

A A A A = s 6 ;{6 . {8 vhq+ ,

{

3?{? {3

d, BcHv i xqd ixqflöq frqwlqxd hq 3B e, BcH{lvwh od uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr S @ +3> 3,B f, BcH{lvwh od uhfwd wdqjhqwh d od judfd gh | @ i+{, hq orv sxqwrv +7/5, | +04/05,B Vroxflöq= d, Gdgr txh i hvwä ghqlgd hq xqd yhflqgdg gh { @ 3/ | 3 5 Grpi vh gheh fxpsolu txh= olp i+{, @ olp i+{, @ i +3, {$3. {$3

Frpr

{5 olp i +{, @ olp {$3. {$3. vhq+{, { @ olp. {  vhq+{, {$3

{ olp. {  olp. {$3 {$3 vhq+{, @ 3  4 @ 3= @

| ho rwur oðplwh odwhudo hv

olp i +{, @ olp {$3 {$3

s 6 6 ;{ . {8 =

| dperv vrq ljxdohv d i+3, vh wlhqh txh

olp i+{, @ olp i+{, @ i +3, {$3. {$3

| sru or wdqwr i hv frqwlqxd hq { @ 3 e, Gdgr txh i hvwä ghqlgd hq xqd yhflqgdg gh { @ 31 Yhdprv vl h{lvwh i+{,  i+3, p @ olp {$3 {3

Sdud srghu fdofxodu hvwh oðplwh/ or kduhprv fdofxodqgr orv oðplwhv odwhudohv olp {$3.

i +{,  i +3, {5 { @ olp. @ olp @ 4> {3 {$3 { vhq+{, {$3. vhq+{,

5:5

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

sru rwud sduwh i +{,  i +3, olp {3 {$3

@ @ @ @ @

s 6 ;{6 . {8 olp { {$3 s 6 {6 +; . {5 , olp { {$3 s { 6 ; . {5 olp { {$3 s 6 ; . {5 olp {$3 s 6 ;

Gdgr txh orv oðplwhv odwhudohv vrq glvwlqwrv whqhprv txh qr h{lvwh p @ olp {$3

i+{,  i+3, > {3

oxhjr qr h{lvwh uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr S s @ +3> 3,1 f, i hvwä ghqlgd hq xqd yhflqgdg gh { @ 7 sxhv i +{, @ { sdud   { |  ? 7> dghpäv +7> 5, s shuwhqhfh d od fxuyd | @ i +{,/ sxhv 7 @ 51 Frpr od shqglhqwh exvfdgd hv s 4 i+{,  i+7, {5 {7 s @ olp @ olp @ olp {$7 {$7 {  7 {$7 +{  7,+ { . 7, 7 {7

h{lvwh od uhfwd wdqjhqwh d od fxuyd hq S @ +7> 5,/ | vhd hfxdflöq hv= 4 |  5 @ +{  7, 7

Hq ho fdvr gho sxqwr +4> 5, whqhprv txh ìo qr shuwhqhfh d od fxuyd | @ i +{, sxhv s s i+{, @ 6 ;  4 @ 6 < 9@ 5

oxhjr qr h{lvwh uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr +4> 5,1

Hmhuflflr 666 Xvdqgr dojheud gh oðplwh | lghqwlgdghv wuljrqrphwulfdv fdofxodu ho oðplwh gh odv vljxlhqwhv ixqflrqhv hq d 41 i+{, @ frw {

51 i+{, @ vhf {

61 i+{, @ fvf {

d 9@ q > q 5 Q

d 9@ 5 . q > q 5 Q d 9@ q > q 5 Q

71 i+{, @ vhq +{ . 6 ,

81 i+{, @ wdq +5{ . ,

91 i+{, @

5 {

4 frv

{

d5U

d@3

d dsursldgr

Hmhuflflr 667 Vhd [  U> s 5 [ 3 > i> j = [ $ U ixqflrqhv wdohv txh olp{$s i +{, @ O | olp{$s j+{, @ P frq O ? P pxhvwuh txh h{lvwh xqd yhflqgdg gh s> Ys wdo txh i +{, ? j+{,

>

;{ 5 +Ys  isj, _ [

D|xgd= Frpr O ? P vh sxhgh wrpdu % @ P 5O | hqfrqwudu sdud hvh % od yhflqgdg Ys

7171 UHFWD WDQJHQWH D OD FXUYD

5:6

Hmhuflflr 668 Vhd olp{$s i+{, @ O a O A 31 Pxhvwuh txh ;{ 5 +Ys  isj, _ [ D|xgd= Xvdu uhvxowdgr dqwhulru

Hmhuflflr 669 Vhd i +{,  j+{,> ;{ 5 [> { 9@ s a olp{$s i +{, @ O a olp{$s j+{, @ P 1 Pxhvwuh txh OP D|xgd= Xvdu whruhpd gho dsuhwöq d +j  i ,+{,

Hmhuflflr 66: Vhd D  U> i = D $ U> s 5 D | vxsrqjdprv txh h{lvwh olp{$s i+{, hqwrqfhv ;{ 5 +Ys  isj, _ D +hv ghflu/ i hv dfrwdgd hq xqd yhflqgdg gh s, Hmhuflflr 66; Vhd

i +{, @



{ {5T 3 {5 9 T

Prvwudu txh olp{$3 i+{, @ 3 | olp{${3 i+{, qr h{lvwh vl {3 9@ 3 D|xgd= Sdud od vhjxqgd sduwh xvdu hmhuflflr 668 | sdud od sulphud sduwh xvdu mi+{,m  m{m

5:7

FKDSWHU 71 OÐPLWHV GH IXQFLRQHV

Fkdswhu 8

Ghulydgd 814

Ghulydgd

Ghqlflöq 814 Vhd L  U> i = L $ U | vhd s xq sxqwr gh dfxpxodflöq Gluhprv txh i hv ghulydeoh hq ho sxqwr s fxdqgr ho vljxlhqwh oðplwh h{lvwh olp {$s

i +{,  i +s, {s

Hq ho fdvr txh ho oðplwh h{lvwd/ ho ydoru gh ho vh ghqrwd sru i 3 +s,/ lqglfdqgr gh hvwd irupd txh ho ydoru ghshqgh gho sxqwr s1 Ho ydoru i 3 +s, hv oodpdgr od ghulydgd gh i hq ho sxqwr s | vh xvdq odv vljxlhqwh qrwdflrqhv1 gi gi gi i 3 +s,> +s,> m{@s > ms g{ g{ g{ Revhuydflöq= 4, i 3 +s, sxhgh vhu fdofxodgr gh rwud irupd |d txh= d, Wrpdqgr k @ {  s vh wlhqh i 3 +s, @ olp k$3

i+k . s,  i +s, k

txh hv rwud gh odv irupdv txh dsduhfh od ghqlflöq gh od ghulydgd hq dojxqrv oleurv1 e, Xvdqgr xqd fdudfwhul}dflöq gh oðplwh yld vxfhvlrqhv vh wlhqh/ txh i hv ghulydeoh hq s vl vöor vl sdud fxdotxlhu vxfhvlöq +{q ,> +{q , 5 L  isj frq olpq$4 {q @ s vh wlhqh txh h{lvwh ho vljxlhqwh oðplwh i 3 +s, @ olp q$4

i +{q ,  i+s, {q  s

f, Jhqhudol}dqgr or dqwhulru/ whqhprv txh wdpelìq whqhprv i 3 +s, @ olp |$e

i +j+|,,  i+s, j+|,  s

grqgh j = [ $ L/ olp|$e j+|, @ s +reyldphqwh e 5 [ 3 , | j+|, 9@ s ;|> | 9@ e 5, i hv ghulydeoh hq s vl sdud od juäfd gh i h{lvwh od uhfwd wdqjhqwh> lqyhuvdphqwh vh sxhgh ghflu= Vl od ixqflöq hv ghulydeoh hq s vh wlhqh txh od uhfwd wdqjhqwh d od juäfd gh i hq +s> i +s,, h{lvwh1 Vl Xg1 frpsdud odv ghqlflrqhv gh uhfwd wdqjhqwh | ghulydgd/ vh sxhgh shufdwdu txh hv or plvpr1 6, Sru surslhgdg gh oðplwh vh wlhqh txh vl h{lvwh i 3 +s,/ ìo hv ýqlfr/ sru hvwr vh sxhgh xvdu ho duwðfxor od hq od ghulydgd/ rod uhfwd wdqjhqwh1 Gh od revhuydflöq 5 vh frqfox|h txh |d whqhprv ydulrv hmhpsorv gh ixqflrqhv ghulydeohv/ hoodv vrq odv gh orv hmhpsorv | hmhuflflrv gh uhfwdv wdqjhqwhv/ dghpäv srghprv qrwdu iäflophqwh txh od shqglhqwh p gh od uhfwd wdqjhqwh d i+{, hq ho sxqwr s4 vdwlvidfh 4 Fruuhfwdphqwh

glfkr hq ho sxqwr

Es> i Es1

5:8

5:9

FKDSWHU 81 GHULYDGD

p @ i 3 +s,

Sru or wdqwr> vl i hv ghulydeoh hq s vh wlhqh txh od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr s hv |  i +s, @ i 3 +s, {s | @ +i 3 +s,,{ . i+s,  i 3 +s,s

Xvdqgr wuljrqrphwuðd vh wlhqh txh ho äqjxor  txh irupd od uhfwd wdqjhqwh d od juäfd gh i hq s +hv ghflu/ hq +s> i+s,,,frq ho hmh gh odv { hvwd gdgr sru=

81414

 @ dufwdq+i 3 +s,,

Hmhpsorv gh ixqflrqhv ghulydeohv

Hmhpsor 66< Yhuhprv txh od ixqflöq i+{, @ 6{ . 8 hv ghulydeoh hq ho sxqwr s @ :1

Vroxflöq=

i +: . k,  i+:, k 6+: . k, . 8  +6  : . 8, @ olp k$3 k 6k @ 6= @ olp k$3 k Ho uhvxowdgr dqwhulru qr gheh vhu h{wudôr |d txh qdwxudophqwh vh wlhqh txh od uhfwd wdqjhqwh d od juäfd gh i +od uhfwd | @ 6{ . 8, hv hood plvpd/ sru or wdqwr odv shqglhqwhv ghehq frlqflglu1 BcFxäo hv od ghulydgd gh i hq rwurv sxqwrvB Sru ho dujxphqwr juäfr txh khprv gdgr hq ho säuudir dqwhulru ghehuðdprv whqhu txh i 3 +s, @ 6 ;s 5 U1 Frpsuxhehor Xg1 xvdqgr od ghqlflöq gh ghulydgd1 i 3 +:, @

olp k$3

Hmhpsor 673 Od ixqflöq j+{, @ 7 +ixqflöq frqvwdqwh, wlhqh frpr juäfd od uhfwd | @ 7 txh hv xqd uhfwd krul}rqwdo +sdudohod do hmh {, vlwxdgd do qlyho 7> reyldphqwh vx shqglhqwh hv fhur/ sru or wdqwr od ghulydgd hq wrgr sxqwr s gheh vhu qxod1 Frpsurehprv hvwr yðd od ghqlflöq1 Vhd s xq qýphur uhdo mr

j+{,  j+s, {s 77 @ olp {$s {  s @ olp 3 @ 3 {$s s Hmhpsor 674 Fdofxohprv dkrud od ghulydgd gh k+x, @ x hq ho sxqwr s @ 71 j3 +s, @

Vl hv txh h{lvwh k3 +7, hvwd gheh vhu ljxdo d k3 +s, @ @ @ @ @

olp {$s

k+x,  k+7, x7 s s x 7 olp x$7 x  7 s s s s x 7 x. 7 s olp s x$7 x  7 x. 7 x7 4 s s olp x$7 x  7 x. 7 4 4 s @ 7 5 7 olp x$7

8141 GHULYDGD

5::

Vl uhshwlprv orv sdvrv dqwhulruhv xq sxqwr mr s +shur duelwudulr, gh U. rewhqguhprv txh 4 4 k3 +s, @ olp s s @ s x$s x . s 5 s

Xwlol}dqgr od juäfd gh k> yhu jxud txh vljxh/ qrwhprv txh od shqglhqwh gh O5 hv pd|ru txh od shqglhqwh gh O4 | d phglgd txh wrphprv s päv fhufd gho ruljhq rewhqguhprv shqglhqwhv/ fdgd yh} pd|ruhv/ khfkr txh srghprv frpsuredu wdpelìq hq od ghulydgd gh k txh k3 +s, hv xqd ixqflöq ghfuhflhqwh hq s +s 5 U. ,/ k3 +s, hv od ixqflöq txh do vhu hydoxdgd hq s qrv gä od shqglhqwh gh odv uhfwdv wdqjhqwhv d od juäfd gh od udð} fxdgudgd1

Ho hmhpsor dqwhulru qrv lqgxfh d ghqlu od ixqflöq ghulydgd% sru dexvr gh ohqjxdmh vh dfrvwxpeud oodpdu vlpsohphqwh od ghulydgd1

Ghqlflöq 815 gh i frpr

5

Vhd i = L $ U xqd ixqflöq ghulydeoh hq wrgr sxqwr gh M1 Vh ghqh od ixqflöq ghulydgd

Gh od ghqlflöq whqhprv txh M  L1

i3 = M s

$ U :$ i 3 +s,

Hq orv hmhpsorv dqwhulruhv whqhprv fdofxodgdv odv ixqflrqhv ghulydgdv i +{, @ 6{ . :> L @ U> M @ U i 3 +{, @ 6 j+{, @ 7> L @ U> M @ U j3 +{, @ 3= s 4 . k3 +{, @ s k+{, @ {> L @ U. 3 > M @ U 5 {

Qrwh txh hq ho ýowlpr fdvr k+3, h{lvwh/ shur k3 +3, qr h{lvwh/ hv ghflu/ M  L1

Hmhpsor 675 Vhd i +{, @ vhq+{, fdofxohprv od ixqflöq ghulydgd gh i 3 +{, Hq orv hmhpsorv gh fäofxor gh uhfwd wdqjhqwh sdud s 5 U rewxylprv i 3 +s, @ olp k$3

5 Xq

vhq+k . s,  vhq+s, @ frv+s, k

s r fr pdv dghodqwh hq yh} gh ghflu od ixqflöq ghulydgd gh

i/

gluhprv vhqfloodphqwh od ghulydgd gh

i1

5:;

FKDSWHU 81 GHULYDGD

Dvð/ od ixqflöq ghulydgd gh i+{, @ vhq+{, hv i 3 +{, @ frv+{,1 Ho vljxlhqwh hmhpsor pxhvwud xqd ixqflöq txh qr hv ghulydeoh hq wrgr sxqwr/ d shvdu gh vhu frqwlqxd hq wrgr U1 Hmhpsor 676 Vhd i +{, @ m{  7m1 Fdofxohprv vx ghulydgd hq wrgr sxqwr txh h{lvwd1

Fäofxor gh od ghulydgd hq {3 1 Vl {  7 ? 3 vh wlhqh txh i +{, @ { . 7 | hq fdvr frqwudulr i +{, @ {  7/ gh hvwd irupd= l, Vl {3 ? 7/ i 3 +{3 , @

olp {${3

@

olp {${3

@

olp {${3

i +{,  i +{3 , {  {3 {  7  +{3 . 7, {  {3 { . {3 @ 4 {  {3

ll, Vl {3 A 7/ i 3 +{3 , @ 4 +fäofxor dqäorjr d orv uhflhq khfkrv,1 lll, Vl {3 @ 7 i +{,  i +{3 , i 3 +7, @ olp {${3 {  {3 Shur frpr i wlhqh glvwlqwr frpsruwdplhqwr fhufd gho 7 sdud fdofxodu hvwh ýowlpr oðplwh ghehprv wrpdu oðplwhv odwhudohv | yhulfdu txh hoorv vrq ljxdohv i+{,  i+7, {73 olp @ olp. @4 {7 {7 {$7. {$7 i +{,  i +7, { . 7  3 @ olp @ 4 olp {7 {7 {$7 {$7

Frpr orv oðplwhv odwhudohv txh ghqhq i 3 +7, vrq glvwlqwrv qr whqhprv od h{lvwhqfld gh i 3 +7,/ dvð od ixqflöq ghulydgd gh i hv i 3 = U  i7j $ U :$

{

i 3 +{, @



4 {A7 4 { ? 7

Qrwdu txh i hv frqwlqxd hq { @ 71 Revhuydflöq= Gh or dqwhulru/ srghprv dupdu txh vl i hv frqwlqxd hq xq sxqwr s/ qr qhfhvduldphqwh i hv ghulydeoh hq d1 Vlq hpedujr ho vljxlhqwh whruhpd qrv dupd txh vl i hv ghulydeoh hq xq sxqwr s/ hqwrqfhv i hv frqwlqxd hq s1 Whruhpd 816 Vl i hv ghulydeoh hq d/ hqwrqfhv i hv frqwlqxd hq d1 Ghprvwudflöq= Ghehprv ghprvwudu txh

olp i +{, @ i +d, {$d

r htxlydohqwhphqwh qrv edvwduðd prvwudu txh= Frpr

olp +i+{,  i+d,, @ 3 {$d

+{  d, i+{,  i+d, olp i+{,  i+d, @ olp +i+{,  i+d,, @ olp olp +{  d, @ i 3 +d,  3 @ 3 {$d {$d {$d +{  d, {$d {d

whqhprv txh hihfwlydphqwh i hv frqwlqxd hq d1

8141 GHULYDGD

5:<

Revhuyh txh ho whruhpd dqwhulru hv gh od irupd s , t/ od ixqflöq i +{, @ m{  7m txh ylprv hq ho hmhpsor 676 txh qr hv ghulydeoh hq { @ 7/ vlq hpedujr hood hv frqwlqxd hq { @ 7/ qrv vluyh gh frqwudhmhpsor sdud od dupdflöq txh= frqwlqxlgdg hq xq sxqwr lpsolfd ghulydelolgdg hq ho sxqwr1 Qrwdflrqhv= Od ghulydgd gh i hq d hv ho oðplwh i +d . k,  i+d, k i + d . k ,  i +d, hv oodpdgd ghulydgd odwhudo l}txlhugd hq d ho oðplwh odwhudo l}txlhugr gh i 3 +d,/ hv ghflu/ olpk$3 k 3 +d,1 Wdpelìq vh ghqh od ghulydgd odwhudo ghuhfkd hq d txh fruuhvsrqgh do oðplwh odwhudo | vh ghqrwd sru i 3 +d,1 ghuhfkr | vh ghqrwd sru i. i 3 +d, @ olp k$3

Hmhuflflr 677 Fdofxodu od ghulydgd gh i+{,/ grqgh ; +{  5,6 {A5 A A ? 5 {  7{ . 7 4 ? {  5 i+{, @ < { @ 4 A A = vhq+{ . 4, { ? 4

Fodudphqwh orv sxqwrv frq lfwlyrv vrq { @ 5 | { @ 4 |d txh hq yhflqgdghv gh hoodv/ i wlhqh glvwlqwr frpsruwdplhqwr | hvwh khfkr wudmr glfxowdghv hq ho hmhpsor dqwhulru1 d, Yhdprv hq sulphu oxjdu vl i hv ghulydeoh hq s @ 5/ sdud hoor qhfhvlwdprv fdofxodu ho vljxlhqwh oðplwh +vl hv txh h{lvwh,= i +{,  i +5, @ i 3 +5, olp {$5 {5 Hvwh oðplwh ghehprv fdofxoduor phgldqwh orv oðplwhv odwhudohv

3 +5, @ olp i+{,  i+5, @ olp +{  5,6  3 @ 3 i. {5 {5 {$5. {$5. i+{,  i+5, {5  7{ . 7  3 @ olp @3 olp {5 {5 {$5 {$5 Gh hvwd irupd whqhprv txh i hv ghulydeoh hq 5 | txh i 3 +5, @ 31 e, Dkrud yhdprv vl i hv ghulydeoh hq ho sxqwr { @ 41 Frpr i+{,  i +4, {5  7{ . 7  < olp . @ olp . @ olp 5 {  8 @ 9 {.4 {.4 {$4 {$4 {$4 | frpr ho rwur oðplwh odwhudo1 olp {$4

i +{,  i +4, vhq+{ . 4,  < @ olp  {.4 {.4 {$4

qr h{lvwh6 vh wlhqh txh i qr hv ghulydeoh hq 4 f, Yhdprv dkrud vl i hv ghulydeoh hq orv sxqwrv d txh vrq pd|ruhv txh 51 Hq hvwh fdvr od ixqflöq wrpd od irupd i+{, @ +{  5,6 1 i +{,  i +d, {d +{  5,6  +d  5,6 @ olp {$d {d ^+{  5,  +d  5,`^+{  5,5 . +{  5,+d  5, . +d  5,5 @ olp {$d {d @ olp +{  5,5 . +{  5,+d  5, . +d  5,5 @ 6+d  5,5 {$d

i 3 +d, @

6 Sdud frq f

' f

olp {$d

q

suredu hvwr edvwd wrpdu od vxfhvlöq {

1

'

 3 n 2q

q

{

< 3

q

> i E{

glyhujh1 Qrwdu txh ho oðplwh hv gh od irupd

f

f

5;3

FKDSWHU 81 GHULYDGD g, Fdofxohprv dkrud od ghulydgd hq ho sxqwr d/ d 5`  4> 5^ olp {$d

i +{,  i +d, {5  7{ . 7  +d5  7d . 7, @ olp @ 5d  7 {$d {d {d

h, Ilqdophqwh/ fdofxohprv i 3 +d, fxdqgr d ? 4 i 3 +d, @ @ @ @ @

@

hv

vhq+{ . 4,  vhq+d . 4, {d vhq+x . d . 4,  vhq+d . 4, olp x$3 x vhq+x, frv+d . 4, . vhq+d . 4, frv+x,  vhq+d . 4, olp x$3 x vhq+x, frv+x,  4 olp +frv+d . 4, . vhq+d . 4, , x$3 x x vhq+x, frv+x,  4 +frv+d . 4,, olp . vhq+d . 4, olp , x$3 x x$3 x frv+d . 4, olp {$d

Gh hvwd irupd/ od ixqflöq ghulydgd +txh frqvhjxlprv sru sduwh | dqdol}dprv hq orv sxqwrv frq lfwlyrv, ; {A5 6+{  5,5 A A ? 3 {@5 i 3 +{, @ 5{  7 4 ? {?5 A A = frv+{ . 4, { ? 4

Dkrud fdofxohprv dojxqdv ghulydgdv gh ixqflrqhv eävlfdv/ hq ho vhqwlgr txh srvwhulruphqwh qrv d|xgduäq d rewhqhu od ghulydgd gh rwudv ixqflrqhv1 Hmhpsor 678 Phgldqwh ho fäofxor gh odv ixqflrqhv ghulydgdv1 Mxvwltxh ho vljxlhqwh olvwdgr1 41 Vl i+{, @ n frqvwdqwh hqwrqfhv i 3 +{, @ 3 51 Vl i+{, @ { hqwrqfhv i 3 +{, @ 4

61 Vl i+{, @ vhq+{, hqwrqfhv i 3 +{, @ frv+{,

71 Vl i+{, @ frv+{, hqwrqfhv i 3 +{, @  vhq+{, s 81 Vl i+{, @ { hqwrqfhv i 3 +{, @ 5s4 { / vl { A 3

91 Vl i+{, @ {q hqwrqfhv i 3 +{, @ q{q4 / vl q 5 Q :1 Vl i+{, @ h{ hqwrqfhv i 3 +{, @ h{

Vroxflöq= Hq wrgrv orv fdvrv d lqglfduä xq hohphqwr jhqìulfr gho grplqlr gh od ixqflöq fruuhvsrqglhqwh1 41

i +{,  i +d, {d nn @ olp {$d {  d 3 @ olp @3 {$d {  d

i 3 +d, @

Dvð/ i 3 +{, @ 3 hq ho fdvr i +{, @ n

olp {$d

8141 GHULYDGD

5;4

51

i +{,  i +d, {d {d @ olp {$d {  d @ olp 4 @ 4 {$d

i 3 +d, @

olp {$d

Sru or wdqwr od ixqflöq lghqwlgdg i +{, @ { wlhqh frpr ixqflöq ghulydgd i 3 +{, @ 41

61 Fdofxodgd dqwhulruphqwh/ hq orv hmhpsorv gh ixqflrqhv ghulydeohv1 71

i 3 +d, @ @ @ @

@

i+d . k,  i+d, k frv+d . k,  frv+d, olp k$3 k frv+d, frv+k,  vhq+d, vhq+k,  frv+d, olp k$3 k frv+d,+frv+k,  4, vhq+d, vhq+k,  olp k$3 k k +frv+d,,3  +vhq+d,,4 @  vhq+d, olp k$3

g frv+{, @  vhq+{, Gh hvwd irupd g{

81 Fdofxodgd dqwhulruphqwh frpr hmhpsor gh ixqflrqhv ghulydeohv/ gh hvwd irupd s 4 + {,3 @ s 5 {

91 i 3 +d, @ @ @ @ @ @

vl { A 3

i +d . k,  i +d, k +d . k,q  dq olp k$3 k   S q dq . qn@4 dqn kn  dq n olp k$3 k   Sq q q  4 kqd . n@5 dqn kn n olp k$3 k  q  [ q q  4 . olp qd dqn kn4 n k$3 n@5 qdq4 . 3 olp k$3

g {q @ q{q4 vl q 5 Q Dvð/ g{ Hvwd iöupxod wdpelìq hv yäolgd sdud srwhqfldv glvwlqwdv gh xq qýphur qdwxudo shur od ghprvwudflöq khfkd hv vöor yäolgd sdud q 5 Q

5;5

FKDSWHU 81 GHULYDGD

:1

i+d . k,  i+d, k hd.k  hd @ olp k$3 k hk  4 d @ h olp k$3 k @ hd  4 @ hd

i 3 +d, @

{ d Gh hvwd irupd gh g{ m{@d @ h

81415

olp k$3

Dojheud gh Ghulydgdv

Whruhpd 817 Vl i hv ghulydeoh hq ho sxqwr d/ hqwrqfhv od ixqflöq i frq  5 U mr wdpelìq hv ghulydeoh hq d | +i ,3 +d, @ ++i ,3 +d,,

Ghprvwudflöq=

gi m{@d g{

i+d . k,  i+d, k i +d . k,  i+d, , @ olp + k$3 k i +d . k,  i +d, @  olp k$3 k @

olp k$3

Frpr i hv ghulydeoh/ ho ýowlpr oðplwh hv vlpsohphqwh od ghulydgd gh i hq ho sxqwr d1 Dvð/ gi gi m{@d @  m{@d g{ g{

txh hv or plvpr txh

+i ,3 +d, @ ++i ,3 +d,,

Hmhpsor 679 Xvdqgr hvwh whruhpd | orv uhvxowdgrv dqwhulruhv whqhprv txh 41 +6 vhq+{,,3 @ 6+vhq+{,,3 @ 6 frv+{,

g +:{7 , @ : g +{7 , @ :+7{6 , @ 5;{6 51 g{ g{

g + frv+{,, @  g +frv+{,, @ + vhq+{,, @  vhq+{, 61 g{ g{

g +
Whruhpd 818 Vl i | j vrq ghulydeohv hq d/ wdpelìq or hv i . j | dghpäv vx ghulydgd vdwlvidfh gi gj g +i . j,+d, @ +d, . +d, g{ g{ g{

Ho whruhpd glfh txh od ghulydgd gh od vxpd hv od vxpd gh odv ghulydgdv1



8141 GHULYDGD

5;6

Ghprvwudflöq=

+i . j,+d . k,  +i . j,+d, g +i . j,+d, @ olp k$3 g{ k i +d . k, . j+d . k,  i+d,  j+d, @ olp k$3 k

Vhsdudqgr orv vxpdqgrv | djuxsäqgrorv sdud irupdu ho oðplwh txh ghqh od ghulydgd +gh fdgd ixqflöq, vh wlhqh i+d . k, . j+d . k,  i +d,  j+d, k i+d . k,  i +d, j+d . k,  j+d, . olp @ olp k$3 k$3 k k @ i 3 +d, . j 3 +d,

+i . j,3 +d, @

olp k$3

txh hv vlpsohphqwh ho uhvxowdgr shglgr/ vöor txh hvwä hvfulwr hq rwud qrwdflöq1

Hmhpsor 67: Xvdqgr hvwh whruhpd whqhprv txh vrq ghulydeohv odv ixqflrqhv> 41 i +{, @

s { . vhq+{,

51 j+{, @ frv+{,  :{6

61 k+{, @ <{ . 7  7 vhq+{,

s 71 o+{, @ 8h{  7{8  8 { | vxv ghulydgdv vrq=

s 41 i 3 +{, @ + {,3 . +vhq {,3 @

5

s4 { . frv+{,

vl { 5 U.

51 j 3 +{, @ +frv,3 +{,  :+{6 ,3 @  vhq+{,  :+6{5 , @  vhq+{,  54{5 vl { 5 U

61 k3 +{, @ +<{,3 . +7,3  7+vhq,3 +{, @ < . 3 . 7 frv+{, @ <  7 frv+{, vl { 5 U s 71 o3 +{, @ 8+h{ ,3  7+{8 ,3  8+ {,3 @ 8h{  53{7 

s8 {

5

vl { 5 U.

Whruhpd 819 Vl i | j vrq ghulydeohv hq d/ wdpelìq or vrq i  j | i @j/ +sdud hvwd ýowlpd j+d, qr qxor, l, +i  j,3 +d, @ i 3 +d,  j+d, . i +d,  j 3 +d,

3 i +d,j3 +d, vl j+d, 9@ 31 ll, +i@j,3 +d, @ i +d,j++dj,+ d,,5

Xvdqgr rwud qrwdflöq od ghulydgd gho surgxfwr hv

| ho fxrflhqwh

gi gj gi  j +d, @ +d,  j+d, . i +d,  +d, g{ g{ g{ gi gj 4 g+i @j, md @ + md  j+d,  i+d,  md , g{ g{ g{ j5 +d,



5;7

FKDSWHU 81 GHULYDGD

Ghprvwudflöq= Od ghprvwudflöq hv vlpsohphqwh xq hmhuflflr gh oðplwhv/ vöor txh hv qhfhvdulr ideulfdu orv oð plwhv gh odv ghulydgdv> hq ho fdvr gho surgxfwr ho wuxfr hv vxpdu | uhvwdu +vxpdu fhur, ho wìuplqr i +d,j+d . k,1 +i  j,+d . k,  +i  j,+d, k i +d . k,j+d . k,  i +d,j+d . k, . i +d,j+d . k,  i+d,j+d, @ olp k$3 k +i +d . k,  i +d,, j+d . k,  j+d, j+d . k, . i +d, @ olp k$3 k k

+i  j,3 +d, @

olp k$3

Frpr j hv frqwlqxd hq d rewhqhprv or exvfdgr

+i  j,3 +d, @ i 3 +d,j+d, . i+d,j3 +d,

Sdud od rwud sduwh ghmduhprv do ohfwru exvfdu hq fxdotxlhu oleur gh fäofxor eävlfr1  Revhuydflöq= Qrwhprv txh ho whruhpd qr glfh txh od ghulydgd gho surgxfwr hv ho surgxfwr gh odv ghulydgdv vlqr txh hv dojr edvwdqwh glvwlqwr/ hvwr wdpelìq rfxuuh frq od ghulydgd gho fxrflhqwh1 Hmhpsor 67; Fdofxodu od ghulydgdv gh= 41 i+{, @ 6{7 frv+{,/ vl { 5 U1

51 i+{, @ wdq+{,/ vl { 9@ @5 . q q 5 ]1 61 i+{, @ frw+{,/ vl { 9@ q q 5 ]1

71 i+{, @ fvf+{,/ vl { 9@ q q 5 ]1

81 i+{, @ vhf+{,/

vl { 9@ @5 . q q 5 ]1

91 i+{, @ +8{5 . 6{7 ,5 vl { 5 U1

s

: :1 i+{, @ 8<{s{67{{8

Vroxflöq=

vl { 5 U. 1

41

i 3 +{, @ +6{7 ,3 frv+{, . 6{7 +frv+{,,3 @ 45{6 frv+{, . 6{7 + vhq+{,, @ 45{6 frv+{,  6{7 vhq+{,

51 i 3 +{, @



vhq+{, frv+{,

3

@ @ @ @

+vhq+{,,3 frv+{,  vhq+{,+frv+{,,3 +frv+{,,5 frv+{, frv+{,  vhq+{,+ vhq+{,, +frv+{,,5 frv+{, frv+{, . vhq+{, vhq+{, +frv+{,,5 4 @ vhf5 +{, frv5 +{,

wdq Dvð djuhjdprv d qxhvwud wdeod gh ghulydgdv d phprul}du gg{ +{, @ vhf5 +{,/ vl { 9@ @5 . q q 5 ]

8141 GHULYDGD

5;8

61 i 3 +{, @

71



frv+{, vhq+{,



4 vhq+{,

3

g fvf +{, @  frw+{, fvf+{,/ vl { 9@ q q 5 ]1 g{ i 3 +{, @

91

+frv+{,,3 vhq+{,  frv+{,+vhq+{,,3 +vhq+{,,5  vhq+{, vhq+{,  frv+{, frv+{, @ +vhq+{,,5 vhq+{, vhq+{, . frv+{, frv+{, @  +vhq+{,,5 4 @  5 @  fvf5 +{, vhq +{, @

frw Gh hvwd irupd gg{ +{, @  fvf5 +{,/ vl { 9@ q q 5 ] +rwud d phprul}du,1

i 3 +{, @

81

3



4 frv+{,

3

+4,3 vhq+{,  4+vhq+{,,3 +vhq+{,,5 4+frv+{,, @ +vhq+{,,5  frv+{, @ +vhq+{,,5 @  frw+{, fvf+{, @

+4,3 frv+{,  4+frv+{,,3 +frv+{,,5 4+ vhq+{,, @ +frv+{,,5 vhq+{, @ +frv+{,,5 @ wdq+{, vhf+{, @

g vhf +{, @ wdq+{, vhf+{,/ vl { 9@ @5 . q q 5 ] g{

i 3 +{, @ +8{5 . 6{7 ,3 +8{5 . 6{7 , . +8{5 . 6{7 ,+8{5 . 6{7 ,3 ^+8{5 . 6{7 ,+8{5 . 6{7 ,`3 @ +43{ . 45{6 ,+8{5 . 6{7 , . +8{5 . 6{7 ,++43{ . 45{6 , @ 5+43{ . 45{6 ,+8{5 . 6{7 , :1 i 3 +{, @ @ Hq uhvxphq vh wlhqh

s s s s +8{:  6 {,3 +< {  7{8 ,  +8{:  6 {,+< {  7{8 ,3 s +< {  7{8 ,5 s s +68{9  5s6 { ,+< {  7{8 ,  +8{:  6 {,+ 5s< {  53{7 ,3 s +< {  7{8 ,5

5;9

FKDSWHU 81 GHULYDGD

Dojheud gh ghulydgdv1

Vl i | j vrq ixqflrqhv ghulydeohv hq d hqwrqfhv i / i . j/ i  j vrq ghulydeohv hq d | vl j+d, 9@ 3 hqwrqfhv i @j hv ghulydeoh hq d1 Dghpäv 41 +i,3 +d, @ i 3 +d,

51 +i . j,3 +d, @ i 3 +d, . j 3 +d,

61 +i j,3 +d, @ i 3 +d,  j+d, . i+d,  j3 +d,

3 3 71 + ij ,+d, @ i +d,j+dj,5+dj, +d,i +d,

j+d, 9@ 31

Frurodulr 81: Vhd j ixqflöq ghulydeoh hq d/ j+d, 9@ 3 hqwrqfhv od ixqflöq 4@j hv ghulydeoh hq d |

g +j,+d, g +4@j,+d, @  g{ g{ +j+d,,5

r

4 j 3 +d, + ,3 +d, @  5 j j +d,

Hmhpsor 67< Frq hvwh frurodulr fdofxohprv od ghulydgd gh i +{, @ {4q > q 5 Q q q{q4 gi +{, @  q 5 @ q.4 = g{ +{ , {

Hmhuflflr 683 Fdofxoh odv ghulydgdv gh i +{, @ Vroxflöq=

81416

4 { . vhq { 5

gi +5{ . frv {, +{, @  5 g{ +{ . vhq {,5

Uhjod gh od Fdghqd

>

j+{, @

4 vhf { . 6 frw {

+wdq { vhf {  6 fvf5 {, gj +{, @  g{ +vhf { . 6 frw {,5

Vljxlhqgr frq whruhpdv txh qrv shuplwdq fdofxodu ghulydgdv gh ixqflrqhv xvdqgr odv |d fdofxodgdv/ yhdprv ho whruhpd gh od ghulydgd gh ixqflrqhv frpsxhvwdv/ sru hmhpsor vdelhqgr txh i+{, @ vhq+{, | j+{, @ <{6 vrq ixqflrqhv ghulydeohv Á Txì srghprv ghflu hq fxäqwr d ghulydelolgdg/ gh +i  j,+{, @ vhq+<{6 , | gh +i  i,+{,B Hmhpsor 684 Fdofxohprv xvdqgr od ghqlflöq/ odv ghulydgd gh od ixqflöq +i  i,+{, @ vhq+vhq {, hq ho sxqwr { @ d1 Vroxflöq=

i 3 +d, @ @ @

@ @

vhq+vhq +{,,  vhq+vhq +d,, {d vhq+vhq +{,,  vhq+vhq +d,, vhq +{,  vhq +d, olp {$d vhq +{,  vhq +d, {d vhq +x,  vhq +e, vhq +{,  vhq +d, olp olp frq x @ vhq { | e @ vhq +d, x$vhq d {$d xe {d frv +e,  frv +d, frq e @ vhq d frv+vhq +d,,  vhq +d, olp {$d

Revhuydflöq= Yhuhprv d frqwlqxdflöq xq uhvxowdgr od uhjod gh od fdghqd txh qrv shuplwluä fdofxodu hvwd | rwudv ghulydgdv +gh xqd ixqflrq frpsxhvwd, gh xqd pdqhud päv vlpsoh1

8141 GHULYDGD

5;:

Whruhpd 81; +Uhjod gh od Fdghqd r Ghulydgd gh od frpsxhvwd, Vhd i = [ $ U/ j = \ $ U/ i+[,  \> d 5 [ _ [ 3 > e @ i+d, 5 \ _ \ 3 Vl h{lvwh i 3 +d, | j3 +e, hqwrqfhv h{lvwh +j  i,3 +d, | r hvfulwd gh rwud irupd

+j  i,3 +d, @ j 3 +e,  j 3 +d, @ j 3 +i +d,,j3 +d,=

gj gi g ji md @ me  md g{ g{ g{ s Hmhpsor 685 Vhdq j+{, @ frv+{,> i +{, @ :{6 . {/ i hv ghulydeoh hq U. | Uhfi  U @ Grpj sru or wdqwr/ sdud d 5 U.

|d txh j3 +{, @  vhq { vh wlhqh

+j  i ,3 +d, @ j3 +i+d,,i 3 +d, s 4 @ j3 +:d6 . d,  +54d5 . s , 5 d

s 4 d,,  +54d5 . s , 5 d s 5 Hmhuflflr 686 Vhd i 3 +{, @ {6 h{ / i+4, @ 3 | j+{, @ +{ . 4,5 . 6 Fdofxoh +j  i ,3 +4,1 +j  i,3 +d, @ + vhq+:d6 .

Vroxflöq= Sru uhjod gh od fdghqd Oxhjr

+j  i ,3 +{, @ j 3 +i +{,,  i 3 +{,

+j  i ,3 +4, @ j 3 +i +4,,  i 3 +4, 5 shur i 3 +4, @ 46 h4 @ h | j3 +{, @ s5+{.4,5 1 5 +{.4, .6 5 4 @ 1 gh grqgh j 3 +i+4,, @ j 3 +3, @ 5s 5 7 Sru or wdqwr/ uhhpsod}dqgr hq 814 +j  i ,3 +4, @

+814,

4 h h@ 5 5

Hmhpsor 687 Vhd j+{, @ vhq+vhq {, grqgh j @ i  i frq i +{, @ vhq { hqwrqfhv j3 +{, @ i 3 +i+{,,i 3 +{, @ frv+vhq {, frv {1 s Hmhpsor 688 Vhd i +{, @ 6{7 . frv+{, . <> { 5 U

Frpr i sxhgh vhu hvfulwd frpr od frpsxhvwd s gh grv ixqflrqhv päv vlpsohv/ frpr vljxh= i +{, @ k+6{7 . frv+{, . <, @ +k  o,+{, grqgh k+{, @ { | o+{, @ 6{7 . frv+{, .
4 @ k3 +o+{,,o3 +{, | frpr k3 +{, @ s 5 { 4 s +45{6  vhq+{,, @ 5 o+{, 45{6  vhq+{, s = @ 5 6{7 . frv+{, . <

Hmhpsor 689 Vhd i +{, @ vhf+5{8 . 7h{ ,1 Fdofxohprv i 3 +{,

5;;

FKDSWHU 81 GHULYDGD Whqhprv txh i +{, @ vhf+k+{,, frq k+{, @ 5{8 . 7h{ oxhjr

g vhf gk mk+{,  m{ g{ g{ @ vhf+{, wdq+{,mk+{,  +43{7 . 7h{ , @ vhf+k+{,, wdq+k+{,,+43{7 . 7h{ , @ vhf+5{8 . 7h{ , wdq+5{8 . 7h{ ,+43{7 . 7h{ ,

i 3 +{, @

Hmhpsor 68: Vhd i xqd ixqflöq ghulydeoh hq { @ d1 Fdofxohprv od ghulydgd gh j+{, @ +i +{,,q > q 5 ]1

Frpr j+{, @ +k  i ,+{, grqgh k+{, @ {q / xvdqgr od uhjod gh od fdghqd whqhprv txh gj md g{

Dvð rewhqhprv od vljxlhqwh irupxod

gj gi mi +d,  md g{ g{ gi @ q+i+d,,q4  +d, g{ @

gi q +{, @ q+i +{,,q4  i 3 +{, g{

txh hq ho fdvr gh i +{, @ 6{5 . frv { qrv txhgd

g+6{5 . frv {,8 @ 8+6{5 . frv {,84 +9{  vhq {, g{ Hmhpsor 68; Od uhjod gh od fdghqd sru lqgxfflöq vh sxhgh jhqhudol}du d od frpsrvlflöq gh päv gh grv ixqflrqhv sru hmhpsor fdofxohprv od ghulydgd gh

Vroxflöq=

i +{, m{ g{

i +{, @ +j  k  m  n  o,+{,

@ @ @ @ @

g+j  k  m  n  o, m{ g{ +j  k  m  n, go mo+{, +{, g{ g{ gn go +j  k  m, mno+{, mo+{, +{, g{ g{ g{ gm gn go +j  k, mm no+{, m+no,+{, mo+{, +{, g{ g{ g{ g{ +j, gk gm gn go mkmno+{, mm no,+{, m+no,+{, mo+{, +{, g{ g{ g{ g{ g{

Hmhpsor 68< Fdofxohprv od ghulydgd gh

i+{, @ vhq+vhf+wdq+{5 . : frv {,7 ,,

Vroxflöq= Xvdqgr od uhjod gh od fdghqd rewhqlgd hq ho hmhpsor 68; whqhprv

g vhq { gi g vhf g wdq {7 @ mvhf+wdq+{5 .: frv {,7 , mwdq+{5 .: frv {,7 m+{5 .: frv {,7 +5{  : vhq {, g{ g{ g{ g{ g{ {5 .: frv {

Fdofxohprv fdgd ghulydgd lqglfdgd | uhdol}dqgr odv hydoxdflrqhv fruuhvsrqglhqwh whqhprv gi g{

@ ^frv+vhf+wdq+{5 . : frv {,7 ,,`^wdq+wdq+{5 . : frv {,7 , vhf+wdq+{5 . : frv {,7 ,` ^vhf5 +{5 . : frv {,7 `^7+{5 . : frv {,6 `^5{  : vhq {`

8141 GHULYDGD

5;<

Hmhuflflr 693 Xvdqgr od uhjod gh od fdghqd/ suxheh txh vl i hv glihuhqfldeoh hq G/ hqwrqfhv +hi +{, ,3 @ hi +{,  i 3 +{,

Vroxflöq= Frqvlghuprv od ixqflöq glihuhqfldeoh hq U

;{ 5 G

j= U $ U { :$ h{

Sru klsöwhvlv i = G $ U hv xqd ixqflöq glihuhqfldeoh hq G/ oxhjr sru ho whruhpd uhjod gh od { :$ i +{, fdghqd j  i hv glihuhqfldeoh hq G |

+815,

+j  i ,3 +{, @ j 3 +i +{,,i 3 +{,

grqgh +j  i ,+{, @ j+i+{,, @ hi +{, | j3 +{, @ h{ 1 Dvð j 3 +i +{,, @ hi +{, 1 Sru or wdqwr hq 815 vh wlhqh +hi +{, ,3 @ hi +{,  i 3 +{, Hmhpsor 694 Fdofxoh od ghulydgd gh odv vljxlhqwhv ixqflrqhv 41 hvhq{

s 51 h {.frv {

Vroxflöq=

41 +hvhq { ,3 @ hvhq {  +vhq {,3 @ hvhq {  frv {

s s 51 +h {.frv { ,3 @ h {.frv { + 5s4 {  vhq {,1

Hmhuflflr 695 Vhdq i | j ixqflrqhv wdohv txh Ghprvwudu txh

G @ i{ 5 U m { 5 Grpi _ Grpj | i +{, A 3j @ 9 !

