Wkìrulh hw sudwltxh ghv lqvwuxphqwv qdqflhuv - Maths-Fi

Transcription

Wkìrulh hw sudwltxh ghv lqvwuxphqwv
qdqflhuv
Dqwrlqh Iudfkrw
Hfroh Sro|whfkqltxh Mdqylhu 5334
Wdeoh ghv pdwlëuhv
4 Ohv surgxlwv
414 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
415 Ohv dfwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41514 Jìqìudolwìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41515 Prgìolvdwlrq ghv sul{ g*dfwlrqv 1 1 1
41516 Frwdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41517 Sul{ g*dfwlrqv hw wkìrulh ìfrqrpltxh
416 Ohv reoljdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41614 Jìqìudolwìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41615 Fdofxo ghv x{ 1 1 1 1 1 1 1 1 1 1 1 1
41616 Txhotxhv h{hpsohv 1 1 1 1 1 1 1 1 1 1
41617 Prgìolvdwlrq ghv sul{ g*reoljdwlrq 1
41618 Wdx{ g*lqwìuíw hw wkìrulh ìfrqrpltxh
41619 Hqfruh txhotxhv qrwlrqv srxu qlu 1
417 Ohv pdwlëuhv suhplëuhv 1 1 1 1 1 1 1 1 1 1 1 1
418 Ohv wdx{ gh fkdqjh 1 1 1 1 1 1 1 1 1 1 1 1 1 1
41814 Prgìolvdwlrq ghv wdx{ gh fkdqjh 1 1
41815 Lqwhusuìwdwlrq ìfrqrpltxh 1 1 1 1 1 1
419 Txhotxhv frqfoxvlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1
41: Uìiìuhqfhv eleolrjudskltxhv 1 1 1 1 1 1 1 1 1
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5 Surgxlwv gìulyìv dfwlrq
514 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
515 Ohv rswlrqv gh edvh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51514 Gìqlwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51515 H{hpsohv gh frqwudw g*rswlrq 0 PRQHS 1 1 1 1
51516 Irupdwlrq ghv sul{ g*rswlrq 1 1 1 1 1 1 1 1 1 1 1
516 G*dxwuhv rswlrqv hw vwudwìjlhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51614 Elqdu| rx gljlwdo rswlrqv 1 1 1 1 1 1 1 1 1 1 1 1
51615 Exoo hw Ehdu vsuhdgv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51616 Vwudggohv hw Vwudqjohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51617 Exwwhu | hw frqgruv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51618 Fdohqgdu vsuhdgv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51619 Zduudqw/ erq gh vrxvfulswlrq 1 1 1 1 1 1 1 1 1 1
5161: Reoljdwlrqv frqyhuwleohv hw uhyhuvh frqyhuwleohv
5161; Surgxlwv ã fdslwdo jdudqwl 1 1 1 1 1 1 1 1 1 1 1 1
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Jorvvdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6 Ohv prgëohv glvfuhwv g*ìydoxdwlrq
614 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
615 Oh fdguh suredelolvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
61514 Od g|qdpltxh gx sul{ gh o*dfwlrq 1 1 1 1 1 1 1 1 1 1
61515 Txhotxhv sursulìwìv gh od g|qdpltxh ghv sul{ 1 1 1
616 O*devhqfh g*rssruwxqlwì g*duelwudjh hw oh sul{ ghv surgxlwv
61614 Od irupxoh g*ìydoxdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
61615 Lqwhusuìwdwlrq ìfrqrpltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1
61616 Xwlolvdwlrq sudwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
617 Od frqyhujhqfh yhuv oh whpsv frqwlqx 1 1 1 1 1 1 1 1 1 1 1 1
61714 Txhotxhv qrwdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
61715 Fkdqjhphqw gh sdudpìwudjh 1 1 1 1 1 1 1 1 1 1 1 1
61716 Frqyhujhqfh yhuv oh whpsv frqwlqx 1 1 1 1 1 1 1 1 1
61717 Txhotxhv uhpdutxhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
61718 Hydoxdwlrq sdu ìtxdwlrq dx{ gìulyìhv sduwlhoohv 1 1
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7 Oh prgëoh gh Eodfn hw Vfkrohv
714 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
715 Od irupxoh gh Eodfn hw Vfkrohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
716 Oh irupdolvph gx whpsv frqwlqx 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
71614 Oh prxyhphqw Eurzqlhq hw ohv ìtxdwlrqv glìuhqwlhoohv vwrfkdvwltxhv
71615 Oh ohpph g*Lw÷ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
71616 O*ìtxdwlrq gh Irnnhu0Sodqfn 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
717 Uhirupxodwlrq gh o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh 1 1 1 1 1 1
718 Dqdo|vh hw xwlolvdwlrq gh od irupxoh gh Eodfn hw Vfkrohv 1 1 1 1 1 1 1 1 1 1 1 1
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8 H{whqvlrqv hw dssolfdwlrqv gx prgëoh gh Eodfn hw
814 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
815 Sulvh hq frpswh ghv glylghqghv 1 1 1 1 1 1 1 1 1 1 1
816 Wdx{ g*lqwìuíw hw yrodwlolwì gìshqgdqwhv gx whpsv 1
817 Rswlrqv gljlwdohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
818 Rswlrqv eduulëuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81814 Gìqlwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81815 Fdofxo gx sul{ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81816 Dxwuhv rswlrqv eduulëuhv 1 1 1 1 1 1 1 1 1 1 1
819 Rswlrqv orrnedfn 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81: Dxwuhv rswlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81; Rswlrqv dpìulfdlqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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1 1 1 1 1
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gìulyìv
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1 1 1 1 1
Vfkrohv
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9 Sul{ }ìur0frxsrq/ wdx{ g*lqwìuíw/ sul{ Iruzdug/ sul{ Ixwxuhv
914 Jìqìudolwìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
915 Rsìudwlrqv dx frpswdqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91514 Gìqlwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91515 Mxvwlfdwlrqv wkìrultxhv gdqv oh fdv glvfuhw 1 1 1 1 1 1 1
91516 Ohv wdx{ g*lqwìuíw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
916 Rsìudwlrqv ã whuph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91614 Gìqlwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91615 Duelwudjh frpswdqw0whuph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91616 Glvwlqfwlrq Iruzdug2Ixwxuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
917 Dqqh{h 4 = Glìuhqfh hqwuh sul{ iruzdug hw sul{ ixwxuh 1 1 1 1 1
91714 Uìvxowdw 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91715 Uìvxowdw 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91716 Uìvxowdw 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
918 Dqqh{h 5 = Frqwudw hxuleru 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
=
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qrwdwlrqv hw
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frqfhswv1
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: Lqwhusrodwlrq hw dqdo|vh idfwrulhooh gh od vwuxfwxuh ghv wdx{1
:14 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:15 Ohv pìwkrghv g*lqwhusrodwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1514 Qrwdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1515 Od prgìolvdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1516 Sulqflshv g*hvwlpdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1517 Ohv sulqflsdohv idploohv gh irqfwlrqv g*lqwhusrodwlrq 1 1 1
:16 O*dqdo|vh idfwrulhooh gh od vwuxfwxuh ghv wdx{ 1 1 1 1 1 1 1 1 1 1 1
:1614 Prwlydwlrqv qdqflëuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1615 Txhotxhv qrwlrqv vxu ohv prgëohv idfwrulhov 1 1 1 1 1 1 1
:1616 Whvw g*k|srwkëvhv qdqflëuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:1617 Hvwlpdwlrq g*xq prgëoh idfwrulho 1 1 1 1 1 1 1 1 1 1 1 1 1
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<14 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<15 Oh fdguh suredelolvwh hw o*devhqfh g*rssruwxqlwìv g*duelwudjh +Khdwk/ Mduurz hw Pruwrq,
<1514 Ohv prxyhphqwv gh od vwuxfwxuh ghv wdx{ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1515 Devhqfh g*rssruwxqlwìv g*duelwudjh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1516 Fkdqjhphqw gh suredelolwì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<16 Oh prgëoh vrxv od suredelolwì ulvtxh0qhxwuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1614 Ohv wdx{ ã whuph lqvwdqwdqìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1615 Surfhvvxv gx wdx{ frxuw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1616 Ohv sul{ }ìur0frxsrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1617 H{hpsoh = iruph gh od frxueh ghv wdx{ gdqv oh fdv g*xqh yrodwlolwì frqvwdqwh 1 1 1
<17 O*lqwurgxfwlrq gh yduldeohv g*ìwdw gdqv ohv prgëohv KMP 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1714 Jìqìudolwìv vxu ohv prgëohv idfwrulhov 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1715 Yrodwlolwì gìwhuplqlvwh hw yduldeohv g*ìwdw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1716 Yrodwlolwìv qrq gìwhuplqlvwhv hw yduldeohv g*ìwdw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<18 H{hpsohv gh prgëohv olqìdluhv ã yrodwlolwì vwrfkdvwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1814 Oh prgëoh ã xq idfwhxu +Fr{/ Lqjhuvroo hw Urvv, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1815 Oh prgëoh gh Gx!h hw Ndq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
<1816 Oh prgëoh txdgudwltxh jdxvvlhq +Ho Ndurxl/ P|qhql hw Ylvzdqdwkdq, 1 1 1 1 1 1
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43 H{hpsohv gh surgxlwv gh wdx{ g*lqwìuíw
4314 Iruzdug udwh djuhhphqw +IUD, 1 1 1 1 1
4315 Frqwudw Ixwxuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4316 Ohv vzdsv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4317 Fds hw Iorru 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4318 Vzdswlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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; Ohv prgëohv gh wdx{ hq whpsv glvfuhw1
;14 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;15 Oh prgëoh gh edvh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1514 Ohv k|srwkëvhv qdqflëuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1515 Oh prgëoh suredelolvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1516 O*devhqfh g*rssruwxqlwì g*duelwudjh 1 1 1 1 1 1 1 1 1 1
;1517 Sursulìwì gh pduwlqjdoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1518 Fkdqjhphqw gh sdudpìwudjh 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;16 Uìvroxwlrq gx prgëoh gdqv xq fdv vlpsoh = oh prgëoh gh Kr hw
;17 Ohv h{whqvlrqv gx prgëoh gh Kr hw Ohh 1 1 1 1 1 1 1 1 1 1 1 1 1
;1714 Oh prgëoh ã yrodwlolwì h{srqhqwlhooh 1 1 1 1 1 1 1 1 1 1 1
;1715 Prgëoh ã soxvlhxuv doìd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1716 Prgëoh gh Eodfn/ Ghupdq hw Wr| 1 1 1 1 1 1 1 1 1 1 1 1
;18 Dssolfdwlrqv ã od ydorulvdwlrq ghv elhqv frqwlqjhqwv 1 1 1 1 1 1
;1814 Sulqflsh g*ìydoxdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;1815 Dxwuhv suredelolwìv 0 Suredelolwìv iruzdug0qhxwuhv 1 1
;19 Sdvvdjh ã od olplwh gdqv ohv prgëohv glvfuhwv 1 1 1 1 1 1 1 1 1 1
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4319 Dqqh{h = h{wudlw gh od grfxphqwdwlrq gx Olh +zzz1olh1frp, 1 1 1 1 1 1 1 1 1 1 46<
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Ohv surgxlwv
414 Lqwurgxfwlrq
Ohv lqvwuxphqwv qdqflhuv vrqw wuëv glyhuv hw ohxu frpsoh{lwì v*dffurñw gh mrxu hq mrxu1 Lo hvw grqf gh soxv
hq soxv gl!floh gh guhvvhu xqh w|srorjlh ghv surgxlwv1 Srxu idluh vlpsoh/ rq glvfxwhud lfl ghv dfwliv gh
edvh = dfwlrq/ reoljdwlrq/ pdwlëuhv suhplëuhv/ ghylvhv1 Gdqv oh fkdslwuh vxlydqw/ rq frpphqfhud ã sduohu
surgxlwv gìulyìv/ f*hvw0ã0gluh ghv frqwudwv grqw ohv x{ qdqflhuv dvvrflìv gìshqghqw +l1h gìulyhqw, ghv
dfwliv gh edvh glvfxwìv lfl1
V*djlvvdqw ghv dfwliv gh edvh/ qrxv grqqrqv lfl ohxu gìqlwlrq hw gìyhorssrqv ohv pdwkìpdwltxhv ìoìphq0
wdluhv tx*lo hvw qìfhvvdluh gh pdñwulvhu dydqw gh srxuvxlyuh1
Udsshorqv tx*xq dfwli rx lqvwuxphqw qdqflhu hvw xq pr|hq g*hhfwxhu ghv wudqvihuwv lqwhuwhpsruhov gh
ulfkhvvh1 Oruvtxh fhv wudqvihuwv gh ulfkhvvh vrqw sduidlwhphqw frqqxv/ rq sduohud g*dfwliv ã uhyhqx {h