3

i +{, , ++i +{,,j+{, ,3 @ +i+{,,j+{,  +j3 +{, oq+i +{,, . j+{, i+{,

Vroxflöq= Vdehprv sru surslhgdg gho orjdulwpr txh oxhjr sru hmhuflflr dqwhulru shur +j+{, oq i+{,,3

;{ 5 G

j +{, +i +{,,j+{, @ hoq+i +{,, @ hj+{, oq i +{,

++i+{,,j+{, ,3 @ hj+{, oq i +{,  +j+{, oq i +{,,3

hv od ghulydgd gh xq surgxfwr | gh frpsrvlflöq gh ixqflrqhv/ oxhjr

Dvð uhhpsod}dqgr hq 816

+j+{, oq i+{,,3 @ j3 +{, oq+i +{,, . j+{,

i 3 +{, i +{,

3

i +{, , ;{ 5 G ++i +{,,j+{, ,3 @ +i+{,,j+{,  +j 3 +{, oq+i +{,, . j+{, i+{,

Revhuydflöq= Gh or dqwhulru vh rewlhqh= +ui +{, ,3 @ ui +{,  i 3 +{,  oq u vl u hv xqd frqvwdqwh srvlwlyd +i+{,u ,3 @ ui +{,u4  i 3 +{, vl i +{, A 31

+816,

5<3

FKDSWHU 81 GHULYDGD

Hmhpsor 696 Fdofxodu odv ghulydgdv gh odv vljxlhqwhv ixqflrqhv1

s 41 i+{, @ +{5 . 5, { 51 j+{, @ 5{

5

.frv

{

61 k+{, @ vhq6 +5{,

Vroxflöq=

s 5 41 i+{, @ h { oq+{ .5, oxhjr dsolfdqgr ho hmhuflflr dqwhulru s s 5 i 3 +{, @ h { oq+{ .5, + {  oq+{5 . 5,,3 s 4 s 5{ @ +{5 . 5, { + s oq+{5 . 5, . { 5 , 5 { { .5

51 j+{, @ h+{

5

{

.frv , oq 5

61 k+{, @ vhq6 +5{,/

Frphqwdulrv

/ dqdorjdphqwh

5 j3 +{, @ h+{ .frv {, oq 5 ++{5 . frv {, oq 5,3 5 @ 5{ .frv { +5{  vhq {, oq 5

k3 +{, @ 6 vhq5 +5{,  +vhq+5{,,3 @ 6 vhq5 +5{,  frv+5{,  5

41 Yhdprv jhrpìwulfdphqwh sru txì ghehprv xvdu od uhjod gh od fdghqd/ sdud hvr xvduhprv odv ixqflrqhv i+{, @ vhq { | j+{, @ vhq+7{,1 gj @ frv 7{/ dvð hq @5/ gj m Vl qr xvdprv od uhjod frphwhprv ho wðslfr huuru gh dupdu txh g{ g{ @5 @ frv 5 @ 41

Dqdol}dqgr orv juäfrv gh i | j hq od jxud txh vljxh/ srghprv qrwdu txh O/ uhfwd wdqjhqwh d od juäfd gh j qr wlhqh od plvpd shqglhqwh txh P / uhfwd wdqjhqwh d i sxhvwr txh odv rqgxodflrqhv gh j+{, +iuhfxhqfld gh rqgd, vrq päv uäslgd txh odv gh i/ reoljdqgr hvwr d whqhu pd|ru shqglhqwh O txh P1 gj @ 7 frv 7{/ yhdprv yld od ghqlflöq frpr vh rewlhqh hvwh uhvxowdgr1 51 Sru uhjod gh od fdghqd g{ gj md g{

j+d . k,  j+d, k vhq+7+d . k,,  vhq +7d, @ olp k$3 k vhq+7d . 7k,  vhq +7d, @ olp k$3 k @

olp k$3

Xvdqgr xqd lghqwlgdg wuljrqrpìwulfd sdud vhq   vhq  whqhprv gj md g{

5 frv+7d . 5k, vhq+5k, k vhq +5k, @ 7 olp frv+7d . 5k, olp k$3 k$3 5k @ 7 frv+7d,  4 @

olp k$3

8141 GHULYDGD

5<4

Iljxuh 814= 61 Vreuh odv glihuhqwhv qrwdflrqhv

Frqvlghuhprv od ixqflöq Y / yroxphq gh xq flolqgur gh dowxud o | udglr u/ hv ghflu/ od ixqflöq yroxphq Y @ u5 o1 Vl qrv slghq fdofxodu od ghulydgd gh Y hq ho sxqwr d/ hv ghflu/ Y 3 +d, ÁTxì ghehprv frqwhvwduB Qr hv or plvpr txh qrv slgdq i 3 +d,/ grqgh i +{, @ 6 vhq {/ |d txh hq hvwh ýowlpr fdvr hvwä fodud od yduldeoh/ shur hq ho fdvr gh Y ÁFxäo hv od yduldeohB ÁGheh vhu wrpdgd u frpr yduldeoh lqghshqglhqwhB ÁR d or phmru oB1 Hq hvwh fdvr od qrwdflöq Y 3 +d, hv dpeljxd | hv frqyhqlhqwh xvdu od qrwdflöq gY mu@d gu

r

gY mo@d go

|d txh gh hvwd irupd txhgd h{soðflwd od yduldeoh frq uhvshfwr d od fxdo vh txlhuh ghulydu1

Phgldqwh od ghqlflöq srghprv yhu txh orv uhvxowdgrv vrq wrwdophqwh glihuhqwhv vl vh wrpd xqd x rwud yduldeoh gY mu@d gu

gY mo@d go

@ @

@

@

@

@

Y +d . k,  Y +d, k +d . k,5 o  d5 o olp k$3 k 5do Y +d . k,  Y +d, olp k$3 k u5 +d . k,  u5 d olp k$3 k 5 u olp k$3

Hq ho fdvr txh xqd ixqflöq wlhqh päv gh xqd yduldeoh +sru hmhpsor i+{> |> }, @ |5 vhq { . } 6 , vh gi dfrvwxpeud d lqglfdu od ghulydflöq frq uhvshfwr d od yduldeoh | sru hmhpsor/ frpr Ci C| d fdpelr gh g| / dvð/ Ci C} lqglfd od ghulydgd gh i frq uhvshfwr d od yduldeoh } vrodphqwh/ txhgdqgr odv rwudv yduldeohv frpr frqvwdqwh +hv ghflu/ vh frpsruwhq frpr qýphur sdud ho prphqwr gh ghulydu1

5<5

FKDSWHU 81 GHULYDGD Hmhpsor 697 Vl i+{> |> }, @ |5 vhq { . } 6 Ci C{ Ci C| Ci C}

@ | 5 frv { . 3 @ 5| vhq { . 3 @ 3 . 6} 5

Hmhpsor 698 Hq ho fdvr gho yroxphq gho flolqgur Y @ u5 o CY @ 5uo Cu

CY @ u5 Co

Hmhpsor 699 Vhd j+{> |> }, @ 6{5 . :{6 vhq | hqwrqfhv whqhprv txh Cj C{ Cj C| Cj C}

@ 9{ . 54{5 vhq | @ 3 . :{6 frv | @ 3

Wrphprv dkrud od ixqflöq i @ 8{5 .vhq {1 Fodudphqwh whqhprv/ sru or dqwhulru/ txh Ci C{ @ 43{.frv { Ci | txh C| @ 3 |d txh hq hvwh ýowlpr fdvr ghehprv shqvdu txh od yduldeoh hv |/ shur i vh frpsruwd frpr frqvwdqwh uhvshfwr d hvwd yduldeoh dvð txh vx ghulydgd hv fhur Ci i +| . k,  i+|, 8{5 . vhq {  +8{5 . vhq {, @ olp @ olp @3 k$3 k$3 C| k k

71 Qrwdflrqhv uhjod gh od fdghqd

Vhdq | @ i +x,> x @ j+{, vxsrqjdprv txh dpedv wlhqh ghulydgdv | txh hv srvleoh kdfhu od frpsxhvwd | @ i +j+{,,1 +hv ghflu/ vl gäq odv klsrwhvlv gh od uhjod gh od fdghqd,1 Hqwrqfhv C| Cy C| @ C{ Cy C|

Qrwdflrqdophqwh ho wìuplqr l}txlhugr gh od ljxdogdg vh uhhuh d od ghulydgd gh | fdofxodgd ghvsxhv gh kdehu khfkr od frpsxhvwd gh i | j +i  j,/ hq fdpelr ho sulphu idfwru gho wìuplqrv ghuhfkr C| vh uhhuh d od ghulydgd gh | dqwhv gh kdfhu od frpsxhvwd1 Cy

Hmhpsor 69: Vhd | xqd ixqflöq gh {/ ghulydeoh frq uhvshfwr d {1 Fdofxohprv odv ghulydgdv txh vh lqglfdq

g +{7 | 6 , @ {7 6| g| . 7{6 |6 41 g{ g{ g +{ vhq |, @ vhq | . { frv | g| 51 g{ g{ g| , g 5 61 g{ +wj {|, @ +vhf +{|,,+| . { g{ g +5{| 6 . { vhq+{|,, @ 5| 6 . 9{| 5 g| . vhq +{|, . { frv+{|,+| . { g| , 71 g{ g{ g{

8141 GHULYDGD

g +s|, @ 81 g{ g +s|, @ 91 g|

5<6

g| s4 | g{

5

s4 |

5

g +vhq | . |6 , @ frv | g| . 6| 5 g| :1 g{ g{ g{ g 6 5 ;1 g{ +vhq | . | , @ frv | . 6| g +vhq | . | 6 , @ 31 <1 g} Hmhuflflr 69; Ghwhuplqdu vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv

41 Vhdq i | j ixqflrqhv ghulydeohv hq `d> e^1 Vl i | j vrq fuhflhqwhv hq `d> e^ hqwrqfhv i . j hv fuhflhqwh hq `d> e^1

51 Vl i hv ghulydeoh hq j+d, | j hv frqwlqxd hq d shur qr ghulydeoh hq d hqwrqfhv i  j qr hv ghulydeoh hq d1

Vroxflöq=

41 Yhugdghur Vl i | j vrq ixqflrqhv fuhflhqwh | ghulydeohv hq `d> e^ hqwrqfhv i 3 +{,  3 | j3 +{,  3 hq `d> e^/ oxhjr +i . j,3 +{, @ i 3 +{, . j 3 +{,  3 ;{ 5`d> e^ Sru or wdqwr i . j hv fuhflhqwh hq `d> e^1

51 Idovr1 Vhdq j+{, @ m{m/ i+{, @ {5 | d @ 3 j hv frqwlqxd hq 3 shur qr ghulydeohv hq 3 | i hv ghulydeoh hq j+3, @ 3/ shur +i  j,+{, @ m{m5 @ {5 od fxdo hv ghulydeoh hq 3

Hmhpsor 69< Hq txì sxqwrv/ h{lvwh od ghulydgd gh od ixqflöq ; {3 ? frv { i+{, @ = s 6 4  5{ { ? 3 Vroxflöq=

d, Vl { A 3/ i +{, @ frv {1 Vdehprv txh frv { hv ghulydeoh sdud wrgr { 5 U hq sduwlfxodu sdud { A 3> i 3 +{, @  vhq+{,1 s e, Vl { ? 3> i +{, @ 6 4  5{1 Hq hvwh fdvr i hv od frpsxhvwd gh ixqflrqhv ghulydeohv | vx ghulydgd i 3 +{, @ 46 +4  5{,5@6  +5,/ sru uhjod gh od fdghqd1

f, Vl { @ 3/ ghehprv dqdol}du sru ghqlflöq od ghulydgd gh i/ |d txh i hvwä ghqlgd gh irupd glvwlqwd sdud { A 3 | sdud { ? 3 i +3, Yhdprv vl h{lvwh i 3 +3, @ olp{$3 i +{{, 3 1 Sdud hvwr fdofxohprv

3 +3, @ i. @ @ @ @

@

i+{,  i+3, {3 frv {  4 olp { {$3. frv5 {  4 olp {$3. {+frv { . 4,  vhq5 { olp. {$3 {+frv { . 4, vhq {  vhq { ,+ , olp + frv { . 4 {$3. { +4,+3, @ 3 olp {$3.

5<7

FKDSWHU 81 GHULYDGD

3 +3, @ 3 sru or wdqwr i. Sru rwud sduwh

i +{,  i +3, {3 s 6 4  5{  4 @ olp { {$3 5 @  = 6

3 +3, @ i

olp {$3

+Sdud fdofxodu ho ýowlpr oðplwh wxylprv txh udflrqdol}du,

3 +3, 9@ i.3 +3, hqwrqfhv qr h{lvwh i 3 +3, | hq frqvhfxhqfld Frpr i ; 5 {?3 A 6 ? 6s +45{,5 3 i +{, @ A =  vhq { {A3

Grpi 3 @ U  i3j +dvð sdud fxdotxlhu sxqwr { 5 U  i3j h{lvwh od ghulydgd gh i ,1

Hmhpsor 6:3 Frqvlghuhprv dkrud od ixqflöq ; 6 { frv+4@{5 , { ? 3 A A A A ? 3 {@3 i+{, @ A A A A = s 5 { . 5{6 { A 3 Ghwhuplqhprv od ghulydgd gh i 1

Vroxflöq=

d, Hv fodur txh sdud { ? 3 od ixqflöq i +{, @ {6 frv+4@{5 , hv ghulydeoh sru dojheud | frpsrvlflöq gh ixqflrqhv ghulydeohv/ oxhjr xvdqgr odv uhjodv gh ghulydflöq whqhprv i 3 +{, @ 6{5 frv+

e, Dqäorjdphqwh sdud { A 3> i+{, @ oxhjr

4 4 5 , . {6 + vhq+ 5 ,,+ 6 , 5 { { {

s {5 . 5{6 hv ghulydeoh sru frpsrvlflöq gh ixqflöqhv ghulydeohv/

4 +5{ . 9{5 , i 3 +{, @ s 5 5 { . 5{6

f, Sdud { @ 3/ ghehprv dqdol}du i 3 +3, phgldqwh od ghqlflöq/ |d txh i wlhqh glvwlqwr frpsruwdplhqwr hq wruqr do fhur/ hv ghflu/ ghehprv fdofxodu i 3 +3, @ olp {$3 i 3 +3, @ .

| ho rwur oðplwh odwhudo

i +{,  i +3, {3

i+{,  i +3, olp @ olp. {3 {$3. {$3

3 +3, @ olp i {$3

s {5 . 5{6 @4 {

i +{,  i +3, {6 frv+4@{5 , @ olp @3 {3 { {$3

8141 GHULYDGD

5<8

i +3, frpr vrq glvwlqwrv/ vh frqfox|h txh ho oðplwh i 3 +3, @ olp{$3 i +{{, 3 qr h{lvwh1 Hq frqvhfxhqfld i qr hv ghulydeoh hq { @ 3 | sru or wdqwr od ixqflöq i 3 +{, hvwä ghqlgd vrodphqwh sdud U  i3j | vx ydoru hv= ; 5 5 5 ? 6{ frv+4@{ , . 5 vhq+4@{ , { ? 3 i 3 +{, @ = {5 s{{.6 {A3 5 .5{6

Hmhpsor 6:4 BcHq txì sxqwrv gho lqwhuydor ^4> 5^ od ixqflöq i +{, @ ^{` . m{  4m hv ghulydeohB

Vroxflöq= Vhsdudqgr ho grplqlr gh i dsursldgdphqwh rewhqhprv od vljxlhqwh ixqflöq ghqlgd sru sduwhv1 ; { 4  { ? 3 A A A A ? 4{ 3{?4 i +{, @ A A A A = { 4{?5

Fdofxohprv dkrud vx ghulydgd

d, Vl 4 ? { ? 3/ i hv ghulydeoh | i 3 +{, @ 4/ dqäorjdphqwh sdud 3 ? { ? 4 | 4 ? { ? 5 vh vdeh txh i hv ghulydeoh | i 3 +{, @ 4 sdud 3 ? { ? 4 | i 3 +{, @ 4 sdud 4 ? { ? 51

3 +4, +vl h{lvwh, txh sru ghqlflöq hv od ghulydgd gh i hq e, Hq { @ 4/ vöor wlhqh vhqwlgr ghwhuplqdu i. 41 i +{,  i +4, {.4 {  4 @ olp @ 4 {$4. { . 4

3 +4, @ i.

olp {$4.

Hv fodur txh hq { @ 3 | hq { @ 4 i hv glvfrqwlqxd/ oxhjr i qr hv ghulydeoh hq hvwrv sxqwrv/ dvð i +{, hv ghulydeoh hq ^4> 5^i3> 4j i 3 +{, @

; ? 4 4  { ? 4> { 9@ 3 =

Hmhuflflr 6:5 Frqvlghuh od ixqflöq i+{, @



4

4?{?5

6 . d{ {  4 {5 . e { A 4

grqgh d> e 5 U Ghwhuplqh d | e gh prgr txh i vhd ghulydeoh hq { @ 41 i +d, 3 3 Vroxflöq= i vhuä ghulydeoh hq { @ 4 vl | vöor vl olp{$4 i +{{, d h{lvwh +hv ghflu i.+4, @ i+4, ,1 i +{,  i +4, {4 6 . d{  +6 . d, @ olp {4 {$4 d+{  4, @ olp {$4 {  4 @ olp d @ d {$4

3 +4, @ i

olp {$4

5<9

FKDSWHU 81 GHULYDGD

Sru rwud sduwh>

i +{,  i +4, {4 {5 . e  +6 . d, @ olp. {4 {$4

3 +4, @ i.

olp {$4.

shur hvwh oðplwh srguld h{lvwlu vöor vl olp{$4. {5 . e  +{ . d, @ 3 shur olp{$4 {5 . e  +6 . d, @ 4 . e  +6 . d,1 3 +4, srguld h{lvwlu vöor vl 4 . e  +6 . d, @ 3 gh grqgh e @ 5 . d sru or wdqwr> oxhjr i.

{5 . e  +6 . d, {4 5 { . 5 . d  +6 . d, @ olp {4 {$4. {5  4 @ olp {$4. {  4 @ olp. { . 4 @ 5 {$4 3 +4, @ i3 +4, +, d @ 5/ oxhjr e @ d . 5 @ 71 Dvð i.

3 +4, @ i.

olp {$4.

5 d, / hv gh od irupd f Revhuydflöq= Vl olp{$4. {5 . e  +6 . d, @ f frq f 9@ 3 hqwrqfhv olp{$4 { .e{+6. 4 3 oxhjr ho oðplwh qr h{vlwh Rwud irupd gh uhvroyhu ho sureohpd/ hv ghwhuplqdu frqglflrqhv sdud txh i vhd frqwlqxd +Vl qr or ixhud i qr vhuðd ghulydeoh,/ gh grqgh Rewhqhprv> olp i +{, @ olp i +{, @ i+4, {$4. {$4 or txh htxlydoh d ghwhuplqdu od frqglflöq [email protected] 3 3 | oxhjr fdofxodu i +4, | i. +4, xvdqgr od frqglflöq e @ 5 . d1

Hmhuflflr 6:6 Vhd



{?3 {8 . {5 . 9 D+frv+6{, . 6 vhq {, . E+{ . 4, {  3 Ghwhuplqdu D | E hq U sdud txh i vhd ghulydeoh hq { @ 31 i+{, @

Vroxflöq= Sulphur ghwhuplqhprv frqglflrqhv sdud txh i vhd frqwlqxd hq { @ 3 olp i+{, @ olp i+{, @ olp i+{, @ i +3, {$3 {$3. {$3

olp i+{, @ olp D+frv+6{, . 6 vhq {, . E+{ . 4, {$3. {$3. @ D . E @ i+3, olp i+{, @ olp {8 . {5 . 9 @ 9 {$3 {$3

Sru or wdqwr D . E @ 9 Fdofxohprv i 3 +3, sru ghqlflöq i 3  +3, @ olp {$3

i +{,  i +3, {

{8 . {5 . 9  +D . E, { {8 . {5 . 9  9 sxhv D . E @ 9 @ olp { {$3 {8 . {5 @ olp { {$3 @ olp {+{6 . 4, @ 3 {$3

@

olp {$3

8141 GHULYDGD

5<:

Dkrud sru od ghuhfkd i 3 . +3, @ olp. {$3

i +{,  i +3, {

shur

olp {$3.

|

D+frv 6{ . 6 vhq {, . E+{ . 4,  +D . E, { D frv 6{ . 6D vhq { . E{  D @ olp. { {$3 D+frv 6{  4, 6D vhq { E{ . . @ olp. { { { {$3

@

olp {$3.

6D vhq { vhq { @ 6D olp. @ 6D { { {$3 E{ @ olp. E @ E olp. {$3 { {$3

qrv idowd fdofxodu olp {$3.

D+frv+6{,  4, {

frv5 +6{,  4 @ D olp. {$3 {+frv+6{, . 4,  vhq5 +6{, @ D olp. {$3 {+frv+6{, . 4, vhq+6{,  vhq+6{,  @ D olp. { frv+6{, . 4 {$3  vhq+6{, vhq+6{, @ 6D olp  olp . . 6{ {$3 {$3 frv+6{, . 4 @ 6D  +4,  3 @ 3

3 +3, @ i3 +3,/ hv ghflu/ 6D . E @ 3/ shur Oxhjr i 3 . +3, @ 6D . E/ i vhuä ghulydeoh hq { @ 3 vl i. D . E @ 9/ oxhjr uhvroylhqgr ho vlvwhpd whqhprv D @ 6 | E @ <1 Hmhuflflr 6:7 Vhd

l, Ghwhuplqdu i 3 +{,1

; A A ?

6{5  9{  ; { ? 4 4  { ? 4 8{7  7{5 i+{, @ 4 {@4 A A = {A4 7{5 frv+{  4, . 6{  9

ll, Fdofxoh ho ydoru gh D grqgh Vroxflöq=

D @ i 3 +5,i 3 +4@5,  i 3 + . 4,

d, Vhd { ? 4 hqwrqfhv i+{, @ 6{5  9{  ;/ i hv ghulydeoh sru dojheud gh ixqflrqhv ghulydeohv | i 3 +{, @ 9{  91

e, Vhd 4 ? { ? 4 hqwrqfhv i+{, @ 8{7  7{5 1 Sru dojheud gh ixqflrqhv ghulydeohv/ whqhprv i 3 +{, @ 53{6  ;{1 f, Vhd { A 4 hqwrqfhv i+{, @ 7{5 frv+{  4, . 6{  91 Sru dojheud gh ixqflrqhv ghulydeohv whqhprv i 3 +{, @ ;{ frv5 +{  4, . 7{5  + vhq+{  4,, . 6

Qrwhprv txh i hv frqwlqxd hq { @ 4 | hq { @ 41

5<;

FKDSWHU 81 GHULYDGD

g, Fdofxohprv i 3 +4, sru ghqlflöq

i+{,  i+4, {.4 7 8{  7{5  4 @ olp . {.4 {$4 6 +8{  8{5 . {  4,+{ . 4, @ olp . {.4 {$4 6 5 @ olp 8{  8{ . {  4 @ 45 {$4.

3 +4, @ i.

olp {$4.

i+{,  i+4, {.4 5 6{  9{  ;  4 @ olp {.4 {$4 6+{ . 4,+{  6, @ olp {.4 {$4 @ olp  6+{  6, @ 45 {$4

3 +4, @ i

3 +4, @ i3 +4, @ 45 Sru or wdqwr i.

h, Idowd fdofxodu i 3 +4,1

3 +4, @ i. @ @ @ @ Frpr olp{$4. 6+{{44, @ 6 | olp {$4. @

@ @ @ @ @ @

olp {$4. olp {$4. olp {$4. olp {$4. olp {$4.

olp {$4

i +{,  i +4, {4 7{5 frv+{  4, . 6{  9  4 {4 7{5 frv+{  4, . 6{  6  7 {4 5 7+{ frv+{  4,  4, 6{  6 . {4 {4 7+{5 frv+{  4,  4, 6+{  4, . {4 {4

7+{5 frv+{  4,  4, {4 +{5 frv+{  4,  4, +{5 frv+{  4, . 4 olp. 7  5 {4 { frv+{  4, . 4 {$4 7 5 +{ frv +{  4,  4, olp 7 {$4. +{  4,+{5 frv+{  4, . 4, +{7 +4  vhq5 +{  4,,  4, olp 7 {$4. +{  4,+{5 frv+{  4, . 4, +{7 vhq5 +{  4,, +{7  4,  7 olp. 5 {$4 +{  4,+{ frv+{  4, . 4, +{  4,+{5 frv+{  4, . 4, {7 vhq+{  4, vhq+{  4, +{  4,+{ . 4,+{5 . 4,   5 7 olp. 5 +{  4, +{ frv+{  4, . 4, {$4 +{  4,+{ frv+{  4, . 4, vhq+{  4, +{ . 4,+{5 . 4, {7 vhq+{  4, 7 olp  olp  5 +{  4, +{ frv+{  4, . 4, {$4. +{5 frv+{  4, . 4, {$4. 7 7 43@; 5

8141 GHULYDGD

5<<

3 +4, @ ; . 6 @ 44 Sru or wdqwr i.

i+{,  i+4, {4 7 8{  7{5  4 @ olp {4 {$4 +8{6 . 8{5 . { . 4,+{  4, @ olp {4 {$4 @ olp 8{6 . 8{5 . { . 4 @ 45 {$4

3 +4, @ i

olp {$4

3 +4, 9@ i3 +4,1 Sru or wdqwr i 3 +4, qr h{lvwh1 Oxhjr i. Dvð ho grplqlr Grpi 3 @ U  i4j1 i 3 +{, @

; ?

9{  9 {  4 4 ? { ? 4 53{6  ;{ = ;{ frv+{  4,  7{5 vhq+{  4, . 6 {A4

ll, i 3 +5, @ 4;/ i+4@5, @ 6@5/ i 3 + . 4, @ ;+ . 4,+4, . 6 oxhjr D @ i 3 +5,i 3 +4@5,  i 3 + . 4, @ 4; 

6 . ;+ . 4, . 6 5

Xqd khuudplhqwd ýwlo | vlpsoh sdud ghwhuplqdu vl xqd ixqflöq hv ghulydeoh hq dojýq sxqwr hv od vljxlhqwh sursrvlflöq txh vh sxhgh ghprvwudu hq edvh do whruhpd gh O*Krslwdo txh yhuhprv päv dghodqwh1 Sursrvlflöq 81< Vl i hv frqwlqxd hq d1

d, Vl orv oðplwhv olp{$d. i 3 +{,/ olp{$d i 3 +{, h{lvwhq | vrq ljxdohv d O 5 U hqwrqfhv i hv ghulydeoh hq d | i 3 +d, @ O1 e, Vl orv oðplwhv olp{$d. i 3 +{, / olp{$d i 3 +{, h{lvwhq +vrq qýphurv uhdohv, | vrq glihuhqwhv hqwrqfhv i qr hv ghulydeoh hq d1 Frqvlghuhprv od ixqflöq gdgd hq ho hmhuflflr 6:7 ; { ? 4 6{5  9{  ; A A ? 4  { ? 4 8{7  7{5 i+{, @ 4 {@4 A A = {A4 7{5 frv+{  4, . 6{  9

Frpr i hv frqwlqxd hq { @ 4/ sxhv dghpäv i 3 +{, @

; A A ?

olp i+{, @ olp i+{, @ 4 {$4. {$4

9{  9 { ? 4 53{6  ;{ 4 ? { ? 4 4 {@4 A A = ;{ frv+{  4,  7{5 vhq+{  4, . 6 {A4

| h{lvwhq olp{$4. i 3 +{,> olp{$4 i 3 +{, sxhv |

olp i 3 +{, @ olp ;{ frv+{  4,  7{5 vhq+{  4, . 6 @ 44 {$4. {$4. olp i 3 +{, @ olp 53{6  ;{ @ 45 {$4 {$4

633

FKDSWHU 81 GHULYDGD

shur olp{$4. i 3 +{, 9@ olp{$4 i 3 +{, oxhjr vhjýq od sursrvlflöq dqwhulru i qr hv ghulydeoh hq { @ 41 Vlq hpedujr> i hv ghulydeoh hq { @ 4 sxhv olp i 3 +{, @ olp 53{6  ;{ @ 45 {$4. {$4.

|

olp i 3 +{, @ olp  9{  9 @ 45 {$4 {$4

| i hv frqwlqxd hq 41 Revhuydflöq= Qrwh txh od klsöwhvlv gh frqwlqxlgdg hv ixqgdphqwdo1 Or srghprv yhu hq ho vljxlhqwh hmhpsor1  8{ . : {  6 i +{, @ 8{ . ; { ? 6

Hq hvwh fdvr vh wlhqh txh olp{$6. i 3 +{, @ 8 @ olp{$6 i 3 +{,/ shur i qr hv ghulydeoh hq { @ 6 sxhv i qr hv frqwlqxd hq { @ 6 Yroylhqgr do hmhpsor  6 . d{ {  4 i +{, @ {5 . e { A 4 i hv frqwlqxd hq { @ 4 vl | vöor vl e @ 5 . d1 Dghpäv sdud { 9@ 4/ vh wlhqh

i 3 +{, @



d {?4 5{ { A 4

Frpr orv oðplwhv olp{$4. i 3 +{, > olp{$4 i 3 +{, h{lvwhq> sxhv olp i 3 +{, @ d> olp. i 3 +{, @ 5 {$4 {$4

hqwrqfhv> i vhuä ghulydeoh hq { @ 4 vl

olp i 3 +{, @ olp i 3 +{, {$4. {$4

hv ghflu/ d @ 5 | frpr e @ 5 . d/ hqwrqfhv e @ 71 Dqdorjdphqwh hq ho hmhuflflr 6:6  {8 . {5 . 9 {?3 i+{, @ D+frv+6{, . 6 vhq {, . E+{ . 4, {  3

i hv frqwlqxd hq { @ 3 vl D . E @ 91 Dghpäv  8{7 . 5{ 3 i +{, @ D+6 vhq+6{, . 6 frv {, . E Orv oðplwhv olp{$3. i 3 +{, | olp{$3 i 3 +{, h{lvwhq sxhv

|

{?3 {A3

olp i 3 +{, @ olp. D+6 vhq+6{, . 6 frv {, . E @ 6D . E {$3. {$3

oxhjr i vhuä ghulydeoh vl gh grqgh D @ 6 | E @ <1

olp i 3 +{, @ olp 8{7 . 5{ @ 3 {$3 {$3 6D . E @ 3 frq D . E @ 9

8141 GHULYDGD

81417

634

Ghulydgd gh xqd ixqflöq Lqyhuvd

Frurodulr 8143 Fäofxor gh ghulydgd gh xqd ixqflöq lqyhuvd Vhd i = [ $ Uhfi  U ixqflöq lqyhuwleoh | ghulydeoh/ ghqrwhprv sru j d od lqyhuvd i 4 / dvð j = Uhfi $ [1 Vl i hv ghulydeoh hq d 5 [ _ [ 3 frq ghulydgd i 3 +d, 9@ 3 | j hv ghulydeoh frqwlqxd hq i +d, @ e1 4 g+i ,4 me @ gi g{ g{ md

Hv ghflu/ od ghulydgd gh od ixqflöq lqyhuvd i 4 hq e vh fdofxod xvdqgr ho lqyhuvr pxowlsolfdwlyr gh od ghulydgd gh i hq d +uhfrughprv txh i 3 +d, hv qýphur uhdo qr qxor,1

Ghprvwudflöq= Vh vdeh txh i  i 4 +{, @ { | txh od ghulydgd gh { hq wrgr sxqwr hv 4/ xvdqgr od uhjod gh od fdghqd

Frpr i+d, @ e

+i  i 4 ,3 +e, @ 4 i 3 +i 4 +e,,  +i 4 ,3 +e, @ 4 i 3 +d,  +i 4 ,3 +e, @ 4

|d txh i 3 +d, hv qr fhur srghprv rewhqhu od ljxdogdg | hq +i 4 ,3 +e, @ r fdpeldqgr gh qrwdflöq

4 i 3 +d,

4 gi 4 me @ 3 g{ i +d,

Hmhpsor 6:8 Fdofxohprv od ghulydgd gh i+{, @ oq+{, hq e 5 U.



Uhfrughprv txh oq+{, hv od ixqflöq lqyhuvd gh h{s+{, @ h{ / dvð oq e @ d hv htxlydohqwh d e @ hd 1 Xvdqgr frurodulr 4 4 4 g oq me @ g h{s @ d @ g{ h e m g{ d Gh hvwd irupd hq qxhvwud wdeod gh ghulydgdv ghehprv dqrwdu 4 g oq { @ g{ {

r

oq3 +{, @

4 {

vl { A 3

Hmhpsor 6:9 Fdofxohprv dkrud od ghulydgd gh dufwdq+{,1 Uhfrughprv txh od lqyhuvd orfdo gh wdq+{, hv dufwdq+{,/ hv ghflu/ sdud { dsursldgrv vh wlhqh dufwdq+wdq+{,, @ {

|

{ @ wdq+dufwj+{,,

Xvdqgr frurodulr hq xq sxqwr e @ wdq+d,/ grqgh vhf5 +d, 9@ 3 whqhprv= g dufwdq 4 4 +{,me @ @ 3 g{ wdq +d, vhf5 +d,

Frq ho remhwr gh ghmdu ho uhvxowdgr hq wìuplqrv gh e xvduhprv od lghqwlgdg wuljrqrpìwulfd Gh hvwd irupd

vhf5 +{, @ wdq5 +{, . 4

635

FKDSWHU 81 GHULYDGD 4 4 g dufwdq @ 5 +{,m{@e @ g{ e .4 wdq5 +d, . 4

Frpr e hud vöor xq uhsuhvhqwdqwh/ whqhprv

dufwdq3 +{, @ Dqäorjdphqwh vh sxhgh suredu txh

4 {5 . 4

duffrwj3 +{, @ 

4 { .4 5

vöor txh dtxð gheh xwlol}du rwud lghqwlgdg wuljrqrpìwulfd1 + fvf5 { @ fwj 5 { . 4, Revhuydflöq= Dkrud txh khprv orjudgr xqd wìfqlfd sdud hqfrqwudu ghulydgd gh ixqflrqhv lqyhuvdv fuhhprv txh hv exhqr kdfhu qrwdu or txh hv ho uhvxowdgr gho whruhpd gh od ixqflöq lqyhuvd1

Whruhpd 8144 +Ixqflöq lqyhuvd, Vhd i = L $ U/ xqd ixqflöq/ L xq yhflqgdg gh d wdo txh i vhd ghulydeoh hq d frq ghulydgd qr qxod +i 3 +d, 9@ 3,1 Hqwrqfhv h{lvwh K yhflqgdg gh d+K  L, | M yhflqgdg gh i +d, wdo txh i = K $ M hv lqyhuwleoh frq lqyhuvd glihuhqfldeoh i 4 = M $ K yhulfd od vljxlhqwh ljxdogdg hv ghflu

gi 4 4 me @ 3 4 g{ i +i +e,

4 gi 4 me @ +i 4 ,3 +e, @ 3 > grqgh d @ i 4 +e, g{ i +d,

Revhuydflöq= Vxsrqjdprv txh i | j vrq ixqflrqhv ghulydeohv | txh +i j, @ Lg hqwrqfhv qhfhvduldphqwh vxv ghulydgdv vrq qr qxodv1 Sru uhjod gh od fdghqd vh wlhqh i 3 +j+{,,j 3 +{, @ 4

| frpr vh wudwd gh od pxowlsolfdflöq gh grv qýphurv uhdohv/ dperv ghehq vhu qr qxorv/ pdv dýq/ xqr gheh vhu ho lqyhuvr pxowlsolfdwlyr gho rwur1 Uhfrughprv txh vhqr hv xqd ixqflöq gh U hq U txh qr hv lq|hfwlyd | wdpsrfr hv vreuh|hfwlyd/ frpr vroxflöq d hvwr ýowlpr srguðdprv frqvlghuduod frpr vhq = U $ ^4> 4`/ shur ho sureohpd gh od lq|hfwlylgdg vhq dýq shupdqhfh1 Hq ghqlwlyd qr vh sxhgh shqvdu txh vhq whqjd lqyhuvd gh ^4> 4` hq U/ shur gg{ +{,m{@ @ 4 txh hv glvwlqwr gh 3/ oxhjr ho whruhpd gh od ixqflöq lqyhuvd qrv jdudqwl}d od h{lvwhqfld gh xqd lqyhuvd1 BcFxäo hv od frqwudglfflöq hqwuh hvwdv ýowlpdv grv iudvhvB Qlqjxqd/ sxhvwr txh ho whruhpd gh od ixqflöq lqyhuvd qr kdeod gh xqd ixqflöq lqyhuvd gh vhq frq grplqlr hq ^4> 4` | uhfruulgr U/ vlqr txh qrv jdudqwl}d od h{lvwhqfld gh xqd lqyhuvd orfdo1 Hvwr vljqlfd txh h{lvwh M yhflqgdg gh i +d, | K yhflqgdg gh d wdo txh i = K $ M wlhqh lqyhuvd1 Hq ho fdvr gh i+{, @ vhq { h{lvw M yhflqgdg gh fhur grqgh i+, @ 3 | K yhflqgdg gh  wdo txh vhq = K $ M wlhqh lqyhuvd1 Frq ho whruhpd gh od ixqflrq lqyhuvd srghprv orjudu yduldu lqyhuvdv orfdohv |d txh edvwd wrpdu sxqwrv {3 txh vdwlvidjdq od klsrwhvlv g vhq { m{3 9@ 3 g{ Hq orv sxqwrv grqgh vh dqxod od ghulydgd gh vhqr +sru hmhpsor hq @5, hv lpsrvleoh hqfrqwudu xqd lqyhuvd orfdo gh hood/ |d txh sdud wrgr lqwhuydor delhuwr txh frqwhqjd ho sxqwr/ vhqr vlhpsuh vhuä qr lq|hfwlyd1 \d txh khprv kdeodgr gh od ixqflöq i +{, @ vhq {/ fdofxohprv od ghulydgd gh i 4 +{, @ dufvhq {/ sdud { dsursldgr hq ho hmhuflflr txh vljxh Hmhuflflr 6:: Fdofxodu od ghulydgd gh i 4 +{, @ dufvhq {

8151 GHULYDGD GH RUGHQ VXSHULRU

636

Vroxflöq= Vhd i +{, @ vhq+{, grqgh vhq =`  @5> @5^$`  4> 4^ hqwrqfhv frvd A 3 vl d 5`  @5> @5^ | +i 4 ,3 +e, @ frv4 d s Vhd } @ frv d hqwrqfhv } 5 @ frv5 d @ 4  vhq5 d1 oxhjr } @ 4  vhq5 d sxhv } A 3 shur d dufvhq e oxhjr s s } @ 4  vhq+dufvhq+e,, } @ 4  e5 Sru or wdqwr

dufvhq3 { @ s

4 4  {5

Hq fdpelr vl frqvlghudprv vhq =`@5> 6@5^$`  4> 4^ hqwrqfhv frv d ? 3 vl d 5`@5> 6@5^ | +i 4 ,3 @ shur s s frv d @  4  vhq5 d @  4  vhq+dufvhq e,,

4 frv

d

sru or wdqwr vl frqvlghudprv i 4 sru dufvhq {/ vh whqguä

4 +dufvhq,3 +{, @ s 4  {5

frqvlghudqgr vhq =`@5> 6@5^$`  4> 4^= Revhuydflöq= Vh dfrvwxpeud d uhvhuydu dufvhq d od lqyhuvd gh od ixqflöq vhqr ghqd hq vx udpd sulqflsdo/ hv ghflu vhq =`  @5> @5^$`  4> 4^ |

4 +dufvhq,3 +{, @ s 4  {5

Dqäorjdphqwh vh fdofxod odv ghulydgd gh odv lqyhuvdv gh frvhqr/ frwdqjhqwh/ vhfdqwh | frvhfdqwh/ ghqlgd hq vx udpd sulqflsdo rewhqlhqgr=

815

4 +dufvhq,3 +{, @ s 4  {5 4 +dufwj,3 +{, @ 4 . {5 4 +dufvhf,3 +{, @ s { {5  4

4 4  {5 4 +duffrwj,3 +{, @ 4 . {5 4 +duffrvhf,3 +{, @ s { {5  4 +duffrv,3 +{, @ s

Ghulydgd gh Rughq Vxshulru

Frqvlghuhprv od ixqflöq i+{, @ 6{7 . 8 vhq5 {/ txh frpr |d khprv ylvwr vx ghulydgd hv gi +{, @ 45{6 . 43 vhq { frv { @ j+{, g{

Srghprv qrwdu txh j wdpelìq hv ghulydeoh | wlhqh frpr ghulydgd d gj +{, @ 69{5 . 43 frv5 {  43 vhq5 { @ k+{,/ qxhydphqwh srghprv fdofxodu od ghulydgd gh hvwd | rewhqhu g{ gk +{,= g{ gi od ixqflöq k hv od ghulydgd gh vhjxqgr rughq Vh glfh txh j hv od ghulydgd gh sulphu rughq | vh dqrwd g{ 5 gi / gk fruuhvsrqgh d od ghulydgd gh whufhu rughq gh i | vh dqrwd gi 6 1 gh i | vh dqrwd g{ 5 g{ g{6

gi vx ghulydgd/ vh ghqh od ghulydgd gh Ghqlflöq 8145 Vhd i xqd ixqflöq ghulydeoh hq `d> e^ | vhd g{ vhjxqgr rughq gh i hq f 5`d> e^ frpr gi +f . k,  gi +f, g{ olp g{ k$3 k

637

FKDSWHU 81 GHULYDGD

5 | vh ghqrwd hvwh oðplwh sru gg{i5 +f,1 Vh glfh txh i wlhqh ghulydgd gh rughq 5 hq `d> e^ vl i wlhqh ghulydgd gh vhjxqgr rughq hq fdgd sxqwr gh `d> e^1 H{lvwlhqgr od ghulydgd gh vhjxqgr rughq gh i hq `d> e^ vh sxhgh ghqlu hq irupd dqdorjd od ghulydgd gh whufhu rughq hq f 5`d> e^ frpr ho vljxlhqwh oðplwh +vl hv txh h{lvwh ,1

g5 i +f . k,  g5 i +f, 5 g{ g{5 olp k$3 k

dqäorjdphqwh vh ghqh od ghulydgd gh rughq q . 4 frpr ho oðplwh txh lqyroxfud od ghulydgd gh rughq q | q.4 vh dqrwduä sru gg{q.4i +f,1

Qrwdflrqhv=

G5 i +f, @ i 33 +f, @

g5 i +f,> g{6

Vh ghqhq orv vljxlhqwhv frqmxqwrv F 3 `d> e^ F 4 `d> e^ F 5 `d> e^ F q `d> e^

@ @ @ @

ii ii ii ii

=`d> e^$ U =`d> e^$ U =`d> e^$ U =`d> e^$ U

= = = =

G6 i +f, @ i 333 +f, @ i i i i

hv hv hv hv

g6 i +f,> g{6

Gq i+f, @ i +q, +f, @

gq i +f,= g{q

frqwlqxd hq `d> e^j ghulydeoh hq`d> e^ frq ghulydgd frqwlqxdj grv yhfhv ghulydeoh hq`d> e^ frq ghulydgd frqwlqxd j q yhfhv ghulydeohv hq`d> e^frq ghulydgd frqwlqxdj

Frq hvwrv frqmxqwrv srghprv ghqlu od vljxlhqwh ixqflöq/ oodpdgd rshudgru glihuhqfldeoh1

Dvð

G = F q `d> e^ $ F q4 `d> e^ gi +{, i :$ Gi +{, @ g{

41 vl i +{, @ 6{5 . vhq {/ Gi +{, @ 9{ . frv { s 4 51 vl j+{, @ oq+{ . 4, . { . 4/ Gj+{, @ {.4 . s 6 61 vl k+{, @ {5 . {6 / Gk+{, @ 56 {4@6 . 6{5 1

s4{.4 1

5

Qrwhprv txh od ixqflöq j 5 F 4 `3> 5^ shur Gj 95 F 4 `  5> 5^1 Dqdorjdphqwh vh sxhgh ghqlu ho rshudgru glihuhqfldo gh rughq n/ Gn

Dvð

Gn = F q `d> e^ $ F qn `d> e^ > q  n n i :$ Gn i+{, @ gg{in +{,

41 vl i +{, @ vhq 8{/ hqwrqfhv G7 i +{, @ 87 vhq 8{

51 vl i +{, @ h{ / hqwrqfhv G53 i +{, @ h{ 1

61 vl i +{, @ {8 / hqwrqfhv G: i +{, @ 3 | G8 i +{, @ 8$1

71 vl i +{, @ h5{ hqwrqfhv G: i+{, @ 5: h5{ 1

Frpr G3 > G4 > G5 > G6 > G7 > G8 > ===/ vrq ixqflrqhv/ vh sxhgh ghqlu od vxpd | pxowlsolfdflöq sru hvfdodu gh hoodv rewhqlhqgrvh rwud ixqflrqhv/ sru hmhpsor1

;G3 . 6G . :G8  7G6 = F q `d> e^ $ F q4 `d> e^ 8 6 gi i :$ ;i . g{ +{, . : gg{i6 +{,  gg{i6 +{, Dtxð vh hqwlhqgh txh G3 = F q `d> e^$ F q `d> e^ | G3 i @ i 1

8151 GHULYDGD GH RUGHQ VXSHULRU

638

Hmhuflflr 6:; Vhd O @ 6G5 . G . 7G3 | i+{, @ {6 . :{ vhq {/ j+{, @ 8{5 . frv {/ k+{, @ oq { . wj {1 Fdofxoh Oi> Oj> Ok1 41

Oi Oi

g5 i gi . . 7i g{5 g{ g @ 6^9{ . +: vhq { . :{ frv {,` . 6{5 . :vhq{ . :{ frv { . 7{6 . 5;{ vhq { g{ @ 4;{ . 6^: frv { . : frv {  :{ vhq {` . : vhq { . :{ frv { . 7{6 . 5;{ vhq { @ 4;{ . 7{6 . +75 . :{, frv { . +: . :{, vhq {

@ 6

51 Oj

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g| | g5 | gh xqd ixqflöq | hq od yduldeoh { ghqlgd lpsoðflwdphqwh sru od Hmhpsor 6<6 Fdofxohprv g{ g{ uhodflöq s {  wj | @ 3 Vroxflöq=

s {  wj |

4 g| s 4  +vhf5 |, s 5 | g{ s 5 | s g| s @ s @ 5 | frv5 | g{ vhf5 |

@ 3@ @ 3

g g{ +81;,

Xvdqgr od uhjod gh od fdghqd vh wlhqh

g| Idfwrul}dqgr sru g{

s 4 s s s 4 g| g5 | @ 5 s frv5 |  5 |5 frv | vhq | s g{5 5 | 5 | g{ s frv5 | g5 | s g| s @ +  5 frv+ |, vhq+ |,, s g{5 | g{

g| | vlpsolfdqgr vh rewlhqh Vxvwlwx|hqgr g{

Hmhuflflr 6<7 Yhultxh txh

g5 | s s s s @ 5 frv7 |  7 | frv6 + |, vhq+ |, g{5 s {  wj | @ 3

vdwlvidfh od hfxdflöq glihuhqfldo +{5 . 4,5

g5 | g| . 5{+{5 . 4, 5@3 5 g{ g{

+81<,

647

FKDSWHU 81 GHULYDGD

Vroxflöq= s Sxhvwr txh { @ wj | vh wlhqh Sru rwur odgr

+{5 . 4,5 @ +wj5

s s s | . 4,5 @ +vhf5 |,5 @ vhf7 |

5{+{5 . 4, @ 5 wj

s s |  vhf5 |

oxhjr sru 8143 / 8144 | sru 81;/ 81< gho hmhpsor dqwhulru vh wlhqh

+8143, +8144,

g5 | g| 5 . 5{+{5 . 4, 5 g{ g{ s s s s s s s s s @ vhf7 |^5 frv7 |  7 | frv6 + |, vhq+ |,` . 5 wj | vhf5 + |^5 | frv5 |`  5 s s s s @ 5  7 | wj | . 7 | wj+ |,  5 @ 3

+{5 . 4,5

Revhuydflöq= Qrwhprv txh wrgdv odv hfxdflrqhv gho hmhpsor dqwhulru sxhghq vhu hvfulwd gh od irupd I +{> |, @ 3 grqgh i hq fdgd fdvr hv 41 I +{> |, @ {| 7 . {5 | . 6{6  | . 5

51 I +{> |, @ oq+{|, . wdq+|5 , . 4  | 6

61 I +{> |, @ | dufwdq+{,  h{|  5 5

71 I +{> |, @ frv+{5 . |,  5| . 6  |{

81 I +{> |, @ vhq+{5 , . |  {| 91 I +{> |, @ {| . | {  5

g| fruuhvsrqgh do fxrflhqwh gh odv ghulydgdv sdufldohv Wdpelìq srghprv qrwdu txh od ghulydgd uhtxhulgd g{ gh od ixqflöq I pxowlsolfdgd sru 41 CI g| C{ @  CI g{ C| Yhdprv hq ho fdvr 41

CI C{ CI C|

Sru or wdqwr

51

@ | 7 . 5{| . <{5 @ 7{| 6 . {5  4

CI 7 5 C{ @  | . 5{| . <{  CI 7{| 6 . {5  4 C|

txh hv mxvwdphqwh od ghulydgd uhtxhulgd hq ho hmhuflflr d, CI C{ CI C|

@ @

4 4 | @ {| { 4  { . vhf5 +| 5 ,5|  6| 5 {|

8161 IXQFLRQHV GHILQLGDV LPSOÐFLWDPHQWH Sru or wdqwr

648

CI C{ @  4  4 . 5| vhf5 +|5 ,  6|5 ,  CI { |4 C|

g| 1 | hvwr hv ljxdo d g{ Hvwr plvpr rfxuuh frq orv hmhuflflrv gh +f, do +i,1 g| / +hq fdgd fdvr, hv srvleoh vl Ilqdophqwh srghprv qrwdu txh od h{lvwhqfld gh g{ CI 9@ 3= C|

Hvhqfldophqwh hvwr hv or txh hvwdeohfh ho whruhpd gh od ixqflöq lpsoðflwd1

Whruhpd 8147 +Ixqflöq Lpsoðflwd, Vhd I = U5 $ U xqd ixqflöq wdo txh 41 H{lvwh xq sxqwr +{3 > |3 , 5 U5 / I +{3 > |3 , @ 3

51 I hv xqd ixqflöq frqwlqxdphqwh glihuhqfldeoh hq xqd yhflqgdg gho sxqwr +{3 > |3 , 5 U5 + sru hmhpsor xq glvfr vlq rulood txh frqwhqjd d +{3 > |3 ,,1 61 CI C| +{3 > |3 , 9@ 31

Hqwrqfhv h{lvwh xqd ixqflöq lpsoðflwd | @ i +{, ghqlgd hq xq lqwhuydor delhuwr L txh frqwlhqh d {3 | fx|r uhfruulgr frqwlhqh d |3 wdo txh |3 @ i +{3 , | sru or wdqwr I +{3 > i +{3 ,, @ 3/ päv dýq I +{> i+{,, @ 3 sdud wrgr { 5 L +hv ghflu qr wdq vöor +{3 > |3 , vdwlvidfh I +{> |, @ 3 vlqr txh wdpelìq orv lqqlwrv sxqwrv +{> i +{,,> { 5 L1 \ qdophqwh/ od phqflrqdgd ixqflöq i hv ghulydeoh hq L | vx ghulydgd hvwä gdgd sru od h{suhvlöq

CI g| C{ @  CI g{ C|

Od ghprvwudflöq qr fruuhvsrqgh d xq fxuvr gh hvwh qlyho/ vlqr txh sdud xqr fruuhvsrqglhqwh d fäofxor hq yduldv yduldeohv1 Sdud fodulfdu xq srfr odv klsöwhvlv/ h{solfduhprv hvwd hq od revhuydflöq | hmhpsor txh vljxh1 Revhuydflöq= Ho frqfhswr gh glihuhqfldelolgdg sdud ixqflrqhv gh grv yduldeohv hv xq srfr päv frpsolfdgr txh kdvwd dkrud | vhjxur txh or yhuhprv hq xq fxuvr gh fäofxor hq yduldv yduldeohv1 Frpr hmhpsor gh ixqflrqhv gh grv yduldeohv frqwlqxdphqwh glihuhqfldeohv phqflrqduhprv odv vljxlhqwhv ixqflrqhv 41 Odv ixqflrqhv srolqrpldohv hq grv yduldeohv vrq frqwlqxdphqwh glihuhqfldeohv

51 Odv ixqflrqhv wdohv frpr vhqr/ frvhqr/ h{srqhqfldo/ orjdulwpr | udl} q0 ìvlpd vrq frqwlqxdphqwh glihuhqfldeohv hq vxv uhvshfwlyrv grplqlrv1

61 Od frpsxhvwd r dojxqd frpelqdflöq dojheudlfd +vxpd/ uhvwd/ surgxfwr/ r glylvlöq gh hoodv, wdpelìq vrq frqwlqxdphqwh glihuhqfldeoh hq vxv grplqlrv gh ghqlflöq1

Hmhpsor 6<8 Odv vljxlhqwhv ixqflrqhv vrq frqwlqxdphqwh glihuhqfldeohv 41 I +{> |, @ {6 | . 7{5 | . |5 { . 4

51 I +{> |, @ vhq+{5 |, . h{| . |{ . 4 s 5 61 I +{> |, @ vhq+h{| , . {6 | . | 5 { . 4 71 I +{> |, @ { oq |

649

FKDSWHU 81 GHULYDGD

Hmhpsor 6<9 Vxsrqjdprv txh od ixqflöq

I +{> |, @ |5 vhq {  h{.| . 4

hv frqwlqxdphqwh glihuhqfldeoh hq U5 / hv ghflu/ vdwlvidfh klsöwhvlv +5, gho whruhpd gh od ixqflöq lpsoðflwd

Ho sxqwr +{3 > |3 , @ +3> 3, vdwlvidfh od hfxdflöq I +{> |, @ 31 Ghwhuplqh vl h{lvwh xqd ixqflöq | @ i+{, ghqlgd lpsoðflwdphqwh sru od hfxdflöq I +{> |, @ 3 | glihuhqfldeoh hq xq lqwhuydor delhuwr gh {3 @ 3 Vroxflöq= Frqvlghudqgr od hfxdflöq I +{> |, @ 3/ hv ghflu/ |5 vhq {  h{.| . 4 @ 3 | ghulydqgr lpsoðflwdphqwh frq uhvshfwr d { vh wlhqh>

5||3 vhq { . | 5 frv+{,  +4 . |3 ,h{.| @ 3

or txh lpsolfd

g| h{.|  |5 frv { @ g{ 5| vhq {  h{.| CI CI Sxhvwr txh C| +3> |, @ 4/ |d txh C| +{> |, @ 5| vhq {  h{.| | I +3> 3, @ 3 sru ho whruhpd gh od ixqflöq lpsoðflwd/ h{lvwh xqd ixqflöq | @ i +{, glihuhqfldeoh hq X +xq lqwhuydor delhuwr gh { @ 3,/ ghqlgd lpsoð flwdphqwh sru od hfxdflöq I +{> |, @ 3 | i +3, @ 31 |3 @

Hmhpsor 6<: Ghwhuplqh +vl h{lvwh, od uhfwd wdqjhqwh d od fxuyd | @ i +{, hq ho sxqwr S @ +3> 3,1 Vroxflöq= Sxhvwr txh S @ +3> 3, vdwlvidfh od hfxdflöq I +{> |, @ 3 | g| g{ hydoxdgd hq S @ +3> 3, h{lvwh/ hqwrqfhv h{lvwh od uhfwd wdqjhqwh hq ho sxqwr +3> 3,/ fx|d hfxdflöq hv |  |3 @ p+{  {3 , g| grqgh +{3 > |3 , @ +3> 3, | p @ g{ m+3>3, @ 4/ hv ghflu/ | @ {

Hmhpsor 6<; Vxsrqjd txh od ixqflöq

I +{> |, @ { . { oq |  vhq+{|, . | 5 

 5

hv frqwlqxdphqwh glihuhqfldeoh hq xqd yhflqgdg gho sxqwr + 5 > 4, +or hv sru revhuydflöq dqwhulru,1 Ghwhuplqh vl h{lvwh xqd ixqflöq { @ i +|, +{ hq ixqflöq gh |, ghqlgd lpsoðflwdphqwh sru od hxdflöq I +{> |, @ 3 glihuhqfldeoh hq xq lqwhuydor delhuwr gh | @ 41 Vroxflöq=

8

Ghulydqgr lpsoðflwdphqwh frq uhvshfwr d | whqhprv

 g @3 m 5 g| 4 g{ g{ g{ . oq | . {  frv+{|,+| . {, . 5| @ 3 , g| g| | g| { g{ @ { frv+{|,   5| , +4 . oq |  frv+{|,, g| | {  5| { frv+{|,  g{ | @ , g| 4 . oq |  | frv+{|,  CI      Sxhvwr CI C{ + 5 > 4, @ 4 9@ 3/ |d txh C{ +{> |, @ 4 . oq |  | frv+{|, | I + 5 > 4, @ 5 . 5 oq 4  vhq+ 5 , . 4  5 @ 3 hqwrqfhv h{lvwh i = X  U $ U | :$ { @ i +|, { . { oq |  vhq+{|, . | 5 

ixqflöq glihuhqfldeoh hq X +xq lqwhuydor delhuwr gh | @ 4,/ ghqlgd lpsoðflwdphqwh sru I +{> |, @ 31 8 Qrwh

txh dkrud vh slgh {

'

i E| / s ru or wdqwr ghe h vdwlvidfhuvh od kls öwhvlv

CI E  >  ' f1 C{ 2

8161 IXQFLRQHV GHILQLGDV LPSOÐFLWDPHQWH

64:

Hmhpsor 6<< Ghwhuplqh n 5 U +vl h{lvwh, gh prgr txh od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd gh | @ i +{, hq ho sxqwr S @ +4> 5, sdvh sru ho ruljhq/ vdelhqgr txh | @ i +{, hv xqd ixqflöq glihuhqfldeoh hq xq lqwhuydor txh frqwlhqh d { @ 4/ ghqlgd lpsoðflwdphqwh sru  | 5 . n vhq+ |, @ {| . 5 5

Vroxflöq= Qrwhprv txh lqghshqglhqwh gho ydoru gh n/ S @ +4> 5, vdwlvidfh od hfxdflöq/  55 . n+vhq+ 5, @ 4  5 . 5 5

Ghulydqgr lpsoðflwdphqwh frq uhvshfwr d {

 g | 5 . n+vhq+ |, @ {| . 5 5 g{   , 5||3 . n frv+ |, |3 @ | . {| 3 5 5 n  , +5| . frv+ |,  {,| 3 @ | 5 5 | g| @ , |3 @ n g{ 5| . 5 frv+ 5 |,  {

hydoxdqgr hvwd ghulydgd hq ho sxqwr S @ +4> 5, whqhprv od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd hq ho sxqwr +4> 5,1 5 g| mS @ p@ g{ 7  n 4 5

Oxhjr hfxdflöq gh od uhfwd wdqjhqwh d od ixqflöq | @ i +{,/ ghqlgd lpsoðflwdphqwh/ hvwä gdgd sru |5@

5 +{  4, 6  n 5

Hvwd uhfwd sdvd sru ho ruljhq vl | vöor vl ho sxqwr +3> 3, vdwlvidfh vx hfxdflöq/ hv ghflu/ 5 @

gh grqgh rewhqhprv ho ydoru gh n1

5 +4, 6  n 5

n@

7 

Hmhpsor 733 Ghwhuplqh ho +r orv, sxqwr+v, grqgh od juäfd gh od uhodflöq {5 . | 5 @ 9{  7| . 8

wlhqh wdqjhqwh+v, sdudohod+v, d od uhfwd | @ 7 . { +juäfdu od vlwxdflöq, Vroxflöq= Ghulydqgr lpsoðflwdphqwh dperv odgrv gh od hfxdflöq

frq uhvshfwr d {/ whqhprv>

{5 . | 5 @ 9{  7| . 8

5{ . 5|| 3 +5| . 7,| 3 |3

@ 9  7| 3 @ 9  5{ 9  5{ > frq 5| . 7 9@ 3 @ 5| . 7

+8145,

64;

FKDSWHU 81 GHULYDGD

Od uhfwd wdqjhqwh d od juäfd hv sdudohod d od uhfwd | @ { . 7 vl | vöor vl | 3 +{3 , @ 4 sdud dojýq +{3 > |3 , shuwhqhflhqwh d od fxuyd1 Frpr +, +,

9  5{3 @4 5|3 . 7 9  5{3 @ 5|3 . 7 |3 @ 4  {3 | 3 +{3 , @

| ho sxqwr +{3 > |3 , gheh fxpsolu od hfxdflöq 8145/ whqhprv sru or wdqwr Uhhpsod}dqgr |3 @ 4  {3 whqhprv

{53 . |35 @ 9{3  7|3 . 8=

{53 . +4  {3 ,5 @ 9{3  7+4  {3 , . 8

gh grqgh

{3 @ 3 ö {3 @ 9

Fxdqgr {3 @ 3/ ho ydoru gh |3 hv 4 | fxdqgr {3 @ 9 vh wlhqh |3 @ 8 Sru or wdqwr orv sxqwrv shglgrv +yhu jxud BB, vrq S4 @ +3> 4,

Qrwhprv txh od uhodflöq

>

S5 @ +9> 8,

{5 . | 5 @ 9{  7| . 8

hv htxlydohqwh d>

+{  6,5 . +| . 5,5 @ 4;

oxhjr/ juäfdphqwh od uhodflöq hv xqd flufxqihuhqfld frq fhqwur hq +6> 5, | odv uhfwdv wdqjhqwhv sdudohodv d | @ 7 . { wlhqhq hfxdflöq |4@{ > |.8@{9 Hmhuflflr 734 Yhultxh vl od ixqflöq | @ i +{, ghqlgd lpsoðflwdphqwh sru +4 . {,h|@{ @ { hv vroxflöq gh {6 |% @ +{| 3  |,5 1

Vroxflöq=

+4 . {,h|@{

|3 {  | h|@{ . +4 . {,h|@{ {5 3 | {| {5

|3 {  |

shur +4 . {,h|@{ @ { hqwrqfhv |3 {  | |3

@ { @ 4

g g{

4  h|@{ +4 . {,h|@{ 4  h|@{ @ {5 +4 . {,h|@{

@

4  h|@{ { | |@{ @ 4h . {

@ {5

ghulydqgr qxhydphqwh hq irupd lpsoðflwd rewhqhprv

|3 {  | |3 {  | g5 i @ |% @ h|@{ . 5 g{ {5 {5

+8146,

8171 UHJOD GH O*KRSLWDO

64<

Sru or wdqwr

|% @ +4  h|@{ ,

shur gh 8146 vh wlhqh

4  h|@{ @

oxhjr uhhpsod}dqgr hq 8147

|3 {  | {5

+8147,

|3 {  | {

| 3 {  | |3 {  |  { { +|3 {  |,5 {5

|% @ |% @

gh grqgh {6 |% @ +|3 {  |,5 1 Hq frqvhfxhqfld od ixqflöq ghqlgd lpsoðflwdphqwh sru +4 . {,h|@{ @ { hv vroxflöq gh od hfxdflöq glihuhqfldo {6 |% @ +{| 3  |,5

817

Uhjod gh O*Krslwdo

Vh glfh txh ho fxrflhqwh ij++{{,, hv gho wlsr 33 hq { @ s + r txh ho oðplwh hv gho wlsr 33 , vl dpedv ixqflrqhv i | j wlhqh oðplwh 3 fxdqgr { wlhqgh d s1 Vl orv oðplwhv gh i+{, | j+{, vrq 4 r 4 fxdqgr { wlhqgh d s/ vh glfh txh ij++{{,, wlhqh od irupd 4 lqghwhuplqdgd 4 4 r ho oðplwh hv gho wlsr 4 Whruhpd 8148 +Uhjod gh O*Krslwdo, Vhd Ys xqd yhflqgdg gh s1 Vhdq i | j ixqflrqhv ghulydeohv hq Ys  isj1 Vl ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 r elhq 4 4 hq { @ s hqwrqfhv i +{, i 3 +{, olp @ olp 3 {$s j+{, {$s j +{,

3 vlhpsuh txh olp{$s ij3 ++{{,, h{lvwd r elhq olp{$s

i 3 +{, @ 41 j3 +{,

Revhuydflöq= Od uhjod gh O*Krslwdo wdpelìq sxhgh dsolfduvh d orv oðplwhv odwhudohv1 Hmhpsor 735 Fdofxohprv xvdqgr O*Krslwdo Vroxflöq= Hvwh oðplwh hv gh od irupd oxhjr fdofxohprv O @ olp{$3

Sru or wdqwr

3{ {3

3 3

sxhv

olp {$3

olp vhq+{, @ 3 {$3

vhq+{, { >

olp { @ 3 {$3

vhq + , + ,

O @ olp {$3

frv+{, @ frv+3, @ 4 4

vhq+{, vhq3 +{, olp @4 @ olp {$3 +{, {$3 +{,3 3 Revhuydflöq= Vl olp{$s ij++{{,, hv gh od irupd 33 | olp{$s ij3 ++{{,, qr h{lvwh/ qr srghprv dupdu txh olp{$s ij++{{,, qr h{lvwh/ frpr yhuhprv hq ho hmhpsor txh vljxh1

653

FKDSWHU 81 GHULYDGD

Hmhpsor 736 Vhd i+{, @ {5 frv+4@{, > olp{$3 ij++{{,, hv gh od irupd 33 shur i 3 +{, olp 3 {$3 j +{,

j+{, @ {1 Frpr olp{$3 i +{, @ 3 | olp{$3 { @ 3 hqwrqfhv

5{ frv+4@{, . {5 + vhq+4@{,,+4@{5 , olp {$3 4 @ olp 5{ frv+4@{, . vhq+4@{, {$3

@

hvwh oðplwh qr h{lvwh/ sxhv olp{$3 5{ frv+4@{, h{lvwh shur olp{$3 vhq+4@{, qr h{lvwh +yhu hmhuflflr 636 fdsðwxor gh oðplwh | frqwlqxlgdg,1 3 Vlq hpedujr/ +d shvdu gh txh olp{$s ij3 ++{{,, qr h{lvwh,/ olp{$s ij++{{,, h{lvwh/ sxhv sru whruhpd gh fhur dqltxlod1 {5 frv+4@{, olp @ olp { frv+4@{, @ 3= {$3 {$3 {

Hmhpsor 737 Fdofxodu

O @ olp { oq { {$3.

{1 Vroxflöq= O @ olp{$3. oq 4@{ { hv Frpr olp{$3. oq { @ 4 +yhu juäfd gh orjdulwpr, | olp{$3. {4 @ 4 hqwrqfhv ho oðplwh olp{$3. oq 4@{ gh od irupd 4 4 oxhjr/ fdofxohprv +vl hv txh h{lvwh, +oq {,3 olp. {$3 +4@{,3

Sru or wdqwr

Hmhpsor 738 Fdofxodu

+oq {,3 olp. { oq { @ olp. @3 {$3 {$3 +4@{,3 orj { O @ olp. {$3 orj+vhq+{,,

Vroxflöq= Hvwh oðplwh hv gho wlsr 4 4 / sxhv olp{$3. orj { @ 4/ olp{$3. orj vhq+{,, @ 4 +orj {,3 olp . {$3 +orj vhq {,3

oxhjr Hmhuflflr 739 Fdofxodu

+4@{, olp {$3. +4@{5 , @ olp { @ 3 {$3.

@

@

olp {$3.

4

{ oq 43 4 frv{ vhq { oq 43 vhq { 4 @ olp @4 {$3 { frv {

orj+{, olp @4 {$3. orj+vhq+{,, d, olp {$3

dufvhq { {

e, olp {$3

 dufvhq { 4@{5 {

Vroxflöq= d, Frpr olp{$3 dufvhq { @ 3 | olp{$3 { @ 3 oxhjr olp{$3 olp {$3

+dufvhq {,3 4 @ olp s @4 {$3 4  {5 {3

dufvhq

{

{ hv gh od irupd 3 / fdofxohprv 3

8171 UHJOD GH O*KRSLWDO

654

Sru od uhjod gh O*Krslwdo whqhprv

olp {$3

e, Fdofxohprv

dufvhq { @4 {

O @ olp h {5 oq+ { , {$3 dufvhq { qrwhprv txh olp{$3 oq+ { , @ oq 4 @ 3 | olp{$3 {5 @ 3/ oxhjr 4

olp {$3

hv gh od irupd 33 / yhdprv/ O4

@

olp {$3

@

olp {$3

dufvhq {

{ oq+ dufvhq { , {5

{ 3 +oq+ dufvhq { ,, +{5 ,3  s { dufvhq {  4{5 { dufvhq { {5 5{

s { 5  dufvhq { { olp  olp 4{ {$3 dufvhq { {$3 5{6 { s 5  dufvhq { @ olp 4{ {$3 5{6

@

ho fxdo qxhydphqwh hv gh od irupd

3 3

 olp {$3

4 s olp {$3 9+4  {5 , 4  {5

@

olp {$3 4 @ 9

s4{{5

sru +d,

 dufvhq {

+5{6 ,3 s4{5 . s { 4{5 4{5 9{5



Dvð sru dojheud gh oðplwh | sru od uhjod gh O*Krslwdo vh wlhqh txh O4 @

3

s44 {5

4 9

| sru frqvljxlhqwh

O @ h4@9

Revhuydflöq= Od uhjod gh O*Krslwdo wdpelìq hv yäolgd vl { $ .4 r { $ 4 | dghpäv qrwh txh olp i+4@{, @ olp i+{, {$4 {$3. olp i+4@{, @ olp i+{, {$4 {$3

Whruhpd 8149 Vhdq i | j ixqflrqhv glihuhqfldeohv sdud wrgr { A Q / grqgh Q 5 U. | vxsöqjdprv txh sdud wrgr { A Q> j 3 +{, 9@ 31 hqwrqfhv d, Vl olp{$4 i +{, @ 3 | olp{$4 j+{, @ 3 +hv ghflu/ ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 , | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 Dqäorjdphqwh vl olp{$4 i+{, @ 4 | olp{$4 j+{, @ 4 | 3 vl olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 e, Vl olp{$4 i+{, @ 3 | olp{$4 j+{, @ 3 +hv ghflu/ ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 , | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1

655

FKDSWHU 81 GHULYDGD

Dqäorjdphqwh vl olp{$4 i +{, @ 4 | olp{$4 j+{, @ 4 | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 Ho whruhpd wdpelìq hv yäolgr vl O vh vxvwlwx|h sru .4 r 4

Hmhpsor 73: Fdofxodu

olp {$4

  5 dufwj { h6@{  4

Vroxflöq= olp{$4   5 dufwj { @ 3 sxhv olp{$4 dufwj { @ 5 1 olp{$4 h6@{  4 @ h3  4 @ 3/ oxhjr ho oð plwh hv gh od irupd 33 / dsolfdqgr O*Krslwdo olp {$4

gdgr txh

+  5 dufwj {,3 +h6@{  4,3

@

@

5 4 5 olp 6@{4.{6 {$4 h + {5 ,

5 {5 olp 6@{ 6 {$4 h +4 . {5 ,

{5 4 @ olp @4 olp {$4 4 . {5 {$4 {45 . 4

| olp{$4 h64@{ @ 4 hqwrqfhv sru dojheud gh oðplwh/

{5 olp 6@{ @4 {$4 h +4 . {5 ,

hq frqvhfxhqfld/ sru O*Krslwdo

olp {$4

Hmhpsor 73; Fdofxodu

5   5 dufwj { @ 6 h6@{  4 {5 olp { {$4 h

Vroxflöq= \d txh olp{$4 {5 @ 4 | olp{$4 h{ @ 4/ dsolfdqgr od uhjod gh O*Krslwdo/ rewhqhprv shur olp{$4 5h{{ hv gh od irupd

sru or wdqwr

4 4

oxhjr

+{5 ,3 5{ olp { 3 @ olp { {$4 +h , {$4 h

+5{,3 5 olp { 3 @ olp { @ 3 {$4 +h , {$4 h {5 olp { @ 3 {$4 h

Revhuyhprv txh hq dojxqrv fdvrv ghehprv dsolfdu päv gh xqd yh} od uhjod gh O*Krslwdo1 Hmhpsor 73< Fdofxodu Hvwh oðplwh hv gho wlsr 44

olp +h{  {,4@{ {$3

5

8171 UHJOD GH O*KRSLWDO

656

Vroxflöq=

Fdofxohprv

5 olp oq+h{  {,4@{ {$3 4 @ olp 5 oq+h{  {, {$3 {

@

O4

hv gho wlsr

+8148,

5 { 4@{5 O @ olp +h{  {,4@{ @ olp hoq+h {, {$3 {$3

3 3

hqwrqfhv yhdprv vl h{lvwh

h{ 4 { h @ olp { {$3 5{ h{  4 @ olp { {$3 +h  {,5{

+oq+h{  {,,3 olp {$3 +{5 ,3 hv gho wlsr 33 / shur +h{  4,3 olp {$3 +5{h{  5{5 ,3 Sru or dqwhulru/ whqhprv

h{ olp { {$3 5{h . 5h{  7{ 4 @ 5

@

5 4 O4 @ olp oq+h{  {,4@{ @ {$3 5

oxhjr hq 8148 xvdqdgr od frqwlqxlgdg gh od ixqflöq h{srqhqfldo vh wlhqh O @ holp{$3 oq+h

Sru or wdqwr

{ {,4@{5

@ h5 4

olp +h{  {,4@{ @ h4@5 {$3 5

Hmhpsor 743 Fdofxodu

olp +  5{,frv { {$+@5,

Hvwh oðplwh hv gho wlsr 33 Vroxflöq=

O@ Fdofxohprv

hoq+5{, olp +  5{,frv { @ olp {$+@5, +@5, $3 O4

olp oq+  5{,frv { {$+@5, @ olp frv { oq+  5{, {$+@5, oq+  5{, @ olp {$+@5, vhf { @

frv {

+8149,

657

FKDSWHU 81 GHULYDGD

hv gho wlsr 4 4 Fdofxohprv olp {$+@5,

+oq+  5{,,3 +vhf {,3

5

5{ olp {$+@5, vhf { wdq { 5 frv5 { @ olp {$+@5, +  5{,vhq{

@

hv gho wlsr 33 / oxhjr dsolfdqgr O*Krslwdo qxhydphqwh

Sru or wdqwr

+5 frv5 {,3 olp {$+@5, ++  5{, vhq {,3

oxhjr hq 8149 vh wlhqh gh hvw irupd oohjdprv d txh Hmhpsor 744 Fdofxodu Hvwh oðplwh hv gho wlsr 43 Vroxflöq=

Fdofxohprv

O4 @

7 frv { vhq { olp {$+@5, +  5{, frv {  5 vhq { @ 3 @

olp oq+  5{,frv { @ 3 {$+@5,

O @ holp{$+@5, oq+5{,

frv {

@ h3

olp +  5{,frv { @ h3 @ 4 {$3 olp +{ . h{ ,4@{ {$4

O @ olp +{ . h{ ,4@{ @ olp hoq+{.h {$4 {$4

{ ,4@{

@

olp oq+h{ . {,4@{ {$4 4 @ olp oq+{ . h{ , {$4 {

O4

shur hvwh oðplwh hv gho wlsr + 4 4 ,/ oxhjr dsolfdqgr O*Krslwdo +oq+{ . h{ ,,3 olp {$4 +{,3 hv gho wlsr

4 4

4.h{ { { @ olp .h {$4 4 4 . h{ @ olp {$4 { . h{

h , | vhuä ljxdo d olp{$4 ++4. {.h{ ,3 vlhpsuh txh h{lvwd hvwh ýowlpr/ {

3

+4 . h{ ,3 olp {$4 +{ . h{ ,3

h{ olp {$4 4 . h{ 4 @ olp {$4 4 . 4@h{ @ 4 @

Hq frqvhfxhqfld/ sru uhjod gh O*Krslwdo vh wlhqh txh

O4 @ olp oq+{ . h{ ,4@{ @ 4 {$4

8171 UHJOD GH O*KRSLWDO oxhjr hq BB rewhqhprv

658

O @ holp{$4 oq+{.h

Sru or wdqwr/

{ ,4@{

@ h4

olp +{ . h{ ,4@{ @ h4 {$4

41 Hmhuflflr 745 Fdofxoh ho ydoru gh orv vljxlhqwhv oðplwhv +O*Krslwdo, { {, 51 olp{$3 h {+4. 5 61 olp{$4 oq{{ q 71 olp{$4 {h{

81 olp{$3. { oq { 91 olp{$4 {+h4@{  4,

:1 olp{$4 +4 . {5 ,{ ;1 olp{$3. {{ s <1 olp{$4 + 5  {5  4,{4

431 olp{$4 +{ . h{ ,5@{ 441 olp{$4 {4@{

4 451 olp{$4 + oq4{  { , 4 { 461 olp{$3 oq+wdq {.4,

{{ 471 olp{$3 vhq h5{ 4 s s 481 olp{$4 {5 . 5  {5  5 491 olp{$3 + {45 ,vhq {

6 4:1 olp{$3. +4  8 frw6 {,wdq {

4;1 olp{$@5 +5 vhf5 { 

4<1 olp{$3. + {4  frw {,

 {,

4 4 vhq

{@5 531 olp{$4 {h {.h{

5 541 olp{$4 +frv+5@{,,{ { {  6{ 551 olp{$3. dufwj vhq 561 olp{$4. +{  4,oq { { 571 olp{$3 dufvhq {

{ +4@{ 581 olp{$3 + dufvhq { , { { 591 olp{$4 6hh{.oq .{

5

,

5:1 olp{$4 +h5 { . vhq{,+4@{,

659

FKDSWHU 81 GHULYDGD

818

81814

Dsolfdflrqhv gh od Ghulydgd Dsolfdflrqhv Jhrpìwulfdv

Od dsolfdflöq jhrpìwulfd päv gluhfwd | fodud reyldphqwh hvwä hq ho khfkr txh od ghulydgd qrv shuplwh hqfrqwudu od shqglhqwh gh od uhfwd wdqjhqwh d flhuwdv juäfdv gh ixqflrqhv/ h{solfdflöq txh khprv gdgr dqwhv gho frqfhswr gh od ghulydgd/ frpr prwlydflöq d hood> shur qr hvwä gh päv kdfhu dojxqrv hmhpsorv Hmhpsor 746 Hqfrqwudu ho sxqwr gh od sduäerod

| @ 6{5 . 5{ . 4@7

fx|d uhfwd wdqjhqwh irupd xq äqjxor @7 frq ho hmh {1

Vroxflöq= wj+@7, @ 4/ dvð od shqglhqwh gh od uhfwd wdqjhqwh gheh vhu 41 g| @ 9{ . 5 @ 4 g{ +, { @ 4@9 p@

Gh hvwd irupd ho sxqwr gh od sduäerod hv +4@9> 3,1

Hmhpsor 747 Xvdqgr ghulydgd hqfrqwudu ho yìuwlfh gh od sduäerod | @ {5 . 7{  ;

Vroxflöq= Hq ho yìuwlfh od wdqjhqwh wlhqh shqglhqwh fhur g| @ 5{ . 7 @ 3 g{

Sru or wdqwr ho yìuwlfh hvwä hq ho sxqwr +5> 45,1

+,

{ @ 5

Hmhpsor 748 Xvdqgr ghulydgd pxhvwuh txh hq xqd flufxqihuhqfld gh fhqwur +k> n, | udglr u1 +{  k,5 . +|  n,5 @ u5

od uhfwd wdqjhqwh hq ho sxqwr S @ +{3 > |3 , hv shushqglfxodu d od uhfwd txh xqh S frq +k> n, +uhfwd udgldo,

Vroxflöq= Vl |3 @ n odv uhfwdv wdqjhqwhv vrq shushqglfxoduhv do hmh { frq hfxdflöq { @ u . k r { @ u  k | od uhfwd udgldo hv | @ n sru or wdqwr vrq shushqglfxoduhv1 Vl |3 9@ n hqwrqfhv +,

5+{  k, . 5+|  n, 5+{  k, g| @ g{ 5+|  n,

g| @3 g{

k 1 Shqglhqwh gh uhfwd wdqjhqwh d od flufxqihuhqfld hq ho sxqwr S @ +{3 > |3 , hv  {|33 n Od hfxdflöq gh od uhfwd wdqjhqwh d od flufxqihuhqfld hq S @ +{3 > |3 , hv= |  |3 @ 

Od hfxdflöq gh od uhfwd udgldo hq ho sxqwr s hv

| fodudphqwh vrq shushqglfxoduhv1

|n @

+{3  k, +{  {3 , |3  n |3  n +{  k, {3  k

8181 DSOLFDFLRQHV GH OD GHULYDGD

81815

65:

Dsolfdflrqhv Iðvlfdv

Frqvlghuhprv ho sureohpd gh hqfrqwudu od yhorflgdg hq fxdotxlhu lqvwdqwh gh xq remhwr txh vh pxhyh vreuh xqd wud|hfwruld uhfwd +prylplhqwr uhfwlolqhr,1 Xqd iöupxod px| xwlol}dgd hq od uhvroxflöq gh sureohpdv dfhufd gho prylplhqwr gh xqd sduwðfxod vreuh xqd oðqhd uhfwd hv g y@ w grqgh y hv od yhorflgdg phgld gxudqwh xq lqwhuydor gh wlhpsr/ w od pdjqlwxg gho lqwhuydor gh wlhpsr | g hv od glvwdqfld wrwdo uhfruulgd1 Hmhpsoltxhprv hvwd vlwxdflöq1 Vxsrqjdprv txh xq dxwrpöylo vdoh gh od flxgdg D d odv 44=33 DP | yldmd d or odujr gh xqd fduuhwhud uhfwd kdvwd xqd flxgdg E txh vh hqfxhqwud d 453 np gh D/ oohjdqgr d odv 4=33 SP1 Dsolfdqgr od iöupxod y @ gw frq g @ 453> w @ 5/ yhprv txh od yhorflgdg phgld gho dxwrpöylo hv yphg @

453 @ 93np@kuv 5

Hvwr vljqlfd txh xq dxwrpöylo 93np@kuv txh pdqwhqjd vx yhorflgdg d 93np@kuv gxudqwh wrgr ho wlhpsr/ wrpdud 5 krudv hq uhfruuhu od glvwdqfld hqwuh odv grv flxgdghv1 Shur/ BcFxäo hv od yhorflgdg +lqvwdqwäqhd, txh pdufö ho yhorfðphwur +sru hmhpsor, d odv 44 = 63kuvB Qrwhprv txh ho dxwrpöylo sxgr kdehu hvwdgr hq uhsrvr +ghwhqlgr hq xq shdmh, r kdehu pdufdgr vx yhorfðphwur :3np@kuv x rwur ydoru1 Sdud ghwhuplqdu od yhorflgdg frq txh ho dxwrpöylo yldmdq flhuwr lqvwdqwh +hq hvwh fdvr d odv 44=63kuv,/ vh qhfhvlwd frqrfhu dojxqd lqirupdflöq vreuh od srvlflöq gho dxwrpöylo douhghgru gh hvd krud1 Sru hmhpsor/ vxsrqjdprv txh d odv 44=63 kuv/ ho dxwrpöylo hvwä d 53 npv gh D | txh d odv 4=68 krudv hvwä d 57/8 npv gh D/ hqwrqfhv hq 8 plqxwrv od glvwdqfld uhfruulgd hv 7/8 npv1 Vxvwlwx|hqgr hq od iöupxod y @ gw whqhprv txh od yhorflgdg phgld gxudqwh hvh lqwhuydor gh wlhpsr hv 8 y @ 87@>93 @ 87np@kuv Shur hvwh ydoru/ qr hv ho ydoru h{dfwr gh od yhorflgdg d odv 44=63kuv/ hv vöor xqd dsur{lpdflöq1 Re0 yldphqwh xqd phmru dsur{lpdflöq gh od yhorflgdg d odv 44=63 kuv vh rewlhqh vl frqvlghudprv lqwhuydorv gh wlhpsr päv shtxhôrv1 Hvwr qrv oohyd d frqvlghudu ho oðplwh gh od yhorflgdg phgld fxdqgr w wlhqgh d 44=63kuv1 Vxsrqjdprv txh vh frqrfh od srvlflöq v+w, gh xq sxqwr S hq wrgr lqvwdqwh hq xq lqwhuydor gh wlhpsr/ hqwrqfhv od yhorflgdg gh S hq ho wlhpsr w @ w3 yhorflgdg lqvwdqwdqhd vh ghqh frpr olp w$w3

v+w, vh frqrfh frpr od ixqflöq gh srvlflöq gh S 1

v+w,  v+w3 , w  w3

Ghqlflöq 814: Vl xq sxqwr S vh pxhyh vreuh xqd uhfwd gh prgr txh vx srvlflöq hq ho wlhpsr w hv v+w,/ frq w 5`w3  > w3 . ^>  A 3/ hqwrqfhv od yhorflgdg phgld gh S hqwuh orv wlhpsrv w3 | w hv y@

v+w,  v+w3 , w  w3

| od yhorflgdg lqvwdqwäqhd y+w3 , gh S hq ho wlhpsr w3 hv y+w3 , @ olp w$w3

v+w,  v+w3 , w  w3

d+w3 , @ olp w$w3

y+w,  y+w3 , w  w3

vlhpsuh | fxdqgr ho oðplwh h{lvwh/ hv ghflu y+w3 , @ v3 +w3 ,1 Od udslgh} gh S do wlhpsr w @ w3 vh ghqh frpr my+w3 ,m | od dfhohudflöq gh S do wlhpsr w3 frpr

65;

FKDSWHU 81 GHULYDGD

hv ghflu/ xvdqgr rwud qrwdflöq

d+w3 , @ y 3 +w3 , @ v33 +w3 , d+w3 , @

gy g5 v mw@w3 @ 5 mw@w3 gw g{

d+w3 , @ y+w b 3 , @ v‚+w3 ,

Hmhpsor 749 Xqd dwohwd fruuh orv flhq phwurv sodqrv9 / gh pdqhud txh od glvwdqfld v+w, txh kd uhfruulgr d orv w vhjxqgrv hvwä gdgd sru 4 v+w, @ w5 . ;w phwurv 8 Fdofxodu od yhorflgdg gho fruuhgru d, hq ho prphqwr gh od vdolgd e, d orv 8 vhjxqgrv gh od vdolgd f, do fux}du od phwd Vroxflöq= d, Qrwhprv txh hq ho prphqwr gh od vdolgd w @ 3/ od srvlflöq gho dwohwd hv v+3, @ 3 phwurv1 Xvdqgr od ghqlflöq/ od yhorflgdg gho dwohwd hq w @ 31 v+w,  v+3, w3 4@8w5 . ;w @ olp w$3 w @ olp 4@8w . ; w$3 @ ;p@vhj

y+3, @

olp w$3

Txh hv htxlydohqwh d fdofxodu y+w,/ ghulydqgr v+w, frq uhvshfwr do wlhpsr | oxhjr hydoxdqgr hq w @ 31 Dvð/ 5 y+w, @ w . ; , y+3, @ ; 8 e, D orv 8 vhjxqgrv gh od vdolgd/ whqhprv 5 y+8, @ 8 . ; @ 43p@vhj 8

f, Do fux}du od phwd/ hv ghflu/ d orv 433 phwurv vh wlhqh/

4 V+w, @ 433 +, w5 . ;w @ 433 +, w @ 43 8

sru or wdqwr d orv 43 vhj ho dwohwd kd fux}dgr od phwd | vx yhorflgdg hv 5 y+43, @ 43 . ; @ 45p@vhj 8

Hmhpsor 74: Grv fduuhwhudv vh fux}dq irupdqgr hq xq äqjxor gh 78 1 Vdelhqgr txh hq xq lqvwdqwh w3 grv dxwrpöylohv D | E glvwdq gho fuxfh 433 npv/ ho dxwrpöylo D dohmäqgrvh frq xqd udslgh} gh ;3 np@kuv | E dohmäqgrvh frq xqd udslgh} gh 93 np2kuv1 Ghwhuplqh od udslgh} frq txh vh vhsdud xqr gho rwur hq ho lqvwdqwh w3 1 +jxud BB, Vroxflöq= Vhd {+w, od glvwdqfld ghvgh ho fuxfh do dxwrpöylo D hq ho lqvwdqwh w/ |+w, od glvwdqfld ghvgh ho fuxfh do dxwrpöylo E/ | v+w, od glvwdqfld hqwuh D | E/ dghpäv yD +w3 ,> yE +w3 , yhorflgdg gh orv dxwrpöylo D/ E uhvshfwlydphqwh hq ho lqvwdqwh w3 1 Sru klsöwhvlv {+w3 , @ |+w3 , @ 433npv / yD +w3 , @ {3 +w3 , @ ;3np2kuv | yE +w3 , @ | 3 +w3 , @ 93np2kuv1 Qrvrwurv exvfdprv gv gw +w3 ,1 9 Wr grv

orv prylplhqwr vrq frqvlghudgrv uhfwlolqhr1

8181 DSOLFDFLRQHV GH OD GHULYDGD

65<

Dsolfdqgr ho whruhpd gho frvhqr do wuläqjxor DRE whqhprv

v5 +w, @ {5 +w, . |5 +w,  5{+w,|+w, frv+78 ,>

hv ghflu/

s v5 +w, @ {5 +w, . | 5 +w,  {+w,|+w, 5

Frq ho remhwr gh rewhqhu gv gw ghulydprv od h{suhvlöq 814: frq uhvshfwr d w whqhprv>

+814:,

gv g{ g| s g{ s g| @ 5{ . 5|  5|  5{ = +814;, gw gw gw gw gw Hydoxdqgr hq w3 od h{suhvlöq 814: rewhqhprv od glvwdqfld hqwuh orv dxwöprylohv hq ho lqvwdqwh w3 1 t s {5 +w3 , . | 5 +w3 ,  5{+w3 ,|+w3 , v+w3 , @ t s 4335 . 4335  54335 v+w3 , @ t s v+w3 , @ 433 5  5 5v

Dvð/ xvdqgr: hvwr hq 814; rewhqhprv od yhorflgdg uhtxhulgd s s 4 5 5 gv s mw@w3 @  433  ;3  433  93, s +433  ;3 . 433  93  gw 5 5 433 5  5 s t s gv 473  :3 5 5 5 @ :3 mw@w3 @ s s gw 5 5 s s Oxhjr orv dxwrpöylohv hq ho lqvwdqwh w3 vh vhsdudq d xqd yhorflgdg gh :3 5  5np2kuv1 Rwud irupd gh gdu hvwd uhvsxhvwd hv= Oxhjr orv dxwrpöylohv hq ho lqwdqwh w3 vh vhsdudq d ud}öq gh s s :3 5  5np2kuv1

81816

Ud}öq gh Fdpelr/ wdvd gh yduldflöq

D frqwlqxdflöq yhuhprv xqd gh odv pýowlsohv dsolfdflrqhv gh od ghulydgd/ od txh uhhuh d yduldflöq r ud}öq gh fdpelr1 Sru hmhpsor/ od yhorflgdg txh kd vlgr ghqlgd dqwhulruphqwh hv od yduldflöq gh od srvlflöq v+w, frq uhvshfwr do wlhpsr +vx udslgh} gh yduldflöq,1 Od dfhohudflöq d+w, gh S do wlhpsr w hv od ud}öq gh fdpelr gh od yhorflgdg frq uhvshfwr do wlhpsr1 Frqvlghuhprv od vljxlhqwh vlwxdflöq jhqhudo> Vhd | @ i+w, grqgh i hv xqd ixqflöq ghulydeoh uhvshfwr do wlhpsr w 41 Od wdvd +r ud}öq, phgld gh yduldflöq gh | @ i +w, hq ho lqwhuydor ^w> w . k` hv i +w . k,  i +w, k