+{hg0lqfrph vhfxulwlhv, hw/ frqfuëwhphqw/ fhod gìvljqhud w|sltxhphqw ohv reoljdwlrqv1 Gdqv fh fdv/ od
txhvwlrq hvvhqwlhooh gh vdyrlu frpelhq ydxw hq hxur g*dxmrxug*kxl od surphvvh +fhuwdlqh, gh uhfhyrlu xq
hxur ã xqh gdwh ixwxuh grqqìh +l1h1 oh sul{ gx whpsv,1 Ohv dfwlrqv/ ohv ghylvhv/ ohv pdwlëuhv suhplëuhv vrqw/ ã
o*lqyhuvh/ ghv h{hpsohv g*dfwliv shuphwwdqw ghv wudqvihuwv gh ulfkhvvh qrq sduidlwhphqw frqqxv dxmrxug*kxl1
Dlqvl/ xqh dfwlrq g*xqh hqwuhsulvh grqqh gurlw ã xqh sduw ghv eìqìfhv ixwxuv gh fhwwh hqwuhsulvh pdlv fh
surw q*hvw sdv frqqx dxmrxug*kxl1 Gdqv fh fdv/ lo | d ghx{ glphqvlrqv = xqh glphqvlrq sul{ gx whpsv hw
xqh glphqvlrq sul{ gx ulvtxh1 Hq hhw/ fhv ghuqlhuv dfwliv lpsoltxhqw ìjdohphqw ghv wudqvihuwv gh ulvtxh
hqwuh lqglylgxv1
O*rxyudjh suìvhqw hvw xqh lqwurgxfwlrq ã fhv txhvwlrqv hq hvvd|dqw gh pdulhu ohv frqvlgìudwlrqv sudwltxhv
vxu o*xwlolvdwlrq gh fhv surgxlwv hw ohv uhsëuhv wkìrultxhv +ydorulvdwlrq hw frxyhuwxuh,1 Oh sduwl sulv hvw gh
qh sdv idluh gh wkìrulh gxuh / fhooh0fl ìwdqw wudlwìh gh idêrq frpsoëwh gdqv ohv frxuv gh Pdwkìpdwltxhv
Dssoltxìhv1
415 Ohv dfwlrqv
41514 Jìqìudolwìv
Xqh dfwlrq +htxlw|/ vwrfn/ vkduh, hvw xq wlwuh gh sursulìwì g*xqh hqwuhsulvh grqqìh1 Hooh jìqëuh grqf
srxu vhv gìwhqwhxuv +ohv dfwlrqqdluhv rx vkduhkroghuv, xq fhuwdlq qrpeuh gh gurlwv/ qrwdpphqw oh gurlw gh
uhfhyrlu xqh sduwlh ghv eìqìfhv ixwxuv hw xq gurlw gh yrwh vxu ohv gìflvlrqv vwudwìjltxhv gh o*hqwuhsulvh1
Rq v*lqwìuhvvh lfl dx{ dfwlrqv frpph dfwli qdqflhu/ f*hvw0ã0gluh xq dfwli txl jìqëuh ghv x{ ixwxuv dsshoìv
Wkìrulh hw sudwltxh ghv lqvwuxphqwv qdqflhuv
glylghqghv1 Ohv glylghqghv uhsuìvhqwhqw wrxw rx sduwlh ghv eìqìfhv vhorq txh ohv dfwlrqqdluhv gìflghqw
gh uhwhqlu rx qrq xqh sduwlh ghv eìqìfhv gdqv o*hqwuhsulvh1 Ohv glylghqghv vrqw yhuvìv sìulrgltxhphqw1
Rq qh uhjdugh sdv lfl ohv udlvrqv txl srxvvhqw xqh hqwuhsulvh ã ìphwwuh ghv dfwlrqv rx ã v*hqghwwhu1 Srxu
idluh vlpsoh/ glvrqv tx*xqh hqwuhsulvh glvsrvh ã vd fuìdwlrq g*xq fdslwdo dssruwì sdu vhv dfwlrqqdluhv1 Fh
fdslwdo q*hvw sdv irufìphqw frwì vrxv iruph g*dfwlrqv vxu ohv pdufkìv gh fdslwdx{1 Wrxwhirlv/ vl o*hqwuhsulvh
hvw vx!vdpphqw lpsruwdqwh/ hooh shxw idluh dssho ã o*ìsdujqh sxeoltxh hw ohv dfwlrqv gh o*hqwuhsulvh
shxyhqw íwuh frwìhv vxu xq pdufkì g*dfwlrqv1 Hqq/ srxu qdqfhu vhv lqyhvwlvvhphqwv/ o*hqwuhsulvh shxw
vrlw ìphwwuh gh qrxyhoohv dfwlrqv/ vrlw v*hqghwwhu +dxsuëv g*xqh edqtxh rx hq ìphwwdqw ghv wlwuhv gh ghwwh/
l1h1 ghv reoljdwlrqv,1 Od txhvwlrq ghv dydqwdjhv frpsduìv ghv ghx{ prghv gh qdqfhphqw qh vhurqw sdv
derugìhv lfl hw uhoëyhqw g*xq frxuv gh qdqfh g*hqwuhsulvh1
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vhv lqyhvwlvvhphqwv1 Od yduldeoh ã fhuqhu hvw oh sul{ gh o*dfwlrq1 Fh sul{ hvw irqfwlrq ghv dqwlflsdwlrqv
ghv lqwhuyhqdqwv vxu oh qlyhdx ghv glylghqghv ixwxuv hw vxu od ydohxu gh uhyhqwh gh o*dfwlrq1 Hq hhw/ xqh
dfwlrq q*d sdv gh whuph gìql dxwuh txh od idloolwh gh o*hqwuhsulvh +rx vd glvsdulwlrq srxu g*dxwuhv udlvrqv
frpph xqh ixvlrq,1 Sdu frqvìtxhqw/ od ydohxu gh uhyhqwh gh o*dfwlrq gdqv 8/ 43 rx 48 dqv q*hvw sdv soxv
frqqxh dxmrxug*kxl txh ohv glylghqghv ixwxuv1 Sdu h{hpsoh/ ohv ydohxuv glwhv gh od Qrxyhooh Hfrqrplh
rqw frwì ã ghv sul{ ìohyìv xqh sduwlh gh o*dqqìh 5333/ pdojuì ghv glylghqghv frxudqwv ìjdx{ ã 3 hw/ srxu
fhuwdlqhv/ qxov ã o*krul}rq gh ghx{ dqv1 Gdqv fh fdv/ od idleohvvh ghv glylghqghv ixw odujhphqw frpshqvìh
sdu o*dqwlflsdwlrq g*xqh lpsruwdqwh ydohxu gh uhyhqwh gh o*dfwlrq1
Oh sul{ g*xqh dfwlrq vxlw grqf xq surfhvvxv doìdwrluh txl uh ëwh dxvvl elhq ohv lqfhuwlwxghv ixwxuhv +hq
sduwlfxolhu vxu ohv surwv gh o*hqwuhsulvh, txh ohv dssuìfldwlrqv tx*hq rqw ohv lqwhuyhqdqwv vxu ohv pdufkìv1
Gh idlw/ oh frxuv g*xqh dfwlrq hvw wuëv odujhphqw lpsuìylvleoh ã sduwlu gh o*lqirupdwlrq sxeoltxhphqw
glvsrqleoh1 Vl who q*ìwdlw sdv oh fdv/ rq srxuudlw frqvwuxluh ghv vwudwìjlhv pdmrulwdluhphqw jdjqdqwhv/ fh
txh gìphqwhqw o*revhuydwlrq frxudqwh hw ohv ìwxghv ìfrqrpìwultxhv1
41515 Prgìolvdwlrq ghv sul{ g*dfwlrqv
Ohv txhvwlrqv g*ìydoxdwlrq ghv surgxlwv gìulyìhv qìfhvvlwhurqw gh suìflvhu od prgìolvdwlrq ghv sul{ g*df0
wlrqv1 Rq shxw wrxwhirlv g*ruhv hw gìmã grqqhu xqh lgìh gh od iruph ghv prgëohv tx*rq uhqfrqwuhud1
Vxssrvrqv xqh vhfrqgh txh oh prqgh hvw gìwhuplqlvwh/ f*hvw0ã0gluh tx*rq frqqdlw oh sul{ ghpdlq gh o*df0
wlrq dlqvl txh oh glylghqgh yhuvì1 Vl rq qrwh V4 oh sul{ ghpdlq/ V3 oh sul{ dxmrxug*kxl hw gly4 oh glylghqgh
yhuvì ghpdlq/ doruv lqyhvwlu V3 hxurv dxmrxug*kxl udssruwh V4 . gly4 ghpdlq1 Sdudooëohphqw/ rq shxw lq0
yhvwlu od píph vrpph vxu xq dfwli vdqv ulvtxh +l1h1 od fdlvvh g*ìsdujqh, hw rewhqlu ghpdlq +4 . u,=V3 vl u
hvw oh wdx{ g*lqwìuíw1 Gdqv xq prqgh gìwhuplqlvwh hw hq qìjoljhdqw ohv frþwv gh wudqvdfwlrq hw ohv wd{hv/
ohv ghx{ sodfhphqwv grlyhqw fròqflghu/ vrlw =
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od iruph =
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rü 4 hvw xq doìd grqw od uìdolvdwlrq qh vhud frqqxh tx*ã od gdwh 41 Od txhvwlrq hvvhqwlhooh hvw grqf gh
vdyrlu txho w|sh gh orl vxlw od yduldeoh %4 1 Rq sduwlud gh fh w|sh gh prgìolvdwlrq srxu ìydoxhu ohv surgxlwv
gìulyìv vxu dfwlrqv1
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irqgdphqwdohv hq dssoltxdqw od irupxoh +414, hw hq idlvdqw ghv k|srwkëvhv vxu ohv glylghqghv ixwxuv
+pìwkrgh glwh GFI/ Glvfrxqwhg Fdvk Iorz,1 Dlqvl/ vl rq vxssrvh txh ohv glylghqghv furlvvhqw ã xq wdx{
j frqvwdqw +vxssrvì lqiìulhxu ã u,/ doruv od ydohxu irqgdphqwdoh v*ìfulw +dyhf W $ .4, =
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fh txl hvw orjltxh frpswh0whqx gh wrxwhv ohv k|srwkëvhv uhvwulfwlyhv idlwhv1 Hooh grqqh qìdqprlqv xq jxlgh
vxu fh txh ghyudlw íwuh oh sul{ gh o*dfwlrq1
41516 Frwdwlrq
Hq sudwltxh/ ghx{ w|shv gh pdufkì fr0h{lvwhqw = ohv pdufkìv glwv uìjohphqwìv hw ohv pdufkìv gh juì ã juì 1 Fh txl ohv glvwlqjxh sruwh vxu ohxu ghjuì g*rujdqlvdwlrq = ohv pdufkìv uìjohphqwìv grlyhqw
vdwlvidluh ã ghv frqwudlqwhv vsìfltxhv g*rujdqlvdwlrq/ gh irqfwlrqqhphqw/ g*dgplvvlrq ghv ydohxuv1 Lov vrqw
g*dloohxuv hq sudwltxh vrxv od wxwhooh g*rujdqlvph gh sodfh +oh Frqvhlo ghv Pdufkìv Ilqdqflhuv hq Iudqfh,1
Ohv pdufkìv gh juì ã juì rqw ghv frqwudlqwhv soxv vrxsohv1 Gdqv od vxlwh/ rq suhqgud o*h{hpsoh ghv pdufkìv
iudqêdlv +rq srxuud vh uìiìuhu dx{ grfxphqwdwlrqv gh zzz1erxuvh0gh0sdulv1iu,1
Ohv lqwhuyhqdqwv +glwv suhvwdwdluhv gh vhuylfhv g*lqyhvwlvvhphqw, vrqw ohv ìwdeolvvhphqwv gh fuìglwv
+edqtxhv/ fdlvvhv g*ìsdujqh hwf1, hw ohv hqwuhsulvhv g*lqyhvwlvvhphqw +h{0vrflìwìv gh erxuvh/ h{0vrflìwìv
gh jhvwlrq,1 Fhwwh rujdqlvdwlrq gdwh g*xqh orl gh 4<<9 ylvdqw ã dgdswhu xqh gluhfwlyh hxursìhqqh vxu
ohv vhuylfhv g*lqyhvwlvvhphqw1 Fh txl fkdqjh sdu udssruw ã od vlwxdwlrq suìfìghqwh hvw od soxv judqgh
olehuwì srxu ohv glìuhqwv dfwhxuv g*h{hufhu ohv glìuhqwhv dfwlylwìv +froohfwh hw wudqvplvvlrq ghv ruguhv/
qìjrfldwlrq/ jhvwlrq gh sruwhihxlooh hwf1,1 Hq sduwlfxolhu/ od qìjrfldwlrq ghv ydohxuv prelolëuhv q*hvw
soxv oh prqrsroh ghv vrflìwìv gh erxuvh1
Ohv pdufkìv uìjohphqwìv iudqêdlv frpsuhqqhqw +hq ghkruv ghv pdufkìv g*rswlrqv +PRQHS, hw ã
whuph +PDWLI, tx*rq yhuud soxv orlq, = oh Suhplhu Pdufkì/ oh Vhfrqg Pdufkì/ hw oh Qrxyhdx Pdu0
fkì1 Hq jurv/ oh Suhplhu Pdufkì hvw uìvhuyì dx{ ydohxuv ghv soxv judqghv vrflìwìv +iudqêdlvhv rx
ìwudqjëuhv,/ oh Vhfrqg Pdufkì dx{ hqwuhsulvhv gh wdlooh soxv idleoh rx ã fhoohv txl rqw yrfdwlrq ã
uhqwuhu xq mrxu vxu oh Suhplhu Pdufkì1 Hqq/ oh Qrxyhdx Pdufkì hvw uìvhuyì dx{ ydohxuv ã iruw
srwhqwlho gh furlvvdqfh +frpph ohv vwduw0xs,1
Ohv wlwuhv vrqw qìjrflìv vrlw dx frpswdqw/ vrlw hq uëjohphqw glìuì +h{0uëjohphqw phqvxho,1 Dx
frpswdqw/ oh sdlhphqw hw od olyudlvrq ghv wlwuhv hvw lppìgldwh1 Hq uëjohphqw glìuì/ oh pìfdqlvph hvw
glìuhqw = oruvtx*xq dfkhwhxu +lqglylgxho, sdvvh xq ruguh g*dfkdw +ã vd edqtxh sdu h{hpsoh,/ o*lqwhu0
pìgldluh wudqvphw o*ruguh g*dfkdw dx qìjrfldwhxu txl dfkëwh ohv wlwuhv dx frpswdqw +grqf sdlhphqw hw
olyudlvrq lppìgldwh dx qìjrfldwhxu,1 Hq uhydqfkh/ o*dfkhwhxu qdo qh sdlh hw q*hvw olyuì tx*dx ghuqlhu
mrxu gh erxuvh gx prlv1 Lo | d grqf xq gìodl hqwuh od frqfoxvlrq gh od yhqwh hw od olyudlvrq ã o*dfkhwhxu
qdo/ o*lqwhupìgldluh dvvxudqw dlqvl oh sruwdjh ghv wlwuhv shqgdqw txhotxhv mrxuv1 Hylghpphqw/ yx
gx f÷wì gh o*dfkhwhxu/ oh qìjrfldwhxu uhqg xq vhuylfh sxlvtxh g*xqh fhuwdlqh idêrq lo dydqfh shqgdqw
txhotxhv mrxuv ohv irqgv txl shuphwwurqw gh sd|hu ohv wlwuhv1 Fh vhuylfh jìqëuh grqf orjltxhphqw
xqh frpplvvlrq sd|ìh sdu o*dfkhwhxu qdo ã o*lqwhupìgldluh1 Oh pìfdqlvph hvw v|pìwultxh gdqv oh
fdv g*xqh yhqwh > gdqv fh fdv/ oh qìjrfldwhxu idlw o*dydqfh ghv wlwuhv1 Wrxv ohv wlwuhv qh vrqw sdv vxv0
fhswleohv g*hqwuhu gdqv fh pìfdqlvph = vhxov ohv wlwuhv gx VEI 453 +lqglfh gh pdufkì frpsuhqdqw
ohv 453 soxv jurvvhv fdslwdolvdwlrqv gx pdufkì iudqêdlv, rx d|dqw xqh wdlooh vx!vdqwh +4 Pgv hxurv
gh fdslwdolvdwlrq/ 4 Pqv hxurv wudlwìv txrwlglhqqhphqw, vrqw ìoljleohv1 Hqq/ od srvlwlrq shxw íwuh
uhsruwìh g*xq prlv ã o*dxwuh1
Fh v|vwëph gh uìjohphqw glìuì glw VUG +vhuylfh uëjohphqw glìuì, shuphw g*dfkhwhu ã fuìglw rx
gh yhqguh vdqv dyrlu ohv wlwuhv1 Soxvlhxuv rsìudwlrqv shxyhqw íwuh idlwhv dx frxuv gx prlv = lo hvw
grqf srvvleoh g*dfkhwhu xq hqvhpeoh gh wlwuhv hq gìexw gh prlv hw gh ohv uhyhqguh dx frxuv gx prlv
vdqv dyrlu ulhq ã gìerxuvhu hw hq uhfhydqw vlpsohphqw oh surw rx od shuwh gh fhwwh frpelqdlvrq
g*rsìudwlrqv/ dxtxho rq uhwudqfkh elhq0vþu ohv frpplvvlrqv gh sruwdjh1
O*ruguh vwdqgdug hvw gx w|sh ruguh dyhf olplwh / l1h1 mh yhx{ dfkhwhu who wlwuh dx sul{ pd{lpxp
gh { hxurv sdu wlwuh rx mh yhx{ yhqguh who wlwuh dx sul{ plqlpxp gh { hxurv sdu wlwuh 1
Oruvtxh { @ .4 srxu xq ruguh g*dfkdw/ hw { @ 3 srxu xq ruguh gh yhqwh/ o*ruguh hvw glw ã wrxw
sul{ 1 Fhv ruguhv vhurqw sulrulwdluhv vxu wrxv ohv dxwuhv1 G*dxwuhv ruguhv h{lvwhqw ìjdohphqw = o*ruguh
Hfroh Sro|whfkqltxh
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wrxw rx ulhq sruwh vxu xqh txdqwlwì qrq iudfwlrqqdeoh ã yhqguh rx ã dfkhwhu dyhf xqh olplwh gh
sul{ grqqìh > ohv ruguhv ã vhxlo gh gìfohqfkhphqw frpsruwhqw xqh olplwh hq ghêd rx dx0ghoã gh
odtxhooh lov vh wudqviruphqw hq ruguhv ã wrxw sul{ > ohv ruguhv dx sul{ gh pdufkì qh frpsruwhqw
dxfxqh lqglfdwlrq gh frxuv1
Fhv pdufkìv vrqw jrxyhuqìv sdu ohv ruguhv1 Soxv suìflvìphqw/ ohv ruguhv vrqw fodvvìv sdu olplwh gh
sul{ hw/ srxu fkdtxh olplwh/ fkurqrorjltxhphqw dx ixu hw ã phvxuh gh ohxu lqwurgxfwlrq1 O*h{ìfxwlrq
ghv ruguhv vh idlw vhorq ghx{ ruguhv gh sulrulwì = vrqw vhuylv hq sulrulwì ohv ruguhv g*dfkdw grqw ohv
olplwhv vrqw ohv soxv ìohyìhv hw lqyhuvhphqw srxu ohv ruguhv gh yhqwh1 Hq fodlu/ rq vhuw g*derug fhx{
txl vrqw suíwv ã dfkhwhu fkhu hw fhx{ txl vrqw suíwv ã yhqguh ã xq sul{ idleoh 1 D olplwh
grqqìh/ ohv suhplhuv duulyìv vrqw vhuylv ohv suhplhuv1
Dydqw o*rxyhuwxuh gx pdufkì +grqf dydqw < k 33 oh pdwlq,/ ohv ruguhv gh yhqwh hw g*dfkdw vrqw plv hq
uhjdug hw xq sul{ +l1h1 oh {lqj, hvw fdofxoì1 Oh sul{ hvw fdofxoì gh whooh vruwh txh o*ruh hw od ghpdqgh
vrlw od soxv ìtxloleuìh srvvleoh / f*hvw0ã0gluh oh sul{ txl pd{lplvh od txdqwlwì gh wlwuhv ìfkdqjìv1
Od qìjrfldwlrq gh ydohxuv prelolëuhv 0 txl ìwdlw dydqw 4<<9 oh prqrsroh ghv vrflìwìv gh erxuvh 0 hvw
gìvrupdlv srvvleoh srxu wrxv ohv lqwhuyhqdqwv1 Yrlfl o*h{hpsoh grqqì gdqv od grfxphqwdwlrq VEI
+zzz1erxuvh0gh0sdulv1iu,
Judskltxh 414 H{hpsohv gh fduqhw g*ruguh
Vl rq qrwh d+o, +uhvs1 y+o,, od txdqwlwì gh wlwuhv ghpdqgìv +uhvs1 rhuwv, srxu od olplwh o/ f*hvw0ã0gluh
ohv ruguhv g*dfkdw +uhvs1 yhqwh, srxu xq sul{ qh ghydqw sdv gìsdvvhu +uhvs1 íwuh lqiìulhxu ã, o/ doruv
oh suhplhu {lqj vh idlw dx sul{ s who txh =
S
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P lq o@s d+o, o@3 y+o,
s
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rü .