51 Od wdvd +r ud}öq, gh yduldflöq gh | @ i+w, frq uhvshfwr d w hv

i +w . k,  i +w, g| @ | 3 +w, @ olp k$3 gw k

Qrwdu txh hvwd ghqlflöq hv dqäorjd do frqfhswr gh yhorflgdg1

Odv xqlgdghv txh ghehq xvduvh hq odv ghqlflrqhv dqwhulruhv ghshqghq gh od qdwxudoh}d gh od fdqwlgdg uhsuhvhqwdgd sru i1 : Frpr

or päv gluhfwr | iäflo gh rewhqhu

vEwf 

xvdqgr

814:/ ho hvwxgldqwh frphwh ho huuru gh kdfhu ho fäofxor/ s huglhqgr gh

ylvwd txh od hfxdflöq 814: oh s huplwh fdofxodu od ghulydgd1 Vh dfrqvhmd do ohfwru d sodqwhdu vlhpsuh orv sureohpdv gh hvwh wls r hq yduldeohv/ vlq hydoxdu kdvwd ghvsxìv gh ghulydu1

663

FKDSWHU 81 GHULYDGD

Hmhpsor 74; Ho yroxphq Y +hq slh6 , gh xqd shtxhôd uhsuhvd gxudqwh od ìsrfd gh ooxyld/ hvwä gdgr sru Y @ 8333+w . 4,5

grqgh w vh plgh hq phvhv | 3  w  61 Od wdvd gh fdpelr gho yroxphq frq uhvshfwr do wlhpsr hv ho xmr lqvwdqwäqhr kdfld od uhsuhvd1 Fdofxoh ho xmr hq orv wlhpsrv w @ 3 | w @ 5 Bc Fxäo hv ho ydoru gho xmr fxdqgr ho yroxphq hv gh 44=583slh6 B

Vroxflöq= Sru ghqlflöq od wdvd gh fdpelr gho yroxphq frq uhvshfwr do wlhpsr hv gY gw 1 Hq hvwh sureohpd gY uhsuhvhqwd ho xmr kdfld od uhsuhvd/ hqwrqfhv ghulydqgr od ixqflöq Y uhvshfwr gho gw wlhpsr whqhprv gY @ 43=333+w . 4, gw hydoxdqgr hq w @ 3 | w @ 5 rewhqhprv ho xmr> gY mw@3 @ 43=333slh6 @phvhv gw gY mw@5 @ 63=333slh6 @phvhv gw

Vl ho yroxphq Y @ 44=583 slh6 hqwrqfhv gh grqgh w @ 3=8 phvhv

44=583 @ 8=333+w . 4,5 gY mw@3=8 @ 48=333slh6 @phvhv gw

Hmhpsor 74< Od Whpshudwxud W +hq judgrv Fhovlxv, gh xqd vroxflöq do wlhpsr w +hq plqxwrv, hvwä gdgd sru 6 W +w, @ 43 . 7w . w.4 sdud 4  w  43

41 Fdofxoh dsur{lpdgdphqwh od wdvd phgld gh fdpelr r yduldflöq gh od whpshudwxud gxudqwh ho lqwhuydor gh wlhpsr ^5> 5=68`1

51 Fdofxoh od wdvd gh yduldflöq r ud}öq gh fdpelr gh W +w, frq uhvshfwr do wlhpsr w hq w @ 5 | w @ <1

Vroxflöq=

41 Vxvwlwx|hqgr w @ 5> k @ 3=68 hq od ghqlflöq vh rewlhqh txh od wdvd phgld gh fdpelr gh W hq ^5> 5=68` hv W +5=68,  W +5,  6=: 3=68 51 Sru ghqlflöq od wdvd gh yduldflöq gh W +w, frq uhvshfwr do wlhpsr hv

Dvð/

W 3 +w, @ 7  W 3 +5, @ 7 

6 +5 . 4,5

6 +w . 4,5

W 3 +<, @ 7 

6 +< . 4,5

Hmhpsor 753 Orv h{wuhprv gh xq deuhydghur gh 6 pw gh odujr wlhqhq od irupd gh wuläqjxor htxloäwhur frq odgrv gh 93fp +shuo,1 Vh vxplqlvwud djxd do deuhydghur d ud}öq gh 53 O2plq1 BcFxäo hv od udslgh} gh fdpelr r yduldflöq gho qlyho gho djxd fxdqgr od surixqglgdg hv gh 53fpB +4O+olwur,@ 4333fp6 ,

8181 DSOLFDFLRQHV GH OD GHULYDGD

664

Vroxflöq= Vhd k+w, ho qlyho gho djxd hq ho lqvwdqwh w/ Y +w, ho yroxphq gho djxd hq ho lqvwdqwh w/ u+w, od orqjlwxg txh vh pxhvwud hq od jxud BB Oxhjr ho yroxphq Y +w, hvwä gdgr sru Y +w, @ 633k+w,u+w,

shur sru whruhpd gh Wkdohv/ hq ho wuläqjxor FGE/ vh wlhqh

u+w, 63 s @ k+w, 63 6

s 5 grqgh 63 6 hv od dowxud gho wuläqjxor htxloäwhur DFE1 Oxhjr u+w, @ ks+w6, | Y +w, @ 633 ks+6w, 1 Dvð/ ghulydqgr frq uhvshfwr do wlhpsr w/ whqhprv gk 633 gY @ 5 s k+w, gw gw 6

gh grqgh Hydoxdqgr fxdqgr k @ 53fp whqhprv

s gk 6 gY @ = gw 933k+w, gw

s gk 6 @  53333 fp6 @plq gw 933  53

Dvð od udslgh} gh fdpelr gho qlyho gho djxd fxdqgr od surixqglgdg hv 53fp hv

s6

8

6

fp6 @plq1

Hmhpsor 754 Od rulood gh xqd slvflqd hv xq uhfwäqjxor gh 93 slh gh odujr | 63 slh gh dqfkr1 Vx surixq0 glgdg dxphqwd xqliruphphqwh gh 7 d < slh hq xq wudpr krul}rqwdo gh 73 slh | ghvsxìv frqwlqýd do plvpr qlyho orv 53 slh uhvwdqwhv frpr vh loxvwud hq od jxud BB/ od fxdo uhsuhvhqwd xqd vhfflöq wudqvyhuvdo1 Od slvflqd vh hvwä oohqdqgr d ud}öq gh 833 jdo2plq gh djxd1 Fdofxoh dsur{lpdgdphqwh od udslgh} gh fdpelr gho qlyho gho djxd hq ho prphqwr hq txh> 41 Od surixqglgdg hq od sduwh päv krqgd hv gh 7 slh +4jdo  3=466:slh6 ,1

51 Od surixqglgdg hv gh 9 slh1 61 Od surixqglgdg hv gh 8 slh1

Vroxflöq=

41 Hq ho hmhpsor 43: gho fdsðwxor gh ixqflrqhv rewxylprv txh od ixqflöq yroxphq hq wìuplqr gh k hv  3?k?8 933k . 453k5 Y +k, @ 933k . 6333 . 4533+k  8, 8  k  <

grqgh k hv ho qlyho gho djxd hq ho odgr päv surixqgr1 Oxhjr sdud fdofxodu od udslgh} gh fdpelr gho qlyho gho djxd frq uhvshfwr do wlhpsr fxdqgr k @ 7 slh frqvlghudprv od ixqflöq yroxphq sdud 3 ? k ? 8> Y +k, @ 933k . 453k5 Dvð/ ghulydqgr frq uhvshfwr do wlhpsr/ sru +uhjod gh od fdghqd, whqhprv

sru or wdqwr

gk gk gY @ 933 . 453  5k gw gw gw gY 4 gk @  gw gw 933 . 573k

665

FKDSWHU 81 GHULYDGD 6 grqgh k @ k+w,/ shur gY gw @ 833  3=466:slh @plq | k @ 7slh oxhjr 4 gk mk@7 @ 833  3=466:   3=375<slh6 @plq gw 933 . 573  7 Dvð od udslgh} gh fdpelr gho qlyho gho djxd hq ho lqvwdqwh hq txh ho qlyho gho djxd hv 7 slhv hv dsur{lpdgdphqwh 3=375<slh@plq1

51 Sdud fdofxodu od ud}öq gh fdpelr fxdqgr k @ 9 vh surfhgh gh pdqhud dqäorjd d od sduwh +d, frq0 vlghudqgr Y +k, @ 933k . 6333 . 4533+k  8, sxhv hvwd ixqflöq hvwä ghqlgd sdud 8  k  <1

61 hmhuflflr sdud ho ohfwru

Hmhpsor 755 Xq wdqtxh föqlfr gh 45 slh gh dowxud | ; gh gläphwur hq od wdsd/ vh oohqd frq djxd d xqd ud}öq frqvwdqwh1 Do hqfrqwudu ho qlyho d phgld dowxud/ od ud}öq gh fdpelr gh hvwd dowxud hv gh 4slh2plq1 Bc Fxäqwr wdugduðd hq oohqduvh ho wdqtxhB

Vroxflöq= Vhd k od dowxud gho qlyho gh djxd | u od orqjlwxg gho udglr gho frqr fxdqgr od dowxud hv k +yhu jxud BB,1 Frpr gY gw @ n grqgh n frqvwdqwh hqwrqfhv Y @ nw . F1 Frpr Y @ 3 hq w @ 3 hqwrqfhv F @ 3 Oxhjr Y @ nw

shur

4 Y @ u5 k 6

oxhjr shur gho wuläqjxor DEF vh wlhqh

+814<,

Y @ nw @

4 5 u k 6

+8153,

u 7 @ 45 k gh grqgh u @ 46 k1 Dvð uhhpsod}dqgr hq + 8153, whqhprv ghulydqgr frq uhvshfwr d w whqhprv

4 4 nw @ + ,5 k6 6 6

4 gk 6k5 5: gw shur do hqfrqwudu ho qlyho gho djxd d phgld dowxud/ hv ghflu/ fxdqgr k @ 9 vh wlhqh> n@

+8154,

gk mk@9 @ 4 gw Dvð hydoxdqgr od h{suhvlöq + 8154, fxdqgr k @ 9>  n @  95  4 @ 7 < Sru or wdqwr Y @ 7w

Sru rwud sduwh ho wdqtxh hvwä oohqr fxdqgr Y @ 46 75 45 @ 97/ hv ghflu ho wdqtxh vh oohqd fxdqgr 7w @ 97 gh grqgh w @ 491 Dvð ho hvwdqtxh wdugduðd hq oohqduvh 49 plqxwrv1

Qr wdq vöor lqwhuhvd odv ud}rqhv gh fdpelr frq uhvshfwr do wlhpsr/ vlqr wdpelìq frq uhvshfwr d rwudv yduldeohv1

8181 DSOLFDFLRQHV GH OD GHULYDGD

666

Ghqlflöq 814; Vhd | @ i +{, grqgh { hv fxdotxlhu yduldeoh

41 Od wdvd phgld gh yduldflöq gh | frq uhvshfwr d { hq ho lqwhuydor ^{> { . k` hv ho fxrflhqwh i+{ . k,  i +{, k

51 Od wdvd lqvwdqwäqhd gh yduldflöq gh | frq uhvshfwr d { hv ho oðplwh gh od ud}öq surphglr fxdqgr k wlhqgh d 3/ hv ghflu/ i +{ . k,  i+{, g| @ olp g{ k$3 k

g| xqlgdghv sru xqlgdg gh Hvwr vljqlfd txh vl od yduldeoh { fdpeld/ hqwrqfhv | fdpeld d ud}öq gh g{ fdpelr gh {1

Hmhpsor 756 Od uhodflöq hqwuh od whpshudwxud I hq od hvfdod Idkuhqkhlw | od whpshudwxud F hq od hvfdod Fhovlxv hvwä gdgd sru 8 F @ +I  65, < Bc Fxäo hv od ud}öq gh fdpelr gh I frq uhvshfwr d FB Vroxflöq= Ghulydqgr frq uhvshfwr d F/ whqhprv

4@

8 gI < gF

gh grqgh od ud}öq gh fdpelr gh I frq uhvshfwr d F hv

gI < @ gF 8

Hmhpsor 757 Ghpxhvwuh txh od wdvd gh fdpelr gho yroxphq gh xqd hvihud frq uhvshfwr do udglr hv ljxdo do äuhd gh od vxshuflh Vroxflöq= Ho yroxphq | ho äuhd gh xqd hvihud hvwäq gdgrv sru odv iöupxodv 7 Y @ u6 6

>

D @ 7u5

grqgh u hv ho udglr1 Hv fodur txh vl ghulydprv Y frq uhvshfwr d u whqhprv gY 7 @ 6u5 @ 7u5 @ D> gu 6

hv ghflu/ od wdvd gh fdpelr gho yroxphq hv ljxdo do äuhd gh od vxshuflh1 Hmhpsorv wðslfrv

Sureohpd gh od vrpeud pwv > ho irfr Sdxolqd/ gh 4> 5 phwurv gh dowr/ fruuh hq od qrfkh dohmdqgrvh gh xq srvwh d od yhorflgdg gh 8 vhj gho srvwh txh loxplqd d Sdxolqd hvwd d :> 8 phwurv gh dowxud1 D phglgd txh vh dohmd gho srvwh od vrpeud gh ohood fuhfh +Áhv yhugdgB r Áuhdophqwh ghfuhfhB,1 Frqwhvwhprv odv vljxlhqwhv suhjxqwdv d10 ÁD txì yhorflgdg fdpeld ho odujr gh od vrpeud gh Sdxolqd B e10 Ho h{wuhpr gh od vrpeud vh dohmd wdpelìq gho srvwh/ od yhorflgdg gh hvwh dohmdplhqwr Á hv od plvpd yhorflgdg gh od qlôdB r Áhv od plvpd yhorflgdg gdgd frpr uhvsxhvwd d od suhjxqwd dqwhulruB1

667

FKDSWHU 81 GHULYDGD

Vhd=

{+w, od glvwdqfld gh od qlôd do srvwh hq ho lqvwdqwh w> v+w, ho odujr gh od vrpeud gh od qlôd hq ho wlhpsr w

Hqwrqfhv od yhorflgdg gh od qlôd hv

g{+w, | hq wìuplqr gh hvwdv yduldeohv/ or txh exvfdprv sdud frqwhvwdu gw

gv+w, gw Or txh ghehprv kdfhu hv hqfrqwudu xqd uhodflöq hq grqgh dsduh}fd v+w, | ghvsxìv ghulydu1 Frpr orv wuläqjxorv DEF | GHE vrq vhphmdqwhv +wlhqhq orv plvprv äqjxorv,/ vh gd od vljxlhqwh uhodflöq hqwuh orv odgrv

od sulphud suhjxqwd hv

| gh hvwd irupd whqhprv

Txh hv or plvpr txh

ghulydqgr d dperv odgrv

DE DF @ GH GE

{+w, . v+w, :> 8 @ 4> 8 v+w, :> 8v+w, @ 4> 8{+w, . 4> 8v+w, 9v+w, @ 4> 8{+w, v+w, @ 3> 58{+w, g{+w, pwv pwv gv+w, @ 3> 58 @ 3> 58  +8, @ 4> 58 gw gw vhj vhj

Od suhjxqwd e vh uhhuh d od ghulydgd gh od vxpd {+w, . v+w, | frpr frqrfhprv fdgd ghulydgd/ od pw = uhvsxhvwd hv qr hv ljxdo d qlqjxqd gh odv grv vxjhulgdv/ od yhorflgdg gho h{wuhpr gh od vrpeud hv 9> 58 vhj np > Vxsrqjdprv dkrud txh Shgulwr hv ho txh fruuh/ shur d 53 ku ÁFxäo fruuh päv uäslgr/ Shgulwr r od qlôdB

8181 DSOLFDFLRQHV GH OD GHULYDGD

668

Vl Shgulwr plgh 468fp> ÁD txì yhorflgdg ho h{wuhpr gh od vrpeud vh dohmd gho srvwhB Sureohpd gh od hvfdohud Xqd hvfdohud gh phwdo frq uxhgdv hq vx edvh hvwd dsr|dgd hq xqd pxudood gh wdo irupd txh vx h{wuhpr vxshulru hvwd d 6pwv gho vxhor1 Ho eortxhr gh vxv uxhgdv oh lpslgh pryhuvh1 Od hvfdohud plgh 7pwv gh odujr | hv shvdgd gh prgr wdo txh fxdqgr vh ghveortxhdq odv uxhgdv od hvfdohud vh ghvol}d hq od pxudood pw = Fxdqgr hvwd d xq phwur gho vxhor Ád txì prylhqgrvh vx h{wuhpr vxshulru d od yhorflgdg gh 433 vhj yhorflgdg vh pxhyhq odv uxhgdv hq hvh lqvwdqwhB Sdud frqwhvwdu hvd suhjxqwd qhfhvlwdprv ghqlu dojxqdv yduldeohv | kdfhu xq gldjudpd gho sureohpd Ho gldjudpd gho sureohpd hv=1

Vhdq=

u+w, od glvwdqfld gh odv uxhgdv d od pxudood | k+w, od glvwdqfld gho h{wuhpr vxshulru gh od hvfdohud do vxhor1

gk pw gu +w, @ 433 | or txh exvfdprv hv hq w3 txh hv ho lqvwdqwh hq txh od hvfdohud hvwä gw vhj gw d 4phwur gho vxhor/ hv ghflu/ k+w3 , @ 4 phwur1 Frpr vh yh/ od pxudood frq od hvfdohud | ho vxhor hq wrgr wlhpsr w irupdq xq wuläqjxor uhfwäqjxor dvl whqhprv ho whruhpd gh Slwäjrudv/ Sru or wdqwr

+k+w,, . +u+w,, @ +7, > 5

5

5



ghulydqgr hq dperv odgrv gh od ljxdogdg +| xvdqgr od uhjod gh od fdghqd, whqhprv gu+w, gk+w, . 5u+w, @3 gw gw k+w, gk+w, gu+w, @ gw u+w, gw

5k+w,

| sru or wdqwr hq ho lqvwdqwh w3

k+w3 , gk 4 gu pw +w3 , @  +w3 , @  433 gw u+w3 , gw u+w3 , vj

Xvdqgr  srghprv rewhqhu u+w3 ,

Q

669

FKDSWHU 81 GHULYDGD

Iljxuh 815= +k+w3 ,, . +u+w3 ,, @ +7, / +4, . +u+w3 ,, @ 49 s u+w3 , @ 48 5

5

5

5

5

Sru or wdqwr odv uxhgdv vh ghvol}dq d od yhorflgdg gh

pw 4 pw gu +w3 , @  s 433  58> ;53 gw vj vj 48 ÁSru txì qrv glö xqd uhvsxhvwd qhjdwlydB

Sru ho vlvwhpd gh uhihuhqfld hohjlgr> frpr od glvwdqfld k+w, glvplqx|h/ od yhorflgdg gh fdlgd gh od hvfdohud pw pw ghehuðd vhu qhjdwlyd/ hv ghflu/ hq yh} gh kdehu sxhvwr hq Q 433 ghehuldprv kdehu sxhvwr 433 = vj vj Sureohpd gh od Vrpeud 5

Hq ho fuxfh gh grv fdoohv/ txh wlhqhq dqfkr 44 phwurv fdgd xqd/ kd| xq srvwh gh 6 phwurv gh dowr hq pw xqd gh odv hvtxlqdv1 Hq od qrfkh Shgulwr/ gh 4> < phwurv/ fdplqd d4> 6 dohmdqgrvh gh od hvtxlqd sru od vj yhuhgd frqwuduld do srvwh | d xq phwur gh od rulood gh od fdooh1

ÁD txì yhorflgdg vh dodujd od vrpeud gh Shgulwr fxdqgr hvwh vh hqfxhqwud d 46 phwurv gh od rulood gh od fdooh txh uhflhq fux}öB Frpr srghprv yhu hq ho gldjudpd hvwh vh sxhgh vlpsolfdu d xq wuläqjxor uhfwäqjxor frpr vljxh

8181 DSOLFDFLRQHV GH OD GHULYDGD

Vhd

66:

s+w, glvwdqfld gh Shgulwr d od edvh gho srvwh +s+w, @ DF, v+w, odujr gh od vrpeud gh Shgulwr +v+w, @ FG, h+w, glvwdqfld gh Shgulwr d od fdooh wudqvyhuvdo d od fdooh grqgh fdplqd +h+w, @ FI ,

Vhd w3 ho lqvwdqwh txh Shgulwr hvwd d 46 phwurv gv ÁTxì exvfdprv B/Ho ydoru gh +w3 ,/ gw gh pw ÁTxì frqrfhprvB +w3 , @ 4> 6 gw vj Frpr whqhprv grv wuläqjxorv uhfwäqjxorv vhphmdqwhv

s+w, . v+w, 6 @ 4> < v+w, +6  4> <,v+w, @ 4> <s+w, gv+w, 4> < gs+w, @ gw +6  4> <, gw gv gs +w3 , @ 4> :5: 6 +w3 , gw gw

Sru or wdqwr qhfhvlwdprv frqrfhu

ghulydgr

gs +w3 ,= Frpr DI F irupd xq wuläqjxor uhfwäqjxor/ whqhprv gw

+s+w,,5 @ +44 . 4,5 . +h+w,,5

5s+w, dvl/ hq ho lqvwdqwh w3

I

gs gh @ 5h+w, gw gw

h+w3 , gh 46 pw gs +w3 , @ +w3 , @ 4> 6 gw s+w3 , gw s+w3 , vj

Hq ho lqvwdqwh w3 > xvdqgr I whqhprv

66;

FKDSWHU 81 GHULYDGD

Gh hvwd irupd orjudprv

Ho jorer gh Shgulwr

5 @ 477 . 49< @ 646 +s+w3 ,,5 @ 477 . +h+w3 ,,s s+w3 , @ 646

46 gs pw +w3 , @ s 4> 6 > | dvð/ gw vj 646

pw 46 pw gv 4> 6 +w3 , @ 4> :5: 6 s  4> 983 3 gw vj vj 646

np > hq ho lqwhulru yd Shgulwr frq xq jorer frq kholr1 Shgulwr vxhowd ku pw ho jorer | hvwh vxeh hq irupd shushqglfxodu d 43 = Yhdprv d txì yhorflgdg vh vhsdud Shgulwr gho jorer vj ghvsxìv gh : vhjxqgrv1 Hvtxhpd gho sureohpd Hq xqd fduuhwhud xq dxwr yd d <3

Vhd w @ 3 ho lqvwdqwh txh Shgulwr vxhowd ho jorer | vhd {+w, @ DE od glvwdqfld hqwuh ho jorer | ho dxwr/ j+w, @ FE od glvwdqfld gho jorer do vxhor/ d+w, @ DF od glvwdqfld gho dxwr do sxqwr gh F / grqgh vh irupd xq äqjxor uhfwr1 g{ ÁTxì exvfdprvB1 Uhvsxhvwd = +:, gw pw pw gj gj > hq sduwlfxodu +w3 , @ 43 > ÁTxì whqhprvB1 Uhvsxhvwd = +w, @ 43 gw vhj gw vhj gd +w,> ho fxdo hvwd hq nloöphwurv sru krud/ dvl txh ghehprv fdpelduod d phwurv vhjxqgrv gw np 4333phwurv phwurv gd +w, @ <3 @ <3 @ 58 gw krudv 6933vhjxqgrv vhjxqgrv Frpr DEF irupd xq wuläqjxor uhfwäqjxor } {5 +w, @ j 5 +w, . d5 +w, g{ gj gd 5{+w, +w, @ 5j+w, +w, . 5d+w, +w, gw gw gw g{ j+:, gj d+:, gd +:, @ +:, . +:, gw {+:, gw {+:, gw j+:, pw d+:, pw g{ +:, @ 43 . 58  gw {+:, vhj {+:, vhj

8181 DSOLFDFLRQHV GH OD GHULYDGD

66<

sru or wdqwr qhfhvlwdprv j+:,> {+:,> d+:, pw gj +w, @ 43 > ghvsxìv gh : vhjxqgr ho jorer hvwd d 43  : @ :3 phwurv gw vhj phwurv gd +w, @ 58 ghvsxìv gh : vhjxqgr ho dxwr hvwä d 58  : @ 4:8 phwurv | xwlol}dqgr } hq w @ : gw vhjxqgrv {5 +:, @ j5 +:, . d5 +:, @ :35 . +4:8, @ 68 858 5

Gh hvwd irupd whqhprv wrgd od lqirupdflöq sdud srqhuod hq  u   4 pw pw g{ +:, @ :3  43 . 4:8  58 gw u 68 858 vhj vhj pw 478 pw 4 g{ +:, @ +:3  43 . 4:8  58 , @ s gw 68 858 vhj 8 vhj

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g{ Or txh exvfdprv hv +w3 ,> grqgh w3 hv ho lqvwdqwh txh Grq iudqflvfr oohjd d P= gw Frpr od xelfdflrq gh Grq Iudqflvfr/ Sdxod | ho fhqwur gho hvfhqdulr hq wrgr wlhpsr w irupdq xq wuläqjxor/ xvduhprv od oh| gho frvhqr {5 +w, @ p5 +w, . 5335  5p+w,  533 frv++w,,

Ghulydqgr hq uhvshfwr do wlhpsr qrv txhgd



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Frpr hq w3 > p+w3 , @ 6 phwurv | +w3 , @

 => xwlol}dqgr orjuduhprv {+w3 , 7

673

FKDSWHU 81 GHULYDGD  {5 +w3 , @ p5 +w3 , . 5335  5p+w3 ,  533 frv++w3 ,, @ 65 . 5335  5  6  533 frv+ , 7 u 4s {+w3 , @ 73 33<  4533 5 5

Gh hvwd irupd K qrv txhgd   4 gp  gp  g g{ 6 +w3 , @ u +w3 ,  533 +w3 , frv+ , . 933 vlq+ , +w3 , gw gw gw 7 7 gw 4s 73 33<  4533 5 5   g{ 4 gp  gp  g 6 +w3 , @ u +w3 ,  533 +w3 , frv+ , . 933 vlq+ , +w3 , gw gw gw 7 7 gw 4s 73 33<  4533 5 5   4s 4 4s g{ 6  3> 5  533  5  3> 5 . 933  5  3> 8 +w3 , @ s s 4 gw 5 5 493 369  7;35 5 5 s   g{ 4 pw +w3 , @ s = 3=9 . 463 5  3=<75 68 s 4 gw vhj 493 369  7;35 5 5 Sureohpd gho wuhq d ydsru= Hq od jxud kd| xqd ydulood gh odujr O md +hq ho sxqwr E,/hq ho dur gh xqd uxhgd gh udglr U ho rwur h{wuhpr gh od yduulood +D, vh pxhyh hq irupd krul}rqwdo +hmh {,1 Od uxhgd urwd md hq vx fhqwur R frqwud ho vhqwlgr gho uhorm d 6 uhyroxflrqhv sru vhjxqgr

Yhdprv od yhorflgdg gho prylplhqwr gho h{wuhpr D1 Vhd {+w, od glvwdqfld ghvgh ho fhqwur R kdvwd ho sxqwr D +RD, hq ho wlhpsr w1 Frq hvwd qrwdflrq g{ +w,= srghprv ghflu txh ex|vfdprv gw Vhd +w, ho äqjxor txh irupd ho hmh krul}rqwdo frq ho udglr txh xqh ho fhqwur frq ho h{wuhpr E gh od ydulood hq ho wlhpsr w= Xwlol}dqgr od oh| gho frvhqr hq ho wuläqjxor RED O5 @ {5 +w, . U5  5{+w,U frv +w,

ghulydqgr hq dperv odgrv +qr roylgdu od ghulydgd gho surgxfwr | od uhjod gh od fdghqd,whqhprv g{ g{ g 5O  3 @ 5{+w, +w, . 5U  3  5 +w,U frv +w, . 5{+w,U vlq +w, +w, gw gw gw

8181 DSOLFDFLRQHV GH OD GHULYDGD

674

g g{ +w, ^5U frv +w,  5{+w,` @ 5{+w,U vlq +w, +w, gw gw g{ +w, ^5U frv +w,  5{+w,` @ 9{+w,U vlq +w, gw g{ 6{+w,U vlq +w, +w, @ gw U frv +w,  {+w, ÁTxì uhodflöq ghehq fxpsolu U | OB ÁHq txh srvlflöq gho äqjxor  vh wlhqh pd|ru yhorflgdgB ÁHq txh prphqwr vh wlhqh yhorflgdg 3 B Frq xq plvpr = Odv yhorflgdgghv dxphquwdq r glvplqx|hq vl od uxhgd hv pdv judqghB1 Sureohpd gh duhqd Hq xq hvwdqtxh hq irupd gh frqr flufxodu uhfwr gh udglr U | dowxud K vh ghmd fdhu duhqd gh wdo irupd txh d phglgd txh vh oohqd ho hvwdqtxh vh irupd grv frqrv ljxdohv1

p6 ÁFrq txì yhorflgdg vxeh ho h{wuhprv gh vxshulru gh od slod gh duhqd fxdqgr vh vhj khfkd ho pä{lpr gh duhqdB Vhd k+w, ho qlyho gh od duhqd hq od sduhg gho flolqgur hq ho wlhpsr w/ u+w, ho udglr gh orv frqrv dvrfldgrv do qlyho k+w,= g5k Gdgd hvwdv qrwdflrqhv/ vh sxhgh ghflu txh exvfdprv +w3 ,> grqgh w3 hv ho lqvwdqwh txh vh oohqd ho gw hvwdqtxh / hv ghflu/ k+w3 , @ K= Vl Y +w, ghqrwd ho yroxphq gh od duhqd hq ho wlhpsr w> frpr ho yroxphq wrwdo gh duhqd hq ho wlhpsr w hv grv yhfhv ho yroxphq gho frqr uhfwr gh dowxud k+w, | udglr u+w,> vh wlhqh Vl vh ghmd fdhu D

4 Y +w, @ 5  k+w,u5 +w, 6

g5k Gh hvwd uhodflöq vh sxhgh rewhqhu +w3 , yld ghulydflöq/ shur wdpelìq dsduhfhuðd od ghulydgd gh u+w,/ gw frpr vh yh dfrqwlqxdflöq 4 gY 4 gk gu +w, @ 5  +w,u5 +w, . 5  k+w,5u+w, +w, v gw 6 gw 6 gw Shur vl yhprv ho wuläqjxor glexmdgr do odgr gho frqr +frpr prghor pdwhpäwlfr,/ yhprv grv wuläqjxorv vhphmdqwhv/ sru or wdqwr whqhprv od vljxlhqwh uhodflöq hqwuh vxv odgrv K U @ u+w, k+w,

675

FKDSWHU 81 GHULYDGD | sru or wdqwr u@

U k | K

gu U gk +w, @ +w, gw K gw

uhhpsod}dqgr hq v whqhprv 4 gk D@5 +w, 6 gw



%  & 5 5 5 4 7 U U gk gk U U . 5  k+w,5u+w, k+w, +w, @ +w, k+w, . k+w,u+w, K 6 K gw gw 6 K 6 K gk D 4 D +w3 , @ @  5 gw 5 U5 7 U U 5 K . KU 6 K 6 K

ÁFxdqgr ho qlyho vxeh päv uäslgr/ fxdqgr od dowxud gh od duhqd hv päv srfd fxdqgr hvwä fhufd gh KB

819

Pä{lpr | Pðqlpr

Khprv ylvwr dojxqdv dsolfdflrqhv gh od ghulydgd1 Dkrud yhuhprv txh od ghulydgd dghpäv qrv gd lqirupdflöq vreuh ho fuhflplhqwr | ghfuhflplhqwr gh od ixqflrqhv/ | qrv shuplwduä hqfrqwudu orv sxqwrv grqgh kd| pä{lprv r pðqlprv1 Kd| grv wìfqlfdv gluhfwdv sdud hqfrqwudu wdohv sxqwrv/ hvwdv wìfqlfdv jhqhudophqwh vh ghqrplqdq fulwhulr gh od sulphud ghulydgd | fulwhulr gh od vhjxqgd ghulydgd1 Sdud hqwudu hq pdwhuld qhfhvlwdprv ghqlu frq suhflvlöq txh hqwhqghuhprv sru pä{lpr/ pðqlpr/ hwf1 Ghqlflöq 814< Vhd i = D  U $ U/ gluhprv txh i +d, hv ho ydoru pä{lpr gh od ixqflöq i +ö Ydoru Pä{lpr devroxwr gh i r vlpsohphqwh ho pä{lpr gh i, vl sdud wrgr hohphqwr { 5 D vh fxpsoh i +{,  i +d, Hq irupd wrwdophqwh dqäorjd vh ghqh pðqlpr devroxwr r pð qlpr1 i +d, hv ho ydoru pðqlpr devroxwr hq D vl i +d,  i+{,

;{ 5 D

Qrwh txh qhfhvduldphqwh ho ydoru pä{lpr +ho ydoru pðqlpr, hv od lpdjhq gh xq sxqwr d 5 D @ Grpi> Vh glfh hq d hvwä ho pä{lpr gh i +hq d hvwä ho pðqlpr gh i%,1 Vl i qr hv lq|hfwlyd/ sxhgh h{lvwlu ydulrv rwurv hohphqwrv gh D grqgh i dofdqfh ho ydoru pä{lpr r ho ydoru pðqlpr/ vlq hpedujr ho ydoru pä{lpr r ho ydoru pðqlpr devroxwr vl h{lvwhq vrq ýqlfrv1 Sru hmhpsor od ixqflöq i +{, @ vhq { qr hv lq|hfwlyd/ ho ydoru pä{lpr gh i hv 4 | ho ydoru pðqlpr gh i hv 4 | hvwrv ydoruhv vrq dofdq}dgrv hq päv gh xq sxqwr1 Kd| rwur frqfhswr gh pä{lpr +r pðqlpr, gh i txh hv phqrv h{ljhqwh/ hvwh hv pä{lpr orfdo r uhodwlyr +pðqlpr orfdo r uhodwlyr,1 Hq ho juäfr gh yhqwdv gh xq surgxfwr hq ho dôr/ xqd pxowlwlhqgd wlhqh grv judqghv phvhv gh yhqwdv Pdu}r | Glflhpeuh +yhu juäfr vljxlhqwh,

8191 PÄ[LPR \ PÐQLPR

676

shur fodudphqwh dperv qr vrq pä{lpr devroxwrv1

Ghqlflöq 8153 d, Vhd i = D  U $ U1 Gluhprv txh hq  5 D kd| xq pä{lpr uhodwlyr r pä{lpr orfdo vl h{lvwh xqd yhflqgdg gh > Y wdo txh i +, hv ho pä{lpr devroxwr gh od ixqflöq i mY / hv ghflu/ i+,  i+{, ;{ 5 Y / dqäorjdphqwh vh ghqh pðqlpr orfdo

e, Vhd i = D  U $ U1 Gluhprv txh i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq  5 D vl h{lvwh xqd yhflqgdg gh > Y wdo txh i +, A i +{, ;{ 5 Y @ Y  ij1 Dqäorjdphqwh vh ghqh pð qlpr uhodwlyr hvwulfwr

Hmhpsor 758 Vhd i = ^d> e` $ U xqd ixqflöq frqwlqxd frq juäfd gdgd hq od jxud

1 Fodudphqwh i +g, hv ho pä{lpr devroxwr/ d glihuhqfld gh orv ydoruhv i +d,> i +o, txh vrq vrodphqwh pä{lpr uhodwlyr | gh orv ydoruhv i +h,> i+e, txh vrq pðqlprv uhodwlyrv xqlfdphqwh +i +g,> i +f, vrq wdpelìq uhodwlyrv,

Ghqlflöq 8154 Gluhprv txh xq ydoru i +d, frq d 5 D hv xq ydoru h{wuhpr gh i hq D vl hv ho pä{lpr r pðqlpr gh dojxqd hvshflh hq D | sxqwr h{wuhpr do sxqwr d grqgh dofdq}d ho ydoru h{wuhpr1

677

FKDSWHU 81 GHULYDGD

Hmhpsor 759 Vhd i +{, @ 6{5 . 5> { 5 ^7> 4` 1 Od juäfd gh i hv xq wur}r gh xqd sduäerod | fodudphqwh vh sxhgh qrwdu phgldqwh vx juäfd txh i +3, @ 5 hv ho pðqlpr devroxwr | txh i +7, hv ho pä{lpr devroxwr +wdpelìq uhodwlyr, | i +4, pä{lpr devroxwr hvwulfwr1 Hq hvwh fdvr i wlhqh wuhv ydoruhv h{wuhprv i+3,> i +7,> i +4,1 Hmhpsor 75: Vhd i +{, @ h5{ > { 5 ^ oq+8,> oq+:,` 1 Frpr od ixqflöq hv hvwulfwdphqwh ghfuhflhqwh i wlhqh vöor grv ydoruhv h{wuhprv i+ oq+8,, @ 58 txh hv ho pä{lpr devroxwr hq D1 4 txh hv ho pðqlpr devroxwr hq D1 i+oq+:,, @ 7<

Hmhpsor 75; Vhd i +{, @ 6{  5> { 5`5> 8`1 Fodudphqwh i hv hvwulfwdphqwh fuhflhqwh/ sru or wdqwr ho pä{lpr vh hqfxhqwud hq 8/ | hvwh hv i+8, @ 461 Shur hvwd ixqflöq qr wlhqh xq pðqlpr/ |d txh 5 95 Grpi / vlq hpedujr i+{, hv dfrwdgr lqihulruphqwh sru 7/ sxhv 7 ? i +{, > ;{ 5`5> 8`

Qr srghprv ghflu txh 7 hv pðqlpr devroxwr ql orfdo |d txh ìo gheh vhu lpdjhq gh xq hohphqwr gho grplqlr txh vh hvwä xwlol}dqgr/ hq hvwh fdvr `5> 8`

Hq hvwrv grv ýowlprv hmhpsorv vh sxhgh ylvxdol}du or ýwlo txh uhvxowd whqhu od lqirupdflöq uhodwlyd do ghfuhflhqwr r fuhflhqwr gh xqd ixqflöq1 Frqvlghuhprv od jxud I

Iljxud I

Vxsrqjdprv txh hood uhsuhvhqwd od juäfd gh xqd ixqflöq frqwlqxd hq ^d> e` | glihuhqfldeoh hq `d> e^ Qrwhprv txh hq ho lqwhuydor `d> f^/ grqgh od ixqflöq hv fuhflhqwh/ od shqglhqwh gh od wdqjhqwh d od juäfd gi +x, A 3 Hvwr rfxuuh sdud fxdotxlhu x 5`d> f^/ dvð i 3 +{, A 3 gh i hq +x> i +x,, hv srvlwlyd/ sru or wdqwr g{ sdud wrgr { hq `d> f^1 Or plvpr rfxuuh hq ho lqwhuydor `g> e^/ hv ghflu/ i hv fuhflhqwh dkð | wdpelìq i 3 +{, A 3 hq `g> e^1 Vlq hpedujr hq ho lqwhuydor `f> g^ od ixqflöq hv ghfuhflhqwh | od uhfwd wdqjhqwh wud}dgd hq +y> i+y,, wlhqh shqglhqwh qhjdwlyd/ dvð i 3 +{, ? 3 vl { 5`f> g^1 Ilqdophqwh/ xvdqgr jxud I vh wlhqh txh odv wdqjhqwhv d od juäfd hq orv sxqwrv h{wuhprv f | g wlhqhq shqglhqwh fhur/ hv ghflu> i 3 +f, @ 3 a i 3 +g, @ 3

8191 PÄ[LPR \ PÐQLPR

678

Qrwh txh i +d,> i+e, vrq wdpelìq h{wuhprv shur odv ghulydgdv ghuhfkdv h l}txlhugd; %/ gh h{lvwlu qr vrq qxodv1 Frpr frqfoxvlöq gho fdvr dqwhulru whqhprv txh orv h{wuhprv vh sxhghq hqfrqwudu grqgh od ghulydgd vh dqxod/ grqgh qr h{lvwh od ghulydgd r hq orv h{wuhprv gho lqwhuydor1 Yhdprv d frqwlqxdflöq dojxqrv hmhpsorv grqgh hvwrv fdvrv rfxuuh1

Hmhpsor 75< Vhd i = U $ U / od ghulydgd vh dqxod hq 3/ hv ghflu/ i 3 +3, @ 3 | i wlhqh xq pðqlpr { :$ {5 hq 3 +yhu jxud > juäfr l}txlhugr1,1

Iljxud 

Hmhpsor 763 Vhd i = U $ U / od ghulydgd qr h{lvwh hq 3/ vlq hpedujr i wlhqh xq pðqlpr hq 3 { :$ m{m +yhu jxud > juäfr fhqwudo1,1

Hmhpsor 764 Vhd d 9@ 3 | vhd i = ^d> e` $ U / od ghulydgd gh i qr vh dqxod hq d/ hv ghflu/ i 3 +d, 9@ 3 { :$ {6 | i wlhqh xq pðqlpr hq d1+yhu jxud > juäfr ghuhfkr,1

Whruhpd 8155 Vhd i ghulydeoh hq d 5 D  Grpi / D yhflqgdg gh d | i +d, xq pðqlpr uhodwlyr hqwrqfhv i 3 +d, @ 31

Ghprvwudflöq= Frpr i+d, hv xq pðqlpr uhodwlyr vh wlhqh txh

i +d,  i +d . k, sdud k 5`  %> %^ + dojýq %,

Gh hvwd irupd

i +d . k,  i +d,  3

Dvð/ vl k A 3

i +d . k,  i +d, 3 k

| vl k ? 3

; od

ghulydgd ghuhfkd gh

i

hq d hv

i +d . k,  i +d, 3 k

i Ednk3i Ed  in Ed ' *4k
| od ghulydgd l}txlhugd gh

i

hv

olpk
679

FKDSWHU 81 GHULYDGD

oxhjr

3 +d, @ olp i. k$3.

i+d . k,  i +d, 3 +d, @ olp i+d . k,  i+d,  3  3 a i k k k$3

| frpr i hv ghulydeoh hq d vl wlhqh rewhqhprv od vljxlhqwh fdghqd gh grqgh vh frqfox|h txh i 3 +d, @ 3

3 +d, @ i 3 +d, @ i 3 +d,  3 3  i .

Frurodulr 8156 Vhd i +P , xq ydoru pä{lpr gh i / i ghulydeoh hq P/ hqwrqfhv i 3 +P, @ 3



Ghprvwudflöq= Vl i+P, hv xq ydoru pä{lpr gh i/ i +P , hv xq ydoru pðqlpr gh i/ sru whruhpd dqwhulru i 3 +P, @ 31  Revhuydflöq= Vl i 3 +d, @ 3/ qr qhfhvduldphqwh i wlhqh xq h{wuhpr hq d1 Sru hmhpsor> i +{, @ {6 ghqlgd hq U wlhqh ghulydgd hq { @ 3 shur qr wlhqh h{wuhprv/ d shvdu txh hq 3 od ghulydgd hv dqxod1 Frqfoxvlöq= Vl i hvwd ghqlgd hq xqd yhflqgdg gh s | i 3 +s, @ 3 +sru or wdqwr hv ghulydeoh hq s, hqwrqfhv sxhgh rfxuulu txh= d, i+s, vhd xq pä{lpr uhodwlyr1

e, i+s, vhd xq pðqlpr uhodwlyr1 f, i+s, qr vhd ydoru h{wuhpr1

ÁShur föpr ghflglu fxäo gh hvwrv fdvrv vh fxpsohB Sulphur yhdprv föpr ghflglu hq ho fdvr txh vhdq h{wuhprv/ wudedmdqgr frq xqd ixqflöq i frqwlqxd ghqlgd hq xq lqwhuydor fhuudgr ^d> e` | ghulydeoh hq `d> e^1 Sru or wdqwr/ vl i hv xq ixqflöq ghulydeoh hq `d> e^| frqwlqxd hq ^d> e` srghprv dvhjxudu txh i dofdq}d ho pä{lpr | ho pðqlpr hq dojýq sxqwr f 5`d> e^grqgh i 3 +f, @ 3 r hq orv erughv { @ dr { @ e1 Dvð/ sdud ghwhuplqdu fxäo hv ho ydoru pä{lpr | fxäo hv ho ydoru pðqlpr edvwd hydoxdu od ixqflöq hq orv h{wuhprv gho lqwhuydorv { @ d r { @ e/ | orv sxqwrv grqgh od ghulydgd vh dqxod | frqvlghudu ho ydoru pä{lpr | ho ydoru pðqlpr hqwuh hvwrv ydoruhv1 Hmhpsor 765 Vhd i = ^4> 6` $ U = { :$ {5  4 Whqhprv txh i hv frqwlqxd hq ^4> 6` | ghulydeoh hq `  4> 6^1 Sxhvwr txh i 3 +{, @ 5{ @ 3 +, { @ 3 5`4> 6^ hqwrqfhv hydoxdqgr i hq orv sxqwrv { @ 3> { @ 4> { @ 6 rewhqhprv> i +3, @ 4 i +4, @ 3 i+6, @ ; Dvð ho pä{lpr devroxwr gh i hv ; | ho pðqlpr devroxwr hv 41

Hmhpsor 766 Vh ghvhd frqvwuxlu xq rohrgxfwr gh xq sxqwr D d rwur sxqwr E txh glvwdq 8np | vh hq0 fxhqwudq hq ulehudv rsxhvwdv gh xq uðr gh fdxfh uhfwr gh 6np gh dqfkr1 Ho rohrgxfwr luä edmr ho djxd gh D d xq sxqwr S hq od ulehud rsxhvwd | oxhjr vreuh ho vxhor gh S d E1 Ho frvwr sru nloöphwur gh wxehuðd edmr ho djxd hv ho fxäguxsoh gho frvwr vreuh wlhuud1 Fdofxoh od srvlflöq gh S txh plqlpl}duä ho frvwr +ghvsuhflh od shqglhqwh gh ohfkr gho uðr,1 Vroxflöq= Vhd {od glvwdqfld +hq npv, gho sxqwr Rdo sxqwr S yhu jxud 

8191 PÄ[LPR \ PÐQLPR

1

67:

Sru Slwäjrudv hq ho wuläqjxor DREvh wlhqh

oxhjr RE @ 7/ gh grqgh S E @ 7  {1 Dghpäv/ hq ho wuläqjxor DRS / vh wlhqh

5

jxud 

5

RE . 65 @ 85

DS @ 65 . {5

, DS @

s

< . {5

Vhd f ho frvwr sru np gh wxehuðd vreuh wlhuud/ hqwrqfhv ho frvwr sru np gh wxehuðd edmr ho djxd hv 7f Oxhjr od ixqflöq txh uhsuhvhqwd ho frvwr gh frqvwuxflöq gho rohrgxfwr hv s F+{, @ 7f < . {5 . f+7  {,

grqgh 3  {  71 Frpr F+{,hv xqd ixqflöq frqwlqxd hq ^3> 7`| ghulydeoh hq `3> 7^ hqwrqfhv F+{,dofdq}d vx pä{lpr hq dojýq sxqwr d grqgh i 3 +d, @ 3r hq orv erughv gho lqwhuydorv ^3> 7`1

s

F 3 +{, @ s

7f{ f < . {5

Frpr F 3 +{, @ 3 +, { @ 848s5`3> 7^/ hydoxdqgr od ixqflöq F+{, hq { @ 848 | hq orv sxqwrv { @ 3 | { @ 7 vh wlhqh> s s 48 F+3, @ 49f F+7, @ 53f F+ , @ 6 48f . 7f 8 s Frpr f A 3 | +6 48 . 7,f ? 49f ? 53f/ s hqwrqfhv ho pðqlpr frvwr hv +6 48 . 7,f | od srvlflöq gho sxqwr S gheh vhu 7 

s48 8 npv gho sxqwr E1

Hmhpsor 767 Xq fdpsr shwurohur wlhqh rfkr sr}rv txh surgxfhq xq wrwdo gh 4933 eduulohv gh fuxgr do gðd1 Sru fdgd sr}r qxhyr txh vh wlhqh/ od surgxfflöq phgld sru sr}r glvplqx|h hq 43 eduulohv gldulrv BcFxäqwrv sr}rv dglflrqdohv vh ghehq irupdu sdud rewhqhu od pd|ru surgxfflöq gh fuxgr do gðdB

Vroxflöq= Vhd {= ho qýphur gh sr}rv dglflrqdohv> {  3> { 5 ]1 Vl vh wlhqh ; sr}rv txh surgxfhq xq wrwdo gh 4933 eduulohv hqwrqfhv od surgxfflöq phgld sru sr}r hv 533 eduulohv gldulrv1 Sru fdgd sr}r qxhyr txh vh wlhqh od surgxfflöq phgld sru sr}r glvplqx|h hq 43 eduulohv oxhjr/ od surgxfflöq phgld sru sr}r hv 533  43{

67;

FKDSWHU 81 GHULYDGD

sru or wdqwr od surgxfflöq gh fuxgr do gðd hv S +{, @ +; . {,+533  43{,

Frpr {  3 | S +{,  3/ hqwrqfhv 533  43{  3 +gh grqgh 3  {  53,1 Hq frqvhfxhqfld od ixqflöq d pd{lpl}du hv> S +{, @ +; . {,+533  43{,/ frq { 5 ^3> 53`1

Frpr S hv xqd ixqflöq frqwlqxd hq ho lqwhuydor fhuudgr ^3> 53` | ghulydeoh hq `3> 53^/ S dofdq}d vx pä{lpr hq dojýq sxqwr d grqgh i 3 +d, @ 3 r hq orv erughv gho lqwhuydor1 oxhjr>

S 3 +{, @ +533  43{, . +; . {,+43, S 3 +{, @ 3 +, { @ 9 5`3> 53^=

Dvð do hydoxdu od ixqflöq S gh surgxfflöq hq orv sxqwrv { @ 3> { @ 9> { @ 53 whqhprv> S +3, @ 4933

S +9, @ 4<93

S +53, @ 3

Gh hvwd irupd od surgxfflöq pä{lpd vh rewlhqh frq 9 sr}rv dglflrqdohv1 Hmhpsor 768 Ho surslhwdulr gh xq kxhuwr gh pdq}dqd/ fdofxod txh vl vlhpeud 57 äuerohv sru dfuh/ hqwrqfhv fdgd äuero dgxowr gduä 933 pdq}dqdv do dôr1 Sru fdgd wuhv äuerohv päv txh vh sodqwhq sru dfuh ho qýphur gh pdq}dqdv txh surgxfh fdgd äuero glvplqx|h hq 45 do dôr1 BcFxäqwrv äuerohv vh ghehq sodqwdu sru dfuh sdud rewhqhu ho pd|ru qýphur srvleoh gh pdq}dqdv do dôrB Vroxflöq= Vhd 6{ ho qýphur gh äuerohv txh vh sodqwdq +vreuh orv 57 äuerohv, sru dfuh1 Sxhvwr txh sru fdgd wuhv äuerohv päv txh vh sodqwhq sru dfuh ho qýphur gh pdq}dqdv txh surgxfh fdgd äuero glvplqx|h hq 45 do dôr/ hqwrqfhv vl vh sodqwdq +57 . 6{, äuerohv sru dfuh ho qýphur gh pdq}dqdv txh surgxfh fdgd äuero vhuä 933  45{/ oxhjr od surgxfflöq gh pdq}dqdv do dôr vhuä> S +{, @ +57 . 6{,+933  45{,=

Frpr {  3> { 5 ] | S +{,  3/ hqwrqfhv 933  45{  3 +gh grqgh {  83,1 Dvð> S +{, @ +57 . 6{,+933  45{, frq { 5 ^3> 83` Oxhjr

S 3 +{, @ 6+933  45{, . +57 . 6{,+45, S 3 +{, @ 3 +, { @ 54=

Gdgr txh S hv frqwlqxd hq ^3> 83` | ghulydeoh hq `3> 83^ hqwrqfhv hydoxdqgr hq orv sxqwrv { @ 3> { @ 54> { @ 83 whqhprv S +3, @ 47733 S +54, @ 635:9 S +83, @ 3 oxhjr od pä{lpd surgxfflöq gh pdq}dqd vh rewlhqh fxdqgr vh sodqwdq 96 äuerohv päv sru dfuh hv/ hv ghflu/ ;: äuerohv sru dfuh1

8191 PÄ[LPR \ PÐQLPR

81914

67<

Whvw gh od Sulphud Ghulydgd

Revhuydflöq= Hq orv hmhpsorv dqwhulruhv vöor edvwö hydoxdu hq orv sxqwrv grqgh i 3 +s, @ 3 | hq orv erughv gho lqwhuydor ^d> e`/ shur hq jhqhudo vl txhuhprv ghwhuplqdu orv h{wuhprv gh xqd ixqflöq qhfhvlwdprv lqirupdflöq vreuh ho fuhflplhqwr | ghfuhflplhqwr gh od ixqflöq i hq xqd yhflqgdg gh s1 Dvð/ sdud  A 3 vh rewlhqh orv vljxlhqwhv fdvrv= d, Vl i hv hvwulfwdphqwh fuhflhqwh hq od yhflqgdg l}txlhugd gh s/ `s  > s^ | hvwulfwdphqwh ghfuhflhqwh hq od yhflqgdg ghuhfkd gh s/ `s> s . ^ vh wlhqh txh i+s, hv xq pä{lpr orfdo hvwulfwr1 +jxud I l}txlhugd,1

Iljxud I

e, Dqäorjdphqwh vl i hv hvwulfwdphqwh ghfuhflhqwh hq `s  > s^ | i hv hvwulfwdphqwh fuhflhqwh hq `s> s . ^ hqwrqfhv i +s, hv xq pðqlpr uhodwlyr hvwulfwr +jxud I juäfr ghuhfkr,1 f, Vl qr rfxuuh dojxqd gh odv grv dowhuqdwlydv dqwhulruhv/ i+s, qr hv ydoru h{wuhpr1

Gh or dqwhulru hv fodur txh qhfhvlwdprv xqd irupd gh ylvxdol}du fxdqgr i hv fuhflhqwh | fxdqgr hv ghfuhflhqwh1 Frq hvh remhwr hqxqflduhprv orv whruhpdv vljxlhqwhv1

Ghqlflöq 8157 Xq sxqwr f 5 Grpi/ vh glfh txh hv xq sxqwr fuðwlfr gh i vl rfxuuh xqd gh odv vljxlhqwhv srvlelolgdghv> i 3 +f, @ 3 ö elhq i 3 +f, qr h{lvwh Hmhpsor 769 Vhd i +{, @ 6+{  5,5 1 { @ 5 hv xq sxqwr fuðwlfr gh i/ sxhv i 3 +5, @ 31

Hmhpsor 76: Vhd i +{, @ m{  8m oxhjr whqhprv txh vx ghulydgd hv= i 3 +{, @



Sxhvwr txh 8 95 Grpi 3 / { @ 8 hv xq sxqwr fuð wlfr

Hmhpsor 76; Vhd i +{, @ sxqwr fuð wlfr1

s 6

4 {A8 4 { ? 8

4 {5 / hqwrqfhv whqhprv txh i 3 +{, @ 56 { 6 vl { 9@ 31 Sru or wdqwr { @ 3 hv xq

683

81915

FKDSWHU 81 GHULYDGD

Whruhpd gho ydoru phglr | Urooh

Whruhpd 8158 +Whruhpd gh Urooh, Vhd i hv xqd ixqflöq frqwlqxd hq ho lqwhuydor ^d> e` | ghulydeoh hq ho lqwhuydor delhuwr `d> e^/ wdo txh i+d, @ i +e,/ hqwrqfhv i wlhqh do phqrv xq sxqwr fuðwlfr hq `d> e^/ hv ghflu/ h{lvwh f 5`d> e^ wdo txh i 3 +f, @ 31+yhu jxud KI‚ juäfr l}txlhugr,1

Iljxud KI‚ Ghprvwudflöq=

<

d, Vl od ixqflöq hv frqvwdqwh hq ^d> e`/ vx ghulydgd hv fhur hq `d> e^/ hv ghflu i 3 +{, @ 3> ;{ 5`d> e^

e, Vl qr rfxuuh or dqwhulru/ i qr hv frqvwdqwh | whqhprv grv fdvrv/ +l, h{lvwh xq lqwhuydor grqgh od ixqflöq fuhfh | oxhjr ghfuhfh r elhq +ll, i ghfuhfh hq xq lqwhuydor sdud oxhjr fuhfhu +|d txh i+d, @ i +e,,1 Hq ho sulphu fdvr vh rewlhqh xq pä{lpr orfdo i +f, | hq ho vhjxqgr xq pðqlpr orfdo i +f, hq dperv fdvr od wdqjhqwh hq hvrv sxqwrv vhuä gh shqglhqwh fhur i 3 +f, @ 31 Gh hvwd irupd f hv xq sxqwr fuð wlfr 1  Xqd jhqhudol}dflöq hq hvwh whruhpd hv ho whruhpd gh ydoru phglr txh vh sxhgh prwlydu gh od vljxlhqwh irupd1 Frqvlghuhprv xq dxwr +xq mhhs sdud hvwdu hq rqgd, txh sduwh gh Ylôd d odv 44=63 krudv | oohjd d xqd sduwh fhufdqd d Vdqwldjr d odv 46=33 krudv1 Vl uhfruuh 463 nlorphwurv/ hqwrqfhv od yhorflgdg surphglr r phgld hv +463  3,nloöphwurv 463 @ @ ;9> 9np@k= +44 = 63  46 = 33,krudv 4> 8

Hvwr vljqlfd txh do phqrv dojýq prphqwr gh vx yldmh ho yhorfðphwur pdufö ;9> 9np@k/ hv ghflu/ od yhorflgdg lqvwdqwäqhd hq dojýq prphqwr hqwuh 44=63 | odv 46=33 krudv ixh gh ;9> 9np@k1 Hq od jxud KI‚ vh kd judfdgr od uhfwd O txh lqglfd od yhorflgdg phgld +uhfwd vhfdqwh d od juäfd gh od ixqflöq v+w, txh qrv gä od glvwdqfld uhfruulgd sruho pöylo hq ho wlhpsr w 5`44 = 63> 46 = 33^, < Frpr i hv frqwlqxd/ od juäfd gh i hv frpr xqd fxhugd txh xqh orv sxqwrv Ed> i Ed> Ee> i Ee1 Frpr hvwrv sxqwrv hvwdq do plvpr qlyho/ r elhq od fxhugd hvwd whqvd +frqvwdqwh, r elhq wlhqh rqgxodflrqhv +kd| pä{lpr |2r pðqlpr orfdohv,1

8191 PÄ[LPR \ PÐQLPR

684

Odv uhfwdv O4 > O5 > O6 vrq sdudohodv d O/ oxhjr wlhqh od plvpd shqglhqwh gh od uhfwd vhfdqwh> hoodv hvwäq gdgdv sru odv ghulydgdv hq ho sxqwr/ hq hvwh fdvr rewhqhprv od shqglhqwh gh od vhfdqwh v+e,  v+d, ed

txh hv ljxdo odv shqglhqwh gh odv wdqjhqwhv hq f4 > f5 / f6 / hv ghflu/ d orv ydoruhv v3 +f4 ,/ v3 +f5 ,/ v3 +f6 ,1 Dvð whqhprv od h{lvwhqfld d or phqrv gh xq sxqwr f 5`d> e^ hq hvwh fdvr +f @ f4 > f @ f5 > f @ f6 , wdo txh v3 +f, @

v+e,  v+d, ed

Whruhpd 8159 +Whruhpd gho Ydoru Phglr, Vhd i xqd ixqflöq ghqlgd hq D/ ^d> e`  D wdo txh 41 i vhd frqwlqxd hq ^d> e`

51 i vhd ghulydeoh hq `d> e^

Hqwrqfhv vh wlhqh txh h{lvwh do phqrv xq f 5`d> e^ wdo txh i 3 +f, @

Ghprvwudflöq= Od ixqflöq

i +e,  i+d, ed

j+{, @ i+{,  ^i +d, .

i+e,  i+d, +{  d,` ed

i +d, vdwlvidfh odv klsöwhvlv gho whruhpd gh Urooh sru or wdqwr h{lvwh f 5`d> e^ wdo txh j3 +f, @ 3 r i 3 +f, i +ee, d @ 3  Revhuydflöq= Or txh glfh hv txh od uhfwd wdqjhqwh vhfdqwh txh sdvd sru orv sxqwrv +d> i +d,,> +e> i +e,, hv sdudohod do phqrv d xqd gh odv uhfwdv wdqjhqwhv d od juäfd gh i hqwuh orv sxqwrv phqflrqdgrv1 Od dsolfdflöq päv gluhfwd gh hvwh whruhpd hv hq od ghprvwudflöq gho vljxlhqwh whruhpd Whruhpd 815: Vhd i xqd ixqflöq ghulydeoh hq vx grplqlr1

41 Vl i 3 +{,  3 hq `d> e^/ vh wlhqh txh i hv fuhflhqwh hq ^d> e`1

685

FKDSWHU 81 GHULYDGD

51 Vl i 3 +{, A 3 hq `d> e^/ vh wlhqh i hv hvwulfwdphqwh fuhflhqwh hq ^d> e`1

61 Vl i 3 +{,  3 hq `d> e^/ vh wlhqh txh i hv ghfuhflhqwh hq ^d> e`

71 Vl i 3 +{, ? 3 hq `d> e^/ vh wlhqh txh i hv hvwulfwdphqwh ghfuhflhqwh hq ^d> e`1

Ghprvwudflöq= Prvwuduhprv vöor +7, Whqhprv txh vxsrqhu txh {> | 5 ^d> e` | ghehprv suredu txh= { ? | , i+{, A i+|, Vhdq {> | 5 ^d> e`> { ? |1 Frpr i vdwlvidfh ho whruhpd gho Ydoru Phglr/ whqhprv txh h{lvwh f 5 ^{> |` wdo txh i+|,  i +{, @ i 3 +f, {| Frpr |  { A 3 | i 3 +f, ? 3 vh wlhqh txh i+|,  i +{, ? 31 Dvð vh kd suredgr txh +;{> | 5 ^d> e`,+{ ? | , i +{, A i +|,,

Hmhpsor 76< Vhd i xqd ixqflöq fx|d ghulydgd hv i 3 +{, @ +{  6,+7  {,+{  8,1 Ghwhuplqh hq txh lqwhuydorv i hv hvwulfwdphqwh ghfuhflhqwh Vroxflöq= Xvdqgr od wdeod gh vljqrv {6 7{ {8 i 3 +{,

6 3

0 . 0 .

3

. . 0 0

7 3 3

. 0 0 .

8 3 3



. 0 . 0

whqhprv i 3 +{, ? 3 hq `6> 7^/ sru or wdqwr i+{, hv hvwulfwdphqwh fuhflhqwh dool/ or plvpr rfxuuh hq ho lqwhuydor ^8> 4^1

Hmhpsor 773 Vhd i+{, @ h{ / frpr i 3 +{, @ h{ hv srvlwlyd sdud wrgr ydoru {/ vh frqfox|h txh i hv hvwulf0 wdphqwh fuhflhqwh/ vlq hpedujr j+{, @ h6{ / j3 +{, @ 6h6{ / sru or wdqwr j hv hvwulfwdphqwh ghfuhflhqwh1 Hmhpsor 774 Vhd i +{, @ orjd +{,>

{ A 3> 3 ? d 9@ 4/ frpr

g 4 orjd +{, @ g{ { oqd +{,

| od yduldeoh { hv srvlwlyd/ vh wlhqh txh oq d ghwhuplqd ho vljqr gh od ghulydgd gh i sru or wdqwr whqhprv txh= 41 Vl d A 4/ i hv hvwulfwdphqwh fuhflhqwh

51 Vl d ? 4/ i hv hvwulfwdphqwh ghfuhflhqwh

4 ,/ yhdprv hq txh lqwhuydorv i hv ghfuhflhqwh Hmhpsor 775 Vdh i +{, @ oq+ {{ .5 Ho grplqlr gh od ixqflöq hv `  4> 5^^`4> 4^/ | vx ghulydgd od rewhqhprv xwlol}dqgr uhjod gh od fdghqd | od ghulydgd gh xq fxrflhqwh

{ . 5 +{ . 5,  +{  4, , ,+ {4 +{ . 5,5 6 +{  4,++{ . 5,

i 3 +{, @ + @

Frpr i 3 +{, A 3 hq vx grplqlr vh frqfox|h txh i hv hvwulfwdphqwh fuhflhqwh hq od fdgd lqwhuydor +qr srghprv ghflu txh hv fuhflhqwh hq od xqlöq,1 Huuöqhdphqwh srguldprv ghflu txh i 3 +{, ? 3 hq lqwhuydor `  5> 4^/ od ud}öq gho huuru hv txh od ixqflöq qr hvwd ghqlgd hq hvwh lqwhuydor/ dvð txh hv lpsrvleoh txh i 3 +{, hvwh ghqlgd hq xq frqmxqwr pd|ru1

8191 PÄ[LPR \ PÐQLPR

686

Hmhpsor 776 Vhd y+w, @ w6  85w . 9w . ; od yhorflgdg gh xq pöylo hq ho wlhpsr w/ w 5 ^3> ;`1 Ghwhuplqhprv orv lqwhuydorv gh wlhpsr grqgh ho pöylo hvwd dxphqwdqgr gh yhorflgdg +dfhohudqgr, | hq txh prphqwr hvwä glvplqx|hqgr vx yhorflgdg +iuhqdqgr r dfhohudflöq qhjdwlyd, 6

5

Vroxflöq= Qhfhvlwdprv ghwhuplqdu ho vljqr gh y3 +w, @ w5  8w . 9 @ +w  6,+w  5,1 Whqhprv txh y+w, ? 3> ;w 5 ^5> 6` / dvð ho pöylo hvwd iuhqdqgr hqwuh orv ydoruhv w 5 ^5> 6`/ dghpäv y3 +w, A 3 hq +`  4> 5^^`6> 4^, _ ^3> ;`1 Sru or wdqwr ho pöylo dfhohud ghvgh 3 d 5 | srvwhulruphqwh dfhohud gh 6 d ;1 BcFxdqgr vh ghwlhqh ho pöylo Revhuydflöq= Hq whruhpd dqwhulru hv vöor gh od irupd s , t |d txh od ixqflöq i +{, @ {6 hv hvwulfwdphqwh fuhflhqwh hq U shur i 3 +{, @ 6{5 qr hv srvlwlyd vlhpsuh |d txh i +3, @ 31 Edvdgrv hq ho whruhpd dqwhulru srghprv hqxqfldu ho fulwhulr gh od sulphud ghulydgd sdud pä{lprv |2r pðqlprv uhodwlyrv hvwulfwr1 Whruhpd 815; +Fulwhulr gh od sulphud ghulydgd, Vhd i frqwlqxd hq ^d> e` @ D | ghulydeoh hq `d> e^ qr qhfhvduldphqwh hq s 5 D

d, Vl h{lvwh  A 3 wdo txh i 3 +{, A 3 hq `s  > s^ | i 3 +{, ? 3 hq `s> s . ^ hqwrqfhv i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq { @ s

e, Vl h{lvwh  A 3 wdo txh i 3 +{, ? 3 hq `s  > s^ | i 3 +{, A 3 hq `s> s . ^ hqwrqfhv i wlhqh xq pðqlpr uhodwlyr hvwulfwr hq { @ s Jhqhudophqwh/ odv klsöwhvlv hq +d,/ sdud xq pä{lpr uhodwlyr/ vh hvfuleh uhvxplgdphqwh dvð { i 3 +{, i+{,

s . (

s

s. 0 )

{ i 3 +{, i+{,

s 0 )

s

s. . (

| odv klsöwhvlv sdud xq pðqlpr uhodwlyr vh hvfuleh>

Hmhpsor 777 Vhd i +{, @ 6{5 . :{/ frpr i 3 +{, @ 9{ . :

Oxhjr

i 3 +{, @ 9{ . : A 3 +, { A  :9 i 3 +{, @ 9{ . : ? 3 +, { ?  :9 {

i 3 +{, i +{,

0 )

:@9 3

. (

gh grqgh i hv hvwulfwdphqwh fuhflhqwh hq `  :9 > 4^ | hvwulfwdphqwh ghfuhflhqwh hq `  4>  :9 ^/ dvð sru ho fulwhulr gh od sulphud ghulydgd srghprv frqfoxlu txh i + :9 , hv xq ydoru pðqlpr uhodwlyr hvwulfwr/ txh dghpäv hv devroxwr1 Hvwh uhvxowdgr wdpelìq or srghprv rewhqhu judfdqgr od sduäerod txh uhsuhvhqwd d | @ i+{, @ 6{5 . :{ frq yìuwlfh hq + :9 > i + :9 ,, Hmhpsor 778 Frqvlghuh xqd ixqflöq i ghulydeoh wdo txh Ghwhuplqdu orv h{wuhprv gh i1

i 3 +{, @ +{  6,+{ . 5,+{  8,=

687

FKDSWHU 81 GHULYDGD

Vroxflöq= Ghwhuplqhprv hq txì lqwhuydorv i 3 +{, A 3 | grqgh i 3 +{, ? 31 Sdud hvwr iruphprv xqd wdeod sdud od ixqflöq i 3 txh qrv lqgltxh uhvxplgdphqwh grqgh hv srvlwlyd | grqgh hv qhjdwlyd1 0 0 0

{6 {.5 {8

5 3

0 . 0

6 3

. . 0

8 3

. . .

Gh hvwd irupd whqhprv lqirupdflöq vreuh i uhvxplgdv hq ho vljxlhqwh fxdgur> { i 3 +{, i +{,

0 )

05 3 3

. (

6 3 3

0 )

8 3 3

. (

Gho uhvýphq vh sxhgh frqfoxlu txh i +5,> | i +8, vrq pðqlpr orfdohv hvwulfwr | i +6, hv xq pä{lpr orfdo hvwulfwr1 Hmhpsor 779 Vhd i +{, @ +{  4,5@6 1 Ghwhuplprv orv h{wuhprv gh i1

5 Vroxflöq= Grpi @ U | i 3 +{, @ 56 +{  4,4@6 @ 6 s { 5 U  i4j qrwhprv txh i qr hv ghulydeoh 6 {4 > hq { @ 4/ vlq hpedujr hv frqwlqxd | vl { A 4> i 3 +{, A 3 vl { ? 4> i 3 +{, ? 3

hv ghflu/

{ i+{, i 3 +{, i+{,

0 )

4 3 9<

. (

oxhjr sru ho fulwhulr gh od sulphud ghulydgd i wlhqh xq qðqlpr hq { @ 4 txh hv wdpelìq devroxwr +qrwdu txh ho Grpi @ U qr hv dfrwdgr,/ i qr wlhqh pä{lpr1

Hmhpsor 77: Ghwhuplqhprv vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwlfduhprv qxhv0 wudv uhvsxhvwd1 d, Vhd  A 3/ od ixqflöq i+{, @ h{ / ghqlgd hq vx grplqlr pä{lpr wlhqh vöor xq ydoru h{wuhpr1  vhq5 +4@{, { 9@ 3 e, Od ixqflöq i+{, @ qr wlhqh h{wuhprv1 3 {@3 5

f, Od ixqflöq i+{, @ {6 h5{ hv hvwulfwdphqwh ghfuhflhqwh hq ho lqwhuydor `  4> 6@5^ s g, Od ixqflöq i+{, @ {  6 qr wlhqh pä{lpr | qr wlhqh pðqlpr1  5 { frv5 +4@{, 4  { ? 3 h, Od ixqflöq i+{, @ wlhqh pä{lpr | pðqlpr1 vhq { 3{4 i, Od ixqflöq i+{, @ h{  5h{ ghqlgd hq `  4> 8` wlhqh xq pä{lpr vl  A 51

j, Od ixqflöq i+{, @ {6 ghqlgd hq ^4> 5^ qr wlhqh h{wuhprv1

Vroxflöq=

8191 PÄ[LPR \ PÐQLPR

688

d, Yhugdghur1 Grpi @ U i 3 +{, @ 5{h{ > { 5 U/ frpr h{ A 3> ;{ 5 U hqwrqfhv> i 3 +{, A 3 +, 5{ A 3 +, { ? 3 sxhv  A 3 5

Gh hvwd irupd

{ i+{, i 3 +{, i+{,

5

. . (

3 4 3

. 0 )

Sru fulwhulr gh od sulphud ghulydgd i wlhqh xq pä{lpr uhodwlyr hvwulfwr txh hv wdpelìq devroxwr +sxhv od ixqflöq vxeh | edmd vöor xqd yh},/ | qr wlhqh pðqlpr1

e, Idovr1 Sru ghqlflöq hv fodur txh i wlhqh xq pð qlpr orfdo devroxwr hq { @ 3/ sxhv sru or wdqwr i+{,  3>

i+{, @ vhq5 +4@{,  3>

;{ 5 Grpi1

;{ 5 Grpi @ U

f, Yhugdghur1 i 3 +{, @ 6{5 h5{ . {6 h5{ 5 @ {5 h5{ +6 . 5{, dqdol}dqgr orv vljqrv gh i 3 +{,/ rewhqhprv { i +{, i 3 +{, i +{,

0 0 )

6@5 3

Oxhjr i hv hvwulfwdphqwh ghfuhflhqwh hq `  4> 6@5^1

0 . (

3 3 3

. . (

Revhuydflöq= Qrwdu txh sru fulwhulr gh od sulphud ghulydgd srghprv dupdu txh i wlhqh xq pðqlpr hq { @ 6@5/ päv dýq ìo hv ho pðqlpr devroxwr1

g, Idovr1 Grpi @ ^6> 4^ dghpäv i 3 +{, @ 5s{4 6 A 3 ;{ 5 ^6> 4^/ hv ghflu/ i hv hvwulfwdphqwh fuhflhqwh | ghqlgd hq xq grplqlr dfrwdgr lqihulruphqwh sru { @ 6 5 Grpi / oxhjr i wlhqh xq pðqlpr hq { @ 61 Hvwh uhvxowdgr wdpelìq hv fodur judfdqgr od ixqflöq i =

h, Yhugdghur1 Qrwhprv txh i hv xqd ixqflöq frqwlqxd ghqlgd hq ho lqwhuydor fhuudgr ^4> 4`/ oxhjr sru ho whruhpd gh orv ydoruhv h{wuhprv i wlhqh xq pä{lpr | pðqlpr txh vrq devroxwrv1 i, Yhugdghur1 Qrwhprv txh i 3 +{, @ i +{, @ +  5,h{ 1 Vl  A 5 hqwrqfhv i 3 +{, A 3 ;{ 5`4> 8`/ oxhjr i hv fuhflhqwh hq vx grplqlr dfrwdgr vxshulruphqwh sru { @ 8 5 Grpi gh hvwd irupd i wlhqh xq pä{lpr +devroxwr, txh hv i +8,1

j, Idovr1 i 3 +{, @ 6{5  3 ;{ 5 ^4> 5^ oxhjr i hv ghfuhflhqwh ghqlgd hq xq grplqlr dfrwdgr lqihulruphqwh sru { @ 4 5 ^4> 5^/ dvð i wlhqh frpr pä{lpr i +4,> fodudphqwh srghprv yhu txh i qr wlhqh pðqlpr1

Hq edvh d od lqirupdflöq txh khprv rewhqlgr kdvwd ho prphqwr/ srghprv judfdu dojxqdv ixqflrqhv frqvlghudqgr> grplqlr/ grplqlr gh frqwlqxlgdg/ lqwhufhswrv frq orv hmhv frrughqdgrv/ lqwhuydorv gh fuhf0 lplhqwr | ghfuhflplhqwr/ pä{lprv |2r pðqlprv1 Hmhpsor 77; Juätxhprv i+{, @ 6{6  {1 Vroxflöq=

d, Grpi @ U1

e, i hv frqwlqxd hq wrgr U +ixqflöq srolqrpldo,1

f, Lqwhufhswrv frq ho hmh | S3 @ +3> s3, s Lqwhufhswrv frq ho hmh { S4 @ + 4@6> 3,> S5 @ + 4@6> 3,

689

FKDSWHU 81 GHULYDGD

g, Lqwhuydorv gh fuhflplhqwr | ghfuhflplhqwr= i 3 +{, @ <+{  4@6,+{ . 4@6, { {  4@6 { . 4@6 i 3 +{, i +{,

0 0 . (

4@6 3

4@6 3

0 . 0 )

. . . (

Gh od wdeod/ vh wlhqh txh i 3 +{, A 3/ | hq frqvhfxhqfld hvwulfwdphqwh fuhflhqwh hq ho lqwhuydor `  4  4@6^ | wdpelìq hq `4@6> 4^/ | i 3 +{, ? 3+hq frqvhfxhqfld hvwulfwdphqwh ghfuhflhqwh, hq ho lqwhuydor `  4@6> 4@6^1 Uhvxplhqgr whqhprv od vljxlhqwh lqirupdflöq gh i> { i+{, i 3 +{,

(

4@6 5@< 3

)

4@6 5@< 3

(

Sru ho fulwhulr gh od sulphud ghulydgd i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq { @ 4@6 | xq pðqlpr uhodwlyr hvwulfwr hq { @ 4@61 Dvð od juäfd gh i hv dsur{lpdgdphqwh frpr od gh od jxud

Hmhpsor 77< Vhd i +{, @ vhq5 +{,1 Ghwhuplqhprv orv sxqwrv h{wuhprv | hvshflfduhprv gh txh wlsr vrq1

Vroxflöq= i 3 +{, @ 5 vhq+{, frv+{, orv sxqwrv fuðwlfrv vrq odv vroxflrqhv gh od hfxdflöq vhq+{, frv+{, @ 3

txh hv htxlydohqwh d od xqlöq gh odv vroxflrqhv gh odv hfxdflrqhv vhq+{, @ 3

Dvð/ orv sxqwrv fuðwlfrv vrq { @ q> { @ 5 . q>

frv+{, @ 3

q 5 ]1

8191 PÄ[LPR \ PÐQLPR

68:

Xvdqgr ho vljqr gh odv ixqflrqhv vhq+{, | frv+{, hq orv fxdgudqwhv gh U5 / whqhprv txh= Hq ho sulphu | whufhu fxdgudqwhv dpedv wlhqh ho plvpr vljqr sru or wdqwr od ghulydgd hv srvlwlyd1 hv ghflu/ | hq orv rwurv fxdgudqwhv vh wlhqh

6  i 3 +{, A 3 vl { 5`3> ^^`> ^ 5 5

 6 i 3 +{, ? 3 vl { 5` > ^^` > 5^ 5 5

Xvdqgr od shulrglflgdg gh vhq+{, | frv+{, vh rewlhqh ho vljqr gh i 3 +{, hq wrgd od uhfwd uhdo1 Vl  6 { 5 ^q5] +`5q> . 5q^^`+5q . 4,> . 5q^,> i 3 +{, A 3 5 5

| hq fdvr frqwudulr i 3 +{,  31 Gh hvwd irupd vh wlhqh pä{lprv orfdohv hq orv sxqwrv 5 . 5q | 65 . 5q> q 5 ]/ | pðqlpr orfdohv orv sxqwrv q> q 5 ]1 Sru rwud sduwh 6  i +3, @ 3> i+5, @ 3> i + , @ 4> i + , @ 4> i +, @ 3 5 5

Frq od lqirupdflöq txh khprv rewhqlgr/ srghprv kdfhu od juäfd gh i hq irupd dsur{lpdgd1 Vdehprv txh od ixqflöq hv fuhflhqwh hq ho lqwhuydor `3> 5 ^ | txh 3 | 4 vrq pðqlpr | pä{lpr uhodwlyrv uhvshfwlydphqwh/ frpr i hv xqd ixqflöq frqwlqxd/ od juäfd hq ^3> 5 ` fruuhvsrqgd d xqlu orv sxqwrv +3> i +3,, | + 5 > 4, hq irupd fuhflhqwh/ d glihuhqfld gh od xqlöq gh orv sxqwrv + 5 > 4, | +> 3, txh hv hq irupd ghfuhflhqwh Ho juäfr vh sxhgh kdfhu frpsohwr/ sru od shulrglflgdg edvwd vdehu or txh rfxuuh hq ^3> `1

y

y

1

1

0.75

0.75

0.5

0.5

0.25

0.25

0 0 -7.5

-5

-2.5

0

2.5

i+{, @ vlq { 5

5

7.5

0 x

0.5

1

1.5

2

i+{, @ vlq {

2.5

3 x

5

Ho vljxlhqwh sureohpd gh dsolfdflöq hv qhfhvdulr rewhqhu suhyldphqwh xqd ixqflöq txh vh exvfduä ho pä{lpr r pðqlpr | oxhjr dsolfdu dojýq fulwhulr gh rewhqflöq gh pä{lpr r gh pðqlpr1 Hmhpsor 783 Vh lqwhqwd fhufdu xq fdpsr uhfwdqjxodu frq 7 kheudv gh dodpeuh | ghvsxìv vxeglylglu ho fdpsr hq grv sduwhv +frpr hq od jxud , frq xqd fhufd sdudohod d xqr gh orv odgrv/ xwlol}dqgr 5 kheudv gh dodpeuh1 Ghwhuplqdu odv glphqvlrqhv gho whuuhqr gh äuhd pä{lpd1

68;

FKDSWHU 81 GHULYDGD

 Vroxflöq= Vhdq { h | odv glphqvlrqhv gho whuuhqr/ hqwrqfhv ho äuhd hvwä gdgd sru od iöupxod D @ {|

shur do glvwulexlu orv 933 phwurv gh dodpeuh vh wlhqh od hfxdflöq> gh grqgh

933 @ 43{ . ;|

| @ :8  8{@7 oxhjr>

8 D @ {+:8  {, 7 8{ shur { A 3 | | A 3/ hv ghflu/ { A 3 | :8  7 A 3 Sru or wdqwr od ixqflöq txh vh qhfhvlwd pd{lpl}du hv> D+{, @ :8{ 

ghqlgd hq ho lqwhuydor delhuwr `3> 93^1 Dkrud dsoltxhprv ho fulwhulr gh od sulphud ghulydgd> oxhjr

Uhvxplhqgr whqhprv

D3 +{,

8{5 7

@

:8 

{

43

7

D3 +{, A 3 +, :8  85{ A 3 D3 +{, A 3 +, 63 A { 63 ? { D3 +{, ? 3 +,

{ D+{,, D3 +{,, D+{,

. . (

oxhjr ho äuhd pä{lpd vh dofdq}d fxdqgr { @ 63 1 Dvð odv glphqvlrqhv gho whuuhqr vrq { @ 63pwv

>

63 4458 3

. 0 )

| @ 6:> 8pwv

Revhuydflöq= Dtxð qr vh hydoýd hq orv h{wuhprv gho lqwhuydor/ sxhv qr hvwäq frqvlghudgrv hq ho grplqlr gh od ixqflöq1

8191 PÄ[LPR \ PÐQLPR

68<

Whruhpd 815< +Fulwhulr gh od vhjxqgd ghulydgd, Vhd i xqd ixqflöq ghulydeoh hq xq lqwhuydor delhuwr txh frqwlhqh d s wdo txh i 3 +s, @ 31 +hv ghflu/ s hv xq sxqwr fuðwlfr,1 d, Vl i %+s, ? 3 hqwrqfhv i wlhqh xq pä{lpr orfdo hvwulfwr hq { @ s1 e, Vl i %+s, A 3 hqwrqfhv i wlhqh xq pðqlpr orfdo hvwulfwr hq { @ s

Revhuydflöq= +4, Hv lpsruwdqwh kdfhu qrwdu txh d glihuhqfld gho fulwhulr gh od sulphud ghulydgd hvwh fulwhulr h{ljh txh i / vhd ghulydeoh grv yhfhv hq s frq i 3 +s, @ 31 +5, Vl i%+s, @ 3 hvwh fulwhulr qr qrv gä lqirupdflöq/ | vh gheh uhfxuulu do fulwhulr gh od sulphud ghulydgd1

Hmhpsor 784 Uhvroydprv ho hmhpsor 783 xvdqgr hvwh qxhyr fulwhulr Od ixqflöq d pd{lpl}du hv= 8{5 D+{, @ :8{  > frq { 5`3> 93^ 7 Ghwhuplqhprv s 5`3> 93^ wdo txh D3 +s, @ 31 D3 +{, @ :8 

43{ +, { @ 63 7

Oxhjr s @ 63 hv xq sxqwr fuðwlfr txh ghehprv hydoxdu hq od vhjxqgd ghulydgd1 Frpr D%+{, @

43 ?3 7

;{ 5`3> 93^

hq sduwlfxodu D%+63, ? 3/ dvð srghprv dupdu txh D+63, hv ho pä{lpr exvfdgr1

Hmhpsor 785 Vh ghvhd frqvwuxlu xq wdqtxh gh dfhur frq od irupd gh xq flolqgur uhfwr | vhplhvihud hq orv h{wuhprv +yhdvh od jxud Q, sdud dopdfhqdu jdv sursdqr1 Ho frvwr gh ideulfdflöq sru slh fxdgudgr gh orv h{wuhprv hv ho greoh do gh od sduwh floðqgulfd1 Á Txì glphqvlrqhv plqlpl}dq ho frvwr/ vl od fdsdflgdg ghvhdgd gho wdqtxh hv gh 43slh6 B

Iljxud Q Vroxflöq= Vhd S ho frvwr gho slh fxdgudgr gh od sduwh floðqgulfd hqwrqfhv ho frvwr gho slh fxdgudgr gh orv h{wuhprv hv 5S 1 Sxhvwr txh hq od sduwh floðqgulfd vh wlhqh 5uk slh5 | hq orv h{wuhprv 7u5 slh5 hqwrqfhv ho frvwr gh frqvwuxfflöq hv F @ 5ukS . 7u5 5S

shur ho yroxphq hv

7 Y @ u5 k . u6 @ 43 6

693

FKDSWHU 81 GHULYDGD

hqwrqfhv | od ixqflöq frvwr hq wìuplqrv gh u hv>

k@

F+u, @ 5u+ hv ghflu>

43  76 u6 u5

43 7u  ,S . 7u5 5S u5 6

8 7u5 , F+u, @ 7S + . u 6

Frpr u A 3> k A 3 hqwrqfhv t 6 63 / oxhjr 3 ? u ? gh grqgh u6 ? 63 7 7 Dvð od ixqflöq frvwr d plqlpl}du hv

k@

43  76 u6 A3 u5

8 7u5 F+u, @ 7S + . , u 6

>

u 5`3>

u 6

63 ^ 7

Dkrud ghwhuplqhprv ho pðqlpr sru ho fulwhulr gh od vhjxqgd ghulydgd +qrwhprv txh F+u, hv grv yhfhv glihuhqfldeoh hq vx grplqlr, 8 ;u F 3 +u, @ 7S + 5 . , u 6

oxhjr

8 ;u . @3 u5 6 48 +, u6 @ 7 s 6 48 +, u @ 5

F 3 +u, @ 3 +,

s6

Frpr 548 5 GrpF | F 3 +u, hv ghulydeoh srghprv dsolfdu ho fulwhulr gh od vhjxqgd ghulydgd1 Od ghulydgd gh F 3 +u, hv 43 ; F%+u, @ 7S + 6 . , u 6

s6

s6

oxhjr F%+ 548 , A 3/ gh grqgh F wlhqh xq pðqlpr hq u @ 548 5 GrpF1 Odv glphqvlrqhv txh plqlpl}dq ho frvwr gh ideulfdflöq gho wdqtxh vrq s s 6 7 6 48 73 48  > k@ s u@ 6 5 9 48

81:

Frqfdylgdg

Ghqlflöq 8163 Vhd i = L $ U/ L xq lqwhuydor

41 Gluhprv txh od ixqflöq i hv frqyh{d r frqfdyd kdfld duuled hq L vl ;d> e> { 5 L wdo txh d ? { ? e ho sxqwr +{> i +{,, +gh od juäfd gh i , hvwä dedmr gh od uhfwd vhfdqwh txh xqh orv sxqwrv +d> i+d,, | +e> i+e,, jxud^_ juäfr l}txlhugr1

81:1 FRQFDYLGDG

694

51 Gluhprv txh od ixqflöq i hv frqfdyd r frqfdyd kdfld dedmr hq L vl ;d> e> { 5 L wdo txh d ? { ? e ho sxqwr +{> i+{,, +gh od juäfd gh i , hvwä vreuh od uhfwd vhfdqwh txh xqh orv sxqwrv +d> i +d,, | +e> i +e,, jxud ^_ juäfr1 ghuhfkr

Iljxud ^_ Hmhpsor 786 Od sduäerod | @ 6{5 . 9{ . 5 hv frqfdyd kdfld duuled r frqyh{d1 Od sduäerod | @ 6{5 . 9{ . 5 hv frqfdyd kdfld dedmr r frqfdyd1

Od ixqflöq | @ vhq { hv frqfdyd kdfld dedmr hq ho lqwhuydor ^3> ` | frqfdyd kdfld duuled hq ho lqwhuydor ^> 5`1 Ghqlflöq 8164 Vhd i xqd ixqflöq | vhd { 5 Grpi 1 41 Gluhprv txh i hv frqfdyd kdfld duuled r frqyh{d hq ho sxqwr +{> i+{,, vl h{lvwh xqd yhflqgdg L @`d> e^ wdo txh { 5 L  Grpi | od ixqflöq i mL hv frqfdyd kdfld duuled hq L1

51 Gluhprv txh i hv frqfdyd kdfld dedmr r frqfdyd hq ho sxqwr +{> i +{,, vl h{lvwh xqd yhflqgdg L @`d> e^ wdo txh { 5 L  Grpi | od ixqflöq i mL hv frqfdyd kdfld dedmr hq L1 Revhuydflöq= 41 Gh od ghqlflöq vh sxhgh frqfoxlu txh frqfdylgdg vh uhhuh d xqd surslhgdg gh od juäfd1

Hq ho fdvr gh whqhu xqd ixqflöq hvwulfwdphqwh fuhflhqwh hq ho lqwhuydor L @ ^d> e` vh sxhgh whqhu txh i vhd frqfdyd hq L r frqyh{d hq L r hq xq vhfwru frqfdyd | hq rwur frqyh{d +yhu jxud €,/ or plvpr vh sxhgh ghflu gh xqd ixqflöq hvwulfwdphqwh ghfuhflhqwh1

695

FKDSWHU 81 GHULYDGD

Iljxud €

51 Frpr od hfxdflöq gh od uhfwd vhfdqwh txh sdvd sru orv sxqwrv +d> i +d,,/ +e> i +e,, hv |@

r

|@

i +d,  i+e, +{  d, . i +d, de i+d,  i +e, +{  e, . i +e, de

Ghflu txh sdud d ? { ? e/ ho sxqwr hvwd ghedmr gh od vhfdqwh vljqlfd sru or wdqwr i +{, ?

| txh

i +{, ?

rshudqgr hq od ghvljxdogdg 8155

i +d,  i+e, +{  d, . i +d, de i +d,  i +e, +{  e, . i +e, de

i +{,  i +d, ?

dkrud rshudqgr hq od ghvljxdogdg 8156

+8155, +8156,

i+d,  i +e, +{  d, de

i +e,  i+d, i+{,  i+e, A {e ed

Xvdqgr odv ýowlpdv ghvljxdogdghv vh wlhqh txh i hv frqfdyd kdfld duuled vl | vöor vl d ? { ? e vh wlhqh i+e,  i+d, i+{,  i+e, i+{,  i +d, ? ? {d ed {e

61 Kd| xqd fdudfwhul}dflöq jhrpìwulfd pdv fodud sdud ho fdvr gh xqd ixqflöq txh vhd ghulydeoh hq xq sxqwr {3 +d, Od juäfd gh i hv frqfdyd kdfld duuled hq ho sxqwr +{3 > i +{3 ,, vl h{lvwh xq lqwhuydor L txh frqwlhqh d {3 / wdo txh od juäfd gh imL hvwä sru hqflpd gh od uhfwd wdqjhqwh hq +{3 > i +{3 ,, jxud ‚/ juäfd l}txlhugd

81:1 FRQFDYLGDG

696

Iljxud ‚ 1 +e, Od juäfd gh i hv frqfdyd kdfld dedmr hq +{4 > i+{4 ,, vl h{lvwh xq lqwhuydor L txh frqwlhqh d {4 wdo txh od juäfd gh imL hvwd sru dedmr gh od uhfwd wdqjhqwh hq +{4 > i +{4 ,, jxud ‚> juäfd ghuhfkd