o@s d+o, hvw od irqfwlrq gh ghpdqgh gh wlwuhv +irqfwlrq gìfurlvvdqwh gh s, hw o@3 y+o, od irqfwlrq
g*ruh gh wlwuhv +irqfwlrq furlvvdqwh gh s,1 Rq yrlw gdqv fhw h{hpsoh txh o*ruh hw od ghpdqgh qh
fròqflghqw sdv sduidlwhphqw hw grqf vhxov ;33 ghv ;83 wlwuhv ghpdqgìv ã 94143 hxurv vrqw vhuylv1
Dx frxuv gh od mrxuqìh/ fkdtxh lqwurgxfwlrq g*xq qrxyho ruguh jìqëuh xqh wudqvdfwlrq srxu shx tx*lo
| dlw xqh ruh rx xqh ghpdqgh frpsdwleoh +hq sul{, hq idfh1 Vxlydqw ohv fdv/ od frqiurqwdwlrq ghv
ruguhv hvw frqwlqxh1 Gdqv g*dxwuhv fdv/ vhxov ghv {lqjv uìjxolhuv rqw olhx1
41517 Sul{ g*dfwlrqv hw wkìrulh ìfrqrpltxh
Ohv sul{ ghv dfwlrqv ydulhqw txdvl0frqwlqþpphqw dx frxuv gx whpsv hw gh idêrq qrq suìylvleoh/ g*rü
o*lgìh g*lqwurgxluh ghv prgëohv doìdwrluhv1 Fhv yduldwlrqv gh sul{ rqw soxvlhxuv ruljlqhv tx*rq shxw yrlu
43
ã sduwlu gh od irupxoh g*dfwxdolvdwlrq +414, = fkdqjhphqw gdqv ohv dqwlflsdwlrqv gh surw ghv hqwuhsulvhv/
prxyhphqwv ghv wdx{ g*lqwìuíw/ fkdqjhphqw g*dssuìfldwlrq vxu od erqqh vdqwì ghv hqwuhsulvhv hwf1 Oh
u÷oh ghv dqdo|vwhv qdqflhuv hvw mxvwhphqw g*lqwìjuhu wrxv fhv sdudpëwuhv gdqv xqh irupxoh gx w|sh +414,
srxu hq gìgxluh fh txh ghyudlw íwuh oh yudl sul{ gh o*dfwlrq +rq sduoh gh frxuv irqgdphqwdo, dq
gh frqvhloohu ohv lqyhvwlvvhxuv srwhqwlhov1 Hq sudwltxh/ rq frqvwdwh vrxyhqw ghv ìfduwv qrwdeohv hqwuh ohv
ìydoxdwlrqv ghv dqdo|vwhv hw ohv sul{ gh pdufkì revhuyìv = wrxw o*duw gh o*dqdo|vwh hvw gh vdyrlu v*lo v*djlw
g*xqh huuhxu g*ìydoxdwlrq gh vd sduw rx xqh huuhxu gx pdufkì txl vhud grqf w÷w rx wdug fruuljìh1 Ohv sul{
ghv dfwlrqv gh vrflìwìv Lqwhuqhw hq 4<<< hw 5333 frqvwlwxhqw xq h{hpsoh w|sltxh gh fh jhquh g*ìfduw1
Lo idxw qrwhu ìjdohphqw txh oh sul{ gh pdufkì grlw whqlu frpswh gh o*dyhuvlrq dx ulvtxh ghv lqwhuyhqdqwv gx
pdufkì hw soxv jìqìudohphqw gh ohxu irqfwlrq g*xwlolwì/ sdudpëwuh tx*lo hvw ìylghpphqw gl!floh g*hvwlphu1
Hq sudwltxh/ ohv dqdo|vwhv udmrxwhqw xqh sulph gh ulvtxh gdqv od irupxoh g*ìydoxdwlrq +414,/ vrxv od iruph
g*xq whuph vxssoìphqwdluh dx gìqrplqdwhxu +l1h1 u.sulph,1 Fhwwh sulph +srvlwlyh, uìgxlw pìfdqltxhphqw
od ydohxu txh o*rq dwwulexh ã xq hxur doìdwrluh gdqv oh ixwxu hw v|qwkìwlvh ã od irlv oh qlyhdx gh ulvtxh
uhvvhqwl sdu ohv lqwhuyhqdqwv gx pdufkì hw ohxu dyhuvlrq srxu fh ulvtxh1
Gx srlqw gh yxh ghv surgxlwv gìulyìv/ oh frxuv gh o*dfwlrq hvw xqh grqqìh hw vrq ìyroxwlrq hvw prgìolvìh
sdu xq surfhvvxv doìdwrluh gdqv ohtxho rq qh fkhufkh sdv ã lqwurgxluh ohv suìrffxsdwlrqv ghv dqdo|vwhv vxu
ohv yduldeohv vxvfhswleohv g*lq xhqfhu ohv frxuv ixwxuv1 O*lgìh hvw g*xqh sduw tx*lo hvw looxvrluh gh suìyrlu
ohv frxuv ixwxuv hw g*dxwuh sduw tx*rq q*d sdv uìhoohphqw ehvrlq gh idluh fhv suìylvlrqv = oh frxuv g*xqh
dfwlrq hvw xqh vrxufh srwhqwlhooh gh ulvtxh 0 txh oh frxuv prqwh rx tx*lo edlvvh 0 hw od yudlh txhvwlrq hvw
gh vdyrlu frpphqw lo lq xh vxu ohv sul{ ghv surgxlwv gìulyìv1
416 Ohv reoljdwlrqv
41614 Jìqìudolwìv
Ohv wlwuhv reoljdwdluhv vrqw ghv wlwuhv gh ghwwh hw qrq gh sursulìwì frpph ohv dfwlrqv1 Fhv wlwuhv v*ds0
sduhqwhqw frpsoëwhphqw ã xqh rsìudwlrq gh suíw 2 hpsuxqw = xqh hqwuhsulvh txl ìphw ghv reoljdwlrqv
hpsuxqwh ghv irqgv dx{ dfkhwhxuv ghv wlwuhv1 Od vhxoh glìuhqfh dyhf ohv rsìudwlrqv edqfdluhv g*hpsuxqw
hw gh suíw uìvlgh gdqv oh fdudfwëuh qìjrfldeoh gh fhv wlwuhv1 O*dfkhwhxu lqlwldo gx wlwuh4 d od olehuwì gh
uhyhqguh vrq wlwuh ã xq dxwuh lqwhuyhqdqw/ fh ghuqlhu ghyhqdqw oh fuìdqflhu gh o*ìphwwhxu gx wlwuh1
Ohv ìphwwhxuv srvvleohv shxyhqw íwuh wuëv glyhuv = Hwdwv/ hqwuhsulvhv/ froohfwlylwìv orfdohv/ irqgv gh fuìdqfhv
hwf1 Ohv x{ jìqìuìv sdu fhv wlwuhv vrqw gh qdwxuh wuëv glìuhqwh ghv wlwuhv g*dfwlrq = gdqv oh fdv ghv
dfwlrqv/ ohv x{ vrqw ghv glylghqghv/ f*hvw0ã0gluh ghv sduwv gx eìqìfh gh o*hqwuhsulvh1 V*djlvvdqw ghv
wlwuhv reoljdwdluhv/ ohv x{ vrqw ghv x{ g*lqwìuíw hw gh uhperxuvhphqw gx fdslwdo1 Fhv x{ vrqw prlqv
doìdwrluhv txh srxu ohv glylghqghv g*dfwlrq +g*rü oh whuph dqjodlv gh {hg0lqfrph vhfxulwlhv,1 Soxv
suìflvìphqw/ o*ìphwwhxu gx wlwuh +l1h1 o*hpsuxqwhxu, q*d hq jìqìudo sdv gh ohylhu srxu prglhu hq frxuv
gh urxwh oh prqwdqw ghv x{ yhuvìv dx{ gìwhqwhxuv gh wlwuh1 Lo v*djlw grqf g*xq frqwudw rü o*hpsuxqwhxu
v*hqjdjh ã uhperxuvhu oh fdslwdo hpsuxqwì dlqvl txh ghv lqwìuíwv vhorq ghv prgdolwìv hw xqh uëjoh gh
{dwlrq gìflgìhv lqlwldohphqw hw lqwdqjleohv sdu od vxlwh +vdxi gìidxw gh o*hpsuxqwhxu,1
Hqq/ oruvtx*lo v*djlw g*xqh hqwuhsulvh hq gl!fxowì qdqflëuh/ ohv fuìdqflhuv vrqw sulrulwdluhv ghydqw ohv
dfwlrqqdluhv1
41615 Fdofxo ghv x{
Ghx{ wlwuhv reoljdwdluhv vh glqvwlqjxhqw sdu od uëjoh gh gìwhuplqdwlrq ghv x{ = reoljdwlrq ã wdx{ yduldeoh/
ã wdx{ {h/ reoljdwlrq lqgh{ìh hwf1 Rq qrwh N oh fdslwdo gh fkdtxh wlwuh +hqfruh dsshoì fdslwdo qrplqdo/
ydohxu idfldoh gx wlwuh/ sulqflsdo, = N uhsuìvhqwh od ghwwh gh o*ìphwwhxu ylv0ã0ylv gx gìwhqwhxu gx wlwuh1
Rq qrwh ìjdohphqw W od gxuìh gx wlwuh/ f*hvw0ã0gluh od gxuìh dx0ghoã odtxhooh ohv lqwìuíwv rqw ìwì sd|ìv hw
oh fdslwdo d ìwì uhperxuvì1 Rq srxuudlw lpdjlqhu xqh gxuìh lqqlh > rq sduoh doruv gh uhqwh shusìwxhooh1
4 Rq sduoh gh pdufkì sulpdluh srxu gìvljqhu od plvh hq yhqwh lqlwldoh ghv wlwuhv/ sxlv gh pdufkì vhfrqgdluh sdu od vxlwh1
Hfroh Sro|whfkqltxh
44
D o*ìplvvlrq gx wlwuh/ vrqw vwlsxoìhv ohv frqglwlrqv gh sdlhphqw ghv lqwìuíwv +l1h1 ohv frxsrqv, hw gh
uhperxuvhphqw gx fdslwdo +l1h1 o*dpruwlvvhphqw gx fdslwdo,1 Hq qrwdqw pw oh x{ yhuvì dx gìwhqwhxu gx
wlwuh ã od gdwh w/ rq d grqf xqh pìfdqltxh gx w|sh =
;
pw
@ lw . dw
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lw
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d
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gx fdslwdo
O*dpruwlvvhphqw gx fdslwdo uhsuìvhqwh od txdqwlwì gh fdslwdo uhperxuvìh ã fkdtxh sìulrgh = o*dpruwlvvh0
phqw hq w hvw grqf sdu gìqlwlrq ìjdo dx fdslwdo uhvwdqw gþ +qrwì Fw4 , hq w 4 prlqv oh fdslwdo uhvwdqw
gþ hq w =
dw @ Nw4 Nw
Ohv lqwìuíwv vrqw wuëv orjltxhphqw fdofxoìv frpph o*dssolfdwlrq g*xq wdx{ g*lqwìuíw vxu oh fdslwdo uhvwdqw
gþ hq gìexw gh sìulrgh/ vrlw =
lw @ fw =Nw4
rü fw hvw dxvvl dsshoì wdx{ gh frxsrq1 Ohv prgdolwìv gh fdofxo ghv x{ vrqw frqqxhv dx gìsduw/ fh txl
vljqlh tx*rq vh grqqh ohv ghx{ irqfwlrqv =
;
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=
dw
@ d +w> }w ,
hw txl shuphw gh fdofxohu ohv x{ jìqìuìv sdu oh wlwuh1 +}w , hvw xqh yduldeoh gh uìiìuhqfh grqw rq yhuud ghv
h{hpsohv sdu od vxlwh +lq dwlrq/ wdx{ gh pdufkì hwf1,1 Gdqv o*lpphqvh pdmrulwì ghv fdv hw ã o*lqyhuvh ghv
hpsuxqwv dffrugìv sdu oh v|vwëph edqfdluh dx{ sduwlfxolhuv/ od irqfwlrq d+w, hvw qxooh vdxi hq W rü hooh
ydxw N = hq fodlu/ oh fdslwdo q*hvw uhperxuvì tx*ã od q +uhperxuvhphqw glw lq0qh, hw vhxov ghv x{
g*lqwìuíw vrqw yhuvìv dydqw W 1
41616 Txhotxhv h{hpsohv
Rq shxw grqqhu txhotxhv h{hpsohv hq sduwdqw ghv soxv fodvvltxhv =
Wlwuh ã wdx{ {h hw ã uhperxuvhphqw lq0qh1 F*hvw oh fdv rü =
;
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=
dw
@ 3 srxu w ? W hw N srxu w @ W
Ohv x{ vrqw grqf frqvwdqwv ìjdx{ ã pw @ f=N hw pW @ +4.f,=N1 Xqh pdmrulwì gh wlwuhv g*reoljdwlrq
irqfwlrqqh vrxv fh vfkìpd1 Dlqvl/ od soxsduw ghv Reoljdwlrqv Dvvlplodeohv gx Wuìvru +RDW, 0 txl vrqw
ohv wlwuhv gh ghwwh ìplv sdu o*Hwdw iudqêdlv 0 hvw ã wdx{ {h hw ã uhperxuvhphqw lq0qh1
Wlwuh ã wdx{ yduldeoh hw ã uhperxuvhphqw lq0qh1 Od vhxoh fkrvh txl fkdqjh hvw txh fw hvw
gìvrupdlv xqh irqfwlrq g*xqh uìiìuhqfh yduldeoh =
fw @ f+}w ,
D wlwuh g*h{hpsoh/ o*RDW WHF 43 dqv hvw xqh reoljdwlrq ã wdx{ yduldeoh fw rü od uìiìuhqfh }w hvw ìjdoh
ã xq wdx{ ã 43 dqv +rq yhuud soxv orlq fh tx*rq hqwhqg sdu oã,1 Frqfuëwhphqw/ hooh vh fdofxoh frpph
o*lqwhusrodwlrq olqìdluh ghv wdx{ gh uhqghphqw ghv ghx{ RDW ohv soxv surfkhv gh od pdwxulwì h{dfwh
gh 43 dqv hw g*xq hqfrxuv vxsìulhxu ã 53 Pgv IUI +6 Pgv hxurv, +fh tx*rq hqwhqg sdu wdx{ gh
4
uhqghphqw vhud fodulì sdu od vxlwh,1 Oh frxsrq hvw yhuvì wulphvwulhoohphqw hw ydxw fw @ +4 . }w , 7 4/
dyhf }w oh wdx{ WHF 43 +gìql ã o*lqvwdqw, fdofxoì +hq jurv, xq wulphvwuh dydqw yhuvhphqw gx frxsrq
+h{dfwhphqw 8 mrxuv rxyuìv dydqw od gdwh gh mrxlvvdqfh gx frxsrq,1 Rq glw txh oh wdx{ hvw suì{ì5 1
5 Srxu soxv g*lqirupdwlrqv vxu oh WHF 43 hw o*RDWl/ yrlu zzz1iudqfhwuhvru1jrxy1iu
45
Wlwuh lqgh{ì hw ã uhperxuvhphqw lq0qh 0 fdv gh o*RDWl1 Sdu udssruw ã od vlwxdwlrq suìfì0
ghqwh/ +}w , hvw oh wdx{ g*lq dwlrq ghv sul{ ghv elhqv hw vhuylfhv +lqglfh LQVHH, hw oh fdslwdo uhperxuvì
hvw jdudqwl frqwuh o*lq dwlrq/ vrlw =
dw @ 3 srxu w ? W hw N= pd{ +L+W ,> 4, srxu w @ W
rü L+w, gìvljqh od kdxvvh ghv sul{ hqwuh od gdwh g*ìplvvlrq +gdwh 3, hw od gdwh w1 Hq fodlu/ od ghwwh gh
o*ìphwwhxu ylv0ã0ylv gx gìwhqwhxu hvw ìjdoh ã xq fhuwdlq fdslwdo N uhydorulvì sdu o*lq dwlrq frqvwdwìh
hqwuh od gdwh g*ìplvvlrq hw od gdwh gh uhperxuvhphqw gx fdslwdo1 Wrxwhirlv/ v*lo | d hx gì dwlrq +l1h1
edlvvh ghv sul{/ L+w, ? 4,/ doruv oh fdslwdo uhperxuvì hvw ìjdo dx fdslwdo qrplqdo N/ fh txl hvw xq
dydqwdjh srxu oh gìwhqwhxu gx wlwuh1 Ohv lqwìuíwv rx ohv frxsrqv vrqw fdofxoìv frpph lw @ f=N=L+w,1
Oh fdofxo gh od yduldeoh L+w, vh idlw ã sduwlu gh o*lqglfh LQVHH ghv sul{ gh frqvrppdwlrq +LSF,/
txl hvw o*lqglfh ghv sul{ kdelwxho gh o*LQVHH/ dyhf wrxwhirlv xqh qxdqfh = o*LSF gx prlv q q*ìwdqw
frqqx tx*hq q gh prlv q . 4/ od uìiìuhqfh L+w, srxu w @ mlëph mrxu gx prlv p hvw ìjdoh ã =
LS Fp6 . mQp4 = +LS Fp5 LS Fp6 ,/ rü Qp hvw oh qrpeuh gx mrxu gx prlv p1
Wlwuh }ìur0frxsrq1 F*hvw xq wlwuh sduwlfxolhu txl qh yhuvh dxfxq x{ vdxi hq W 1 Fhv wlwuhv rqw xqh
judqgh lpsruwdqfh wkìrultxh fdu lov yrqw mrxhu oh u÷oh g*xqh edvh fdqrqltxh srxu o*hvsdfh ghv wlwuhv
ã x{ {hv1 Gdqv oh fdv g*xq }ìur0frxsrq/ rq d grqf =
pw @ 3 srxu w ? W hw pW @ 4
Hq sudwltxh/ ohv }ìur0frxsrq dssdudlvvhqw sdu gìphpeuhphqw g*reoljdwlrqv h{lvwdqwhv +rq sduoh gh
Vwulsv,1 Hq Iudqfh/ xq judqg qrpeuh g*RDW vrqw gìphpeuìhv ghsxlv od suhplëuh rsìudwlrq hq mxlq
4<<41
41617 Prgìolvdwlrq ghv sul{ g*reoljdwlrq
Rq v*lqwìuhvvh lfl ã ghv wlwuhv reoljdwdluhv grqw ohv x{ vrqw sduidlwhphqw frqqxv ã o*dydqfh píph vl/ gdqv
ohv h{hpsohv suìfìghqwv/ rq d yx txh fh q*ìwdlw sdv wrxmrxuv oh fdv +WHF 43/ RDWl hwf1,1 Vl rq qrwh p4 /111/
pW fhv x{ {hv/ xq wlwuh reoljdwdluh shxw vh frqfhyrlu frpph xq sruwhihxlooh +l1h1 xqh frpelqdlvrq
olqìdluh, gh wlwuhv }ìur0frxsrq1 Hq gìqlvvdqw xq wlwuh }ìur0frxsrq frpph xq wlwuh txl yhuvh 4 hxur ã xqh
gdwh w ixwxuh/ doruv xq wlwuh reoljdwdluh hvw xq sruwhihxlooh frpsrvì gh p4 wlwuhv }ìur0frxsrq g*ìfkìdqfh
4/ 111/ pW wlwuhv }ìur0frxsrq g*ìfkìdqfh W 1
Sdu frqvìtxhqw/ oh sul{ g*xqh reoljdwlrq v*ìfulw =
S @ p4 =E+3> 4, . == . pW =E+3> W ,
rü E+3> w, hvw oh sul{ g*xq wlwuh }ìur0frxsrq dxmrxug*kxl +gdwh 3, txl yhuvh xq hxur ã od gdwh w1 Rq
dsshooh dxvvl fhv sul{ gh }ìur0frxsrq Glvfrxqw Idfwruv1 Lov shuphwwhqw gdqv xq ghx{lëph whpsv gh
gìqlu od qrwlrq gh wdx{ g*lqwìuíw1 Lo hvw lpsruwdqw gh qrwhu txh ohv glìuhqwhv qrwlrqv gh wdx{ g*lqwìuíw
tx*rq uhqfrqwuhud qh vrqw tx*xqh pdqlëuh g*h{sulphu oh sul{1 Dlqvl/ rq gìqlw glìuhqwhv qrwlrqv gh wdx{
g*lqwìuíw frpph =
4
E+w> W , @
^4 . u+w> W ,`W w
rx =
4
E+w> W , @
^4 . +W w,=u+w> W ,`
rx hqfruh =
E+w> W , @ 4 +W w,=u+w> W ,
Fhv wurlv qrwlrqv +uhvshfwlyhphqw wdx{ g*lqwìuíw dfwxdulho/ srvw0frpswì/ suì0frpswì, vrqw fhuwhv
ìtxlydohqwhv dx suhplhu ruguh pdlv shxyhqw lqgxluh ghv ìfduwv vxevwdqwlhov qxpìultxhphqw1 Lo hvw grqf
lpsruwdqw gh vdyrlu/ srxu fkdtxh pdufkì/ od frqyhqwlrq gh wdx{ g*lqwìuíw xwlolvìh/ od idêrq gh frpswhu
ohv gxuìhv +hq mrxuv/ prlv/ iudfwlrq g*dqqìhv hwf1,1
Dx wrwdo/ hq suhqdqw od frqyhqwlrq dfwxdulhooh/ oh sul{ g*xqh reoljdwlrq dxmrxug*kxl +gdwh 3, v*ìfulw =
S @
Hfroh Sro|whfkqltxh
pW
p4
. === .
W
^4 . u+3> 4,`
^4 . u+3> W ,`
+415,
46
irupxoh glwh irupxoh g*dfwxdolvdwlrq txl hvw gdqv oh píph hvsulw txh od irupxoh gh ydorulvdwlrq ghv
dfwlrqv +414,1
Ohv pdufkìv gìqlvvhqw ìjdohphqw od qrwlrq gh wdx{ gh uhqghphqw dfwxdulho W UD +|lhog0wr0pdwxulw| rx
lqwhuqdo udwh ri uhwxuq hq dqjodlv, g*xqh reoljdwlrq sdu od irupxoh =
S@
pW
p4
. === .
+4 . W UD,
^4 . W UD`W
Vl rq frpsduh fhwwh irupxoh dyhf od irupxoh g*dfwxdolvdwlrq +415,/ rq shxw wrxmrxuv frqvlgìuhu txh oh WUD
hvw xqh vruwh gh pr|hqqh srqgìuìh ghv wdx{ }ìur0frxsrq1 Xqh irlv gh soxv/ ohv glìuhqwhv qrwlrqv gh wdx{
g*lqwìuíw qh vrqw tx*xqh idêrq g*h{sulphu od qrwlrq gh sul{/ txl hooh hvw qrq0dpeljxh1
Hqq/ vl rq v*lqvsluh gh fh txl d ìwì glw vxu od prgìolvdwlrq ghv sul{ g*dfwlrqv/ rq shxw vh sodfhu gdqv xq
prqgh gìwhuplqlvwh rü xq lqyhvwlvvhxu gìwlhqw ã od gdwh 3 xqh vrpph E+3> W , hxurv1 Lo shxw vrlw od sodfhu
vxu xqh sìulrgh hw lo rewlhqgud =
E+3> W ,= ^4 . u+3> 4,`
vrlw dfkhwhu xq wlwuh }ìur0frxsrq g*ìfkìdqfh W +grqw oh sul{ dxmrxug*kxl hvw mxvwhphqw E+3> W ,, hw txl
ydxgud E+4> W , hq 41 Vl oh prqgh hvw gìwhuplqlvwh/ fhv ghx{ vwudwìjlhv grlyhqw íwuh ìtxlydohqwhv =
E+4> W , @ E+3> W ,=+4 . u+3> 4,,
rx hqfruh =
E+4> W , @
E+3> W ,
E+3> 4,
Dlqvl/ oh sul{ ghpdlq vh gìgxlw sdu uìfxuuhqfh ghv sul{ dxmrxug*kxl1 Frpph oh prqgh q*hvw sdv gìwhupl0
qlvwh/ rq prgìolvhud o*ìyroxwlrq ghv sul{ hq udmrxwdqw xq doìd %4 gdqv fhwwh ìtxdwlrq1
41618 Wdx{ g*lqwìuíw hw wkìrulh ìfrqrpltxh
Ohv wdx{ g*lqwìuíw vrqw gìqlv lfl ã sduwlu ghv sul{ +ghv wlwuhv ã x{ {hv whov txh ohv reoljdwlrqv,1 Lo v*djlw
elhq gh od píph qrwlrq lqwxlwlyh kdelwxhooh = u+w> w.4, sdu h{hpsoh gìvljqh elhq oh wdx{ gh uhqghphqw g*xq
sodfhphqw g*xq hxur vxu od sìulrgh +w> w . 4,1 Fh wdx{ phvxuh od suìiìuhqfh ghv lqglylgxv srxu oh suìvhqw1
Od suìiìuhqfh qdwxuhooh gh wrxw lqglylgx hvw ìylghpphqw gh frqvrpphu dxmrxug*kxl soxw÷w txh g*dwwhqguh
ghpdlq/ qh vhudlw0fh txh sdufh txh o*lqglylgx d xqh suredelolwì qrq qxooh g*íwuh pruw ghpdlq1 Ohv sul{
ghv }ìur0frxsrq txl phvxuhqw h{dfwhphqw fhwwh lgìh +l1h1 oh sul{ g*xq hxur ghpdlq ydxw prlqv tx*xq hxur
glvsrqleoh dxmrxug*kxl, grlyhqw grqf íwuh lqiìulhxuv ã 41 Gh idêrq ìtxlydohqwh/ ohv wdx{ g*lqwìuíw grlyhqw
íwuh srvlwliv1 Hqq/ ghv wdx{ g*lqwìuíw ìohyìv vljqdohqw xqh iruwh suìiìuhqfh srxu oh suìvhqw = ohv lqglylgxv
h{ljhqw xq gìgrppdjhphqw ìohyì +l1h1 ghv lqwìuíwv ìohyìv, srxu qh sdv frqvrpphu dxmrxug*kxl hw uhsruwhu
fhwwh frqvrppdwlrq ã soxv wdug1
Oruvtx*rq frqvlgëuh ghv wdx{ g*lqwìuíw vxu xq sodfhphqw prqìwdluh +l1h1 rq ìfkdqjh ghv hxurv g*dxmrxug*kxl
frqwuh ghv hxurv ghpdlq,/ rq sduoh doruv gh wdx{ qrplqdo1 Oruvtxh oh elhq q*hvw sdv od prqqdlh pdlv xq
dxwuh elhq/ rq sduoh gh wdx{ uìho1 O*ru hvw xq h{hpsoh = rq shxw lpdjlqhu xq }ìur0frxsrq txl yhuvh xq
judpph g*ru hq W hw grqf oh sul{ gh fh wlwuh h{sulpì hq judpph g*ru dxmrxug*kxl shuphw gh gìqlu xq wdx{
g*lqwìuíw uìho1 Ohv ìfrqrplvwhv dsshoohqw wdx{ uìho od txdqwlwì = wdx{ qrplqdo prlqv wdx{ g*lq dwlrq +ghv
sul{ gh frqvrppdwlrq ghv pìqdjhv/ hq jìqìudo,/ fh txl hvw krprjëqh ã od qrwlrq suìfìghqwh ã frqglwlrq
gh qh soxv frqvlgìuhu o*ru pdlv xq sdqlhu uhsuìvhqwdwli gh od frqvrppdwlrq ghv pìqdjhv +ã o*lqvwdu gx
sdqlhu gh o*LQVHH,1
Rq sduoh gh frxueh sdu whuph ghv wdx{ g*lqwìuíw rx soxv vlpsohphqw frxueh ghv wdx{ g*lqwìuíw
rx hqfruh jdpph ghv wdx{ g*lqwìuíw1 Fhv whuphv gìvljqhqw od frxueh revhuyìh ã xq lqvwdqw w hw txl
grqqh oh wdx{ g*lqwìuíw u+w> w . k, srxu k yduldqw gh 3 ã o*lqql +hq sudwltxh 63 dqv,1
Frpph srxu ohv dfwlrqv/ ohv sul{ ghv wlwuhv }ìur0frxsrq +hw grqf ohv sul{ ghv reoljdwlrqv, ydulhqw frqvwdp0
phqw1 Dlqvl/ fkdtxh srlqw gh od frxueh ghv wdx{ erxjh g*xq lqvwdqw ã o*dxwuh1 D o*lqyhuvh ghv dfwlrqv/ rq
q*d sdv ddluh ã xq surfhvvxv doìdwrluh gh glphqvlrq 4 +l1h1 oh sul{ g*xqh dfwlrq, pdlv ã xq surfhvvxv
47
gh glphqvlrq lqqlh +l1h1 ohv sul{ ghv }ìur0frxsrq srxu wrxwhv ohv pdwxulwìv,1 Qrxv yhuurqv txh ohv lgìhv
qdqflëuhv xwlolvìhv srxu prgìolvhu xq srlqw +fdv ghv dfwlrqv, rx xqh frxueh +fdv ghv wdx{ g*lqwìuíw, vrqw
irqgdphqwdohphqw ohv píphv1 Hq uhydqfkh/ oh wudlwhphqw pdwkìpdwltxh vhud glìuhqw hqfruh txh/ gdqv
o*ìfudvdqwh pdmrulwì ghv prgëohv/ rq uìgxlw od glphqvlrq lqqlh ghv prxyhphqwv gh od frxueh ghv wdx{ ã
ghv glphqvlrqv qlhv1 Dlqvl/ rq frqvlgìuhud sdu h{hpsoh txh/ oruvtxh od frxueh ghv wdx{ g*lqwìuíw erxjh/
rq shxw uìgxluh fh prxyhphqw 0 dx 4hu ruguh 0 ã xq prxyhphqw gh wudqvodwlrq/ dx 5lëph ruguh/ ã xq
prxyhphqw gh shqwlfdwlrq rx g*dsodwlvvhphqw/ dx 6ëph ruguh ã xqh yduldwlrq gh od frqfdylwì +gìulyìh
vhfrqgh gh od frxueh, hwf1 Vxlydqw o*ruguh dxtxho rq v*duuíwh/ oh sureoëph dxud xqh glphqvlrq soxv rx
prlqv judqgh pdlv wrxmrxuv qlh/ fh txl vlpsolh judqghphqw ohv fkrvhv111
D o*lqyhuvh ghv dfwlrqv/ lo q*| d sdv g*lqfhuwlwxgh vxu ohv x{ ixwxuv +kruplv wrxv ohv fdv g*reoljdwlrqv ã
wdx{ yduldeoh/ lqgh{ìhv vxu o*lq dwlrq hwf1, hw ohv yduldwlrqv gh sul{ vrqw grqf hvvhqwlhoohphqw jrxyhuqìhv
sdu ohv yduldwlrqv ghv sdudpëwuhv ghv lqglylgxv/ qrwdpphqw od suìiìuhqfh srxu oh suìvhqw sdu udssruw dx{
gdwhv ixwxuhv whooh tx*hooh vh uh ëwh gdqv o*ìtxloleuh ruh 0 ghpdqgh gh wlwuhv1 Wrxwhirlv/ od vlwxdwlrq hvw
soxv frpsoh{h txh fhod1 Ohv wdx{ g*lqwìuíw qh uh ëwhqw sdv xqltxhphqw od suìiìuhqfh srxu oh suìvhqw pdlv
ìjdohphqw od srolwltxh prqìwdluh1 Hq hhw/ od edqtxh fhqwudoh g*xq sd|v +od Edqtxh Fhqwudoh Hxursìhqqh
srxu od }rqh hxur/ od Ihghudo Uhvhuyh srxu ohv Hwdwv0Xqlv, shxw frqwu÷ohu ohv wdx{ g*lqwìuíw frxuwv +l1h1
oh wdx{ g*lqwìuíw srxu xq sodfhphqw g*dxmrxug*kxl ã dxmrxug*kxl soxv txhotxhv mrxuv, fdu/ ìwdqw fdsdeoh
gh fuìhu gh od prqqdlh/ hooh shxw suíwhu dxwdqw tx*hooh yhxw hw dx wdx{ g*lqwìuíw tx*hooh vrxkdlwh +dyhf
ghv olplwhv lqvwlwxwlrqqhoohv elhq0vþu,1 Gdqv fhv frqglwlrqv/ hooh frqwu÷oh dvvh} elhq ohv wdx{ frxuwv pdlv
prlqv ohv pdwxulwìv soxv ìohyìhv1 Sdu frqvìtxhqw/ ohv prxyhphqwv gh od frxueh ghv wdx{ g*lqwìuíw uh ëwh/
srxu ohv pdwxulwìv orqjxhv/ o*ruh hw od ghpdqgh gh wlwuhv hw grqf hq ghuqlëuh dqdo|vh od suìiìuhqfh srxu
oh suìvhqw ghv lqglylgxv/ hw/ srxu ohv pdwxulwìv frxuwhv/ od srolwltxh prqìwdluh1
Ohv fkrvhv vrqw hqfruh soxv frpsoltxìhv txdqg rq vrqjh tx*lo q*hvw sdv srvvleoh gh vìsduhu ohv ghx{
skìqrpëqhv1 Lov vhurqw gh idfwr olìv = hq hhw/ xq lqglylgx txl yhxw sodfhu vrq dujhqw vxu sdu h{hpsoh
43 dqv shxw wrxmrxuv dfkhwhu xq wlwuh }ìur0frxsrq gh pdwxulwì 43 dqv/ rx dowhuqdwlyhphqw dfkhwhu xq
}ìur0frxsrq gh pdwxulwì xq dq dyhf o*lgìh gh oh uhyhqguh o*dqqìh surfkdlqh srxu hq udfkhwhu xq dxwuh gh
pdwxulwì xq dq hwf1 Fhv ghx{ sodfhphqwv vrqw gh pdwxulwì 43 dqv pdlv vrqw wuëv glìuhqwv fdu rq qh vdlw sdv
dxmrxug*kxl frpelhq ydxgud gdqv xq dq oh }ìur0frxsrq gh pdwxulwì xq dq1 Hq fodlu/ rq qh frqqdlw sdv od
frxueh ghv wdx{ txl vh uìdolvhud gdqv xq dq1 Wrxwhirlv/ v*djlvvdqw gh sodfhphqwv ã 43 dqv/ lo | d fhuwdlqhphqw
xq olhq = hq pr|hqqh/ ohv uhqghphqwv qh grlyhqw sdv íwuh wurs glìuhqwv1 Gdqv oh suhplhu fdv/ oh wdx{ gh
4 ^u+3> 4, . u+4> 5, . === . u+<> 43,`
uhqghphqw dqqxho hvw u+3> 43,/ gdqv oh vhfrqg fdv/ lo hvw soxw÷w ìjdo ã 43
+dx suhplhu ruguh,1 Vl wrxv ohv wdx{ frxuwv u+w> w . 4, vrqw soxw÷w jrxyhuqìv sdu od srolwltxh prqìwdluh hw
ohv wdx{ orqjv u+3> 43, sdu o*ruh hw od ghpdqgh/ hw vl hq soxv rq dmrxwh o*lgìh txh u+3> 43, qh grlw sdv íwuh
4 ^u+3> 4, . u+4> 5, . === . u+<> 43,`/ doruv rq hvw irufì gh frqvwdwhu txh ohv
wurs glìuhqw gh o*hvsìudqfh gh 43
ghx{ skìqrpëqhv +suìiìuhqfh srxu oh suìvhqw hw srolwltxh prqìwdluh, lqwhudjlvvhqw1
O*lgìh txh ohv wdx{ orqjv grlyhqw íwuh surfkhv g*xqh fhuwdlqh hvsìudqfh gh od pr|hqqh ghv wdx{ frxuwv
lqwhupìgldluhv hvw ã od edvh ghv udlvrqqhphqwv ghv ìfrqrplvwhv hw v*dsshooh k|srwkëvh gh od wkìrulh
ghv dqwlflsdwlrqv1 Hq h{sorlwdqw fhwwh k|srwkëvh/ xqh kdxvvh ghv wdx{ g*lqwìuíw uìyëoh txh ohv pdufkìv
dqwlflshqw ghv kdxvvhv ghv wdx{ frxuwv sdu od edqtxh fhqwudoh +l1h1 xq uhvvhuuhphqw gh od srolwltxh prqì0
wdluh,/ hw lqyhuvhphqw hq fdv gh edlvvh +l1h1 xq dvvrxsolvvhphqw gh od srolwltxh prqìwdluh,1 Gh idlw/ fhwwh
julooh g*dqdo|vh uhqg dvvh} elhq frpswh ghv whqgdqfhv gh irqg ghv prxyhphqwv gh wdx{ g*lqwìuíw1
Hqq/ lo idxw uhpdutxhu txh ohv wdx{ g*lqwìuíw frpsruwhqw xqh glphqvlrq vxssoìphqwdluh ulvtxh gh
frqwuhsduwlh / f*hvw0ã0gluh xq ulvtxh txh o*ìphwwhxu gx wlwuh 0 o*hpsuxqwhxu 0 idvvh gìidxw hw vrlw grqf gdqv
o*lpsrvvlelolwì gh uhperxuvhu vd ghwwh1 Fh ulvtxh hvw qìjoljhdeoh srxu ohv ìplvvlrqv gh od soxsduw ghv Hwdwv
+rq sduoh gh ulvtxh vrxyhudlq, pdlv lo o*hvw prlqv srxu g*dxwuhv dfwhxuv frpph ohv froohfwlylwìv orfdohv1 Lo
o*hvw wuëv qhwwhphqw prlqv srxu erq qrpeuh g*hqwuhsulvhv ìphwwulfhv = oh wdx{ g*lqwìuíw txh grlyhqw sd|hu
fhv ìphwwhxuv hvw doruv xqh irqfwlrq furlvvdqwh gh ohxu ulvtxh gh gìidxw1 Hq hhw/ ohv dfkhwhxuv gh wlwuhv
h{ljhqw xq wdx{ g*lqwìuíw g*dxwdqw soxv ìohyì txh o*ìphwwhxu gx wlwuh d xq ulvtxh gh qh mdpdlv uhperxuvhu
vd ghwwh +ã o*h{wuíph/ rq sduoh gh pdufkì ghv kljk |lhogv ,1 Rq gìfrpsrvh doruv ohv wdx{ g*lqwìuíw