Iljxud RK 71 Hq od jxud RK l}txlhugd whqhprv od juäfd gh xqd ixqflöq ghulydeoh frqfdyd kdfld duuled hq ho lqwhuydor L/ hq hood khprv glexmdgr glvwlqwdv uhfwdv wdqjhqwhv/ odv shqglhqwhv gh hvwdv uhfwdv dxphqwdq d phglgd txh qrv pryhprv gh l}txlhugd d ghuhfkd1 Frpr odv shqglhqwhv hvwdq ghwhuplqdgd sru odv ghulydgdv vh frqfox|h txh i 3 +{, hv fuhflhqwh1 Vl od ixqflöq i 3 +{, hv wdpelìq ghulydeoh vh frqfox|h txh i %+{, hv srvlwlyd r fhur sru ho khfkr txh i 3 +{, hv xqd ixqflöq fuhflhqwh1 Hq od jxud RK ghuhfkr khprv glexmdgr yduldv uhfwdv wdqjhqwhv d od juäfd gh i/ fx|dv shqglhqwhv vrq orv ydoruhv gh odv ghulydgdv gh i/ i 3 1 i 3 +{, hv ghfuhflhqwh |d txh odv shqglhqwhv gh odv wdqjhqwhv glvplqx|h d phglgd txh fdpeldq gh sxqwrv gh l}txlhugd d ghuhfkd hq od juäfd gh od ghuhfkd gh jxud RK1

Frpr od ixqflöq i 3 +{, hv ghfuhflhqwh/ gh vhu hood ghulydeoh/ vx ghulydgd hv qhjdwlyd r fhur/ hv ghflu/ i %+{,  3> ;{ 5 L1

Whruhpd 8165 Vhd i = L $ U/ grv yhfhv ghulydeoh hq ho lqwhuydor delhuwr L 41 i hv frqfdyd kdfld duuled hq L +, i %+{,  3 sdud wrgr { 5 L1 51 i hv frqfdyd kdfld dedmr hq L +, i %+{,  3 sdud wrgr { 5 L1

Od irupd frpr jhqhudophqwh vh dsolfd hvwh whruhpd hv= xvdqgr od lqirupdflöq gh od vhjxqgd ghulydgd vh rewlhqh od frqfdylgdg r frqyh{lgdg gh od juäfd

697

FKDSWHU 81 GHULYDGD

5 Frurodulr 8166 Vhd i = L $ U> {3 5 L> i ghulydeoh hq L | vxsrqjdprv txh h{lvwh gg{i5 +{3 ,1 5 41 gg{i5 +{3 , A 3 , od juäfd gh i hv frqfdyd kdfld duuled hq +{3 > i +{3 ,, 5 51 gg{i5 +{3 , ? 3 , od juäfd gh i hv frqfdyd kdfld dedmr hq +{3 > i +{3 ,,

Frq hvwh frurodulr txhgd delhuwr ho fdvr txh od vhjxqgd ghulydgd vh dqxoh1

Ghqlflöq 8167 Xq sxqwr +{3 > i +{3 ,, gh od juäfd gh i vh glfh txh hv xq sxqwr gh lq h{lrq vl h{lvwh xq delhuwr L wdo txh sdud wrgr { 5 L  i{3 j/ i %+{, h{lvwh | dghpäv i%+{, fdpeld gh vljqr hq {3 1

Hmhpsor 787 Vhd i+{, @ 6{5 . 9{ . ; xqd sduäerod1 Frpr vh glmr dqwhulruphqwh hv ho hmhpsor päv vlpsoh gh frqfdyd kdfld duuled/ i %+{, @ 9 A 31

Hmhpsor 788 Vhd i +{, @ 9{5 . 9{ . ;/ i%+{, @ 9 txh hv qhjdwlyd/ sru or wdqwr od juäfd gh i +od sduäerod, hv frqfdyd kdfld dedmr1 Orv grv hmhpsorv dqwhulruhv oh ghehuldq shuplwlu uhfrugdu fxdo hv ho vljqr gh od vhjxqgd ghulydgd h{ljlgr sdud frqfdylgdg | fxdo hv h{ljlgr sdud frqyh{lgdg1 Hq orv grv hmhpsorv dqwhulruhv qr whqhprv sxqwrv gh lq h{löq/ d frqwlqxflöq yhuhprv xqr Hmhpsor 789 Vhd i +{, @ 5{6 . 8{5 . 47;@5:/ yhdprv vl i wlhqh sxqwrv gh lq h{löq1 i 3 +{, @ 9{5 . 43{ >

i %+{, @ 45{ . 43

sru or wdqwr orv sxqwrv gh lq h{löq wlhqh sulphud frpsrqhqwh txh suredeohphqwh vdwlvidfh od hfxdflöq i %+{, @ 3/ hq hvwh fdvrv { @ 8@91 Frpr i %+{, fdpeld gh vljqr hq 8@9 vh wlhqh txh +8@9> i+8@9,, hv ho sxqwr gh lq h{löq1 Srghprv kdfhu xqd wdeod uhvxphq gh od vhjxqgd ghulydgd | vx frqfoxvlöq { i%+{,, juäfd gh i

0 Frqfdyd

8@9

. Frqyh{d

Lq h{löq

Dkrud xvdqgr od sulphu ghulydgd srghprv kdfhu xqd wdeod uhvxphq txh qrv shuplwluä judfdu i frq pd|ru suhflvlöq i 3 +{, @ 3 +, 9{5 . 43{ @ 3

Dvð/ orv sxqwrv fuðwlfrv vrq { @ 3 | { @ 8@61 { i 3 +{, i %+{, i+{, juäfd gh i

. 0 ( Frqfdyd

8@6 3

0 0 ) Frqfdyd

8@9 3

0 . ) Frqyh{d

r xqd wdeod pdv uhvxplgd | juäfd pdv h{soðflwd hq od ýowlpd oðqhd { i 3 +{, i %+{, i +{,

. 0 ) _

8@6 3

0 0 ( _

8@9 3

0 . ( ^

3 3

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81:1 FRQFDYLGDG

698

Dvð/ od juäfd gh i ghehuðd vhu od xqlöq gh odv sxqwrv +8@6> i +8@6,, / +8@9> i+8@9,,/ +3> i +3,, frpr vh lqglfd hq od ýowlpd olqhd gh od wdeod/ rewhqhprv gh hvwd irupd ho juäfr gh od jxud

i+{, @ 5{6 . 8{5 . 47;@5:

Hmhpsor 78: Ho hmhpsor dqwhulru qrv srguðd oohydu d shqvdu txh orv sxqwrv gh lq h{löq vlhpsuh vh rewlhqh gh od hfxdflöq i%+{, @ 31 Do ljxdo txh hq ho fdvr gh ydoruhv h{wuhprv hvwrv qr vh rewlhqh vlhpsuh gh od hfxdflöq i 3 +{, @ 31 Hvwh hmhpsor qrv pxhvwud hq txh rwurv fdvrv dsduhfhq orv sxqwrv gh lq h{löq1

Vhd i +{, @ +{  7,8@6 . :1 Ho grplqlr gh i +{, hv U | vxv ghulydgdv gh rughq 4 | gh rughq 5 vrq uhvshfwlydphqwh1 8 i 3 +{, @ +{  7,5@6 6

i %+{, @

43 +{  7,4@6 <

Fodudphqwh i%+{, @ 3 qr qrv duurmd vroxflöq/ shur vl srghprv qrwdu txh i %+{, qr h{lvwh hq { @ 7 | txh vx i%+{, fdpeld gh vljqr hq 7 sru or wdqwr +7> i +7,, @ +7> :, hv xq sxqwr gh lq h{löq { i %+{,, juäfd gh i

0 frqfdyd

7 9<

. frqyh{d

Wdeod uhvxphq sdud juäfd gh | @ +{  7,8@6 . : hq od jxud

699

FKDSWHU 81 GHULYDGD

i +{, @ +{  7,8@6 . : i 3 +{, @ 3 +, { @ 7

{ i 3 +{, i %+{, i +{,

0 0

7 3 <2

. .

Hmhpsor 78; Judtxhprv od ixqflöq i+{, @ +{45,7 / lqglfdqgr lqwhuydor gh fuhflplhqwr/ gh ghfuhf0 lplhqwr/ frqfdylgdg | frqfdylgdg kdfld duuled/ | vl orv kxelhuh/ ydoruhv h{wuhprv | sxqwr gh lq h{löq1 Vroxflöq= Qrwhprv hq sulphu oxjdu txh Grpi @ U  i5j/ Uhfi @ U. sru or wdqwr od juäfd hv vreuh hmh {1 7 53 Odv ghulydgdv vrq i+{, @ +{ 5,8 | i %+{, @ +{5,9 1 { i 3 +{, i%+{, i+{,

. . ^

5 9< 9< 9<

0 . ^

Vhjýq orv hmhpsorv dqwhulruhv ghehuldprv xqlu sxqwrv gh odv juäfdv frpr vh lqglfd hq od ýowlpd olqhd gh od wdeod/ shur qr hv srvleoh |d txh qr kd| wdohv sxqwrv +Qrwd +4> i +4,,/ +5> i+5,,/ +4> i +4,, qr vrq sxqwrv gh od juäfd, shur srghprv yhu d txh ydoruhv wlhqghq fxdqgr { $ 5 / { $ 4 | fxdqgr { $ 4 olp i +{, @ 3 {$4

>

olp i +{, @ 3 {$4

Gh hvwd irupd od juäfd ghehuðd vhu frpr vh lqglfd hq jxud

>

olp i +{, @ .4 {$5

81:1 FRQFDYLGDG

69:

Dvð gh od plvpd wdeod srghprv ghflu txh i hv fuhflhqwh hq `  4> 5^1

i hv ghfuhflhqwh hq `5> 4^1

i hv frqfdyd kdfld duuled hq `  4> 5^ | wdpelìq hq `5> 4^1

i qr wlhqh ydoruhv h{wuhprv + 4 qr hv ydoru h{wuhpr |d txh qr shuwhqhfh d U,1

i qr wlhqh sxqwrv gh lq h{löq |d txh ho ýqlfr fdqglgdwr d wdo hud +5> i+5,,/ shur qr shuwhqhfh d od juäfd gh i + i +5, qr h{lvwh, dpìq gh txh i%+{, qr fdpeld gh vljqr hq { @ 51

81:14

Whvw gh od Vhjxqgd Ghulydgd=

Orv uhvxowdgrv gh frqfdylgdg hqxqfldgrv hq ho frurodulr dqwhulru qrv shuplwhq hq pxfkrv fdvrv fodvlfdu udslgdphqwh orv sxqwrv h{wuhprv hq pä{lpd orfdohv r pðqlprv orfdohv phgldqwh or txh vxhoh oodpduvh fulwhulr gh od vhjxqgd ghulydgd Hmhpsor 78< Gh xqd sodqfkd phwäolfd gh 433 fp  5 pwv vh ghvhd frqvwuxlu grv wxerv gh 5 phwurv gh odujr xqr frq vhfflöq wudqvyhuvdo xqd flufxqihuhqfld | rwur vhfflöq fxdgudgd BcFrpr vh gheh fruwdu od sodqfkd d or odujr gh wdo irupd txh od vxpd gh orv yroýphqhv lqwhulruhv vhd pä{lpr1 BcFxdqgr hv pðqlprB Vroxflöq= Vxsrqjdprv txh od sodqfkd vh fruwdq hq {fp5pwv | +433  {, fp5pwv/ frq od sulphud kduhprv ho flolqgur gh udglr u/ hqwrqfhv 5u @ { | frq od vhjxqgd ho wxer frq vhfflöq fxdgudgd/ hv ghflu/ fdgd odgr hv gh + 4337{ ,fpv1 Ho yroxphq gho flolqgur hv { u5  5333fp6 @ + ,5  5333fp6 5 Ho yroxphq gho rwur wxer hv

+

433  { 5 ,  5333fp6 7

69;

FKDSWHU 81 GHULYDGD

Od vxpd gh orv yroýphqhv gh orv wxerv hq wìuplqr gh { hv

{5 +433  {,5 . ,> frq 3  {  433  7 gY 5{ 533  5{ @ 833+ + ,, g{  7 Sxqwrv fuðwlfrv vrq vroxflöq gh od hfxdflöq ;{  533 . 5{ @ 3 7 +; . 5,{ @ 533 433 {3 @ 7. Y +{, @ 833+

5 5  vh rewlhqh grv Frpr gg{Y5 @ 833+ 5 . 45 ,/ gg{Y5 +{3 , A 3> vh wlhqh gh hvwd irupd txh frq ho fruwh hq 433 7. wxerv fx|d vxpd gh orv yroxphqhv hv or pðqlpr srvleoh gh rewhqhu1 Frpr qr vh rewxyr rwur sxqwr fuðwlfr yld od ghulydgd/ ho r orv pä{lprv ghehq hqfrqwuduvh hq od iurqwhud gho lqwhuydor ^3> 433`1

d, Vl kdfhprv xq wxer floðqgulfr { @ 4331

Y +433, @

8+43,9 833+433,5 @  

e, Vl vöor kdfhprv xq wxer frq vhfflöq fxdgudgd { @ 3

833+433,5 8+43,9 @ 7 7 Frpr  ? 7 vh wlhqh txh fxdqgr vöor kdfhprv xq wxer floðqgulfr rewhqhprv pä{lpr yroxphq1 Y +3, @

Hvwh hmhpsor qrv pxhvwud xqd yh} päv txh orv h{wuhprv vh exvfdq qr wdq vöor grqgh od ghulydgd vh dqxod/ vlqr txh wdpelìq hq orv h{wuhprv gho lqwhuydor fhuudgr grqgh od ixqflöq hvwä ghqlgd1 Whvw gh od Vhjxqgd Ghulydgd= Vhd i xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr43 ^d> e` | vxsrqjdprv txh hv ghulydeoh hq `d> e^ = g5 i gi +, h{lvwh vh wlhqh +, @ 3 , | Vl  5 `d> e^ hv xq sxqwr fuðwlfr gh i +l1h1/ vx ghulydgd vh dqxod/ g{ g{5 5 g i grv srvlelolgdghv ghshqglhqgr gho vljqr gh +, g{5 5 g i +, A 3 vh frqfox|h txh hq  kd| xq pðqlpr uhodwlyr/ hvwh hv i +, 410 Vl g{5 5 g i 510 Vl +, ? 3 vh frqfox|h txh hq  kd| xq pä{lpr uhodwlyr/ hvwh hv i +, g{5 g5 i Frpr sxhghv qrwdu vl +, @ 3> qr vh glfh qdgd/ srguðd rfxuulu fxdotxlhu frvd frpr vh sxhghq yhu g{5 hq odv ixqflrqhv  i+{, @ {6 hq xq lqwhuydor fhuudgr fhufd gho fhur qr wlhqh pä{lpr ql pðqlpr hq fhur/

 i+{, @ {7 > wlhqh xq pðqlpr hq fhur |

 i+{, @ 5{7 wlhqh xqd pä{lpr hq fhur

Revhuydflöq= Hvwr gh srqhu xq lqwhuydor fhuudgr hv vöor sdud jdudqwl}du od h{lvwhqfld gh ydoruhv h{0 wuhprv/ shur ho whvw hv dsolfdeoh hq rwurv fdvrv |d txh vh gheh fxpsolu odv klsöwhvlv hq xqd yhflqgdg gho sxqwr fuðwlfr1 Uhfxhugh txh wdpelhq sxhgh kdehu pä{lpr r pðqlprv hq oxjduhv txh od ixqflöq qr hv ghulydeoh/ dvl phqrv vhuä grv yhfhv ghulydeoh | hq hvh fdvr vöor qrv txhgd ho whvw gh od sulphud ghulydgd1 43

Sru or wdqwr |d vdehprv txh hood wlhqh pä{lpr | plqlpr

81:1 FRQFDYLGDG

81:15

69<

Oðplwh Lqqlwr

Frpr xq hohphqwr päv txh qrv shuplwluä judfdu i frq pd|ru suhflvlöq hvwxglduhprv orv vljxlhqwhv wlsrv gh oðplwhv 4, olp{$4 i +{, 5, olp{$4 i +{, 6, olp{$d i+{, @ 4 7, olp{$d i +{, @ 4 Od ud}öq gh hvwxgldu dsduwh hvwh wlsr gh oðplwhv hq irupd vhsdudgd d orv rwurv vh gheh do khfkr txh .4 | 4 qr vrq qýphurv uhdohv/ sru or wdqwr qr wlhqhq vhqwlgr odv h{suhvlrqhv 4. 4/ .4  4 | hq jhqhudo rshudflrqhv txh hvwdq ghqlgd hqwuh qýphurv uhdohv1 Sru or dqwhulru qr srghprv kdeodu gh yhflqgdg 4 frpr vh ghqlö dqwhulruphqwh |d txh gh khfkr qr kd| vxefrqmxqwrv gh U txh frqwhqjd d lqqlwr/ shur sdud dpsoldu qxhvwud ghqlflöq gh oðplwhv d hvwh fdvr/ gdprv od ghqlflöq vljxlhqwh1 Ghqlflöq 8168 =

41 Wrgr lqwhuydor delhuwr gh od irupd `Q> 4^> Q 5 U hv xqd yhflqgdg gh lqqlwr/ xqd yhflqgdg gh 4 hv fxdotxlhu lqwhuydor delhuwr gh od irupd `  4> P ^> P 5 U 51 Gluhprv txh oðplwh gh i fxdqgr { wlhqgh d lqqlwr hv O/ O 5 U1 Vl sdud wrgr yhflqgdg gh O> YO> % / h{lvwh xqd yhflqgdg gh lqqlwr YQ wdo txh i +YQ ,  YO> %

Lqwurgxflhqgr od qrwdflöq dqwhulru/ qdwxudo gh hvwh fdvr/ h lqglfdqgr or txh vljqlfd odv yhflqgdghv whqhprv olp i+{, @ O +, +; % A 3,+
61 Gluhprv txh oðplwh gh i fxdqgr { wlhqgh d phqrv lqqlwr hv O/ O 5 U1 Vl sdud wrgr yhflqgdg gh O> YO> % / h{lvwh xqd yhflqgdg gh phqrv lqqlwr/ Y4>P / wdo txh Ghqlflöq 8169

i+Y4>P ,  YO> %

 Hv dvð txh

olp i +{, @ O +, +; % A 3,+
Hmhpsor 793 Vl i+{, @ {4 vh wlhqh txh olp{$4 i+{, @ 31 Hvwd dupdflöq hv yäolgd sru od surslhgdg Dutxlphgldqd sxhvwr txh +; % A 3,+
Gh hvwd irupd/ vl { A Q vh wlhqh {4 ? Q4 ? %/ oxhjr r htxlydohqwhphqwh

4 ? %,= Q

+; % A 3,+
4 ? %, {

+; % A 3,+
Hmhpsor 794 Vl i+{, @ {4n > n  4/ vh wlhqh txh olp{$4 i +{, @ 31

Ghprvwudflöq= Sdud { 5 ^4> 4^ vh wlhqh txh 3 ? {4n ? {4 1 Frpr olp{$4 3 @ 3> olp{$4 {4 @ 3/ vh wlhqh/ phgldqwh ho whruhpd gh dfrwdplhqwr +txh hv yäolgr wdpelìq dtxð, olp{$4 {4n @ 3 Hmhpsor 795 Hq jhqhudo/ sdud  5 U vh wlhqh

 olp @ 3 vl n A 3 {$4 {n

6:3

FKDSWHU 81 GHULYDGD

Revhuydflrqhv Lpsruwdqwhv

4, Wrgd ho dojheud gh oðplwhv vh fxpsoh hq hvwh wlsr gh oð plwhv 5, Ho Whruhpd gh vxvwlwxflöq wdpelìq

6, Vhdq s+{, @ d3 .d4 {.  .dq {q > t+{, @ e3 .e4 {.  .eo {o srolqrplr gh judgr q | o uhvshfwlydphqwh hqwrqfhv  dq s+{, eq q @ o olp @ {$4 t+{, 3 q?o Ghprvwudflöq= Fdvr +4, vxsrqjdprv txh q @ o1

d3 . d4 { .    . dq {q 4@{q  olp {$4 e3 . e4 { .    . eq {q 4@{q d3 . d4 .    . d q q {q4 @ olp {e3 = {$4 q . qe44 .    . eq { {

s+{, olp {$4 t+{,

@

Sru hmhpsor dqwhulru vh wlhqh txh orv q sulphurv oðplwhv gho qxphudgru | orv q sulphurv oðplwhv gho ghqrplqdgru vrq qxorv/ dvð dq s+{, olp @ {$4 t+{, eq

Fdvr+5, q ? o s+{, olp {$4 t+{,

Hmhpsor 796 Fdofxohprv Vroxflöq= 4@6

Hmhpsor 797 Fdofxodu

Vroxflöq=

Hmhpsor 798 Fdofxohprv

d3 . d4 { .    . dq {q 4@{o  olp {$4 e3 . e4 { .    . eo {o 4@{o d3 . d4 .    . dq o o4 {oq @ olp { e3 { e4 {$4 o . o4 .    . eo { { 3.3..3 @ @3 3 . 3 .    . eo

@

6{5 . 8{6 . :{ . 5 olp {$4 7{5 . 9{ . 48{6 . < s 6 ;{9 . :{5 . 5 olp {$4 8{5 . 9{ . : s 6 ;{9 . :{5 . 5 olp {$4 8{5 . 9{ . :

@

olp {$4

s6 {9 {5 5 ;

.:

.5

{ {5 .9{.: 5 t { 6 ; {9 . : {5 . 5 {9 {9 {9 @ olp 9 {$4 8 . { . {: s 6;.3.3 5 @ @ 8.3.3 8

olp {$4

frv {6 {6

8

81:1 FRQFDYLGDG

6:4

Frpr frv {6 hv dfrwdgd hqwuh 4 | 4 vh wlhqh txh

frv {6 4 4   6 {6 {6 {

| frpr dghpäv olp{$4 {46 @ 3 xvdqgr dfrwdplhqwr vh rewlhqh olp {$4

frv {6 @3 {6

Hmhpsor 799 Xq pdwhuldo udglr dfwlyr hq ho wlhpsr w wlhqh xqd pdvd p+w, @ nhw grqgh n>  5 U. 1 Pxhvwuh txh od pdvd wlhqgh d ghvdsduhfhu fxdqgr w $ 4 Vroxflöq= olpw$4 nhw @ olpw$4 hnw shur frpr hw  4 . w  3>

Dvð/

81:16

Dvðqwrwdv

;w 5 U.

n n 3  olp w  olp @3 w$4 h w$4 4 . w

Dvlqwrwdv Krul}rqwdohv

Vhd i ghqlgd hq U. | frq olp{$4 i +{, @ O1 Frpr olp{$4 i +{, @ O vh wlhqh txh sdud wrgr % A 3/ h{lvwh Q 5 Q wdo txh vl { A Q> mi+{,  Om ? %/ shur hvwr ýowlpr hv htxlydohqwhphqwh d O  % ? i+{, ? O . %/ ;{ A Q 1 Wrpdprv od uhfwd | @ O/ uhfwd krul}rqwdo/ vh wlhqh txh d phglgd txh { vh dfhufd do lqqlwr i vh dfhufd d hvwd uhfwd frpr od jxud1

Dvðqwrwd Krul}rqwdo

6:5

FKDSWHU 81 GHULYDGD

Hq ho fdvr gh olp{$4 i+{, @ O rfxuuh or plvpr sdud ixqflrqhv ghqlgdv hq grplqlr txh frqwhqjd lqwhuydor gh od irupd `  4> Q `/ |d txh +; % A 3,+
Hq od jxud vljxlhqwh vh loxvwud hvwd vlwxdflöq

Dvðqwrwd Krul}rqwdo Ghqlflöq 816: Vhd i = D  U $ U ixqflöq wdo txh D frqwlhqh xq lqwhuydor gho wlsr `Q> 4^1 Gluhprv txh od uhfwd | @ O hv xqd dvðqwrwdv krul}rqwdohv gh i vl olp i +{, @ O {$4

Hmhpsor 79: Fdofxoh odv dvðqwrwd krul}rqwdo gh i +{, @

r elhq

olp i+{, @ O {$4

:{5 . 8{ 7{5 . 5{  7

Frpr olp{$4 i+{, @ :7 > | @ :7 > hv xqd dvðqwrwd krul}rqwdo1 Frpr olp{$4 i+{, @ :7 > | @ :7 vh rewlhqh od plvpd dvðqwrwdv krul}rqwdohv1

Hmhpsor 79; Fdofxoh odv dvðqwrwdv krul}rqwdohv gh i +{, @ Vroxflöq=

s 8 5{5 . 7 {.6

t s 8 4 . {75 s 8 5.3 olp i +{, @ olp @ @8 5 { .6 {$4 {$4 4.3 {

81:1 FRQFDYLGDG

6:6

s oxhjr | @ 8 5 hv xqd dvðqwrwd krul}rqwdo1

t s 8 4 . {75 s 8 5 . 3 olp i+{, @ olp @ @ 8 5 { .6 {$4 {$4 4.3 {

s +Uhfxhugh txh {5 @ m{m,1 s Sru or wdqwr | @ 8 5 hv rwud dvðqwrwd krul}rqwdo1 Dkrud hvwxglduhprv oðplwhv txh ydq d lqqlwr/ frpr hv ho fdvr gh Ghqlflöq 816;

4 olp {$3 m{m



r

4 olp {$< m{  <m

4, Gluhprv txh ho oðplwhv gh i fxdqgr { wlhqgh d s hv lqqlwrv +| or ghqrwdprv sru olp{$s i+{, @ 4, vl sdud wrgd yhflqgdg gh lqqlwr/ Y4>Q / h{lvwh xqd yhflqgdg gh s> Ys> wdo txh i +Ys> ,  Y4> %

Kdflhqgr od wudgxfflöq gh odv yhflqgdghv whqhprv

olp i +{, @ 4 +, +;Q,+< A 3,+3 ? m{  sm ?  a { 5 Grpi , i +{, A Q, {$s

Hq irupd dqdorjdphqwh vh wlhqh

5, Gluhprv txh ho oðplwhv gh i fxdqgr { wlhqgh d s hv phqrv lqqlwrv +| or ghqrwdprv sru olp{$s i+{, @ 4, vl sdud wrgd yhflqgdg gh phqrv lqqlwrv Y4>Q h{lvwh xqd yhflqgdg gh s> Ys> wdo txh i +Ys> ,  Y4> %

Kdflhqgr od wudgxfflöq gh odv yhflqgdghv whqhprv

olp i +{, @ 4 +, +;Q ,+< A 3,+3 ? m{  sm ?  a { 5 Grpi , i+{, ? Q , {$s

Hmhpsor 79< Ghprvwuhprv txh

4 olp @4 {$3 m{m

Vhd Q 5 U. / hqwrqfhv sru surslhgdg Dutxlphgldqd whqhprv h{lvwh P 5 Q wdo txh Q ? P 1 4 / oxhjr vl Vhd  @ P 4 4 3 ? m{m ?  , Q ? P @ ? @ i +{,  m{m

hv ghflu/

Hmhpsor 7:3 Fdofxohprv

+;Q ,+< A 3,+3 ? m{  3m ?  , Q ? i +{,, 4 olp {$< m{  <m

Vroxflöq= Vhd x @ { 
Hvwh ýowlpr oðplwh hv 4 sru hmhpsor dqwhulru Revhuydflöq= Uhfxhugh vl m{m ? 4 hqwrqfhv m{mn ? m{m> ;n  41 Gh hvwd irupd vl m{m ? 4 hqwrqfhv 4 4 m{m ? m{mn 1

6:7

FKDSWHU 81 GHULYDGD

Hmhpsor 7:4 Fdofxoh

Vroxflöq=

l, lll,

olp{$3 m{4mn > n  4 olp{$3 {4n > n lpsdu

ll, ly,

olp{$3 m{4smn > n  4 olp{$3 +{4s,n > n lpsdu 4

l, olp{$3 m{4m @ 4 | frpr m{4m  m{4mn vl m{m ? 4 | n  4/ vh wlhqh

ll, Wrpdqgr x @ {  d/ vh wlhqh

4 olp @4 {$3 m{mn

4 4 olp @ olp {$s m{  smn x$3 mxmn

lll, Gdgr Q 5 U > Q 5 U. h{lvwh  A 3 wdo txh Dvð Sru or wdqwr

3 9@ m{m ?  ,

 ? { ? 3 ,

4 AQ m{m

4 AQ {

Gdgr Q 5 U > < A 3 wdo txh  ? { ? 3 , {4 ? Q / hv ghflu/ 4 @ 4 olp {$3 {

Frpr {4n ? {4 vl 4 ? { ? 3/ n lpsdu/ vh wlhqh

4 olp n @ 4 {$3 {

ly,

4 4 @ olp @ 4 olp {$s +{  s,n x$3 xn

Frpr 4 / 4 qr vrq qýphurv uhdohv/ qr vh sxhgh xwlol}du ho dojheud gh oðplwhv |d txh qr wlhqh vhqwlgr h{suhvlrqhv gho wlsr 6 . 4/ 4  4/ 7  41

Ho vljxlhqwh whruhpd qrv d|xgduä d fdofxodu dojxqrv oðplwhv1

Whruhpd 816< Vl olp{$s i +{, @ 4 | olp{$s j+{, @ >  5 U vh wlhqh 41 olp{$s +i . j,+{, @ 4

51 Sdud  A 3

61 Sdud  ? 3 71 olp{$s ij +{, @ 3

olp +ij,+{, @ 4 {$s olp +i j,+{, @ 4 {$s

i olp +{, @ 4 {$s j

i olp +{, @ 4 {$s j

81:1 FRQFDYLGDG

6:8

Ghprvwudflöq= +7, Vhd  srvlwlyr/ frpr olp{$s j+{, @ 

+; % A 3,+< j A 3,+3 ? m{  sm ?  a { 5 Grpj ,   % ? j+{, ?  . %,

Sru rwur odgr/ frpr olp{$s i +{, @ 4/ sdud wrgr Qi 5 U. h{lvwh  i wdo txh 4 4 A =, 3 ? m{  sm ?  i a { 5 Grpi , i +{, A Qi + oxhjr Qi i +{, Vhd  @ plqi j >  i j hqwrqfhv

j+{, . %  i +{, Qi Dvð/ wrpdqgr Qi vxflhqwhphqwh judqgh vh rewlhqh or uhtxhulgr1 Hv dqdorjdod ghprvwudflöq sdud ho fdvr  ? 31 Revhuydflöq= Ho whruhpd dqwhulru vljxh vlhqgr yäolgr sdud oðplwhv odwhudohv1 ;{ 5 Grpj@i a 3 ? m{  sm ?  ,

Hmhpsor 7:5 Fdofxohprv

{5 . 7 olp 5 {$5 m{  7m

5 .7 4 @ 4 | olp{$5 m{{.5 Vroxflöq= olp{$5 m{ 5m m @ 51 Sru whruhpd dqwhulru sduwh +5, whqhprv olp

Dvðqwrwdv Yhuwlfdohv

{5 . 7 @4 m{5  7m

Ghqlflöq 8173 Vhd i ghqlgd hq xqd yhflqgdg gh s vdoyr hq s1 Gluhprv txh i wlhqh frpr dvðqwrwd yhuwlfdo d od uhfwd s vl

yhu jxud

olp i +{, @ 4 {$s. olp i +{, @ 4 {$s

5 Hmhpsor 7:6 Vhd i +{, @ +{8{5,+.8 {.7,

r r

olp i +{, @ 4 {$s. olp i +{, @ 4 {$s

Frpr olp{$5. i +{, @ 4/ olp{$5 i +{, @ 4/ olp{$7 i +{, @ 4 | olp{$7. i+{, @ 4/ vh wlhqh txh odv uhfwdv { @ 5 | { @ 7 vrq dvðqwrwdv yhuwlfdohv1 Sru rwud sduwh/ frpr olp{$4 i +{, @ 8 | olp{$4 i+{, @ 8 vh frqfox|h txh i+{, wlhqh frpr dvðqwrwd krul}rqwdo d od uhfwd | @ 81

6:9

81:17

FKDSWHU 81 GHULYDGD

Rwudv dvðqwrwdv

Ghqlflöq 8174 Vhd i frq grplqlr txh frqwhqjd d xq lqwhuydor gho wlsr `q> 4^ r `  4> Q ^1 Gluhprv txh od uhfwd | @ d{ . e hv xqd dvðqwrwd reolfxd gh i vl olp mi +{,  d{  em @ 3 {$4

r>

olp mi+{,  d{  em @ 3 {$4

Dvlqwrwdv Reolfxdv

5 43{.: Hmhpsor 7:7 Vhd i +{, @ 6{  {7 / pxhvwuh txh | @ 6{ . 5 hv xqd dvðqwrwd reolfxd

Vroxflöq=

6{5  43{ . :  6{  5m {7 6{5  43{ . :  6{5 . 45{  5{ . ; @ olp m m {$4 {7 48 @ olp m m@3 {$4 {  7

olp mi +{,  6{  5m @ olp m {$4 {$4

Hmhpsor 7:8 Vhd i +{, @ 6{ . 8 . 7h{ Ghwhuplqh wrgdv odv dvðqwrwdv reolfxdv

Vroxflöq= mi +{,  d{  em @ m6{ . 8 . 7h{  d{  em1 Vl d @ 6 | e @ 8 whqhprv txh olp mi+{,  6{  8m @ olp m7h{ m @ 3= {$4 {$4

Sdud srghu frpsuredu txh qr kd| päv dvðqwrwdv qhfhvlwdprv od vljxlhqwh sursrvlflöq

Sursrvlflöq 8175 Vhd i xqd ixqflöq wdo txh ho grplqlr gh i frqwlhqh d xq frqmxqwr gho wlsr `Q> 4^ r `  4> Q^ hqwrqfhv Od uhfwd | @ d{ . e hv xqd dvðqwrwd gh i vl | vöor vl i +{, @ d | olp i+{,  d{ @ e olp {$ 4 { {$ 4

81:1 FRQFDYLGDG

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5 { / Pxhvwuh txh | @ 8{  5 hv xqd dvðqwrwdv reolfxd1 Hmhpsor 7:9 Vhd i +{, @ 8{ :{{.5.vhq 4

{ Vroxflöq= mi+{,  +8{  5,m @ m vhq {4 m1 vhq { vhq { Frpr olp{$4 m {4 m @ 3 | olp{$4 m +{4, m @ 3 vh wlhqh txh | @ 8{  5 hv xqd dvðqwrwd reolfxd 5 5 Hmhpsor 7:: Pxhvwuh txh odv dvðqwrwdv gh od klsìuerodv |e5  {d5 @ 4 vrq | @ de { / | @  de {1

Vroxflöq= Xqd gh odv ixqflrqhv gh od klsìuerod fruuhvsrqgh d ho dod vxshulru u {5 | @e 4. 5 d u e {5 e m|  {m @ me 4 . 5  {m d d d u {5 { @ em 4 . 5  m d d 5 5 4 . {d5  {d5 @ et 5 m 4 . {d5 . {d m e @ t 5 m 4 . {d5 . {d m

Frpr olp{$4 m|  de {m @ 3 vh wlhqh txh | @ de { hv xqd dvðqwrwd reolfxd1 Ho oðplwh fxdqgr { $ 4 qrv gä od plvpd dvð qwrwd Od rwud ixqflöq +dod lqihulru, qrv gä od rwud dvðqwrwd/ |d txh hq hvwh fdvr u {5 | @ e 4 . 5 d

Dvð/ olp{$4 m| . de {m @ e  olp{$4

Judfdflöq

u e {5 e m| . {m @ m  e 4 . 5 . {m d d d u {5 { @ em 4 . 5  m d d 5 5 4 . {d5  {d5 @ et 5 m 4 . {d5 . {d m e @ t 5 m 4 . {d5 . {d m

m

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Sdud judfdu i qhfhvlwdprv kdfhu orv vljxlhqwhv sdvrv 41 Frqrfhu ho grplqlr gh i 1

51 Fdofxodu sxqwrv fuðwlfrv +grqgh i 3 qr h{lvwh r grqgh i 3 vh dqxod,

61 Ghwhuplqdu orv lqwhuydorv gh fuhflplhqwr | ghfuhflplhqwr +vljqr gh i 3 ,1

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FKDSWHU 81 GHULYDGD

71 Ghwhuplqdu grqgh i % qr h{lvwh r grqgh i % vh dqxod1

81 Ghwhuplqdu od frqfdylgdg gh i +vljqr gh i %,

91 Dvðqwrwdv yhuwlfdohv +hqfrqwudu d wdo txh olp{$d i +{, @ 4,1 :1 Dvðqwrwdv krul}rqwdohv +olp{$ 4 i+{, @ O,1

;1 Dvðqwrwdv Reolfxdv +hqfrqwudu p | e wdo txh olp{$ 4 mi+{,  p{  em @ 3,1

7 6 Hmhpsor 7:; Judtxh i+{, @ {45 . {9  6{5 Od ghulydgd gh sulphu rughq gh i hv= i 3 +{, @

s s {6 {5 { 6 . 6 66 6  6 66 .  9{ @ +{  ,+{  , 6 5 6 7 7

Od ghulydgd gh vhjxqgr rughq i hv=

{ i 3 +{, i %+{, i+{, i+{,

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41 Grpi @ U

51 Lqwhufhswrv frq orv hmhv frrughqdgrv Vl { @ 3 hqwrqfhv | @ 31

Vl | @ 3 hqwrqfhv 3 @ {h{ gh grqgh { @ 3/ oxhjr ho ýqlfr sxqwr lqwhufhswr frq orv hmhv hv S +3> 3,1

61 Lqwhuydor gh fuhflplhqwr | lqwhuydor gh ghfuhflplhqwr1

| 3 @ i 3 +{, @ h{ . {h{ @ h{ +4 . {, Sxhvwr txh h{ A 3> ;{ 5 U hqwrqfhv

i 3 +{, A 3 +, 4 . { A 3 +, { A 4

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71 Pä{lpr |2r Pðqlpr

4.{ h{ 3 i +{, i +{,

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Ho sxqwr pðqlpr hv Splq @ +4> i+4,, @ +4> h4 ,1 Revhuydu txh i qr wlhqh pä{lpr1

81 Frqfdylgdg/ frqyh{lgdg | sxqwr gh lq h{löq

| 33 @ i 33 +{, @ h{ . +4 . {,h{ @ +5 . {,h{

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FKDSWHU 81 GHULYDGD Hq hvwh fdvr i 33 hv srvlwlyd hq ho lqwhuydor `  5> 4^ | qhjdwlyd hq `  4> 5^1 Hvwr vh uhvxphq hq od wdeod vljxlhqwh 5.{ h{ 33 i +{, i +{,

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91 Dvðqwrwdv

Sxhvwr txh i hvwä ghqlgd hq wrgr U/ hv fodur txh i qr wlhqh dvðqwrwdv yhuwlfdohv1 Ghwhuplqdu odv srvleohv dvðqwrwdv reolfxdv x krul}rqwdohv1 Vhd | @ p{ . e sdud { A 3

i+{, {h{ @ olp @ olp h{ @ 4 p @ olp {$4 { {$4 { {$4

oxhjr qr h{lvwh dvðqwrwdv sdud { A 31 Vhd | @ p{ . e sdud { ? 31

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i +{, {h{ p @ olp @ olp @ olp h{ @ 3 {$4 { {$4 { {$4

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oxhjr Grpi @ U  ^4> 3`1

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shur

olp i +{, @ olp. h{ oq+4.4@{, {$3. {$3 @ holp{$3. { oq+4.4@{, olp { oq+4 . 4@{, @ olp. {$3. {$3

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61 Dvðqwrwdv1

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olp x+4 . x,4@x x$3. @ olp. x  olp. +4 . x,4@x x$3 x$3 @ 3h@3

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71 Ho lqwhufhswr e @ olp i +{, @ olp {$4 {$4



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{  4 @ olp  h{ oq+4.4@{, olp 4. { {$4 {$4

shur olp{$4 { oq+4 . 4@{, @ 4 sxhv/ olp{$4 { @ 4 | olp{$4 +4 . 4@{, @ 41

Sru or wdqwr olp4 ${ oq+4 . 4@{, @ 4

Hvwr vljqlfd txh i wlhqh xqd dvðqwrwd yhuwlfdo hq { @ 4 +fxdqgr { wlhqgh d 4 sru od l}txlhugd/ odv lpdjhqhv wlhqghq d lqqlwr,1 91 Gh od juäfd gh | @ i 3 +{, vh revhuyd txh i 3 +{, A 3 hq `  4> 4^ | hq `3> 4^/ oxhjr i hv hvwulfwdphqwh fuhflhqwh hq hvwrv lqwhuydorv1 Dghpäv i 3 hv fuhflhqwh hq `4> 4^ | ghfuhflhqwh hq`3> 4^/ dvð i %+{, A 3 hq `  4> 4^ | i %+{, ? 3 hq `3> 4^ Sru or wdqwr i hv frqyh{d hq `  4> 4^ | frqfdyd hq `3> 4^1 Hq uhvxphq vh wlhqh od vljxlhqwh wdeod i3 i% i

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Hmhuflflr 7;5 Vhd i xqd ixqflöq wdo txh od juäfd gh i 3 hv

81:1 FRQFDYLGDG

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| @ i+{, Ghwhuplqdu vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1

41 Od ixqflöq i hv hvwulfwdphqwh ghfuhflhqwh hq `3> 4^1

51 Od juäfd gh od ixqflöq i hv frqyh{d vöor hq `  4> 4^1

61 Od juäfd gh od ixqflöq i hv frqfdyd vöor hq `  4> 4^ | hq `4> 4^1 71 Od ixqflöq i wlhqh xq pðqlpr hq { @ 41

81 Od ixqflöq i wlhqh xq pä{lpr hq { @ 31

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Gh grqgh vh wlhqh od vljxlhqwh lqirupdflöq i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq { @ 6@5/ qr wlhqh pðqlpr +txh sdvd vl vh uhvwuljh d xq lqwhuydor fhuudgr,/ wuhv sxqwrv gh lq h{löq S4 +4> i+4,,>

S5 +3> i+3,,>

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41 Qrwh txh sxhgh whqhuvh

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G @ Grp U r elhq H @ FrgU>

hq hvwh fdvr/ od uhodflöq uhfleluä xq qrpeuh päv hvshfðfr1

51 Revhuyh dghpäv txh vl4 U hv xqd ixqflöq/ gheh whqhuvh

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+;{ 5 G,+<$| 5 H,+| @ i+{,,

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Hmhpsor 7;8 Od uhodflöq

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s=

U5 $ U +{> |, $ {  |

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J5 $ J +d> e, $ d  e

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Hq ho hmhpsor dqwhulru/ od uhodflöq vhuðd

| hq ho hmhpsor 7;8 5

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] ghqrwd ho frqmxqwr gh orv qýphurv hqwhurv1

6 Hq

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4 @ i+ > q, = q 5 ]j q

4 j {

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hv ghflu/ frpr xq vxefrqmxqwr gho surgxfwr GH txh fxpsoh od surslhgdg

D14

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Vhd f 5 U xqd frqvwdqwh md/ ho frqmxqwr

O @ i+{> |, 5 U5 = { @ fj

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E15

E1514

Wudqvirupdflöq gh Frrughqdgdv

Wudvodflrqhv hq frrughqdgdv fduwhvldqdv

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| @y.n

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E1515

+x . 6,5 . +y  8,5 @ 7 {5 . | 5 @ 7

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| @ xvhq . y frv 

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E151 WUDQVIRUPDFLÖQ GH FRRUGHQDGDV

6<4

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d{5 . 5k{| . e|5 . 5j{ . 5i| . f @ 3

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Dvð @ dufwj4 @ 7 |

s s s 5 5 5 { @ x y @ +x  y, 5 5 5 s s s 5 5 5 | @ x .y @ +x . y, 5 5 5

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E1516

5

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e, Gdgr ho sxqwr S frq frrughqdgdv sroduhv +u> ,/ odv frrughqdgdv fduwhvldqdv vrq +{> |, grqgh { @ u frv >

| @ uvhq

Hmhpsor 7<6 Frqvlghuhprv ho sxqwr S frq frrughqdgdv fduwhvldqdv +4> 4,1 Ghwhuplqhprv vxv frrugh0 qdgdv sroduhv1 Frpr s s +4,5 . 45 @ 5 | u @ 6  @ dufwj+4, @ 7 s 6 vh wlhqh txh odv frrughqdgdv sroduhv gh +u> , @ + 5> 7 ,1

Od uhfwd { @ | hq frrughqdgdv fduwhvldqdv wrpd xqd irupd wrwdophqwh glvwlqwd hq frrughqdgdv sroduhv1 Hq hihfwr { @ | +, u frv  @ uvhq +, wj @ 4

Dvð/ orv vljxlhqwhv frqmxqwrv uhsuhvhqwd or plvpr

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6<5

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E16

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ge vl f  d 9@ 3 fd

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Od hfxdflöq gh xqd uhfwd krul}rqwdo +hv ghflu/ gh shqglhqwh 3, | txh sdvd sru ho sxqwr +d> e, hv |@e

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Vl ho äqjxor txh irupd od uhfwd frq ho hmh R[ hv  | od uhfwd sdvd sru ho sxqwr +d> e,/ hood wlhqh frpr hfxdflöq |  e @ wj+,+{  d, |d txh wj @ p+jxud l}txlhugd,1

Iljxud 

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E1614

6<6

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Od glvwdqfld hqwuh orv sxqwrv S4 @ +d> e, | S5 @ +f> g, gh U5 od ghqluhprv | od ghqrwduhprv frpr vljxh s +d  f,5 . +e  g,5 =@ g+S4 > S5 , @ S4 S5

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r htxlydohqwhphqwh

5

@ TS5

5

. TS4

5

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r elhq

g+S4 > S5 , @

s +d  f,5 . +e  g,5

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Revhuydflöq= devroxwr=

s s +43  5,5 . +7  +5,,5 @ 433 @ 43

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d, Vlphwuðd= Sdud wrgr sdu gh sxqwrv S4 > S5 5 U5

g+S4 > S5 , @ g+S5 > S4 ,

e, Ghvljxdogdg wuldqjxodu= Sdud wrgr wuðr gh sxqwrv S4 > S5 > S6 5 U5 vh wlhqh f, Vhd S4 > S5 5 U5 hqwrqfhv

g+S4 > S6 ,  g+S4 > S5 , . g+S5 > S6 , g+S4 > S5 , @ 3 +, S4 @ S5

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s +d  f,5 . +e  g,5 @ 3

sru or wdqwr +d  f,5 @ 3 a +e  g,5 @ 31 Gh dtxðvh wlhqh d @ f a e @ f/ hv ghflu/ +d> e, @ +f> g,