vxu fh pdufkì hq xqh sduwlh wdx{ g*lqwìuíw sxu hw xqh sduwlh ulvtxh gh gìidxw 1
41619 Hqfruh txhotxhv qrwlrqv srxu qlu
Rq grqqh lfl hq yudf txhotxhv whuphv xwlolvìhv oruvtx*rq sduoh g*reoljdwlrqv =
Hfroh Sro|whfkqltxh
48
od gxudwlrq g*xqh reoljdwlrq hvw gìqlh frpph =
G@
p4
5=p5
W=pW
+4.W UD, . +4.WUD,5 . === . ^4.WUD`W
S
f*hvw0ã0gluh od pr|hqqh srqgìuìh ghv gdwhv ixwxuhv 4/111/ W / od vrpph ghv srqgìudwlrqv ìwdqw ìjdoh
ã 41 Fhwwh txdqwlwì hvw krprjëqh ã xqh gxuìh hw hvw h{dfwhphqw ìjdoh ã W srxu xq }ìur0frxsrq1 Lo
hvw idfloh gh yìulhu txh f*hvw dxvvl od gìulyìh gx orjdulwkph gx sul{ g*xqh reoljdwlrq sdu udssruw dx
wdx{ gh uhqghphqw dfwxdulho W UD +dx vljqh suëv hw ã xq idfwhxu 4 . UD suëv,1 Sdu frqvìtxhqw/ G
v*dsshooh dxvvl vhqvlelolwì gx sul{ > hooh phvxuh gh frpelhq +hq srxufhqwdjh, oh sul{ gx wlwuh ydulh 0
dx 4hu ruguh 0 oruvtxh W UD ydulh1 Txh fhod vljqlh0w0lo B Vxssrvrqv txh od frxueh ghv wdx{ vrlw sodwh
dxmrxug*kxl/ l1h1 u+3> w, @ u @ frqvwdqwh +srxu wrxw w,/ doruv W UD @ u hw G phvxuh oh srxufhqwdjh gh
yduldwlrq gx sul{ txdqg od frxueh ghv wdx{ g*lqwìuíw d xq prxyhphqw gh wudqvodwlrq1 G phvxuh grqf
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surfkdlq frxsrq v*dsshooh oh frxsrq frxux1
frqyhqwlrqv gh gdwh = hq sudwltxh/ lo hvw wuëv lpsruwdqw gh vdyrlu frpphqw ohv gxuìhv vrqw phvxuìhv1
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417 Ohv pdwlëuhv suhplëuhv
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hw ghv wdx{ gh fkdqjh6 1
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shx gh jhqv rqw uìhoohphqw ehvrlq g*dfkhwhu xqh fdujdlvrq gh sìwuroh111, = ohv wudqvdfwlrqv frqfhuqhqw
vxuwrxw ohv surgxlwv gìulyìv +frqwudwv ã whuph txh qrxv yhuurqv sdu od vxlwh,1 Ohv lqwhuyhqdqwv fkhufkhqw
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frxyulu frqwuh xqh kdxvvh ghv sul{ gx sìwuroh/ xq surgxfwhxu gh pdòv txl yhxw suhqguh xqh dvvxudqfh
frqwuh xqh edlvvh gx sul{ gx pdòv hwf1 Frpph o*hqvhljqh od wkìrulh ìfrqrpltxh/ o*lqwurgxfwlrq g*xqh
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g*xqh dfwlrq hvw oh wdx{ gh fkdqjh gh o*hxur 2 dfwlrq/ f*hvw0ã0gluh oh sul{ hq hxur srxu dftxìulu xq
wlwuh g*dfwlrq1 Ghylvhv hw dfwlrqv vrqw ghv qxpìudluhv/ f*hvw0ã0gluh xq ìwdorq gh ydohxu1 Dlqvl/ frpph ohv
sul{ g*dfwlrqv/ ohv wdx{ gh fkdqjh xfwxhqw dx frxuv gx whpsv1 Hq rxwuh/ od prgìolvdwlrq vhud vlplodluh1
41814 Prgìolvdwlrq ghv wdx{ gh fkdqjh
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uìho pdlv hq pr|hqqh vhxohphqw1 Fh jhquh g*k|srwkëvh uhoëyh gx píph hvsulw txh o*k|srwkëvh
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grlyhqw dyrlu/ vrxv fhuwdlqhv frqglwlrqv/ oh píph uhqghphqw1 Fh w|sh gh udlvrqqhphqw vhud irupdolvì sdu
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xqh rsìudwlrq txl shuphw gh idluh xq jdlq wrxmrxuv srvlwli vdqv ulhq lqyhvwlu1 Glw gh fhwwh idêrq/ lo sdudñw
mxvwlì gh vxssrvhu txh fh w|sh g*rsìudwlrqv q*h{lvwh sdv rx/ oruvtx*hoohv h{lvwhqw/ hoohv glvsdudlvvhqw
udslghphqw dx surw ghv suhplhuv lqwhuyhqdqwv txl v*hq vrqw dshuêx1 Fhwwh k|srwkëvh hvw grqf g*dxwdqw
soxv mxvwlìh txh oh pdufkì irqfwlrqqh elhq1
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ã0gluh ohv dfwliv grqw ohv x{ jìqìuìv vrqw ghv irqfwlrqv 0 sdv qìfhvvdluhphqw frqwlqxv 0 ghv sul{ ghv dfwliv
gh edvh1
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phqw lqqlhv hw lo hvw grqf gl!floh g*hq idluh xqh suìvhqwdwlrq h{kdxvwlyh1 Hq rxwuh/ ohv surgxlwv gìulyìv
pìodqjhqw frxudpphqw glìuhqwv w|shv g*dfwliv gh edvh = dfwlrq hw reoljdwlrq/ wdx{ gh fkdqjh hw wdx{
g*lqwìuíw hwf1 Lo hvw grqf gh soxv hq soxv duelwudluh gh vìsduhu surgxlwv gìulyìv dfwlrq hw surgxlwv gìulyìv
wdx{ g*lqwìuíw/ vìsdudwlrq txh qrxv ihurqv wrxw gh píph1
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rqw sulv xqh whooh sodfh ghsxlv od fuìdwlrq gx suhplhu pdufkì rujdqlvì hq 4<:6 +Fklfdjr Erdug Rswlrqv
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phqw/ ohv surgxlwv gìulyìv shuphwwhqw gh vh frxylu frqwuh fhuwdlqv ulvtxhv/ gh od píph idêrq tx*xq frqwudw
g*dvvxudqfh dxwrpreloh frxyuh frqwuh ohv gìjåwv frqvìfxwliv ã xq dfflghqw dxwrpreloh1 Hq hhw/ xq frqwudw
g*dvvxudqfh +uhvs1 xq dfwli gìulyì, hvw xq surgxlw frqwlqjhqw txl jìqëuh xq x{ oruvtx*xq ìyëqhphqw who
tx*xq dfflghqw +uhvs1 xqh yduldwlrq gx sul{ g*xqh dfwlrq/ xqh edlvvh ghv wdx{ g*lqwìuíw hwf1, vh uìdolvh1 Ru/
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oxl grqqh od srvvlelolwì gh v*dvvxuhu1
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iruwhphqw lpsruwdwulfh +gh sìwuroh sdu h{hpsoh, vhud wuëv h{srvìh ã xqh kdxvvh gx groodu +sdu udssruw ã
od prqqdlh gh vrq sd|v gh uìvlghqfh, doruv tx*ã o*lqyhuvh xqh hqwuhsulvh h{sruwdwulfh +yhuv ohv Hwdwv0Xqlv,
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surgxlwv sìwurolhuv gdqv vrq surfhvvxv gh surgxfwlrq pdlv txl h{sruwh yhuv ohv Hwdwv0Xqlv = xqh edlvvh
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ohxu shuphwwhqw gh v*dvvxuhu pxwxhoohphqw1
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lov shuphwwhqw ã ghv djhqwv gh vh uhqfrqwuhu srxu idluh ghv ìfkdqjhv pxwxhoohphqw eìqìtxhv1 Wrxwhirlv/
fhv ìfkdqjhv qh vrqw sdv ghv elhqv sk|vltxhv pdlv ghv txdqwlwìv gh ulvtxh 1
515 Ohv rswlrqv gh edvh
51514 Gìqlwlrqv
Gìqlwlrq 4 Xqh rswlrq vwdqgdug g*dfkdw +l1h1 fdoo, hvw xq gurlw g*dfkhwhu xq dfwli grqqì ã xq sul{
frqyhqx ã o*dydqfh/ ã xqh gdwh ixwxuh grqqìh1
Lo v*djlw grqf g*xq gurlw hw qrq g*xqh reoljdwlrq = oh gìwhqwhxu gh o*rswlrq d wrxmrxuv oh gurlw gh qh sdv
h{hufhu vrq rswlrq/ sdu h{hpsoh hw vxuwrxw vl o*h{huflfh qh oxl hvw sdv surwdeoh1 Oh sul{ frqyhqx ã
o*dydqfh v*dsshooh oh sul{ g*h{huflfh +l1h1 oh vwulnh hq dqjodlv, = f*hvw oh idlw tx*lo vrlw frqqx ã o*dydqfh
txl grqqh gh od ydohxu ã o*rswlrq1 Hq hhw/ xqh rswlrq txl grqqhudlw oh gurlw g*dfkhwhu xqh dfwlrq ghpdlq
dx sul{ gh pdufkì ghpdlq q*d ìylghpphqw dxfxqh ydohxu1 Od gdwh ixwxuh grqqìh v*dsshooh od gdwh
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sdv vrq rswlrq/ uhêrlw grqf 3 hxur/ hw frqvwdwh h{ srvw tx*lo d gìshqvì hq sxuh shuwh oh sul{ g*dfkdw
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g*dvvxudqfh v*lo q*d sdv hx g*dfflghqw shqgdqw od sìulrgh1
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gh vhv rswlrqv1
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rq shxw dfkhwhu xqh rswlrq gh yhqwh rx od yhqguh txdqg rq hq gìwlhqw1 Pdwkìpdwltxhphqw/ oh suro gh
jdlq hvw ìylghpphqw oh vxlydqw =
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rswlrqv vxu glìuhqwv wlwuhv g*dfwlrqv hw ìjdohphqw vxu ghv lqglfhv1 O*lqglfh FDF 73 v|qwkìwlvh o*ìyroxwlrq
ghv 73 soxv jurvvhv fdslwdolvdwlrqv gh od Erxuvh gh Sdulv hw grqf ìyroxh dx mrxu oh mrxu frpph oh ihudlw
q*lpsruwh txhooh dfwlrq1 Rq shxw grqf frqvlgìuhu ghv rswlrqv vxu fhw lqglfh = fh w|sh g*rswlrq shuphw
grqf gh vh frxyulu rx gh vsìfxohu vxu xqh kdxvvh rx xqh edlvvh jìqìudoh gx pdufkì ghv dfwlrqv +wrxw dx
prlqv ghv 73 soxv jurvvhv fdslwdolvdwlrqv,1 D wlwuh g*h{hpsoh/ yrlfl od qrwlfh h{solfdwlyh gh fhwwh rswlrq
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Dfwli vrxv0mdfhqw = O*lqglfh FDF 73/ frpsrvì ghv 73 ydohxuv ohv soxv uhsuìvhqwdwlyhv ghv glìuhqwv
vhfwhxuv g*dfwlylwì uhsuìvhqwìv vxu oh suhplhu pdufkì gh od Erxuvh gh Sdulv1 Od jhvwlrq gh o*ìfkdqwloorq ghv
vrflìwìv uhsuìvhqwìhv gdqv o*lqglfh FDF 73 hvw sodfìh vrxv o*dxwrulwì g*xq frqvhlo vflhqwltxh g*h{shuwv lq0
gìshqgdqwv txl hq gìflgh oh uhqrxyhoohphqw oruvtxh oìyroxwlrq gx pdufkì rx ohv prglfdwlrqv lqwhuyhqdqw
gdqv oh fdslwdo ghv vrflìwìv oh qìfhvvlwhqw1
O*lqglfh FDF 73 hvw fdofxoì hq frqwlqx sdu Hxurqh{w Sdulv VD hw glxvì wrxwhv ohv 63 vhfrqghv1 O*dwwhqwlrq
ghv grqqhxuv g*ruguhv hvw dsshoìh vxu oh idlw txh oruvtxh ohv ydohxuv uhsuìvhqwdqw soxv gh 68 ( gh od
fdslwdolvdwlrq gh o*lqglfh FDF 73 qh vrqw prphqwdqìphqw sdv vxvfhswleohv g*íwuh frwìhv +vrlw gx idlw g*xq
lqflghqw whfkqltxh/ vrlw gx idlw gh phvxuhv gh uìvhuydwlrq gh frwdwlrq dhfwdqw ohv ydohxuv gh o*ìfkdqwloorq,/
xq ìfodluhxu gh whqgdqfh hvw vxevwlwxì ã o*lqglfh = o*lqirupdwlrq sxeolìh gdqv fhv flufrqvwdqfhv h{sulph od
yduldwlrq gh frxuv ghv vhxohv ydohxuv txl irqw o*remhw g*xqh frwdwlrq1
O*h{wudsrodwlrq gh fhwwh yduldwlrq ã fhooh gh o*lqglfh wrxw hqwlhu qh shxw íwuh hhfwxìh tx*dyhf suxghqfh
hw hq irqfwlrq g*k|srwkëvhv txl vrqw gh od vhxoh uhvsrqvdelolwì gh ohxuv dxwhxuv1
Xqlwì gh qìjrfldwlrq = O*xqlwì gh qìjrfldwlrq hvw frqvwlwxìh g*xq frqwudw gdqv ohtxho fkdtxh srlqw
g*lqglfh hvw dhfwì g*xqh ydohxu gh 4 hxur1 Od wdlooh gx frqwudw hvw ìjdoh ã od ydohxu gh o*lqglfh { 4 hxur
hw vd ydohxu dx frxuv gh o*rswlrq { 4 hxur1
Hfkhorq gh frwdwlrq = O*ìfduw plqlpdo hqwuh ghx{ frwdwlrqv hvw {ì ã 3/43 srlqw g*lqglfh/ vrlw 3/4 hxur
sdu frqwudw1
Hfkìdqfhv ghv rswlrqv = Ohv rswlrqv shxyhqw íwuh qìjrflìhv mxvtx*dx ghuqlhu mrxu gh erxuvh gx prlv
g*ìfkìdqfh1 Ohv qìjrfldwlrqv sruwhqw vxu kxlw ìfkìdqfhv jolvvdqwhv = 6 phqvxhoohv/ 6 wulphvwulhoohv gx f|foh
pduv/ mxlq/ vhswhpeuh/ gìfhpeuh hw 5 vhphvwulhoohv gx f|foh pduv/ vhswhpeuh1