+f,++, Vl S4 @ S5 > g+S4 > S5 , @ 3 wulyldophqwh

S4 @ S5

6<7

DSSHQGL[ E1 JHRPHWUÐD DQDOÐWLFD Od glvwdqfld hqwuh ho sxqwr +d> e, | od uhfwd D{ . E| . F @ 3 hv mDd . Ee . Fm s D5 . E 5

Ho äqjxor  hqwuh odv uhfwdv D{ . E| . F @ 3/ G{ . H| . I @ 3 hvwä gdgr sru wj @ Dvðhoodv vrq sdudohodv vl | vöor vl r hoodv vrq shushqglfxoduhv vl | vöor vl

DH  GE DG . EH

DH  GE @ 3 DG @ EH

Vl odv shqglhqwhv gh odv uhfwdv vrq p4 | p5 uhvshfwlydphqwh vh whqguä txh ho äqjxor hqwuh hoodv hvwä gdgr sru p5  p4 wj @ 4 . p4 p5 Odv uhfwdv vhuäq sdudohodv vl p4 @ p5 | shushqglfxoduhv vl p4 p5 @ 4

E17

E1714

Föqlfdv

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F @ i+{> |, 5 U5 = g++{> |,> S , @ uj=

g++{> |,> S , @ u hv htxlydohqwh d

s +{  k,5 . +|  n,5 @ u>

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+E14,

{5 . D{ . | 5 . E| . F @ 3

+E15,

|/ phgldqwh od uhvroxflöq gh orv elqrplrv fxdguäwlfrv vh sxhgh oohydu d xqd h{suhvlöq frpr od vljxlhqwh

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hv xqd flufxqihuhqfld | sdud hvrv f +dsursldgrv,/ ghwhuplqhprv ho udglr | ho fhqwur fruuhvsrqglhqwh1

+E16,

E171 FÖQLFDV

6<8

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5

@ 47: . 6  f f @ 83  6

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E1715

Sduäerod

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Iljxud K

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Dvð/ od sduäerod txhgd ghqlgd sru od hfxdflöq r elhq

+{  3,5 . +|  s,5 @ +| . s,5 {5 @ 7s|

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6<9

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@ 7s+|  n, @ 7s+{  k,

+E17, +E18,

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y @|n

x5 @ 7sy

y 5 @ 7sx

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uhsuhvhqwd xqd flufxqihuhqfld frq fhqwur hq ho ruljhq | udglr Sru hmhpsor vl f @ 3 vh wlhqh od flufxqihuhqfld gh udglr 4

s 4  f5 +jxud BB0d,1

3 @ i+{> |, 5 U5 = {5 . |5 @ 4j

Vl f @ 4@7 vh wlhqh od flufxqihuhqfld gh udglr

s

48

7

4@7 @ i+{> |, 5 U5 = {5 . |5 @

Vl f @ 4@5 vh wlhqh od flufxqihuhqfld gh udglr Vl f @ 4 vh wlhqh ho sxqwr

s

6

48 j 49

5

6 4@5 @ i+{> |, 5 U5 = {5 . | 5 @ j 7

4 @ i+{> |, 5 U5 = {5 . |5 @ 3j @ i+3> 3,j s Hq jhqhudo sdud fdgd f 5 ^3> 4` vh wlhqh od flufxqihuhqfld gh udglr 4  f5 s f @ i+{> |, 5 U5 = {5 . | 5 @ + 4  f5 ,5 j

Hvwdv flufxqihuhqfldv vh ghqrplqdq Fxuydv gh qlyho sdud lqglfdu txh do qlyho f +hq ho hmh }, vh hqfxhqwud od flufxqihuhqfld f +jxud QQ l}txlhugd ,1 Vl } 5 Uhfi @ ^3> 4` hqwrqfhv odv suhlpäjhqhv gh } vrq orv +{> |, 5 U5 txh shuwhqhfhq d od flufxqihuhqfld {5 . | 5 @ 4  f5 grqgh } @ f Orv sxqwrv gh od vxshuflh txh uhsuhvhqwdq od juäfd gh } @ i+{> |, vrq gh od irupd

hv ghflu/

+{> |> i +{> |,, frq +{> |, 5 Grpi

+{> |> f, 5 U6 wdo txh {5 . | 5 @ 4  f5 frq f 5 ^3> 4`

jxud QQ

F151 DOJHEUD GH IXQFLRQHV

73:

Hq od pd|ruðd gh orv fdvrv/ sdud judfdu od vxshuflh qrv edvwd glexmdu dojxqrv vxefrqmxqwrv gh Judii 1 Hq ho hmhpsor s dqwhulru qrv edvwd judfdu odv fxuydv gh qlyho +jxud QQ fhqwur, do qlyho f/ sdud rewhqhu od judfd gh } @ 4  {5  | 5 +jxud QQ0ghuhfkd,1 s Revhuydu txh od juäfd gh i/ hv ghflu od vxshuflh txh uhsuhvhqwd d } @ 4  {5  | 5 hv od plwdg gh od hvihud xqlwduld frq fhqwur hq ho ruljhq1 s Dqäorjdphqwh/ srghprv rewhqhu odv juäfdv gh odv vxshuflhv= } @ 7{5 . | 5 oodpdgd frqr hoðswlfr | } @ {5 . | 5 oodpdgr sduderorlgh +jxud I,

I

1

F15

Dojheud gh ixqflrqhv

Do ljxdo txh ixqflrqhv uhdohv gh xqd yduldeoh/ vh ghqh ho äojheud gh ixqflrqhv gh grv yduldeohv +r päv yduldeohv, Vl i | j vrq ixqflrqhv uhdohv gh grv yduldeohv wdo txh G @ Grpi _Grpj 9@ !/ hqwrqfhv h{lvwhq ixqflrqhv i . j> i  j> i  j frq grplqlr G | od ixqflöq i@j frq grplqlr i+{> |, 5 G = j+{> |, 9@ 3j grqgh 4, +i . j,+{> |, @ i+{> |, . j+{> |,

5, +i  j,+{> |, @ i+{> |,  j+{> |,

6, +i  j,+{> |, @ i +{> |,  j+{> |,

{>|, 7, +i @j,+{> |, @ ij++{>| ,

Hmhpsor 836 Vl i+{> |, @ oq+{ . |, | j+{> |, @ Grpi Grpj

s

{ vrq ixqflrqhv hqwrqfhv

@ i+{> |, = { . | A 3j> @ i+{> |, = {  3j

oxhjr G @ Grpi _ Grpj 9@ !/ sru or wdqwr h{lvwh od ixqflöq i . j ghqlgd hq G s +i . j,+{> |, @ oq+{ . |, . {

Revhuydflöq= Pxfkrv gh orv frqfhswrv ghqlgrv sdud ixqflrqhv gh xqd yduldeoh +frpr sru hmhpsor orv xwlol}dgrv hq ho äojheud gh ixqflrqhv,/ wdpelìq vh ghqhq sdud ixqflrqhv gh grv r päv yduldeohv/ shur qr hv srvleoh jhqhudol}du wrgrv hoorv d hvwh wlsr gh ixqflrqhv1 Xqr gh hvwrv hv ho frqfhswr gh ixqflöq fuhflhqwh | ghfuhflhqwh/ ho txh qr vh sxhgh ghqlu sdud ixqflöq gh grv yduldeohv/ |d txh U5 qr hv xq frqmxqwr rughqdgr +qr wlhqh vljqlfdgr/ sru hmhpsor/ od vljxlhqwh h{suhvlöq +{> |, ? +{3 > | 3 ,,4 1 4 Hv

ghflu/ qr kd| xq vljqlfdgr txh vhd frpsdwleoh frq odv rshudflrqhv gh U2 1

73;

DSSHQGL[ F1 IXQFLRQHV UHDOHV GH GRV YDULDEOHV

Sru rwud sduwh/ sdud ixqflrqhv uhdohv i | j gh grv yduldeohv qr srghprv ghqlu od frpsrvlflöq j  i sxhv Uhfj _ Grpi @ !1 Vlq hpedujr/ vl frqvlghudprv odv ixqflrqhv i = D  U5 $ U | j = E  U $ U

hqwrqfhv wlhqh vhqwlgr ghqlu od frpsrvlflöq j  i vl Uhfi _ Grpj 9@ !/ grqgh

Grpj  i @ i+{> |, 5 Grpi = i+{> |, 5 Grpjj | +j  i ,+{> |, @ j+i +{> |,,

Qrwh txh vl sdud wrgr +{> |, 5 Grpi> Uhfi  Grpj hqwrqfhv Grpj  i @ Grpi 1

Hmhpsor 837 Vhd

i = D  U5 +{> |,

hqwrqfhv

j = U $ U { $ {5

D @ Grpi @ i+{> |, 5 U5 = {|  3j

| oxhjr Grp+j  i, @ Grpi | Dqäorjdphqwh=

Hmhpsor 838 Vhd j= hqwrqfhv

$ U s $ {|

Uhfi @ U.  Grpj @ U

s +j  i,+{> |, @ j+i+{> |,, @ j+ {|, @ {| hq D

$ U U5 +{> |, $ +{  d,5 . +| . e,5

+i  j,+{> |, @ i ++{  d,5 . +|  e,5 , @

i = U. 3 {

$ sU { $

s +{  d,5 . +|  e,5 frq +{> |, 5 U5

Vhd V @ i  j/ grqgh GrpV @ Grpi  j @ Grpj @ U5 Qrwhprv txh od ixqflöq s V+{> |, @ +{  d,5 . +|  e,5

uhsuhvhqwd od glvwdqfld gh xq sxqwr fxdotxlhud +{> |, hq ho sodqr d xq sxqwr gdgr +d> e,1

Dsshqgl{ G

Sureohpdv Plvfhoäqhrv Sduwh L L1 Ghwhuplqh fxäohv gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh { s  s 54 5 , 41 +;{> | 5 U,+{ . | 9@ 3 , +{.| , {5 .| 5 { .| s 5 51 +;{ 5 U,+ 7{ @ 5m{m,

61 +;{ 5 U,+{5 @ 49 +, { @ 7 b { @ 7,

4 71 +;{,+;|,+{> | 5 U. ,+ {|  {4 ,

81 Vhdq d> e 5 U1 Vl d ? e hqwrqfhv 91 i{ 5 U @

mm mm {

{

4

d e

?4

A 4j @ i{ 5 U @ m{m A m4  {mj

4 :1 +;{ 5 U  i3j, +{ ? 43 , 43 ? 4, sd5.e5 {  4, ;1 +;d 5 U  i3j,+;e 5 U,+ d

<1 +;d 5 U. ,+;e 5 U. ,+;f 5 U. ,+d  e , d  ef,

{m  m{m, 431 +;{> | 5 U  i3j, + {5m. |5

441 +;{> | 5 U  i3j, +6m{m  m|m ,

s m5{m 5  45 , { .|

451 +;{> | 5 U  i3j, +{5  | 5 , mm{|mm  4, t s s {  4j @ i{ 5 U @ {  4  {j 461 i{ 5 U @ 4 { 471 Vhdq d> e 5 U1 Vl d ? e hqwrqfhv 4 ?

481 i{ 5 U @

mm mm {

5

{

e

d

A 4j @ i{ 5 U @ m{m A m5  {mj

491 +;{ 5 U  i3j, +{  : ? 3 , 4: A {4 , s 5 5 { .| 4:1 +;{ 5 U  i3j,+;| 5 U,+ {  4, 4;1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

hv ydfðr

4<1 +;{ 5 U, +{5  {7 ,

531 vxsi{ 5 U@{7 . 6{5  7 @ 3j @

{ s ?3 {  {

s 5

541 Ho ðqpr gh i{ 5 U@ 5{{.8 A 4j hv 08 551 Ho vxsuhpr gho frqmxqwr D @ i4 

4

q

mq 5 Qj hv 31 73<

743

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L 561 Vl orj{ +8, ? 4 hqwrqfhv vh wlhqh txh 8 ? {1

571 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

581 5 orj 5 . +4 .

orj8 {  4 hv ^6> 4^ orj8 +4@6,

4

s , orj 6  orj + { 6 . 5:, @ 3/ od hfxdflöq dqwhulru qr wlhqh vroxflöq srvlwlyd

591 Vl orjd e @ 6 | orjd f @ 5 hqwrqfhv orjf e5 @ 6 | ef @ d8

5:1 i{ 5 Um

5{

4 A 4j @ i{ 5 Um 4 A orj5 +{,j 5 5;1 Sdud wrgr d> e 5 U.  i4j vh wlhqh orje +d5 ,  orje +d6 ,1 s 5<1 vxsi{ 5 U@{7 . {5  9 @ 3j @ 5 orj +{,

631 +;d 5 U. , +orj+d, ? orj+d5 ,, 641 Od lqhfxdflöq

orj4@5 +d{5 . 4, ? orj4@5 +d,

wlhqh frpr frqmxqwr vroxflöq wrgrv orv uhdohv vl | vöor vl d 5`3> 4^

651 +;e 5 U.  i4j, +orje +{,  4 , {  e,

661 Vl i hv lq|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 671 Vhd i+{, @ {5  4 > j+{, @ { . 4/ hqwrqfhv ho Grp+i  j, @ U

681 +;d 5 U. , +orj+d, ? orj+d6 ,,

691 +;e 5 U.  i4j, +orje + {4 ,  4 , 4  {e,

6:1 Vl i hv el|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 6;1 Vhd i+{, @ {5 . 4> j+{, @ {  4/ hqwrqfhv ho Grp+i  j, @ U 6<1 Ho ydoru pðqlpr gh od ixqflöq i+{, @ {+{  6, hv <@7

731 Vhd i = G $ ^6> 4^ { :$ h{  4 hqwrqfhv ho grplqlr pä{lpr gh i hv ^ oq+7,> 4^ s 741 Vhdq i +{, @ 5  { > j+{, @ oq+{5 . <, ixqflrqhv hqwrqfhv ho grplqlr pä{lpr gh od ixqflöq j  i hv `  4> 44^1 751 Vl i hv xqd ixqflöq lq|hfwlyd hqwrqfhv i hv hvwulfwdphqwh ghfuhflhqwh 761 Od vroxflöq gh od lqhfxdflöq orj{ +{ . 4@5, ? 4 hv ydfðr1

771 Vl i +{, @ oq+{ . 7, | j+{, @ h{ . 6 vrq ixqflrqhv hqwrqfhv +j  i ,+{, @ { . : | vx grplqlr hv U

LL1 Frpsohwdflöq=

41 Ho frqmxqwr vroxflöq gh od lqhfxdflöq s {5  3 hv1111111111111111111 m{  5m  m4  {m 51 Ho frqmxqwr vroxflöq gh od lqhfxdflöq s {5  3 hv 111111111111111 m6  {m  m{  5m 61 Od lqhfxdflöq

5 5 4 6  + ,8{.5{ 
744 71 Ho frqmxqwr vroxflöq gh od lqhfxdflöq u

81 Od lqhfxdflöq

4  m{  4m hv11111111111111 {5  5{ . 4  8{.5.{5 5 4 6 
wlhqh frpr vroxflöq d orv U vl d 511111111111111111111111

91 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

m{  4m6 +m{  5m,8 A 3 hv 111111111111111 m{5 . 5m

:1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

+{5 . :{ . 43,+{5  6{ . 7, ? 3 hv 11111111111111111111 m{ . 6m

;1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq u

6 4  hv 111111111111111111111 {5  9{ . < 7

<1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq u

4  4 hv 1111111111111 { . 7{ . 7 5

431 Vhdq d 5 U  5 U. ho frqmxqwr i{ 5 U m m{  dm ? j hv ho lqwhuydor 11111111111111111 441 Vl { 5 U. > | 5 111111111111111111 } 5 U hqwrqfhv vh yhulfd

451 Od sursrvlflöq

} }  {| {

+ . 4,{5  7{ . +  4, A 3 +;{ 5 U,

hv yhugdghud vl | vöor vl  5111111111111111

461 Od sduäerod {5 5{d| 5d.| @ 6/ frq fhqwur hq +4> 5, vh deuh kdfld duuled vvl d 51111111111111

471 Od hfxdflöq gh od uhfwd txh sdvd sru ho fhqwur gh od holsvh 5{5  7{ . 6| 5 . 9| @ 7 | hv sdudohod d od uhfwd 5{ . 6| @ 4 hv 11111111111111 481 Od uhodflöq {5  | 5  7{ . 5| . 7 @ 3 uhsuhvhqwd od föqlfd 1111111111111 frq fhqwur hq 11111111111111

491 Vhd DEFG ho uhfwäqjxor frq odgrv sdudohorv d orv hmhv lqvfulwrv hq od holsvh 49{5 . 7| 5 @ 971 Hqwrqfhv od ixqflöq txh uhsuhvhqwd ho äuhd gho uhfwäqjxor hq ixqflrq gh } hv= 11111111111111111

745

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L

4:1 Od uhodflöq {5  | 5 . 7{  5| . 7 @ 3 uhsuhvhqwd od föqlfd 111111111111111 frq fhqwur hq1111111111111 s 4;1 Ho grplqlr gh od ixqflöq i+{, @ orj+{5  9,  4 hv 1111111111111 s 4<1 Ho grplqlr gh od ixqflöq i+{, @ orj+{5  4,  4 hv 1111111111111111

531 Vhd i = G $ ^h5 > h4 ^> i +{, @ + 5:4h ,6{4 hqwrqfhv ho grplqlr pä{lpr gh i hv 111111111111 541 Vhd i +{, @ orj +{, | j+{, @ {5 . e{  e ixqflrqhv/ hqwrqfhv ho frqmxqwr gh wrgrv orv e 5 U wdo txh ho grplqlr gh i  j hv ydfðr hv 11111111111111111 s 551 Vhd i+{, @ orj++{  6,5 , | j+{, @ 7  {/ hqwrqfhv ho grplqlr pä{lpr gh j  i hv 111111111 561 Ho grplqlr gh od ixqflöq s s 4  4  {5 i +{, @ {5 hv 11111111111111111111 571 Od uhodflöq |5 @ 5  { ghqh xqd ixqflöq el|hfwlyd hq 11111111111111111   5 { >{ ? 3 { >{ ? 4 hqwrqfhv j  i +{, hv 11111111111111111111 581 Vhd i+{, @ | j+{, @ s { >{  4 5{ > {  3 591 Gdgd odv ixqflrqhv i | j ghqlgdv sru   s {.4 { ?3 {4 4{8 i+{, @ j+{, @ {.5 { 3 { . : {A8 +d, Ho Uhfruulgr gh i hv 111111111111 +e, Ho Grplqlr gh j  i hv 1111111111111111  +f, +j  i ,+{, @

5:1 Grv wxehuðdv wdugdq 9 krudv hq oohqdu xqd slvflqd1 Xqd vrod od oohqduðd 8 krudv päv gh sulvd txh od rwud1 Hqwrqfhv fdgd wxehuð d wdugduðd lqglylgxdophqwh hq oohqdu od slvflqd 111111111111111 | 1111111111111111 uhvshfwlydphqwh 5;1 Vhd i = `3> 4^i4j $ U 4 { :$ { .6 4 hqwrqfhv ho uhfruulgr hv 111111111111111

G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV

746

5<1 Vhd i = U  i4j $ U  i4j {.4 { :$ 4{ xqd ixqflöq el|hfwlyd hqwrqfhv i 4 +{, hv ljxdo d 1111111111111111

631 Od ixqflöq txh uhsuhvhqwd ho yroxphq +hq ixqflöq gh k, gho flolqgur flufxodu uhfwr lqvfulwr hq xq frqr gh 53fp gh dowxud | 8fp gh udglr/ frpr hq od jxud/ hv i+k, @1111111111111 | vx grplqlr hv 11111111111111111 641 Ho grplqlr gh od ixqflöq i +{> |, @

G14



oq+4

{

{.|

|,

hv ho frqmxqwr 1111111111111 fx|d juäfd hv 11111111111

Uhvsxhvwdv d orv Hmhuflflrv Sursxhvwrv

Vroxflrqhv gh od vhfflöq 41814 41



 4 3 +d, Idovr1 + ,  4 4  4 3 +f, Idovr1 + , 5 4  3 3 +h, Idovr1 + , 3 4

  4 3 +e, Yhugdghur1 + , 3 4  3 3 +g, Idovr1 + ,  3 4 3 3 +i, Idovr1 + , 4 4

Vroxflrqhv gh od vhfflöq 41815 41 < shuvrqdv1

51 Ho suhflr gho sulphu | gho vhjxqgr yroxphq hv gh ' 6=333 1 61 6 phwurv | 9 phwurv/ uhvshfwlydphqwh1

71 Ho sulphu vdfr wlhqh ;3 nlorjudprv | ho vhjxqgr 93 nlorjudprv1 81 Ho shglgr vh frpsohwduä hq 5: gðdv1

91 4: nlorjudprv gh freuh | : nlorjudprv gh }lqf1 :1 Ho qýphur exvfdgr hv 971 ;1 Ho qýphur exvfdgr hv 571

<1 Od yhorflgdg gho sulphu wuhq hv gh 83 np2k | od gho vhjxqgr 73 np2k1

431 54 np1

441 Od yhorflgdg gho flfolvwd hv gh 45 np2k | od gho frfkh 63 np2k1 451 < krudv ghvsxìv gh od vdolgd gho wuhq uäslgr1

461 Od yhorflgdg gho sulphu flfolvwd hv gh 47 np2k/ | od gho vhjxqgr 4; np2k> ho sulphur hvwä d 6 krudv gho hqfxhqwur hq od fduuhwhud/ ho vhjxqgr/ 5 krudv1 471

< 7

olwurv sru plqxwr> 6 olwurv sru plqxwr1

481 Ho sulphu wudedmdgru wdugduä 47 gðdv/ ho vhjxqgr 44 gðdv hq uhdol}du lqglylgxdophqwh ho wudedmr1 491 Od sulphud pätxlqd shuirud 5 phwurv gh wýqho sru gðd/ | od vhjxqgd

6 5

phwurv sru gðd1

4:1 Od sulphud fxdgulood sxhgh uhdol}du od uhsdudflöq frpsohwd hq 43 gðdv/ | od vhjxqgd hq 48 gðdv1 4;1 Ho sulphu reuhur sxhgh kdfhu ìo vror ho wudedmr hq 45 krudv> ho vhjxqgr/ hq 57 krudv1 4<1 Od sulphud wxehuðd oohqd od slvflqd hq 43 krudv> od vhjxqgd/ hq 48 krudv1

747

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L

531 Od uxhgd pd|ru gd 533 usp | od phqru 933 usp1

541 Ho sulphu prwru frqvxph 93 judprv gh frpexvwleoh sru krud> ho vhjxqgr/ 7; judprv gh frpexvwleoh sru krud1

551 Od juýd gh pd|ru srwhqfld ghprud 57 krudv hq uhdol}du vrod ho wudedmr frpsohwr/ plhqwudv txh od gh phqru srwhqfld ghprud 69 krudv1 561 Od dowxud hv k  68> 89fp vx udglr u  4<> 4
Vroxflrqhv gh od vhfflöq 41816

41 S4 +{> |, @ +5> 3, S5 +{> |, @ +3> 5, 51 S4 +{> |> }, @ +5> 5> 5,

61 S4 +{> |, @ +6> 4, S5 +{> |, @ +4> 6,

71 S4 +{> |, @ +7> 6, S5 +{> |, @ +7> 6,

: > 47 , 81 S4 +{> |> , @ +3> 4> 4, S5 +{> |> , @ + 67 >  49

s s s s 91 S4 +{> |> , @ + 55 > 5> 47 , S5 +{> |> , @ + 55 >  5>  47 , s s s s S6 +{> |> , @ + 55 > 5>  47 , S7 +{> |> , @ + 55 >  5> 47 ,

:1 S @ + 46 > 46 > 46 > 4< ,

;1 S @ +7> 7> 7> 4, s s s s <1 V @ i+ 6> 6> ,> + 6>  6> , =  5 Uj

431 S4 +{> |> }> , @ + 65 > 5>

8 5

> 4, S5 +{> |> }> , @ + 65 > 5>  85 > 4,

441 S4 +{> |> }> , @ +4> 4> 3> 5, S5 +{> |> }> , @ +4> 4> 3> 5, S6 +{> |> }> , @ +3> 3> 4> 4, S7 +{> |> }> , @ +3> 3> 4> 4, s s s s 451 S4 +{> |> }> , @ + s 8@6> 8@6> s 8@6>s 8@45,s S5 +{> |> }> , @ + 8@6>  8@6>  8@6>  8@45,

Vroxflrqhv gh od vhfflöq 41817

Or lqglfdgr hqwuh sduìqwhvlv +, hv vöor xqd d|xgd sdud xqd fruuhfwd mxvwlfdflöq 41 +d, +f, +h, +j, +l, +n, 51

+d, +g, +j, +m,

I dovr=+{ @ 4> | @ 5,> +e, I dovr=+{ @ 5> | @ 4, I dovr=+{ @ 4@6> | @ 4@5,> +g, I dovr=+{ @ 4@;, Y hugdghur= +i, I dovr=+{ @ 4,> Y hugdghur= +k, I dovr=+{ @ 4> | @ 3,> I dovr=+V @ i6j,> +m, I dovr=+V @`  4> 3` ^ ^5> 4^, I dovr=+U 9@ U  i5j,> +o, Y hugdghur= Yhugdghur1 Idovr1 Idovr1 Idovr1

+e, +h, +k,

Yhugdghur1 Yhugdghur1 Yhugdghur1

61 +d, ^4> 3` ^ ^4> 4^1 +e, ^3> 4`1

+f, +i, +l,

Yhugdghur1 Yhugdghur1 Idovr1

+f, ^3> 4`1 +g, +{ 5 ^4> 4^a| 5 U, b +{ @ 3 a | 5 U  i3j,1

+i, `4> 4^1

G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV

748

71 +d, V @`  4@6> s 4^1 +e, Vs@`5> <^1 +f, V @`3> 4^1 +g, V @`  4> 3^^`4@:> 4^1 s s s s s +h, V @`4@5> :9 46 ^^`4> :.9 46 ^1 +i, V @`  4>  45^^`  8> 3^^` 8> 7  44^^`45> 7 . 44^= +j, V @ ^3> 4^1 +k, V @ ^3> 4^ +l, Vs @ ! +m, V @`  4> 5^ +n, V @ ^9> 4^ +o, V @ ! +p, V @`  4> 9@46^i3j +q, V @ ^ 45 :6 > 5` ^ ^4> 6` +r, V @ ^5> 657` +s, V @ !1 +t, V @ i4j +u, V @ ! +v, V @ i5j s s 81 +d, V @ !1 +e, V @ i4j1 +f, V @ i4  6> 5j1 +g, V @ U1 +h, V @`  4> 6@5`^`4@7> 4^1

+i, V @ U`4@7> 6^1

+j, V @`  4> 5^ +k, V @`  4> 5` ^ ^3> 6=8^1

+l, V @ U  i3j +m, V @`  :> 4^ +n, V @ ^4@6> 6` s s +o, V @`3> 4` +p, V @`  :> 5^ +q, V @ ^ 8> 8` ^ i45j

+r, V @ ^6> :`1

+s, V @ ^4> 5` +t, V @ U +u, V @ i3j

91 D|xgd= Hvfulelu frpr hfxdflöq gh vhjxqgr judgr | txh ho glvfulplqdqwh hv xqd vxpd gh fxdgudgr +s  t,5 | 7d5 1

s

:1 ^ 4;43 ;7 > 4^

;1 +d, p @ :1 +e, p @ 5<@:1 s <1 n @ 781

s +f, `4> 8^ +g, p @ 44 6 44 +h, `8> 4^1

431 !

441 +d, Vxsuhpr= 9 <> 4> 9 <> 4> 9 <1 Lqpr= 04 / 3/ 04/ 3 / 04 +e, Vxsuhpr= 9 <> 9 h{lvwv> 9 <> 9 <> 9 <1 Lqpr= 4@5> 9 <> 4@5> 9 <> 4@51

451 +d, vxs D @ :@6 +e, ! +f, :@6

461 D|xgd= ghpxhvwuh txh { . vxs D hv frwd vxshulru gh { . D | dvð wdpelìq txh vxs+{ . D,  { hv frwd vxshulru gh D1

471 Q qr hv dfrwdgr | Q  ]1

q  481 D|xgd= +l, Frqvlghuh q5.6 +Surs1 Dutxlphgldqd,1

5q

q

@ 5 oxhjr 5 hv frwd vxshulru1 +ll, +; A 3,+
491 D|xgd= +l, {q  3> ;q oxhjr 3 hv frwd lqihulru1 +ll, Sru Dutxðphghv


5

5q .6

4.5q





5

5q

4.5q

 q | frpr Q qr hv dfrwdgr hqwrqfhv D wdpsrfr or hv1

Vroxflrqhv gh od vhfflöq 51:14 41 d, f, h, j,

4

Q

^4> 4^> ^3> 4@5` s e, ^5> 7@6`> ^3> 4@ 6` g, s i, ^4@5> 6@5`> ^3> 5` ^4> 4^> ^3> 4^ k,

U  i4j> U s i7j ^4> 4^> ^ 5> 4^ `  4> 5` ^ ^3> 4^> U.  U  i6j> U  i4j

51 d, `  ;> 3^ e, ^4> 5^ f, ^49> 97^ g, ^<> 5934^ h, Qr h{lvwh i, `  4> 6^^`  5> 5^^`8> 4^ 61 d, ^5> 4^ e, `3> 4@6^f, `  4> 4^ g, ^4> 4^ h, ^3> 4^

?  frq

4

Q

9

q.6

A

? % 5

6q.4

1

749

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L

Vroxflrqhv gh od vhfflöq 51:161 41 D+{, @



5{+O

7{,

6

51 D+{, @ +{  45 ,+ 4;  56 , {

61 w+{, @

s5

.

{ .49 4;



{

43

63

s 71 F+{, @ 6  437 +7  {, . 8  437 < . {5 ; 5=83{ 3  {  53 A A ? 5=33{ 53 ? {  83 81 | @ 4=:8{ 83 ? {  433 A A = 4=83{ 433 ? {

91 | @

{ 45

. 3=43+{  45

{ , 45

:1 E+{, @ +533 . 53{,+;333  833{,  5333+533 . 53{, ;1 X +{, @ +4533 . 433{,+57  {,

<1 Y +{, @ {+49  5{,+54  5{,

431 Y +{, @

P{



{

6

s { 5`3> P ^

>

; 7d5 . ;d{ A A ? 6
7

451 Y +u, @ u5 +45  6u,>

3  { ? 7d { @ 7d 7d ? { ? :d { @ :d

u 5`3> 7^1

461 S +{, @ +; . {,+533  43{,>

471 L+{, @ +583 . {,+;  3=34{,>

481 Plq F @ 6{ . 7| V1D1

5{. 6| 5{. 5| ;{. 5| 3{/

   3

69 5; 65 |

9{. 7{. 8{. 3

   3

87 7; 83 |

491 Pd{ } @63{ . 53| V1D1

6| 9| 8| {/

4:1 Pd{ } @6333{ . 5333| V1D1

{. 5| 5{. | 0{. | | 3  {/

    3

9 ; 4 5 |

{ 5 ^3> 53` _ ] { 5 ^3> 783`1

G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV

74:

4;1 d,Pd{ } @533F . 433P V1D1

F. 5P 5F. 418P F P 3  F/

    3

433 454 83 78 P

Ho öswlpr F@83/ P@47> frqwulexflöq pä{lpd 44173310 e, Pd{ } @533F . 433P V1D1

F. 5P 5F. 418P F P 3  F/

    3

433 463 83 78 P

Ho öswlpr gh hvwh qxhyr sureohpd hv= F@83/ P@53> frqwulexflöq pä{lpd 45133310

Vroxflrqhv gh od vhfflöq 51:17 41 Wrgdv vrq el|hfwlydv

5 51 d, D @`  4> 3^^`4> 4^ E @ U  i5> 5j i 4 +{, @ {5{7 {  4 e, D @ ^3> 4^i4j E @`  4> 3^^`4> 4^ i +{, @ + {4 ,5 61 d, Grp+j  i , @ ^6@5> 4^ Grp+i  j, @ U.  e,

f,

Grp+j  i , @ ^7<@7> 4^ Grp+i  j, @ ^4> 4^

Grp+j  i , @`  4> 6  Grp+i  j, @`4> 4^

s s 43` ^ ^6 . 43> 4^

g, Grp+j  i , @ ^<> 4^ qr h{lvwh od frpsxhvwd gh i r j h,

Grp+j  i , @ i3j Grp+i  j, @ ^4> 4^

e,

Grp+j  i , @ ^7<@7> 4^

f,

Grp+j  i , @`  4> 6 

g, Grp+j  i , @ ^<> 4^ h,

Grp+j  i , @ i3j

s s 43` ^ ^6 . 43> 4^

74;

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L

71 d,

j  i+{, @ i  j+{, @

e,

j  i+{, @ i  j+{, @

f,

j  i+{, @

= s

{  4

9{ . 49

5{ . 6  4 { A 4

; ? 9{ . 4:

{4

= s 5 {.4 {A4

; ?

s 9 {  47

= s s 5 {:4 ; s ? 5 6{ . :  :

= ss 5 {4:

3  {  49 { A 49

:@6  {  4 {A4

; ? 6{5 . 4;{ . :

{ 5`> ^i3j

= s

5 {s . 9{  4 { 5` s  4> ` ^ ^> 4^= grqgh  @ 6  43  @ 6 . 43

i  j+{, @

g, j  i+{, @

h,

; ?

; ? <{5 . 93{ . <4 {  4 { 9@ :@6 =

; ?

s {.7 {8 43

{A4

{@<

= s 7 s{  5  4 { A <

; ss A 6{ . :  5 5@6  {  4 ? i  j+{, @ t A = ss{  4  5 {  433 s j  i+{, @ 6 4  m{m . :  4  {  4

i  j+{, @

; s ? 4  m6{ . :m ;@6  {  5 =

s s 5 {

4?{7

; 4;{ . 6< { 5`  4> ;@6` A A A A ? s 6{ . ; { 5`  ;@6> 4` 81 j  i+{, @ A A A A s = 7{ { 5`4> 4^

91 e, k+{, @ 46 +{5 . 6{  4, | ho grplqlr hv U

f, Grp @ ^6@5> 4^ Uhf @ ^<@7> 4^> i 4 +{, @

 s 6.

f*, Grp @`  4> 6@5` Uhf @ ^<@7> 4^> i 4 +{, @

<.7|

 s 5

<.7|

6

5

G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV :1 Grpk @ ^5> 4^> k4 +{, @ orj5 +{5 . 7,

;1 Grpi @ U > Uhfi @ U. > Qr10

<1 d, Grpi @`3> 8@7^> Uhfi @ U> Grpj @ U> Uhfj @`  4> 4^

e, Grp+i  j, @ U. > +i  j,+{, @ orj8@7 +4 . 6h{  7h5{ , s 8 5849+8@7,{ f, Grpi @`3> 8@;^ Uhfi @`  4> 5^> i 4 +{, @ ; s 8. 5849+8@7,{ f*, Grpi @`8@;> 8@7^ Uhfi @`  4> 5^> i 4 +{, @ ;

Vroxflrqhv gh od vhfflöq 51:18 41 +d, +e, +f,

51 +d, +e, +f,

+g, +h,

+i,

+j,

61 +d, +e, +f,

+g,

8 orj s 6 6s 85

orj sd676 5 s{5|.f orj | s 7 f{ { @ 5@6

{ @ 4:@7 { @ 4@6

Ho frqmxqwr vroxflöq hv ydfðr s { @ 8 . 6 s { @ :5 { @ 5

{ @ 3> 34 > { @ 3> 334 { @ 5 > { @ 3> 58

{ @ 5 > { @ 46 54 4.s54 4s54 { @ 43 5 > { @ 43 5

+h,

{ @ 7

+j,

{ @ 3 > { @ 4

+i,

+k, +l,

+m,

{ @ 4 > { @ 8 { @ orj: ;

{ @ orj4>38 5

{ @  orj8 7 > { @  orj8 9

+n,

{ @

+p,

{ @ 5

+o,

+q, +r,

+s,

71 +d, +e, +f,

4 orj6 +8@6, 5 s { @ 8

{ @ 5

: { @ orj89 : + 8;4: , { @ 57

s

s

` 65 6: > 4^^`7> 6.5 6: ^ t t `  4>  87 ` ^ ^ 87 > 4^

d A 4 `6> 9` > d ? 4 ^9> 4^

74<

753

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L +g,

`3> 6^

+i,

^

+h,

+j,

+k,

+l,

`3> 4< `^` 46 > 4` ^ ^6> 4^

s

5

5

> 4^

`  4> 4^

`  4>  67 ^

`  4> orj: ;`

s

+m,

`3> 5

+o,

`6> 4^

+n, +p, +q,

5

`4@7> 4^

s

^^`5

5

> 4^

`3> 6^^^ 44 > 7^= 6

`5> 4^

4 81 +d, Frpr 3 ? { 9@ 4 | 3 ? | 9@ 4 Vl ts 9@ 4 hqwrqfhv odv vroxflrqhv vrq { @ +dt e, ts4 +e,

+f,

{ @ 83 > | @ 53

+g,

{ @  45 orj 7< 6 > | @

+i,

{ @ 6@5 > | @ 4@5

+h,

+j,

+k,

+l,

+m,

91 +d, +e,

+f,

4 7

orj 56 +59 @68 ,

{ @ 5orj9 < > | @ orj9 <  5orj9 <

{ @ 8> | @ 9 r { @ 89 > | @ 4

Ho frqmxqwr vroxflöq hv ydfðr1 6 4 4 6 { @ d5 > | @ d5 r { @ d5 > | @ d5

{ @ + 65 ,6 > | @ + 65 ,5 r { @ 4> | @ 4 { @ 65 > | @

4 5

l, 47/5 dôrv ll, 55/8 dôrv ' 719<6

' 75<13<9

Vroxflrqhv Sureohpdv Plvfhoäqhrv L1

LL1

410 910 4410 4910 5410 5910 6410 6910 7410

I I Y Y I Y Y I I

510 :10 4510 4:10 5510 5:10 6510 6:10 7510

I I Y I I I I I I

610 ;10 4610 4;10 5610 5;10 6610 6;10 7610

41 `6@5> 4^1

51 `8@5> 4^^i5j1 61 ^8@;> 4^1

71 `  4> 3` ^ ^5> 4^ 81 ^6@;> 4^1

Y I Y Y I I I I I

710 <10 4710 4<10 5710 5<10 6710 6<10 7710

I I I I I Y I Y I

810 4310 4810 5310 5810 6310 6810 7310

I I I I I I I I

4 | @ +des , ts4

G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV 91 U  i4> 5j

:1 `  8> 5^^i6j1

;1 ^8@6> 46@6`  i6j1 <1 ^6> 4`  i5j1

431 `d  > d . ^1

441 `3> 4`1 s 451 ` 8> 4^1

461 `4> 4^1

471 | . 4 @  56 +{  4,1

481 klsìuerod frq fhqwur +5> 4,1 s 491 D+{, @ 7{ 49  7{5 1

4:1 klsìuerod frq fhqwur +5> 4, 4;1 `  4> 7^^`7> 4^1 s s 4<1 `  4>  44^^` 44> 4^1 531 ` orj5: +6,> orj5:h +6h,`1

541 ^7> 3`1

551 `  <:> 436^i6j1 561 ^4> 4`  i3j1 571 `  4> 5`1

; 5 {?3 ? { 5 3  { ? 4@5 7{ 581 +j  i ,+{, @ = s 5{ {  4@5 591 +d, U  ^4> 5^ +e, U. 3

+f, +j  i ,+{, @

5:1 43 | 481

 s {.4 { 5 ^3> 6` { . 8 { 5 U.  ^3> 6`

5;1 Uhfi @`  4> 5^^`6> 4^ 5<1 i 4 +{, @



{

4

{.4

1

631 i +k, @ +8  k7 ,5 k> 641 i+{> |, 5 U

5

k 5`3> 53^1

= 4  {  | A 3> { . | 9@ 3j

754

755

DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L

Dsshqgl{ H

Hmhuflflrv Sursxhvwrv Oðplwh gh vxfhvlrqhv

Hmhuflflr 839 Kdoodu orv oðplwhv gh odv vxfhvlrqhv 41 4>  45 > 46 >    > +4,q

q4

q > 51 4 > 6 >    > 5q 4 5

7

5

>

61 3=7> 3=76> 3=766> 3=766> 3=76666>   

Hmhpsor 83: Ghprvwudu ho vljxlhqwh oðplwh olpq$4

s qq 4

$

@3

Ghprvwudflöq= Sdud hoor sulphur yhuhprv od ghvljxdogdg vljxlhqwh Vhd n | q vljxlhqwh qýphurv qdwxudohv wdo txh 4  n ? q1 Dvð 3 ? q  n | 3  n  4 oxhjr 3  q  n  4 | 3  n  4 Sru or wdqwr 3  +q  n  4,+n  4, 3  qn  q  n5 . 4 q  4  +q  n,n

or dqwhulru hv ydolgr sdud 4  n ? q1

+q  4,q4 

q\ 4 +q  n,n n@4

+q  4,q4  +q  4,$  +q  4,$

Dvð khprv ghprvwudgr

s s s +q  4,  q4 ++q  4,$,5 q  4  q4 +q  4,$

| sru whrupd gh dfrwdplhqwr whqhprv

4 4 3 s s q q q$ 4 olp s @ 3 q$4 q q$ 756

757

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

s s s Hmhpsor 83; Gdgd od vxfhvlöq 5> 5 5> Fdofxodu ho oðplwh gh od vxfhvlöq

t s s 5 5 5>   

Vroxflöq= Srghprv qrwdu txh hvwä vxfhvlöq hvwd ghqlgd sru uhfxuuhqfld sru od vljxlhqwh irupxod= d4 @

s 5

dq @

s 5dq4 =

Xvdqgr hvwd lqirupdflöq srghprv ghprvwudu txh hvwd vxfhvlöq hv fuhflhqwh | dfrwdgd1 Sulphur yhdprv txh hv dfrwdgd/ fodudphqwh 3 ? dq / dghpäv srghprv ghprvwudu sru lqgxfflöq txh dq ? 5/ sdud hoor yhprv orv vljxlhqwhv= l, d4 ? 5 ll, dq4 ? 5 , 5dq4 ? 7 hv ghflu / dq ? 5 Vhjxqgr yhdprv txh od vxfhvlöq hv hvwufwdphqwh fuhflhqwh u s 5dq 5 dq.4 @ @ dq dq dq shur sru od sulphud sduwh ylprv txh dq ? 5/ hv ghflu/ 4 ? d5q / oxhjr u s 5 dq.4 @ A 4@4 dq dq

| frq hoor hqwrqfhv whqhprv txh dq.4 A dq 1 Xvdqgr orv grv uhvxowdgrv dqwhulruhv whqhprv txh od vxfhvlöq hv frqyhujhqwh/ dvð/ vhd O @ olpq$4 dq A 3 O5

Sru or wdqwr olpq$4 dq @ 51

olp d  d q$4 q q @ olp 5dq4 q$4 @ 5 olp dq4 q$4 @

@ 5O

s s s Hmhuflflr 83< Gdgd od vxfhvlöq 5> 5 . 5> Fdofxodu ho oðplwh gh od vxfhvlöq

Hmhuflflr 843 Ghprvwudu orv vljxlhqwhv oðplwhv

t s s 5 . 5 . 5>   

41 olpq$4 5qq @ 31

51 olpq$4 5q$ @ 31 q

n 61 olpq$4 qdq @ 3/ frq d A 41 q 71 olpq$4 dq$ @ 31 81 olpq$4 qqq$ @ 31 91 olpq$4 orjdq+q, @ 3/ frq d A 41

Hmhuflflr 844 Ghprvwudu txh orv vljxlhqwhv oðplwhv qr h{lvwh1 41 olpq$4 vhqq1

q frv+q@5,1 51 olpq$4 4 . q.4

758 61 olpq$4 vhf+q@5,1 71 olpq$4 + +q4, . q



4.+

4,

q

5

1

Hmhpsor 845 Fdofxodu olpq$4 Vroxflöq=

olpq$4 Yhdprv sulphur olpq$4

t 7

s7 q6 q vhq q .

+

q.4

+

$,,

s7 q6 q vhq q .

+

q.4

+

s

7 6 +vhq +q$,, @ olpq$4 q .qq.4  44@q @q t4 4 7 q . q6 +vhq +q$,, @ olpq$4 4 4. q

$,,

4 4 4 q . q6 sdud hoor uhfrughprv txh olpq$4 qs @ 3/ oxhjr

4 4 olp . 6 @3 q$4 q q

| sru or wdqwr

olp q$4

u 7

4 4 @ 3> . q q6

dghpäv sru od surslhgdg gho fhur dqltxlod whqhprv u 4 7 4 olp . vhq+q$, @ 3> q$4 q q6 sru rwur odgr whqhprv txh

4 olp +4 . , @ 4= q$4 q

dvð olpq$4

s7 q6 q vhq q .

+

q.4

+

$,,

@ olpq$4 @ 34 =

t4 7

q . q6 +vhq +q$,, 4

4.