Sul{ g*h{huflfh = Ohv sul{ g*h{huflfh vrqw ghv ydohxuv vwdqgdug {ìhv sdu dmrxw rx vrxvwudfwlrq gh 83
srlqwv g*lqglfh vxu ohv ìfkìdqfhv phqvxhoohv +ìfkìdqfhv glwhv udssurfkìhv,/ 433 srlqwv vxu ohv ìfkìdqfhv
wulphvwulhoohv hw 533 srlqwv vxu ohv ìfkìdqfhv vhphvwulhoohv sdu udssruw dx sul{ g*h{huflfh ã sdulwì1 Ohv
sul{ g*h{huflfh vrqw glìuhqwv srxu ohv rswlrqv g*dfkdw hw ohv rswlrqv gh yhqwh1 Q1E1 Srxu ohv ìfkìdqfhv
rxyhuwhv ã od qìjrfldwlrq dydqw oh 4hu iìyulhu 4<<</ ohv lqwhuydoohv uhvwhqw {ìv ã 483 srlqwv mxvtxã fh
tx*hoohv ghylhqqhqw ghv ìfkìdqfhv udssurfkìhv1 Oruv gh o*rxyhuwxuh g*xqh ìfkìdqfh/ ghv vìulhv vrqw fuììhv
+rswlrqv g*dfkdw hw rswlrqv gh yhqwh, dx{ sul{ g*h{huflfh ohv soxv surfkhv gh od ydohxu gh o*lqglfh/ xq ã
sdulwì hw ghx{ hq0ghkruv ghv frxuv1
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gh 533 srlqwv g*lqglfh/ fuìdwlrq g*rswlrqv g*dfkdw dx{ sul{ g*h{huflfh 7 333/ 7 533/ 7 733 hw g*rswlrqv gh
yhqwh dx{ sul{ g*h{huflfh 7 333/ 6 ;33/ 6 9331
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sul{ g*h{huflfh hq ghkruv ghv frxuv +h{1 = .20 433 srlqwv srxu ohv vìulhv grqw ohv sul{ g*h{huflfh vrqw fuììhv
ã lqwhuydoohv gh 533 srlqwv g*lqglfh,/ gh qrxyhoohv vìulhv vrqw fuììhv dq tx*lo | dlw hq shupdqhqfh/ hw vxu
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vxu fkdfxq ghv w|shv g*rswlrqv +rswlrqv gdfkdw hw rswlrqv gh yhqwh,1
H{huflfh ghv rswlrqv = D o*ìfkìdqfh/ ohv rswlrqv gdqv ohv frxuv vrqw h{hufìhv dxwrpdwltxhphqw/ vdxi
lqvwuxfwlrq frqwudluh gx grqqhxu g*ruguhv1 O*h{huflfh g*xqh rswlrq dfkhwìh grqqh olhx ã o*dvvljqdwlrq
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gh o*rswlrq h{hufìh hw od ydohxu gh o*lqglfh gh oltxlgdwlrq/ ydorulvìh sdu oh qrpeuh gh frqwudwv h{hufìv hw
o*xqlwì gh qìjrfldwlrq +4 hxur,1 O*lqglfh gh oltxlgdwlrq txl vhuw gh uìiìuhqfh ã o*h{huflfh dxwrpdwltxh ghv
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k 73 hw 49 k 33 oh mrxu gh o*ìfkìdqfh +| frpsulv od suhplëuh ydohxu gh o*lqglfh glxvìh dsuëv 49 k 33,1
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whuph ihuph vxu o*lqglfh FDF 73 hvw {ìh ã .20 5:8 srlqwv g*lqglfh sdu udssruw dx frxuv gh frpshqvdwlrq
gh od yhlooh1 Oruvtxh fhwwh olplwh hvw iudqfklh vxu xqh ghv ghx{ ìfkìdqfhv ohv soxv surfkhv/ ohv frwdwlrqv
shxyhqw íwuh prphqwdqìphqw vxvshqgxhv vxu ohv frqwudwv g*rswlrqv hw ohv frqwudwv ã whuph ihuph frwìv
vxu oh PRQHS hw sruwdqw vxu o*lqglfh FDF 731 Lo shxw/ sdu dloohxuv/ íwuh surfìgì ã xq dssho gh jdudqwlh
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H{hpsoh = vl oh frxuv gh frpshqvdwlrq gh od yhlooh hvw gh 9 833 = oh vhxlo gh uìvhuydwlrq srwhqwlhooh ã
od kdxvvh hvw 9 833 . 5:8@ 9 ::8 > oh vhxlo gh uìvhuydwlrq srwhqwlhooh ã od edlvvh hvw 9 833 0 5:8 @ 9
5581 Gh píph/ fh glvsrvlwli gh frxsh0flufxlw shxw wurxyhu ã v*dssoltxhu hq fdv gh gìvìtxloleuh gx pdufkì
hqwudñqdqw xqh uìvhuydwlrq ghv frwdwlrqv vxu xq ìfkdqwloorq gh ydohxuv gh olqglfh uhsuìvhqwdqw hqvhpeoh
soxv gh :8( gh od fdslwdolvdwlrq gh o*lqglfh FDF 731
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Gìs÷w gh jdudqwlh = Oh sulqflsh uhwhqx frqvlvwh ã joredolvhu ohv srvlwlrqv gìwhqxhv sdu xq píph
grqqhxu g*ruguhv hq rswlrqv FDF 73 ã pdwxulwì frxuwh hw orqjxh +frqwudwv gh 4 hxur hw frqwudwv gh
urpsxv, hw ã ydorulvhu oh sruwhihxlooh dlqvl rewhqx vxu od edvh g*xqh vhxoh hw píph k|srwkëvh g*ìyroxwlrq
gh o*lqglfh1 Ohv grqqhxuv g*ruguhv gìwhqdqw xqh srvlwlrq joredoh qhwwh yhqghxu vh yrlhqw dsshohu xq gìs÷w
gh jdudqwlh dmxvwì txrwlglhqqhphqw/ g*xq prqwdqw uhsuìvhqwdqw od ydohxu oltxlgdwlyh od soxv gìidyrudeoh
gh ohxu srvlwlrq gdqv o*k|srwkëvh g*xqh yduldwlrq gh od ydohxu gh o*lqglfh gh .20 633 srlqwv sdu udssruw ã
o*lqglfh gh fo÷wxuh gx mrxu1 O*dpsohxu gh fhwwh yduldwlrq shxw íwuh dmxvwìh sdu od fkdpeuh gh frpshqvdwlrq
hq irqfwlrq ghv frqglwlrqv gx pdufkì1
Od jdudqwlh shxw íwuh frqvwlwxìh g*hvsëfhv hq hxurv rx hq ghylvhv/ gh Erqv gx Wuìvru +EWI hw EWDQ,/
g*RDW/ gh W1Eloov/ gh Exqgv/ gh wlwuhv vxssruw rx gh wlwuhv hqwudqw gdqv od frpsrvlwlrq gh o*lqglfh FDF
73/ ghv lqglfhv Grz Mrqhv VWR[[VP vxssruwv g*lqvwuxphqwv qdqflhuv frwìv vxu oh PRQHS/ g*dfwlrqv
gh VLFDY rx gh sduwv gh irqgv frppxqv grqw od olvwh hvw duuíwìh sdu FOHDUQHWVEI VD/ hw gh wrxw
dxwuh dfwli djuìì sdu FOHDUQHWVEI VD1 Ohv dfwliv dgplv hq jdudqwlh/ dxwuhv txh ohv hvsëfhv hq hxurv/
vrqw ydorulvìv txrwlglhqqhphqw ã ohxu frqwuhydohxu hq hxurv vdjlvvdqw ghv ghylvhv/ ã ohxu ydohxu gh pdufkì
rx ã ohxu ydohxu qrplqdoh v*djlvvdqw ghv wlwuhv gh fuìdqfh/ ã ohxu ydohxu gh pdufkì vdjlvvdqw ghv wlwuhv
gh fdslwdo/ hw ã ohxu ydohxu oltxlgdwlyh v*djlvvdqw ghv RSFYP/ dyhf dssolfdwlrq oh fdv ìfkìdqw g*xq wdx{
g*dedwwhphqw {ì sdu FOHDUQHWVEI VD hq irqfwlrq gh od qdwxuh hw gh od pdwxulwì ghv wlwuhv1
Rujdqlvdwlrq ghv wudqvdfwlrqv = Ohv frwdwlrqv v*hhfwxhqw hq frqwlqx gh < k 35 ã 4: k 63 +od frwdwlrq
ghv vìulhv duulydqw ã ìfkìdqfh fhvvh ã 49 k 33 oh mrxu gh o*ìfkìdqfh,1
Iudlv = Ohv qìjrfldwlrqv hq rswlrqv ã orqj whuph vxu o*lqglfh FDF 73 hhfwxìhv vxu oh PRQHS grqqhqw
olhx ã od shufhswlrq sdu Hxurqh{w Sdulv VD g*xqh frpplvvlrq gh qìjrfldwlrq gh 3/35 hxur sdu frqwudw/ hw
g*xqh frpplvvlrq gh frpshqvdwlrq g*xq prqwdqw yduldeoh hq irqfwlrq gx prqwdqw ghv fdslwdx{ qìjrflìv1
Ohv wdx{ dssolfdeohv dx prqwdqw gh fkdtxh qìjrfldwlrq vrqw ohv vxlydqwv =
Mxvtx*ã 483 333 hxurv = 3/48 (
gh 483 333 hxurv ã :83 333 hxurv = 3/43 (
gh :83 333 hxurv ã 4 833 333 hxurv = 3/38 (
dx0ghoã gh 4 833 333 hxurv = 3/358 (
Oh prqwdqw gh od frpplvvlrq gh qìjrfldwlrq shuêxh sdu rsìudwlrq qh shxw íwuh vxsìulhxu ã 4 ( ghv
fdslwdx{ wudlwìv1 Ohv frpplvvlrqv vrqw vrxplvhv ã od WYD vhorq od uìjohphqwdwlrq hq yljxhxu1 Qrwd = ohv
fdudfwìulvwltxhv gìfulwhv gdqv od suìvhqwh fkh vrqw fhoohv txl v*dssoltxhqw ã frpswhu gx 6 qryhpeuh 5333
hw vrqw vxvfhswleohv g*íwuh prglìhv1
51516 Irupdwlrq ghv sul{ g*rswlrq
Ohv fkdslwuhv vxlydqwv v*lqwìuhvvhurqw ã o*ìydoxdwlrq ghv rswlrqv1 Od txhvwlrq vhud gh vdyrlu txho olhq gh
frkìuhqfh lo grlw | dyrlu hqwuh oh sul{ g*xqh dfwlrq hw oh sul{ g*xqh rswlrq grqw oh vrxv0mdfhqw hvw fhwwh
dfwlrq1 Wrxwhirlv/ rq shxw gëv ã suìvhqw idluh od olvwh ghv yduldeohv hw sdudpëwuhv vxvfhswleohv g*lq xhqfhu
oh sul{ =
od ydohxu gh o*dfwlrq dxmrxug*kxl >
od gdwh g*h{huflfh = vl sdu h{hpsoh od gdwh g*h{huflfh hvw wuëv surfkh/ od ydohxu gh o*rswlrq vhud surfkh
gh vrq sd|r >
oh sul{ g*h{huflfh = soxv oh sul{ g*h{huflfh hvw ìohyì/ prlqv xqh rswlrq g*dfkdw ã gh fkdqfhv g*íwuh
h{huêdeoh dyhf surw1 Oh sul{ g*xqh rswlrq g*dfkdw hvw grqf xqh irqfwlrq gìfurlvvdqwh gx sul{ g*h{hu0
flfh1
od yrodwlolwì gh o*dfwlrq = fh sdudpëwuh vh phvxuh jurvvr0prgr frpph o*ìfduw0w|sh ghv yduldwlrqv gh
frxuv gh o*dfwlrq1 Lo phvxuh o*lqfhuwlwxgh vxu od ydohxu gh o*dfwlrq1 Soxv fhwwh yrodwlolwì hvw iruwh/ soxv
oh sul{ gh o*dfwlrq d gh fkdqfhv gh sdvvhu dx0ghvvxv rx hq0ghvvrxv gx sul{ g*h{huflfh1
ohv glylghqghv gdqv oh fdv g*xqh dfwlrq txl yhuvh ghv glylghqghv1
Lo h{lvwh xqh uhodwlrq hqwuh oh sul{ g*xqh rswlrq g*dfkdw hw oh sul{ g*xqh rswlrq gh yhqwh/ glwh sdulwì
fdoo0sxw1 Hq hhw/ frqvlgìurqv od vwudwìjlh frqvlvwdqw ã dfkhwhu xqh rswlrq g*dfkdw hw ã yhqguh xqh rswlrq
gh yhqwh/ fhv ghx{ rswlrqv d|dqw píph sul{ g*h{huflfh hw píph gdwh g*h{huflfh1 Oh sd|r uhêx ã od gdwh
g*h{huflfh v*ìfulw grqf =
pd{+VW N> 3, pd{+N VW > 3,
vrlw/ gdqv wrxv ohv fdv =
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Ru/ fh sd|r shxw íwuh dwwhlqw hq dfkhwdqw o*dfwlrq dxmrxug*kxl +srxu od uhyhqguh hq W srxu xq sul{ VW ,
hw hq ìphwwdqw xq }ìur0frxsrq gh qrplqdo N1 Od ydohxu dxmrxug*kxl +gdwh w, gh fhwwh vhfrqgh vwudwìjlh
hvw grqf =
Vw N=E+w> W ,
Ghx{ vwudwìjlhv jìqìudqw h{dfwhphqw ohv píphv sd|rv dx{ píphv gdwhv grlyhqw dyrlu oh píph frþw
+devhqfh g*rssruwxqlwì g*duelwudjh,/ hw grqf =
F+w> W> N, S +w> W> N, @ Vw N=E+w> W ,
rü F+w> W> N, +uhvs1 S +w> W> N,, hvw oh sul{ dxmrxug*kxl +gdwh w, g*xqh rswlrq g*dfkdw +uhvs1 gh yhqwh, gh
sul{ g*h{huflfh N hw gh gdwh g*h{huflfh W 1
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Dlqvl/ sdu h{hpsoh/ xqh rswlrq g*dfkdw gh sul{ g*h{huflfh 78133 hxurv hw gh gdwh g*h{huflfh pduv 5334
ydodlw 4135 hxurv oh ; gìfhpeuh 53331
516 G*dxwuhv rswlrqv hw vwudwìjlhv
Od vhfwlrq suìfìghqwh d gìfulw oh fdv gh edvh/ f*hvw0ã0gluh fhoxl gh o*rswlrq vwdqgdug g*dfkdw rx gh yhqwh1
Rq shxw elhq ìylghpphqw lpdjlqhu xqh lqqlwì gh sd|r srvvleohv1 Rq v*lqwìuhvvh lfl ã txhotxhv h{hpsohv
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51614 Elqdu| rx gljlwdo rswlrqv
Xqh rswlrq g*dfkdw +uhvs1 gh yhqwh, elqdluh rx gljlwdo hvw xqh rswlrq txl yhuvh 4 hxur vl oh frxuv gh
o*dfwlrq vrxv0mdfhqwh hvw dx0ghvvxv +uhvs1 hq ghvvrxv, g*xq sul{ g*h{huflfh grqqì ã od gdwh g*h{huflfh1
Pdwkìpdwltxhphqw/ oh sd|r v*ìfulw =
4 +VW A N,
srxu o*rswlrq g*dfkdw hw =
4 +VW ? N,
srxu o*rswlrq gh yhqwh1 Hq frpsdudlvrq/ o*rswlrq vwdqgdug gh od vhfwlrq suìfìghqwh v*ìfulydlw =
+VW N, =4 +VW A N,
Frpph suìfìghpphqw/ lo h{lvwh xqh uhodwlrq gh sdulwì fdoo0sxw txl v*ìfulw lfl =
F+w> W> N, S +w> W> N, @ E+w> W ,
hq frqvhuydqw ohv píphv qrwdwlrqv1
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Rswlrq g*dfkdw elqdluh
51615 Exoo hw Ehdu vsuhdgv
Fhv rswlrqv vrqw ghv frpelqdlvrqv g*rswlrqv vwdqgdugv1 Soxv suìflvìphqw/ fhwwh vwudwìjlh vxssrvh g*dfkhwhu
hw gh yhqguh ghx{ rswlrqv gh píph w|sh hw gh píph gdwh g*h{huflfh/ pdlv gh sul{ g*h{huflfh glìuhqwv1
Oh exoo vsuhdg fruuhvsrqg ã xq dfkdw hw xqh yhqwh g*xqh rswlrq g*dfkdw/ vrlw xq sd|r =
pd{+VW N4 > 3, pd{+VW N5 > 3,
dyhf N5 A N4 hw/ srxu oh ehdu vsuhdg +ìplvvlrq g*xqh rswlrq gh yhqwh hw dfkdw g*xqh rswlrq gh yhqwh , =
pd{+N4 VW > 3, pd{+N5 VW > 3,
dyhf N5 ? N4
Ohv judskltxhv gh fhv vwudwìjlhv prqwuhqw txh ohv sd|r v*dssurfkhqw ghv sd|rv ghv rswlrqv elqdluhv1
Gdqv oh fdv gx exoo vsuhdg/ o*lqyhvwlvvhxu v*dwwhqg ã xqh kdxvvh gx frxuv gx wlwuh vrxv0mdfhqw +hw lqyhuvhphqw
srxu oh ehdu vsuhdg,1
Hfroh Sro|whfkqltxh
58
50
40
30
20
10
0
100
140
cours
Exoo vsuhdg
50
40
30
20
10
0
100
140
cours
Ehdu vsuhdg
51616 Vwudggohv hw Vwudqjohv
Oh vwudggoh hvw xq sruwhihxlooh frpsrvì g*xqh rswlrq g*dfkdw hw g*xqh rswlrq gh yhqwh gh píph sul{
g*h{huflfh/ vrlw xq sd|r =
pd{+VW N> 3, . pd{+N VW > 3,
Frpph rq oh frqvwdwh vxu oh judskltxh txl uhsuìvhqwh fh sd|r/ fhwwh vwudwìjlh vhud fkrlvlh sdu xq
lqyhvwlvvhxu txl shqvh txh oh frxuv gx vrxv0mdfhqw yd ydulhu vljqlfdwlyhphqw vdqv vdyrlu h{dfwhphqw gdqv
59
txho vhqv1 D o*lqyhuvh/ xq lqyhvwlvvhxu txl dqwlflsh txh oh frxuv uhvwhud vwdeoh dxwrxu gh od ydohxu gh N
v*hqjdjhud gdqv od vwudwìjlh rssrvìh1
Oh vwudqjoh hvw h{dfwhphqw gx píph w|sh pdlv ohv sul{ g*h{huflfh gh o*rswlrq g*dfkdw hw gh o*rswlrq gh
yhqwh vrqw glìuhqwv1
250
200
150
100
50
0
100
cours
Vwudggoh
51617 Exwwhu | hw frqgruv
Oh exoo vsuhdg srxydlw v*dssduhqwhu ã xqh vruwh gh glìuhqfldwlrq ã o*ruguh 4 gh o*rswlrq hxursìhqqh sdu
udssruw dx sul{ g*h{huflfh1 Rq rewhqdlw dlqvl xqh irqfwlrq dqdorjxh ã xqh irqfwlrq gh Khdylvlgh1 Lfl/ rq
v*lqwìuhvvh ã o*ruguh 51 Lo v*djlw g*xqh vwudwìjlh Exwwhu | grqw oh sd|r hvw gx w|sh =
pd{+VW +N . N, > 3, 5 pd{+VW N> 3, . pd{+VW +N N, > 3,
f*hvw0ã0gluh o*dfkdw gh 5 rswlrqv g*dfkdw hw o*ìplvvlrq g*xqh rswlrq g*dfkdw1 Wuëv orjltxhphqw/ oh sd|r d
xqh iruph gh Gludf1 Fhwwh vwudwìjlh vhud fkrlvlh sdu xq lqyhvwlvvhxu txl dqwlflsh txh oh frxuv gx vrxv0mdfhqw
uhvwhud wuëv surfkh gh N hw /qh erxjhud sdv ql gdqv xq vhqv ql gdqv xq dxwuh1
Xq frqgru hvw xqh vwudwìjlh Exwwhu | pdlv dyhf 7 sul{ g*h{huflfh hw qrq 6 frpph suìfìghpphqw1
Hfroh Sro|whfkqltxh
5:
30
20
10
0
100
cours
Exwwhu |
51618 Fdohqgdu vsuhdgv
Ohv fdohqgdu vsuhdgv vrqw ghv vwudwìjlhv lpsoltxdqw ghv rswlrqv gh glìuhqwhv gdwhv gh pdwxulwìv1 Fhv vwud0
wìjlhv shxyhqw íwuh xwlohv oruvtxh o*lqyhvwlvvhxu dqwlflsh xq prxyhphqw gx vrxv0mdfhqw +hw ìyhqwxhoohphqw
oh vhqv gh fh prxyhphqw, ã xq prphqw suìflv gdqv oh ixwxu/ sdu h{hpsoh ã o*rffdvlrq gh od sxeolfdwlrq
ghv uìvxowdwv gh o*hqwuhsulvh/ g*lqirupdwlrqv pdfur0ìfrqrpltxhv/ gh gìflvlrqv hwf1
51619 Zduudqw/ erq gh vrxvfulswlrq
Ohv zduudqwv hw ohv erqv gh vrxvfulswlrq vrqw ghv dfwliv srvvìgdqw xqh glphqvlrq rswlrqqhooh1 Ohv
zduudqwv vrqw ìplv sdu xq ìwdeolvvhphqw qdqflhu doruv txh ohv erqv gh vrxvfulswlrq vrqw ìplv sdu ghv
vrflìwìv1 Ohv sulqflsdohv glìuhqfhv vrqw h{soltxìhv gdqv oh grfxphqw fl0ghvvrxv txh od Vrflìwì Jìqìudoh
grqqh ã vhv folhqwv1
5;
Hfroh Sro|whfkqltxh
5<
5161: Reoljdwlrqv frqyhuwleohv hw uhyhuvh frqyhuwleohv
Ohv reoljdwlrqv frqyhuwleohv vrqw ghv reoljdwlrqv ã uhperxuvhphqw lq0qh txl yhuvhqw xq frxsrq frpph
wrxwh reoljdwlrq pdlv txl shxyhqw íwuh frqyhuwlhv hq dfwlrqv dydqw od pdwxulwì gh o*reoljdwlrq1 Oruvtx*lo |
d frqyhuvlrq hq dfwlrqv/ oh wlwuh reoljdwdluh glvsdudñw1 Ohv sd|r jìqìuìv sdu fh wlwuh vrqw grqf =
;
f
hq w vl A w/ 3 vlqrq
?
=
+4 . f, hq W vl A W / 3 vlqrq
rü hvw od gdwh gh frqyhuvlrq/ f oh frxsrq hw W od gdwh gh pdwxulwì gh o*reoljdwlrq1 D fhv x{/ lo idxw
elhq0vþu dmrxwhu o*rsìudwlrq gh frqyhuvlrq hooh0píph +vl ? W , txl frqyhuwlw fhv reoljdwlrqv hq wlwuhv
g*dfwlrqv gh od vrflìwì1 Od gdwh gh frqyhuvlrq hvw xqh yduldeoh gh frppdqgh gx gìwhqwhxu gh o*reoljdwlrq
frqyhuwleoh = lo o*h{hufh oruvtxh oh frxuv gh o*dfwlrq hvw vx!vdpphqw kdxw srxu tx*lo vrlw lqwìuhvvdqw gh
frqyhuwlu o*reoljdwlrq hq dfwlrq1
Xqh reoljdwlrq uhyhuvh frqyhuwleoh hvw/ frpph suìfìghpphqw/ xqh reoljdwlrq ã uhperxuvhphqw lq0qh hw
txl yhuvh xq frxsrq1 Wrxwhirlv/ oh uhperxuvhphqw hvw frqglwlrqqì ã o*ìyroxwlrq gx vrxv0mdfhqw1 D pdwxulwì/
vl oh frxuv gx vrxv0mdfhqw hvw vxsìulhxu ã xq sul{ gh uìiìuhqfh/ oh fdslwdo gh o*reoljdwlrq hvw hqwlëuhphqw
uhperxuvì hq hvsëfh1 Gdqv oh fdv frqwudluh/ +l1h1 o*dfwlrq hvw lqiìulhxuh dx sul{ gh uìiìuhqfh,/ o*reoljdwlrq hvw
uhperxuvìh hq dfwlrqv/ hq sudwltxh srxu xqh ydohxu lqiìulhxuh dx fdslwdo gh o*reoljdwlrq +sul{ gh uìiìuhqfh
? qrplqdo,1 Ohv frxsrqv vrqw orjltxhphqw soxv ìohyìv txh srxu xqh reoljdwlrq fodvvltxh srxu frpshqvhu
oh idlw txh oh fdslwdo q*hvw sdv jdudqwl1 Oh ulvtxh sulv sdu o*lqyhvwlvvhxu hvw grqf txh oh frxuv gh o*dfwlrq
edlvvh wurs1 O*lqyhvwlvvhxu fkrlvlud fhwwh vwudwìjlh v*lo dqwlflsh xqh kdxvvh gx frxuv gh o*dfwlrq1
63
D wlwuh g*h{hpsoh/ rq wurxyhud fl0ghvvrxv g*dxwuhv h{hpsohv soxv vrsklvwltxìv wluìv ghv ruhv Fuìglw
O|rqqdlv +zzz1fozduudqwv1frp,1
H{hpsohv zzz1fozduudqwv1frp
Hfroh Sro|whfkqltxh
64
65
Hfroh Sro|whfkqltxh
66
67
Hfroh Sro|whfkqltxh
68
69
Hfroh Sro|whfkqltxh
6:
5161; Surgxlwv ã fdslwdo jdudqwl
Fh vrqw ghv dfwliv txl jdudqwlvvhqw xq qlyhdx gh fdslwdo hw2rx xq wdx{ g*lqwìuíw plqlpxp jdudqwl hw grqw oh
uhqghphqw gìshqg gh o*ìyroxwlrq g*xq vrxv0mdfhqw grqqì1 Yrlfl txhotxhv h{hpsohv +zzz1fozduudqwv1frp, =
6;
Fhwwh lgìh shxw íwuh gìfolqìh hq surgxlwv hqfruh soxv vrsklvwltxìv +zzz1fozduudqwv1frp, =
Hfroh Sro|whfkqltxh
6<
73
Hfroh Sro|whfkqltxh
74
75
517 Jorvvdluh
Oh yrfdexodluh olì dx{ rswlrqv hvw ulfkh hw wuëv dqjor0vd{rq1 Yrlfl xq jorvvdluh srxu v*| uhwurxyhu =
Sulph 0 Suhplxp = oh sul{ gh o*rswlrq1 Lo v*djlw gh od píph whuplqrorjlh tx*hq dvvxudqfh +rq sduoh
gh sulph g*dvvxudqfh,/ fh txl hvw orjltxh gdqv od phvxuh rü xqh rswlrq hw soxv jìqìudohphqw xq dfwli
gìulyì shxyhqw íwuh yxv frpph xq rxwlo gh frxyhuwxuh frqwuh xq ulvtxh1
Vrxv0mdfhqw 0 Xqghuo|lqj = o*lqvwuxphqw qdqflhu txl hqwuh gdqv od gìqlwlrq gx sd|r gh o*rswlrq1 Lo
shxw v*djlu g*xqh dfwlrq/ g*xq wlwuh g*reoljdwlrq/ g*xqh pdwlëuh suhplëuh/ g*xq shuplv g*ìplvvlrq gh
srooxhu/ hwf1
Sul{ g*h{huflfh 0 Vwulnh = vhxlo gh uìiìuhqfh hqwudqw gdqv od gìqlwlrq gx sd|r gh o*rswlrq +yrlu ohv
h{hpsohv grqqìhv gdqv fh fkdslwuh,1 Lo v*djlw hq sudwltxh gx vhxlo gh gìfohqfkhphqw ã sduwlu gxtxho
oh gìwhqwhxu gh o*rswlrq h{hufh vhv gurlwv +g*dfkhwhu rx gh yhqguh,1
Gdwh g*h{sludwlrq 0 H{sludwlrq = Gdwh dx0ghoã gh odtxhooh o*rswlrq h{sluh1
Hq pdujh gh fh yrfdexodluh/ lo idxw qrwhu ohv qrwlrqv vxlydqwhv tx*rq uhwurxyhud gdqv ohv fkdslwuhv vxlydqwv =
Od ydohxu lqwulqvëtxh 0 Lqwulqvlf ydoxh fruuhvsrqg dx sd|r tx*rq uhfhyudlw vl o*rswlrq ìwdlw rx srxydlw
+fdu oh gìwhqwhxu gh o*rswlrq q*hq d sdv qìfhvvdluhphqw oh gurlw, íwuh h{hufìh dxmrxug*kxl1
Hfroh Sro|whfkqltxh
76
Od ydohxu0whpsv rx Wlph ydoxh hvw od glìuhqfh hqwuh oh sul{ gh pdufkì gh o*rswlrq dxmrxug*kxl hw
vd ydohxu lqwulqvëtxh1 Fhod fruuhvsrqg dx vxssoìphqw gh ydohxu gþ dx idlw txh o*rswlrq vhud h{hufìh
gdqv oh ixwxu1
Gdqv od prqqdlh 0 Lq wkh prqh| = vh glw g*xqh rswlrq txl d xqh ydohxu lqwulqvëtxh srvlwlyh1 Hq fodlu/
vl od gdwh g*h{huflfh ìwdlw dxmrxug*kxl/ oh gìwhqwhxu gh o*rswlrq h{hufhudlw vrq gurlw1
D od prqqdlh 0 Dw wkh prqh| = vh glw g*xqh rswlrq srxu odtxhooh oh vrxv0mdfhqw hvw surfkh gx sul{
g*h{huflfh +l1h ydohxu lqwulqvëtxh surfkh gh 3,1
Hq ghkruv gh od prqqdlh 0 Rxw ri wkh prqh| = vh glw g*xqh rswlrq grqw od ydohxu lqwulqvëtxh hvw qxooh1
Fhod qh vljqlh sdv srxu dxwdqw txh oh sul{ gh pdufkì gh o*rswlrq hvw qxo = o*rswlrq d jìqìudohphqw
xqh ydohxu0whpsv qrq qxooh1
77
6
Ohv prgëohv glvfuhwv g*ìydoxdwlrq
614 Lqwurgxfwlrq
Ohv fkdslwuhv suìfìghqwv rqw ìwì frqvdfuìv ã od ghvfulswlrq ghv dfwliv gh edvh g*xqh sduw hw ghv dfwliv
gìulyìv g*dxwuh sduw1 Sdupl fhx{0fl/ qrxv qrxv lqwìuhvvrqv lfl dx pdufkì ghv dfwlrqv hw gh vhv surgxlwv
gìulyìv/ píph vl od iurqwlëuh hqwuh surgxlwv gìulyìv dfwlrqv/ wdx{ g*lqwìuíw rx fkdqjh hvw hq sudwltxh xq
shx rxh1
Od sureoìpdwltxh hvw od vxlydqwh = ìwdqw grqqì ohv sul{ ghv dfwliv gh edvh +l1h1 ohv dfwlrqv,/ txhoohv ydohxuv
grlyhqw suhqguh ohv sul{ ghv surgxlwv gìulyìv frqvwuxlwv ã sduwlu gh fhv dfwliv vrxv0mdfhqwv B Lo v*djlw grqf gh
gìwhuplqhu ghv uhodwlrqv gh frkìuhqfh hqwuh ohv sul{ ghv dfwlrqv hw fh txh ghyudlhqw íwuh ohv sul{ ghv dfwliv
gìulyìv gh fhv dfwlrqv1 Fhwwh gìpdufkh vh irqghud srxu o*hvvhqwlho vxu o*k|srwkëvh g*devhqfh g*rssruwxqlwì
g*duelwudjh/ gìmã ìyrtxìh soxv kdxw hw vxu odtxhooh qrxv uhylhqgurqv derqgdpphqw1
Gdqv xq ghx{lëph whpsv hw xqh irlv wurxyìh od uhodwlrq hqwuh sul{ gh o*dfwli gìulyì hw sul{ gx vrxv0
mdfhqw/ qrxv ìwxglhurqv ohv txhvwlrqv gh frxyhuwxuh1 Hq hhw/ ohv dfwliv gìulyìv vrqw g*derug ghv frqwudwv