4

q

s s s s q Hmhpsor 846 Fdofxodu olpq$4 + 6 7 6 ; 6    5 6,1

Vroxflöq= Sdud srghu fdofxodu hvwh oðplwh/ qhfhvlwdprv xvdu od surslhgdg gh srwhqfld1 s s s s 4 4 4 q olpq$4 + 6 7 6 ; 6    5 6, @ olpq$4 6+5 . 7 .. 5q, @ olp  q$4 6



4 +4@5,q.4 4 +4@5,

 olpq$4  @ 6 4 @ 6+ 4+4@5, ,



4 +4@5,q.4 4 +4@5,

@ 65 @ <=



Qrwhprv txh od surslhgdg ixh xvdgd sdud xqd h{srqhqfldo gh edvh frqvwdqwh1

Hmhpsor 847 Fdofxodu Vroxflöq= Vhd

{ { { olp frv+ , frv+ ,    frv+ q ,= q$4 5 7 5

{ { { D @ olp frv+ , frv+ ,    frv+ q ,= q$4 5 7 5

759

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV Uhfrughprv odv vljxlhqwhv lghqwlgdghv wuljrqrphwulfdv 5 frv  frv  @ frv+ . , . frv+  ,> 5vhq frv  @ vhq+ . , . vhq+  ,>

dvð hqwrqfhv whqhprv

{ 4 { frv+ , frv+ , @ 5 7 5

sru or wdqwr

frv+ {5 , frv+ {7 , frv+ {; , @ @ @

Sru lqgxfflöq srghprv ghprvwudu txh=

  6{ { frv+ , . frv+ , > 7 7

 6{ { , frv+ { , 7 ;  frv+ 67{ , . frv+ frv+ 7 , frv+ {; ,  frv {7 frv+ {; , 5   4 frv+ :;{ , . frv+ 8;{ , . frv+ 6;{ , . frv+ {; , > 7 4 5 4

q4

5 [4 { 4 +5n . 4,{ { { frv+ , frv+ ,    frv+ q , @ q4 frv = 5 7 5 5 5q n@3

Yroydprv do hmhuflflr ruljlqdo

D @ olpq$4 frv+ {5 , frv+ {7 ,    frv+ 5{q , S5q4 4 {, @ olpq$4 5q44 n@3 frv+ +5n5.4, q q , S5q4 4 vhq + {@ 5 {, @ olpq$4 5q44 vhq +{@5q , n@3 frv+ +5n5.4, q S5q4 4 { ,vhq+ { , frv+ +5n5.4, @ olpq$4 5q4 vhq4 +{@5q , n@3 q 5q q  4 S 5  4 +5n5,{ 4 5n{ @ olpq$4 5q vhq +{@5q , n@3 +vhq+ 5q ,  vhq+ 5q ,, q @ olpq$4 5q vhq 4+{@5q , +vhq+ +5 5q5,{ ,  vhq+ 55q{ ,, q4 4,{@5q4 , q4 , +{@5 . vhq @ olpq$4 vhq ++55q vhq+{@ 5q , 5q vhq +{@5q , q4 q4 q q4 , q 5 +{@5 5 @ olpq$4 vhq ++5 {4,{@5 ,  vhq{@ . vhq{@  vhq{@  +{@5q , 5q4 +{@5q , @ vhq{ { =

Hmhuflflr 848 Fdofxodu orv vljxlhqwhv oðplwhv

q 41 olpq$4 433333 q5 .4 1

6 q.8 1 51 olpq$4 6q5q.7 6 .9 s s 61 olpq$4 + q . 4  q,1 s s 71 olpq$4 + 6 q . 4  6 q,1 s s 81 olpq$4 + 7 q5 . 4  q,1

91 olpq$4 :1 olpq$4

;1 olpq$4 <1 olpq$4

s6 q5 vhq q +

q.4

q.4q

+



+

q .4 q q1

5 5

5,

5,



$,

1

q q.4 1

.6 .6

d d5 ..dq frq +mdm ? 4> mem ? 4,1 e e5 ..eq q4 431 olpq$4 + q4  q5 . q6     . +4,q q ,1 4. .

4. .

q4 4

5

75: 5 5 5 4, 441 olpq$4 + q4 5 . q5 5 .    . +q q5 ,1 5 5 5 451 olpq$4 + q4 6 . q6 6 .    . +5qq64, ,1 4 461 olpq$4 + 45 . 565 . 586 .    . 5q5 q ,1

471 olpq$4 + sq45 .4 . sq45 .5 . sq45 .6 .    . sq45 .q ,1

481 olpq$4 + 445 . 546 .    . q+q4.4 , s s s s q 491 olpq$4 + 5 7 5 ; 5    5 5, 4:1 olpq$4

4;1 olpq$4 4<1 olpq$4

q, 1

5.6vhq +

t

t 6

q

7.

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91 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

m{  4m6 +m{  5m,8 A 3 hv 111111111111111 m{5 . 5m

:1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

+{5 . :{ . 43,+{5  6{ . 7, ? 3 hv 11111111111111111111 m{ . 6m

s{ s {

8 .4

5

5

hv 51

778 ;1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq u

6 4  hv 111111111111111111111 {  9{ . < 7 5

<1 Ho frqmxqwr vroxflöq gh od lqhfxdflöq u

4  4 hv 1111111111111 { . 7{ . 7 5

431 Vhdq d 5 U>  5 U. ho frqmxqwr i{ 5 U m m{  dm ? j hv ho lqwhuydor 11111111111111111 441 Vl { 5 U. > | 5 11111111111111111/} 5 U hqwrqfhv vh yhulfd

451 Od sursrvlflöq hv yhugdghud vvl  5111111111111111

} }  {| {

+ . 4,{5  7{ . +  4, A 3

461 Odv vroxflrqhv gho vlvwhpd

{ | } 5{5 . 7| 5 . } 5

vrq 1111111111

@ 5{ @ 7| @ } @ ;

471 Grv wxehuldv wdugdq 9 krudv hq oohqdu xqd slvflqd1 Xqd vöod od oohqduðd 8 krudv päv gh sulvd txh od rwud hqwrqfhv fdgd wxehuld wdugduðd lqglylgxdophqwh hq oohqdu od slvflqd 111111111111111 | 1111111111111111 uhvshfwlydphqwh LLL10 Ghvduuroor

41 Vl ho udglr gho flolqgur uhfwr glvplqx|h hq xq 43( plhqwudv txh vx dowxud dxphqwd hq xq 45(1 hq txh wdqwr sru flhqwr yduðd> d, ho yroxphq gho flolqgur

Yf @ u5 k Df @ uk

e, ho äuhd odwhudo gho flolqgur1

51 Uhvroyhu

Fdsðwxor 5

{5  | 5 @ 5+{|  6, 5+6  {|, @ +{5  | 5 , 5 6{ |  9{  | 6 @ 3

L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh 41 Vl orj{ +8, ? 4 hqwrqfhv vh wlhqh txh 8 ? {1

51 Ho frqmxqwr vroxflöq gh od lqhfxdflöq

orj8 {  4 hv ^6> 4^ orj8 +4@6,

779

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

s 61 5 orj 5 . +4 . 54{ , orj 6  orj + { 6 . 5:, @ 3/ od hfxdflöq dqwhulru qr wlhqh vroxflöq srvlwlyd

71 Vl orjd e @ 6 | orjd f @ 5 hqwrqfhv orjf e5 @ 6 | ef @ d8 81 i{ 5 Um

{, A 4j @ i{ 5 Um 4 A orj5 +{,j

4 orj5 +

91 Sdud wrgr d> e 5 U.  i4j vh wlhqh orje +d5 ,  orje +d6 ,1 s :1 vxsi{ 5 U@{7 . {5  9 @ 3j @ 5 ;1 +;d 5 U. , +orj+d, ? orj+d5 ,, <1 Od lqhfxdflöq

orj4@5 +d{5 . 4, ? orj4@5 +d,

wlhqh frpr frqmxqwr vroxflöq wrgrv orv uhdohv vvl d 5`3> 4^

431 +;e 5 U.  i4j, +orje +{,  4 , {  e,

441 Vl i hv lq|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 451 Vhd i +{, @ {5  4 > j+{, @ { . 4/ hqwrqfhv ho Grp+i  j, @ U 461 +;d 5 U. , +orj+d, ? orj+d6 ,,

471 +;e 5 U.  i4j, +orje + {4 ,  4 , 4  {e,

481 Vl i hv el|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 491 Vhd i +{, @ {5 . 4> j+{, @ {  4/ hqwrqfhv ho Grp+i  j, @ U

4:1 Ho ydoru pðqlpr gh od ixqflöq i+{, @ {+{  6, hv <@7 4;1 Vhd i = G {

$ ^6> 4^ :$ h{  4

hqwrqfhv ho grplqlr pä{lpr gh i hv ^ oq+7,> 4^ s 4<1 Vhdq i+{, @ 5  { > j+{, @ oq+{5 . <, ixqflrqhv hqwrqfhv ho grplqlr pä{lpr gh od ixqflöq j  i hv `  4> 44^1

531 Vl i hv xqd ixqflöq lq|hfwlyd hqwrqfhv i hv hvwulfwdphqwh ghfuhflhqwh

541 Od vroxflöq gh od lqhfxdflöq orj{ +{ . 4@5, ? 4 hv ydflr1 551 Vl i+{, @ oq+{ . 7, | j+{, @ h{ . 6 vrq ixqflrqhv hqwrqfhv +j  i ,+{, @ { . : | vx grplqlr hv U

561 Vl i | j vrq ixqflrqhv el|hfwlydv hqwrqfhv ho surgxfwr i  j hv el|hfwlyd 571 Vl  A 3/ od ixqflöq i +{, @ oq+{5 . 4, wlhqh pðqlpr devroxwr1 LL Frpsohwdflöq=

41 Od uhfwd | @ 8{ . e qr lqwhuvhfwd d od sduäerod | @ 5{5  5{ . 6 vl e 51111111111111

51 Od sduäerod {5  5{  d|  5d . | @ 6/ frq fhqwur hq +4> 5, vh deuh kdfld duuled vl | vöor vl d 51111111111111

61 Od hfxdflöq gh od uhfwd txh sdvd sru ho fhqwur gh od holsvh 5{5  7{ . 6|5 . 9| @ 7 | hv sdudohod d od uhfwd 5{ . 6| @ 4 hv 11111111111111 71 Od uhodflöq {5  | 5  7{ . 5| . 7 @ 3 uhsuhvhqwd od föqlfd 1111111111111 frq yìuwlfh hq 11111111111111

77: 81 Vhd DEFG ho uhfwäqjxor frq odgrv sdudohorv d orv hmhv lqvfulwr hq od holsvh 49{5 . 7|5 @ 971 Hqwrqfhv od ixqflöq txh uhsuhvhqwd ho äuhd gho uhfwäqjxor hq ixqflöq gh { hv= 11111111111111111

{

91 Od uhodflöq {5  |5 . 7{  5| . 7 @ 3 uhsuhvhqwd od föqlfd 111111111111111 frq fhqwur hq1111111111111

:1 Sdud d 511111111| e 5 111111111111 od ixqflöq +sduäerod, i+{, @ {5 . d{ . e hv ghfuhflhqwh hq ^5> 4^ s ;1 Ho grplqlr gh od ixqflöq i +{, @ orj+{5  9,  4 hv 1111111111111 s <1 Ho grplqlr gh od ixqflöq i +{, @ orj+{5  4,  4 hv 1111111111111111

431 Od vroxflöq gh od lqhfxdflöq

orj4@6 +m{m  5,  orj< +6, orj4@6 7

hv 111111111111111

441 Vl od ixqflöq i vh ghqh frpr i = `  5> 4^i7j $ { :$

hqwrqfhv ho uhfruulgr gh od ixqflöq i hv 1111111111111

U

m{m 4

7

451 Vhd i = G $ ^h5 > h4 ^> i +{, @ + 5:4h ,6{4 hqwrqfhv ho grplqlr pä{lpr gh i hv 111111111111

461 Vhd i +{, @ orj +{, | j+{, @ {5 . e{  e ixqflrqhv/ hqwrqfhv ho frqmxqwr gh wrgrv orv e 5 U wdo txh ho grplqlr gh i  j hv ydflr hv 11111111111111111 s 471 Vhd i+{, @ orj++{  6,5 , | j+{, @ 7  {/ hqwrqfhv ho grplqlr pä{lpr gh j  i hv 111111111 481 Ho grplqlr gh od ixqflöq hv 11111111111111111111 491 Frqvlghuh od ixqflöq i +{, @ e 5 111111111111111



s s 4  4  {5 i +{, @ {5

d{5 . 4 {  3 hqwrqfhv i hv lq|hfwlyd vl | vöor vl d 5 11111111111111 | {5 . e { ? 3

4:1 Od uhodflöq |5 @ 5  { ghqh xqd ixqflöq el|hfwlyd hq 11111111111111111   5 { >{ ? 3 s{ > { ? 4 hqwrqfhv j  i +{, hv 11111111111111111111 4;1 Vhd i+{, @ | j+{, @ 5{ > {  3 { >{  4 s 4<1 Ho frqmxqwr gh orv sxqwrv { wdo txh i+{, @ +{ . 5,+{  6, wlhqh xq ydoru pðqlpr hv 1111111111111111

77;

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

531 Vhd i = `3> 4^i4j $ { :$

hqwrqfhv ho uhfruulgr hv 111111111111111 U . 6 {4 4

541 Vhd i = U  i4j $ U  i4j xqd ixqflöq el|hfwlyd hqwrqfhv i 4 +{, hv ljxdo d 1111111111111111 {.4 { :$ 4{

551 od ixqflöq i+{, @ 5{  m{m ghqlgd hq ^5> 8` wlhqh xq pä{lpr devroxwr hq { @11111111111 | xq ydoru pðqlpr devroxwr ljxdo d 11111111111

561 Od ixqflöq txh uhsuhvhqwd ho yroxphq +hq ixqflöq gh k, gho flolqgur flufxodu uhfwr lqvfulwr hq xq frqr gh 53fp gh dowxud | 8fp gh udglr/ hv i +k, @1111111111111 | vx grplqlr hv 11111111111111111

571 Ho grplqlr gh od ixqflöq i+{> |, @

{|, hv ho frqmxqwr 1111111111111 {.| | od juäfd gh hvwh frqmxqwr hv 11111111111

581 Vl i+{, @ h

oq+4

s {5 1 4

+d, i hv hvwulfwdphqwh fuhflhqwh hq 111111111111111

+e, i hv hvwulfwdphqwh ghfuhflhqwh hq 111111111111111 +f, i dofdq}d vx ydoru pä{lpr hq 11111111111111

+g, Ho ydoru pðqlpr gh i hv 111111111111111

LLL10 Ghvduuroor

41 Od rulood gh xqd slvflqd irupd xq uhfwäqjxor gh 73 slhv gh odujr | 53 slhv gh dqfkr1 Vx surixqglgdg dxphqwd xqliruphphqwh gh 6 d : slhv hq xq wudpr krul}rqwdo gh 57 slhv | ghvsxìv frqwlqýd do plvpr qlyho orv uhvwdqwhv 48 slhv/ frpr vh loxvwud hq od jxud/ od fxdo pxhvwud xqd vhfflöq wudqvyhuvdo1 Vl od slvflqd vh hvwä ydfldqgr/ dofdq}dqgr xq qlyho k hq ho odgr päv surixqgr1 Ghwhuplqh ho yroxphq Y +k, gho djxd hq ixqflöq gh od dowxud k/ hvshflfdqgr vx grplqlr1

51 Od frpsdôld gh duulhqgr gh dxwrv Mlphqh}%/ duulhqgd fkhyhwwh d ' 431833 sru gðd | 73 shvrv sru nloöphwur uhfruulgr/ plhqwudv txh od frpsdôld Dudqfleld% duulhqgd ho plvpr wlsr gh dxwr d ' ;<33 sru gðd | d 93 shvrv sru nloöphwur1 +d, Vl { uhsuhvhqwd ho qýphur gh nlorphwurv uhfruulgr hq xq gðd/ hqfxhqwuh od ixqflöq FD +{,txh uhsuhvhqwd ho frvwr sru duuhqgdu xq fkhyhwwh hq od frpsdôld Dudqfleld% hq hvh gðd1

+e, Fxdqwrv nlorphwurv vh gheh uhfruuhu hq ho gðd sdud txh ph frqyhqjd duuhqgdu ho dxwr gh od frpsdôld Mlphqh}%1

61 Uhvroyhu od vljxlhqwh lqhfxdflöq

71 Ghwhuplqh ho grplqlr gh od ixqflöq

Fdsðwxor 6

orj6{ +<{, . orj6 +{6 ,  5 i = G  U $ ^5> 4^ m { :$ m{{.5 4

L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh

q 41 Vl olpq$4 dq @ O hqwrqfhv olpq$4 dq vhq q h{lvwh1

51 Vl idq j hv xqd vxfhvlöq ghfuhflhqwh hqwrqfhv olpq$4 dq h{lvwh

77< 61 Od vxfhvlöq idq j hv frqyhujhqwh grqgh  6 q > q  5333 d, dq @ 4 q > q A 5333

e, dq @



q6

s s s 71 Od vxfhvlöq idq j @ i+ q . 4  q, q . 6j frqyhujh d 4@5

4

q

> q A 5333 > q  5333

LL Frpsohwdflöq=

41 Ho ydoru gho oðplwh

olp q$4

hv 11111111111111111

51 Vhd dq @ 6q {5q hqwrqfhv q

vvl { 5 111111111111111

61 Ho ydoru gho oðplwh hv 71 Ho oðplwh



q5 q.4

q

6q . s 7 49q7 . 8

   dq.4  ?4 olp  q$4 dq 

olp q$4



 5q 5q . frv+q$, s hv 111111111111111 . 6 5q.4 . 4 q6 . 8

olp q$4

81 Ho ydoru gho oðplwh olpq$4

vhq +q,.+

91 Ho ydoru gho oðplwh olpq$4

6



4,

+5,q . frv+q, @ ============== 6q . 7

q .86q.4

q.8q

5+q.4,

mvhq

.

q 7.8q

<

q ,m

+<

hv 1111111111111111111

hv 1111111111111111111

5q 4 / hqwrqfhv ho juäfr gh i hv 1111111111 :1 Vhd i+{, @ olpq$4 {{5q  .4

;1 Ho oðplwh

+ , olp h q6 4 @ ============ q$4

q5 .4

q hv 111111111111 s s 431 Ho oðplwh olpq$4 +q+
vhq +q,

451 Ho ydoru gho oðplwh vljxlhqwh hv 111111111111111

s 6 ;q6 . 5q frv +4@q, olp . q q$4 q 7 .8

LLL10 Ghvduuroor

41 Vhd

i = `4> 4^ $ U s { :$ orj8 + {  4,

j = U  i3j $ { :$

U

{ 4 8{ 8{ 8

783

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV +d, Ghwhuplqh vl i hv el|hfwlyd + vl qr or hv uhvwulqjlu i gh prgr txh or vhd, | hqfxhqwuh i 4 +hvshflfdqgr vx grplqlr,

+e, Ghwhuplqh od ixqflöq j  i +hvshflfdqgr vx grplqlr, +f, Fdofxoh orv vljxlhqwhv oðplwhv +vl h{lvwhq,

olp j+q, q$4

Fdsðwxor 7

olp +j  i,+{, {$5

L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh 41 Vhd i +{, xqd ixqflöq frqwlqxd hq U | i+q, @ q5 / dghpäv/ vhd olp j+{, @ 8> {$+4@5,

hqwrqfhv vh sxhgh ghflu txh 51 Ho ydoru gh olpq$4

5q.4

5



@q, hv 5 1 8

frv+4

q .87q

6

olp +i  j,+{, @ 8= {$+4@5,.

4 61 olpq$4 +q5 . 8q,vhq+ q5 .8 q, @ 4

71 Vhd i | j ixqflrqhv1 Vl olp{$d i+{, h{lvwh | olp{$d j+{, qr h{lvwh hqwrqfhv olp{$d ij++{{,, qr h{lvwh1 81

91 Ho oðplwh olp{$8@5 ^6{` h{lvwh1

:1 Ho oðplwh olp{$3

olp {$6

m{  7m  4 qr h{lvwh 6{

@{, hv 4 +{, hv 4 ;1 Ho oðplwh olp{$ vhq +{, +{ , hv 4 <1 Ho oðplwh olp{$ vhq +{ , vhq +4@{, +4

 / dvl vh wlhqh m{  5m ?  , 431 Sdud ghprvwudu txh olp{$5 {5 @ 7> gdgr  srvlwlyr/ edvwd wrpdu  @ m{.5 m m{5  7m ? 1 441 Vhd



6{ . : {  4 {5 .8{.7 { ? 4 5 { 4 Ghwhuplqh ho ydoru gh yhugdg gh od vljxlhqwh dupdflrqhv i +{, @

+d, olp{$4 i+{, @ 7 +e, olp{$4 i+{, @ 7

+f, i hv frqwlqxd hq wrgr U

+{, hv  451 Ho ydoru gho oðplwh olp{$4 vhq{ 4

461 Ho oðplwh olp{$3 vhq5 {  oq+ {45 , qr h{lvwh1

471 Vhdq i | j ixqflrqhv1

Vl olp{$d i+{, h{lvwh | olp{$d j+{, qr h{lvwh hqwrqfhv olp{$d i +{,j+{, qr h{lvwh1

784 481 olp{$

vhq +6{, vhq +5{,

@ olp{$3

vhq +6{, vhq +5{,

491 olp{$3 vhq+ {4 . , qr h{lvwh1

4:1 olp{$d i | olp{$d j qr h{lvwhq/ hqwrqfhv olp{$d + ij , qr h{lvwh 1

4;1 Ho ydoru gho oðplwh olp{$7 ^{`m{ . 7m hv 31

4<1 Vl oðplwh gh i fxdqgr { $ d h{lvwh | oðplwh gh j fxdqgr { $ d qr h{lvwh/ hqwrqfhv olp{$d +i . j, qr h{lvwh1

531 Wrgdv odv ixqflrqhv vljxlhqwhv vrq frqwlqxdv hq K s +d, i +{, @ {> K @ U.  i4j 5 4 > K @ U  i4j1 +e, i +{, @ {{.4  6 {4 > K @ U. +f, i +{, @ vhq +{, { ?4 { + vhq +4@+{4,, { 9@ 4 4@+{4, 541 Od ixqflöq i +{, @ hv frqwlqxd hq 41  {@4 LL Frpsohwdflöq=

+{, hv 111111111111 41 Ho ydoru gho oðplwh olp{$7 +{vhq 7, frv+{,

51 Ho oðplwh olp{$8

{8,5  vhq 6+{8,5 hv 111111111111 {8, {5 43{.58

+

4.frw5 +

s

d{mdm > d ? 3 hv 111111111111 61 Ho oðplwh olp{$d d{ d5

71 Ho ydoru gho oðplwh olp{$3

{5

{, hv 11111111111111111

4.frw5 +

81 Orv ydoruhv gh D | E wdo txh od ixqflöq

; ?



vhq 5+{

4,

{4 i +{, @ D{5 . 5 = E{ . 8

>{ A 4 >{ @ 4 >{ ? 4

uhvxowd frqwlqxd hq wrgrv orv uhdohv vrq D 5 1111111111111 E 5 11111111

91 Vhd

i+{, @

 {6 .5: {3 {5 < 6 frv+{ . , { A 3

Ghqlgd hq xq grqlplr dsursldgr/ vxefrqmxqwr gh U

+d, Ho pä{lpr grplqlr gh frqwlqxlgdg gh i hv 111111111

+e, i wlhqh glvfrqwlqxlgdg uhsdudeohv hq orv sxqwrv 111111111

+f, Ho frqmxqwr gh orv sxqwrv hq txh i wlhqh glvfrqwlqxlgdg luuhsdudeohv hv 111111111111111

s

em 5 :1 Vhd i+{, @ {{m e5 / frq e ? 3/ hqwrqfhv ho ydoru gho oðplwh gh i +{, fxdqgr { wlhqgh d e hv 111111111111 ; ? 6{5 . 8 { A 4 { . 5 { @ 4 ;1 Vhd i+{, @ Hqwrqfhv orv ydoruhv gh  |  txh shuplwhq txh od ixqflöq i +{, = 46{ .  { ? 4 vhd frqwlqxd hq { @ 4 vrq  @11111111111 |  @ 111111111111

785

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

<1 Orv ydoruhv gh D / E txh shuplwhq txh

; ?

> {A4 5D{ . {5 E+{  4, . 8 > { @ 4 i+{, @ = 8{ . E > {?4

vhd frqwlqxd hq { @ 4 vrq 11111111111111

431 Vhd d 5 U wdo txh

; d+4{5 , ? vhq { i +{, @ = {.d

od ixqflöq i hv frqwlqxd hq { @ 4 vvl d @ ====================

441 Vhdq d> e 5 U wdo txh

i+{, @

; A ? A =

3?{?4 {4

 vhq{+e{, 4 {5 d5 {d

vhq +d{,

{

i vhuä frqwlqxd hq { @ 3 vvl d @ ==========> e @ ===========

{?3 {@3 { A 3> { 9@ d

LLL10 Ghvduuroor ; 4frv+{@6, >{ ? 3 A { A A A ? 5{ . 5d . 4 > { @ 3 grqgh d> e vrq qýphurv uhdohv mrv= 41 i+{, @ A A A A = ds{5 .{6 .e >{ A 3 { +d, Fdofxoh ho oðplwh odwhudo ghuhfkr gh i hq 31

+e, Fdofxoh ho oðplwh odwhudo l}txlhugr gh i hq 31 +f, Fdofxoh orv oðplwh odwhudohv gh i hq 041

+g, Hqfxhqwuh orv ydoruhv gh d | e/ sdud txh i +{, vhd frqwlqxd hq 04 +h, Hqfxhqwuh orv ydoruhv gh d | e/ sdud txh i +{, vhd frqwlqxd hq 3

Fdsðwxor 8

L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh

41 Vl j qr hv ghulydeoh hq d shur frqwlqxd hq d | i hv ghulydeoh hq d hqwrqfhv ho surgxfwr i  j qr hv ghulydeoh hq d1

51 Vhdq  A 3/ od ixqflöq i +{, @ h{ wlhqh vöor xq pä{lpr hq orv uhdohv1 5

61 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd ghqlgd lpsoðflwdphqwh sru hq ho sxqwr +4@5> 3, hv 5{  5| @ 41

| @ dufvhq+{|,

71 Od ixqflöq i+{, @ +{  4,5@6 qr wlhqh pðqlpr s 81 Od ixqflöq i+{, @ {  5 qr wlhqh pä{lpr | qr wlhqh pðqlpr1

91 Vl j+{, @ vhf+8{, . { | i 3 +{, @ 6h5{ hqwrqfhv +i  j,3 +3, @ 6h5

786 :1 Vl od yhorflgdg gh fuhflplhqwr gh od edvh gh xq wuläqjxor hv gh 6 p2vhj | od dowxud 7 p2vhj/ hqwrqfhv od yhorflgdg gh fuhflplhqwr gh vx äuhd hq ho lqvwdqwh txh fxdqgr vx edvh hv gh 45 phwurv | vx dowxud gh 53 phwurv hv gh 87 p5 2vhj

;1 Wrgd ixqflöq dfrwdgd wlhqh h{wuhprv s <1 Od ixqflöq i +{, @ 7 4  { qr wlhqh h{wuhpr1 LL Frpsohwdflöq=

41 Gdgd odv ixqflrqhv i | j ghqlgdv sru  {.4 {?3 i+{, @ {.5 {3

j+{, @

+d, Ho Uhfruulgr gh i hv 111111111111

 s {4 4{8 { . : {A8

+e, Ho Grplqlr gh j  i hv 1111111111111111  +f, +j  i ,+{, @

+g, Od ghulydgd gh od ixqflöq j  i hq ho sxqwr 5 hv 11111111111

51 Vl i +{, @

 s 6

4  d{ {  3 vhq{ . e { A 3

+d, i hv frqwlqxd hq { @ 3 vl | vöor vl d 5 11111111111111 | e 5 111111111111111

+e, i hv ghulydeoh hq { @ 3 vl | vöor vl d 5 11111111111111 | e 5 111111111111111

61 Ho ydoru gho oðplwh

olp +frv {,fvf { hv 11111111111111111111 {$3

71 81 Frqvlghuh j+{, @



olp +{5 . frv +5{,, {$3 vhf { . wdq { 3 ? { ? @5 s {  @5 {  @5

5

{ hv===========

fvf +6 ,

ho ydoru gho oðplwh

olp +h+@5{, wdq+{, . j+{,,hv 111111111 {$@5

91 Vl i 3 +{, @ h  / j+{, @ frv+ 55 , hqwrqfhv +i  j,3 +3, @111111111111111 7{

:1 Vl

hqwrqfhv i 3 +{, @ 111111111111111

{

; 6 { . oq+{, { A 4 A A A A ? 5{ . 4 {@4 i +{, @ A A A A = frv+5{, {?4

6 s s6 {7+{ , hv j3 +{, @11111111111111111111 ;1 Od ghulydgd gh j+{, @ + {,frv { . dufvhq

hqwrqfhv

787

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

<1 Vl

; 5{ frv +{, A A A A ? 6{ i +{, @ A A A A = s 6 { . 7{5

hqwrqfhv i 3 +{, @ 111111111111111

431 Od ghulydgd gh

i +{, @

hv 11111111111111

441 Od ghulydgd gh

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s

5 6 vhf +{,  h{ orj8 +{6 . 4,

i +{, @ vhq+{5 . 4,dufwdq+{.5,

hv 1111111111111111

451 Od ghulydgd gh

i +{, @



vhq+{  5, { A 5 {5 {4

461 Vhd i +{, @ orj6 { hqwrqfhv ho grplqlr gh od ghulydgd gh i +{, hv Grpi 3 @ 111111111111111

gi @ 5{ . 4 | j+{, @ frv5 { . vhq{/ hqwrqfhv ho ydoru gh +i  j,3 +, 471 Vhd i xqd ixqflöq frq ghulydgd g{ hv 11111111111111111111 s 481 Vhd K+{, @ 6 i +hd{ oq+{,,/ frq i +{, xqd ixqflöq ghulydeoh hq U/ hqwrqfhv K 3 +{, hq U. hv 1111111111111111

491 Vhd i +{, @ +{5 . 4,vhq +{, / hqwrqfhv i 3 +{, hv 11111111111  {.d {A3 4:1 Vhd i +{, @ hqwrqfhv ho ydoru d txh kdfh ghulydeoh od ixqflöq i hq @7 hv vhq+{, {  3 111111111111111

4;1 Vhd i +{,@ +{5 . 47,dufvhq +{

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gi m @ hqwrqfhv g{ {@4

s6 gi hv ljxdo d111111111111111 4<1 Vhd i +{, @ +vhq+{7 . 5 ,, { .8 sdud { 5`  @;> @;^ hqwrqfhv g{ 531 Od ghulydgd gh | @ +{5{ ,6{ frq { 5 U.  i4j hv 1111111111 541 Vhd i +{, od ixqflöq ghqlgd sru=

i +{, @

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Od ixqflöq hv ghulydeoh hq  vl orv ydoruhv gh D | E vrq 1111111111111

551 Vhd d 5 U > e> f 5 U

; s ? +{  ,6 . 5d{ f s i+{, @ = frv+ +{7 . 5 ,@5, . e{ d

s { ? s {@  s {A 

s  sdud orv vljxlhqwhv ydoruhv gh d> e> f111111111111111 s +e, i hv ghulydeoh hq  sdud orv vljxlhqwhv ydoruhv gh d> e> f 111111111111111 ; 6 {A3 ? { vhq+{, { @ 3 561 Vhd i +{, @ hqwrqfhv od ghulydgd gh i hv 11111111111111111111111 = {?3 7{5 +d, i hv frqwlqxd hq

788 571 Vhd 6{5 | 5 . wdq+| 6 , @ 4/ hqwrqfhv hq xq frqmxqwr dsursldgr od ghulydgd gh | frq uhvshfwr d { hv= g| g{ @1111111111111

581 Gdgd od uhodflöq vhq+ 7 frv+{6 , . | 5  7, @ gi m hqwrqfhv od ghulydgd gh g{ {@3 hv1111111111

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hood ghqh fhufd gho fhur xqd ixqflöq ghulydeoh i /

591 Vhd K+{, @ {I 4 +{,  frv+I 4 +{,,/ frq I 3 +{, 9@ 3 sdud wrgr { hqwrqfhv od ghulydgd gh K+{, hv K 3 +{, @1111111111111

5:1 Orv odgrv gh xq wuläqjxor gh shuðphwur pðqlpr frq äuhd md D/ dvð frpr wdpelìq md vx edvh e vrq 1111111111111111 5;1 Od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd gh od ixqflöq gdgd sdudphwulfdphqwh sru  { @ 6+w  4,5 . 8 | @ 9w6  vhq+w, > w 5 U hq ho sxqwr +;> 3, hv 1111111111111111

5<1 Od ixqflöq i +{, @ +{5  4,7 hv fuhflhqwh hq ho vljxlhqwh frqmxqwrv 1111111111111111

631 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd { oq | . | 5 @ vhq+{|, hq +@5> 4, hv 11111111111111111 s 641 Od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd gh i+{, @ { . 5 hq ho sxqwr +4> 6, hv1111111111111

651 Od ghulydgd gh xqd ixqflöq i +{, hv i 3 +{, @ 6{ . :vhq+{  4,+{  5, . ; | i +4, @ 5> i+5, @ 6 hqwrqfhv +i 4 ,3 +5, @111111111111111

661 Vhd i +{, @ vhq+6{, frv+5{, . 4/ frq { 5`  @;> @;^/ hqwrqfhv od ghulydgd gh od ixqflöq lqyhuvd hq ho 4 sxqwr 4/ hv ghflu/ gig{ +4, hv 1111111111111111

671 vhd i +{, @ 5vhq+{, frv+{,> { 5`@7> 6@7^ hqwrqfhv od ghulydgd gh od ixqflöq lqyhuvd hq ho sxqwr 4 3/+ gig{ +3,, hv 111111111111111111 681 Ho oðplwh

hv 111111111111111111

4 olp + ,wdq+{, {$3. {

691 Ho ydoru gho oðplwh olp{$3 +4 . 6{,+4@5{, hv 1111111111

6:1 Ho ydoru gho oðplwh olp{$4 +4  {,frv+{@5, hv 11111111111111

6;1 Gdgd od fxuyd {| . | 6 {5 @ 4 hq irupd lpsoðflwd1 Fxdqwrv sxqwrv gh od ixqflöq | @ i +{, wlhqh uhfwd wdqjhqwh sdudohod d hmh { 111111111111111

6<1 Xq mduglq hv glvhôdgr frq od irupd gh xq vhfwru flufxodu gh äqjxor  | udglr u frpr hq od jxud1 Ho mduglqhur fxhqwd frq vhploodv vhpeudu Dp5 1 Sdud ghwhuplqdu odv glphqvlrqhv gho mdugðq gh wdo irupd txh vh jdvwh or phqrv srvleoh hq fhufdor hqwrqfhv od ixqflöq d pðqlpl}du hq od yduldeoh u hv 111111111111111 731 Ho uhfwäqjxor DEFG gh äuhd pä{lpd lqvfulwr hqwuh od juäfd gh | @ {6 / ho hmh R[ +xq odgr vreuh hvwh hmh, | od uhfwd { @ 5 wlhqh äuhd ljxdo d 11111111111111

741 Ho uhfwäqjxor DEFG gh äuhd pä{lpd lqvfulwr hqwuh od sduäerod | @ 7{5 / ho hmh R[ +xqr gh orv odgrv hq hvwh hmh, | od uhfwd { @ 9 wlhqh äuhd ljxdo d 111111111111111

751 Vhd s+{, @ {6 . d{5 . e{ . f xq srolqrplr frq frhflhqwhv d> e| f/ hqwrqfhv s+{, wlhqh xq pä{lpr orfdo hq { @ 4 | xq pðqlpr orfdo hq { @ 6 vl vxv frhflhqwhv vrq ljxdohv d 11111111111111111111

789

DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV

761 Grv wuhqhv vh dohmdq gh xq fuxfh txh irupd xq äqjxor gh <3 d yhorflgdghv gh 493np@k | 453np@k1 Fxdqgr ho sulphu wuhq hvwd d ;3np gho fuxfh hqwrqfhv od yhorflgdg frq txh vh hvwdq dohmdqgr orv wuhqhv hv gh 111111111111111 5,vhq { vrq +hv, 1111111111111111 771 Odv dvðqwrwdv yhuwlfdohv gh i +{, @ {+{6 .{5 9{

781 Orv sxqwrv gh lq h{löq gh | @ oq+ { {.4 5 , vrq 111111111 5

791 Od juäfd gh i+{, @ {5 wlhqh lqqlwdv uhfwdv wdqjhqwhv +xqd sru fdgd sxqwr,1 Grv gh hvwd uhfwdv wdqjhqwh sdvd sru ho sxqwr +4> 3,/ xqd gh hood hv od wdqjhqwh d rwur sxqwr1 Ho sxqwr hv 11111111111 7:1 Od juäfd gh od ixqflöq i +{, @ +{5  4,7 wlhqh sxqwrv gh lq h{löq hq 1111111111111

7;1 Od juäfd gh i +{, @ +{5  4,7 hv 11111111111111111111

5 {5 .4, hqwrqfhv 5{ 7<1 Vhd i +{, @ {5{4 > i 3 +{, @ +{ i%+{, @ 5+6 5 4,5 > +{5 4,6

+d, i hv fuhflhqwh hq 111111111111111

+e, i hv frqfdyd +kdfld duuled, hq 1111111111111 +f, i wlhqh dvðqwrwd yhuwlfdo hq 111111111111

+g, i wlhqh dvðqwrwd krul}rqwdo hq 1111111111111 831 Odv dvðqwrwdv gh | @

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vrq 1111111111

841 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd | @ i+{, ghqlgd lpsolflwdphqwh sru od uhodflöq s wdq+{|7 ,  5 oq+ |, @ 3

hq +3> 4, hv 1111111111

851 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd +5{ . 6|,5  +6{  5|,5 @ 57 hq +4> 4, hv 11111111111111111 861 Od ixqflöq i+{, @ { hv hvwulfwdphqwh fuhflhqwh hq 11111111111111111

871 Od ixqflöq | @ +vhq{,5 . { ghqlgd hq ^3> ` wlhqh xq pä{lpr hq { @ 111111111111111111111

881 Od ixqflöq i+{, @ {5 h{ hv hvwulfwdphqw ghfuhflhqwh vöor hq ho lqwhuydor 1111111111111

891 Od yhorflgdg gh ghvydorul}dflöq gh xq htxlsr sru ho xvr gho plvpr hv sursruflrqdo hq fdgd prphqwr d vx frvwr uhdo |1 Od h{suhvlöq txh uhsuhvhqwd ho hqxqfldgr dqwhulru grqgh n hv xqd frqvwdqwh srvlwlyd | w hv ho wlhpsr hv 1111111111111 LLL10 Ghvduuroor

41 Ghulyh

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s 6 {5 wdq+{5 ,

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61 Xwlolfh od hfxdflöq gho Rydor gh Fdvvlql1

g| +d, Hqfxhqwuh g{

lll, | @ orj{5 +6{,

6{ . dufvlq +frv +{5 ,, s wdq +frv :{, . {

+{5 . |5 . 7,5  49{5 @ 69>

78: +e, Bc Vh s sxhgh hqfrqwudu odv hfxdflrqhv gh od uhfwd wdqjhqwh do rydor gh Fdvvlql hq orv sxqwrv +4> 5, | +5> 5,B +f, Hqfxhqwuh od uhfwd wdqjhqwh dq orv sxqwrv dqwhulruhv hq orv txh vh sxhgdq1

71 Vhdq  5 U> e 5 U1

Ghwhuplqdu vl od ixqflöq | @ hd{ . {hd{ hv vroxflöq gh od hfxdflöq glihuhqfldo g5 | g| . d5 | @ 3>  5d g{5 g{

d frqvwdqwh

81 Vh ghvhd frqvwuxlu xq wdqtxh gh dfhur frq od irupd gh xq flolqgur flufxodu uhfwr | vhplhvihudv hq orv h{wuhprv +yhu od jxud, sdud dopdfhqdu jdv sursdqr1 Ho frvwr sru slh fxdgudgr gh orv h{wuhprv hv ho greoh gho gh od sduwh floðqgulfd/ grqgh ho frvwr gho phwur fxdgudgr gho flolqgur hv S 1 BcTxì glphqvlrqhv plqlpl}dq ho frvwr vl od fdsdflgdg ghvhdgd hv gh 43 slh6 B Dh @ 7u5 D|xgd= Yf @ u5 k Yh @ 76 u6 Df @ 5uk 91 Xq fdpshvlqr wlhqh glqhur sdud frpsudu 4833 phwurv gh dodpeuh gh sxdv/ wdo fdpshvlqr ghvhd fhufdu frq 8 kheudv gh dodpeuhv xq whuuhqr uhfwäqjxodu hq vx suhglr/ hq fdvr txh { uhsuhvhqwh xq odgr gho uhfwäqjxor/ ghwhuplqh od ixqflöq D+{, äuhd gho whuuhqr fhufdgr frq hvwh dodpeuh hq whuplqrv gh {1 Phgldqwh/ od juäfd gh od ixqflöq D+{, hqfxhqwuh odv glphqvlrqhv gho whuuhqr frq pä{lpd äuhd

s

:1 Vh ghvhd frqvwuxlu xq hvwdqtxh hq irupd föqlfd +uhfwr, frq fdsdflgdg sdud ;3336 5 olwurv1 vl 4 olwurv s gh lpshuphdelol}dqwh fxeuh  6p5 gh vxshuflh1 BcFxdohv ghehq vhu odv glphqvlrqhv gho frqr sdud txh vh xvh xqd fdqwlgdg pðqlpd gh lpshuphdelol}dqwh B BcFxäqwrv olwurv vh ghehq frpsudu sdud slqwdu wdo frqrB s D|xgd= Vxshuflh gh xq frqr gh udglr u | dowxud k hv u u5 . k5 | vx yroxphq hv 46 u5 k1 Fxlgdgr frq odv xqlgdghv

;1 Grv iäeulfdv D | E txh vh hqfxhqwudq d 7 ploodv xqd gh od rwud/ hplwhq kxpr frq sduwðfxodv txh frqwdplqdq ho dluh gh od uhjlöq 1 Vxsrqjd txh ho qýphur gh sduwðfxodv suryhqlhqwh gh fdgd iäeulfd hv gluhfwdphqwh sursruflrqdo d od fdqwlgdg gh kxpr h lqyhuvdphqwh sursruflrqdo do fxer gh od glvwdqfld ghvgh od iäeulfd1 Bc Txì sxqwr hqwuh D | E whqguä od phqru frqwdplqdflöq vl od iäeulfd D hplwh ho greoh gh kxpr txh od iäeulfd E B <1 Fdofxodu odv glphqvlrqhv gho frqr lqyhuwlgr +gh prgr txh orv hmhv frlqflghq, gh yroxphq pä{lpr txh vh sxhgh lqvfulelu hq xq frqr gh dowxud d fp | udglr e fp1 Fxäo hv ho yroxphq pä{lpr1

431 Xqd hvfdohud gh 53 slhv gh odujr hvw dsr|dgd frqwud xqd pxudood1 Ho slh gh od hvfdohud hv duudvwudgr/ dohmäqgrod gh od pxudood d ud}öq gh 5 slhv sru vhjxqgr BcD txì yhorflgdg vh ghvol}d kdfld dedmr ho h{wuhpr vxshulru gh od hvfdohud hq ho lqvwdqwh txh ho slh gh hood/ hvwä d 45 slhv gh od pxudood B

441 Frqvlghuh i+{, @ +4 . {4 ,{ +d, Ghwhuplqh Grpi

+e, Fdofxoh olp{$3. i +{,

+f, Fdofxoh p @ olp{$4 i +{{, +D|xgd = Fdofxoh olpx$3. x  i+4@x,

+g, Fdofxoh e @ olp{$4 i+{,

+h, Bc Txì sxhgh ghflu gh orv oðplwhv +6, | +7, fxdqgr { $ 4B +vrq ljxdohv r glvwlqwrv,1

+i, Judtxh | @ i +{,/ frqvlghudqgr od lqirupdflöq rewhqlgd dqwhulruphqwh | vdelhqgr txh odv juäfdv gh i 3 +{, | i 33 +{, vrq

451 Vhd i od ixqflöq frqwlqxd `0:/53` vdoyr hq orv sxqwrv 06 | 43 | txh vdwlvidfh or vljxlhqwh

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461 Frqvlghuh/ sdud ho dqäolvlv | od juäfd gh od ixqflöq i {4,6 > i 3 +{, @ +{4,5 +{7, | i 33 +{, @ 45+{4, 1 i+{, @ +5+{ 5,5 +{5,6 +{5,7 Lqglfdu hq od wdeod vljxlhqwh= +d, +e, +f, +g,

Ghwhuplqh ho Grpi Orv ydoruhv gh { grqgh od juäfd fdpeld gh frpsruwdplhqwr1 Orv vhfwruhv gh fuhflplhqwr | ghfuhflplhqwr orv vhfwruhv grqgh od juäfd hv frqfdyd kdfld duuled | grqgh hv frqfdyd kdfld dedmr1 Sxqwrv Lpsruwdqwh Vljqr gh i 3 +{, Vljqr gh i 33 +{, Fuhflplhqwr | Ghfuhflplhqwr Frqfdyd | frqyh{d

+h, Odv dvðqwrwdv yhuwlfdohv vrq +i, Odv dvðqwrwdv reolfxdv vrq +j, Judtxh

471 Frqvlghuh/ sdud ho dqäolvlv | od juäfd gh od ixqflöq i/ {6 > i 3 +{, @ {5 +{6, | i 33 +{, @ 9{ 1 i+{, @ +{ 4,5 +{4,6 +{4,7 Lqglfdu hq od wdeod vljxlhqwh= +d, +e, +f, +g,

Ghwhuplqh ho Grpi Orv ydoruhv gh { grqgh od juäfd fdpeld gh frpsruwdplhqwr1 Orv vhfwruhv gh fuhflplhqwr | ghfuhflplhqwr orv vhfwruhv grqgh od juäfd hv frqfdyd kdfld duuled | grqgh hv frqfdyd kdfld dedmr1 Sxqwrv Lpsruwdqwh Vljqr gh i 3 +{, Vljqr gh i 33 +{, Fuhflplhqwr | Ghfuhflplhqwr Frqfdyd | frqyh{d

78< +h, Odv dvðqwrwdv yhuwlfdohv vrq +i, Odv dvðqwrwdv reolfxdv vrq

+j, Judtxh