shuphwwdqw gh vh frxyulu frqwuh fhuwdlqv ulvtxhv olìv dx vrxv0mdfhqw = lo qh vx!w grqf sdv gh frqqdñwuh
frpelhq xq dfwli gìulyì grlw ydorlu dxmrxug*kxl pdlv lo idxw ìjdohphqw vdyrlu hq txrl lo shuphw gh vh
frxyulu frqwuh ghv doìd olìv dx vrxv0mdfhqw1
Hqq/ qrxv qrxv uhvwuhlqgurqv gdqv fh fkdslwuh ã ghv prgìolvdwlrqv rü oh whpsv hw ohv doìd vrqw glvfuhwv1
O*dydqwdjh gh fh w|sh gh prgëoh hvw ohxu wulyldolwì pdwkìpdwltxh/ fh txl shuphw gh vh frqfhqwuhu vxu ohv
lqwxlwlrqv qdqflëuhv soxv txh vxu ohv sureoëphv pdwkìpdwltxhv1 Hq rxwuh/ rq prqwuhud tx*rq shxw ohv
idluh frqyhujhu yhuv ghv prgëohv rü oh whpsv hvw frqwlqx1 Odvw exw qrw ohdvw/ fhv prgëohv glvfuhwv vrqw
wuëv frxudpphqw xwlolvìhv hq sudwltxh hw frqvwlwxhqw xq rxwlo rsìudwlrqqho lqfrqwrxuqdeoh1
615 Oh fdguh suredelolvwh
61514 Od g|qdpltxh gx sul{ gh o*dfwlrq
Gdqv wrxv ohv sureoëphv g*ìydoxdwlrq/ lo idxw hq suhplhu olhx vh grqqhu xq fdguh suredelolvwh srxu gìfuluh
ohv dfwliv gh edvh +l1h1 ohv vrxv0mdfhqwv,1 Uhsuhqdqw o*ìedxfkh gh prgìolvdwlrq hqylvdjìh dx gìexw gh fhw
rxyudjh/ oh sul{ g*xqh dfwlrq shxw vh yrlu frpph oh sul{ txl dssdudlwudñw vl oh prqgh ìwdlw gìwhuplqlvwh
+hw vl ohv glylghqghv hw wdx{ g*lqwìuíw ixwxuv ìwdlhqw frqqxv gëv dxmrxug*kxl, dxtxho rq dmrxwh xq doìd
srxu lqwurgxluh o*lgìh txh oh prqgh q*hvw mxvwhphqw sdv gìwhuplqlvwh1 Rq dydlw rewhqx dlqvl ã sduwlu g*xqh
orjltxh g*duelwudjh =
Vw . glyw Vw4 @ uw4 =Vw4 . %w
rü %w hvw xq doìd +l1h1 xq fkrf qdqflhu , txl vh uìdolvh hqwuh w 4 hw w1 Pdwkìpdwltxhphqw/ lo hvw
soxv vlpsoh gh idluh xq fkdqjhphqw gh sdudpìwudjh hw gh frqvlgìuhu soxw÷w xq prgëoh pxowlsolfdwli gh od
iruph =
4 . uw4
=Kw
Vw @ Vw4 =
4 . w
rü =
Vw hvw oh sul{ gh o*dfwlrq hq w >
w
w hvw oh wdx{ gh glylghqgh gìql sdu w @ gly
Vw
uw4 hvw oh wdx{ g*lqwìuíw srxu od sìulrgh doodqw gh w 4 ã w >
Kw hvw xqh yduldeoh doìdwrluh1
Srxu vlpsolhu oh prgëoh/ qrxv vxssrvhurqv ohv k|srwkëvhv vxlydqwhv =
oh wdx{ gh glylghqgh hw oh wdx{ g*lqwìuíw vrqw vxssrvìv frqvwdqwv dx frxuv gx whpsv >
o*doìd K vxlw xqh orl gh Ehuqrxlool/ f*hvw0ã0gluh tx*lo ydxw x + xs , dyhf xqh suredelolwì s hw g +
grzq , dyhf xqh suredelolwì 4 s >
ohv yduldeohv +Kw ,w vrqw lqgìshqgdqwhv hqwuh hoohv hw lghqwltxhphqw glvwulexìhv1
Hq fodlu/ rq qh v*lqwìuhvvh sdv lfl dx{ txhvwlrqv olìhv dx fdudfwëuh doìdwrluh ghv glylghqghv hw ghv wdx{
g*lqwìuíw pdlv vhxohphqw dx fdudfwëuh doìdwrluh gx sul{ gh uhyhqwh gh o*dfwlrq1 Sdu dloohxuv/ oh prgëoh gh
Ehuqrxlool glw hqfruh prgëoh elqrpldo hvw oh soxv vlpsoh tx*rq sxlvvh hqylvdjhu1 Fh prgëoh suredelolvwh
frqvlgëuh grqf lpsolflwhphqw tx*lo qh shxw | dyrlu txh ghx{ fkrfv = xq fkrf kdxvvlhu hw xq fkrf
edlvvlhu 1 Dlqvl/ qrxv rewlhqgurqv xq prgëoh vdqv dxfxqh frpsoh{lwì pdwkìpdwltxh hw txl srxu0
wdqw dssruwh o*hvvhqwlho ghv lqwxlwlrqv qdqflëuhv/ lqwxlwlrqv tx*rq uhwurxyhud gdqv oh fdguh ghv prgëohv
pdwkìpdwltxhv hq whpsv frqwlqx1
Srxu qlu/ hq qrwdqw uh od txdqwlwì whooh txh =
4 . uh @
4.u
4.
txl ydxw grqf dx suhplhu ruguh u / oh prgëoh g|qdpltxh gh vrxv0mdfhqw v*ìfulw =
;
? Vw4 = +4 . uh, =x dyhf suredelolwì s
Vw @ Vw4 = +4 . uh, =Kw @
=
Vw4 = +4 . uh, =g dyhf suredelolwì 4 s
+614,
txl hvw od g|qdpltxh gh edvh txh qrxv frqvlgìurqv srxu o*dfwlrq1 Lo v*djlw grqf g*xqh ìtxdwlrq gh uì0
fxuuhqfh dyhf wrxwhirlv xq whuph doìdwrluh1 Txdqg qrxv vhurqv hq whpsv frqwlqx/ fhwwh ìtxdwlrq vhud
uhpsodfìh sdu xqh ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh1
61515 Txhotxhv sursulìwìv gh od g|qdpltxh ghv sul{
Rq shxw wudydloohu o*ìtxdwlrq suìfìghqwh srxu gìgxluh txhotxhv uìvxowdwv1 Hq sduwlfxolhu/ rq shxw ìfuluh
o*ìtxdwlrq +614, hq gìurxodqw od uìfxuuhqfh/ vrlw =
w4
w \K
Vw @ V3 = +4 . uh, =
wn
n@3
+615,
gh whooh vruwh txh oh sul{ gh o*dfwlrq dxmrxug*kxl dssdudñw frpph od uìvxowdqwh gh wrxv ohv fkrfv erxuvlhuv
sdvvìv1
Rq shxw qrwhu ìjdohphqw txh od orl gh Vw frqglwlrqqhoohphqw ã V3 hvw xqh orl elqrpldoh txl suhqg ohv
ydohxuv V3 = +4 . uh,w =xm =gwm dyhf xqh suredelolwì =
w
=sm = +4 s,wm
m
w
rü
@ m $+ww$ m,$
m
Rq shxw ìjdohphqw uhpdutxhu txh =
79
Sursrvlwlrq 4 Vl rq lpsrvh frqyhqwlrqqhoohphqw txh g ? x/ doruv rq d =
g?4?x
Lo v*djlw oã g*xq uìvxowdw qdqflhu hw qrq pdwkìpdwltxh1 Hq hhw/ vxssrvrqv x> g A 41 Xq lqyhvwlvvhxu ã
od gdwh w 4 shxw hpsuxqwhu od vrpph Vw4 srxu dfkhwhu xqh dfwlrq1 Oh elodq gh fhwwh rsìudwlrq hq w hvw
oh vxlydqw =
uhperxuvhphqw gh o*hpsuxqw = Vw4 =+4 . u,
uhyhqwh gh o*dfwlrq/ vrpph ã odtxhooh rq dmrxwh oh glylghqgh yhuvì = Vw =+4 . ,
Dx wrwdo/ o*lqyhvwlvvhxu q*d ulhq gìerxuvì hq w 4 sxlvtxh o*dfkdw gh o*dfwlrq v*hvw idlw sdu oh eldlv g*xq
hpsuxqw hw uhêrlw Vw =+4 . , Vw4 =+4 . u, hq w/ vrpph txl hvw wrxmrxuv srvlwlyh vl x> g A 41 Lo v*djlw
fodluhphqw g*xqh rssruwxqlwì g*duelwudjh fdu fhwwh vwudwìjlh hvw vdqv ulvtxh/ qh qìfhvvlwh sdv gh plvh gh
irqgv lqlwldoh/ hw udssruwh ã wrxv frxsv xqh vrpph srvlwlyh1 Vl rq vxssrvh 0 frpph qrxv oh ihurqv wrxw oh
whpsv 0 txh oh pdufkì ìolplqh lqvwdqwdqìphqw fhv rssruwxqlwìv/ qrxv srxyrqv dgphwwuh tx*hoohv q*h{lvwhqw
grqf sdv hw txh x> g qh shxyhqw sdv íwuh vlpxowdqìphqw vxsìulhxuv ã 41 Lqyhuvhphqw sdu oh píph w|sh
g*dujxphqw/ rq qh shxw dyrlu x> g ? 41
616 O*devhqfh g*rssruwxqlwì g*duelwudjh hw oh sul{ ghv surgxlwv gìulyìv
61614 Od irupxoh g*ìydoxdwlrq
Oh fdguh suredelolvwh ìwdqw srvì/ qrxv qrxv lqwìuhvvrqv dx sul{ ghv surgxlwv gìulyìv grqw oh vrxv0mdfhqw
vxlw xqh g|qdpltxh gx w|sh suìfìghqw1 Hq rxwuh/ srxu vlpsolhu/ qrxv vxssrvrqv w @ 41 O*ìtxdwlrq +614,
v*ìfulw grqf =
V4 @ V3 = +4 . uh, =K4
Lo hvw qdwxuho gh vxssrvhu gdqv xq suhplhu whpsv txh oh sul{ gx surgxlw gìulyì hvw xqh irqfwlrq gx sul{
gx vrxv0mdfhqw hw gx whpsv1 Dlqvl/ F4 / oh sul{ gx surgxlw gìulyì hq 4 suhqgud ghx{ ydohxuv srvvleohv vhorq
od uìdolvdwlrq gh o*doìd1 Rq qrwhud F4>x hw F4>g fhv ghx{ ydohxuv1
Rq shxw doruv idluh o*rsìudwlrq vxlydqwh ã od gdwh 3 =
ìplvvlrq g*xqh rswlrq g*dfkdw >
dfkdw g*xqh txdqwlwì gh wlwuhv g*dfwlrq vrxv0mdfhqwh >
sodfhphqw rx hpsuxqw g*xqh vrpph +vl A 3/ lo v*djlw g*xq sodfhphqw hw g*xq hpsuxqw gdqv oh
fdv frqwudluh,
O*lqyhvwlvvhxu grlw gìerxuvhu +hqwuh jxloohphwv fdu fhwwh vrpph shxw íwuh qìjdwlyh, =
F3 =V3 hw uhêrlw hq 4 =
F4 . =V4 =+4 . , . =+4 . u,
Ru/ g*dsuëv oh fdguh suredelolvwh gìfulw soxv kdxw/ fhwwh vrpph qh shxw suhqguh txh ghx{ ydohxuv =
;
dyhf xqh suredelolwì s
? F4>x . =V3 =+4 . u,=x . =+4 . u,
=
F4>g . =V3 =+4 . u,=g . =+4 . u,
dyhf xqh suredelolwì 4 s
Lo hvw wrxmrxuv srvvleoh gh fkrlvlu srxu txh fh sruwhihxlooh vrlw vdqv ulvtxh1 Lfl/ fhod vljqlh txh ohv ghx{
ydohxuv suìfìghqwhv vrqw ohv píphv = hq fodlu/ txhotxh vrlw oh fkrf erxuvlhu hqwuh 3 hw 4/ oh sruwhihxlooh
d xqh ydohxu lqgìshqgdqwh gh fh fkrf1 Rq glw tx*rq d frqvwuxlw xq sruwhihxlooh gh frxyhuwxuh1 Fh
sdudpëwuh grlw grqf yìulhu =
4 F4>x F4>g
=
@
4 . u V3 =+x g,
hw hvw wrwdohphqw fdofxodeoh hq w 41
Hfroh Sro|whfkqltxh
7:
Vl o*lqyhvwlvvhxu xwlolvh fhwwh ydohxu srxu frqvwuxluh vrq sruwhihxlooh/ doruv lo d frqvwuxlw gh idfwr xq sruwh0
ihxlooh vdqv ulvtxh1 Vl gh soxv lo d fkrlvl who txh o*lqyhvwlvvhphqw lqlwldo vrlw qxo/ l1h1 F3 =V3 @ 3
doruv/ dyhf fhv ghx{ ydohxuv gh hw +hw xqltxhphqw dyhf fhv ghx{ ydohxuv,/ rq grlw dyrlu sdu devhqfh
g*rssruwxqlwì g*duelwudjh =
F4 . =V4 =+4 . , . =+4 . u, @ 3
txl wudgxlw tx*xq sruwhihxlooh vdqv ulvtxh hw txl qh qìfhvvlwh dxfxq lqyhvwlvvhphqw lqlwldo qh grlw sdv
udssruwhu dxwuh fkrvh txh 3 hxur1
Hq uhpsodêdqw hw sdu ohxu ydohxu/ rq rewlhqw/ dsuëv fdofxov/ od uhodwlrq vxlydqwh =
F3 @
4
= +=F4>x . +4 , =F4>g ,
4.u
dyhf =
4g
xg
Soxv jìqìudohphqw/ vl rq qrwh Fw oh sul{ gx surgxlw gìulyì ã od gdwh w/ rq d =
@
Fw @
4
=Hw +Fw.4 ,
4.u
+616,
rü Hw +=, gìvljqh o*hvsìudqfh frqglwlrqqhooh +vdfkdqw Vw hw vrq sdvvì, fdofxoìh dyhf od suredelolwì hw qrq od suredelolwì s1 Rq d grqf od sursrvlwlrq vxlydqwh =
Sursrvlwlrq 5 Vrxv o*k|srwkëvh g*devhqfh g*rssruwxqlwìv g*duelwudjh/ lo h{lvwh xqh suredelolwì whooh txh
ohv sul{ dfwxdolvìv ghv dfwliv gìulyìv vrlhqw pduwlqjdohv1
Hq hhw/ vl rq frqvlgëuh oh surfhvvxv ]w @
surfhvvxv yìulh =
Fw
+4.u,w
doruv rq yrlw wrxw gh vxlwh txh/ vrxv od suredelolwì / fh
]w @ Hw +]W ,
srxu wrxw w ? W 1 Hq rxwuh/ hvw elhq xqh suredelolwì fdu hvw frpsulv hqwuh 3 hw 4/ uìvxowdw txl gìulyh
gluhfwhphqw gh od sursulìwì g ? 4 ? x1
H{hpsoh 4 O*ìydoxdwlrq g*xqh rswlrq vwdqgdug hxursìhqqh shxw doruv vh idluh hq uhpdutxdqw tx*ã od
gdwh g*h{huflfh =
.
FW @ +VW N,
hw grqf =
Fw @
.
W w =Hw +VW N,
+4 . u,
4
Ru/ frpph rq o*d uhpdutxì gdqv od vhfwlrq suìfìghqwh/ od orl gh VW frqglwlrqqhoohphqw ã Vw hvw xqh orl
elqrpldoh1 Frqfuëwhphqw/ g*dsuëv +615, =
W
w4
\
KW n
VW @ Vw = +4 . uh,W w =
n@3
g*rü =
Fw @
vrlw hqfruh =
Vw
=Hw
+4 . ,W w
#W w4
\
n@3
KW n N
$.
Vw = +4 . uh,W w
#
$.
W[
w N
W w
W
w
m
m
m
W
w
m
=
x =g
Fw @
= = +4 ,
m
+4 . ,W w m @3
Vw = +4 . uh,W w
Vw
Fhwwh irupxoh shuphw grqf g*h{sulphu oh sul{ gh o*rswlrq hq w hq irqfwlrq gh w hw Vw 1
7;
+617,
61615 Lqwhusuìwdwlrq ìfrqrpltxh
Od irupxoh g*ìydoxdwlrq +616, hvw irqgdphqwdoh = hooh h{sulph txh oh sul{ g*xq dfwli shxw vh yrlu frpph
o*hvsìudqfh ghv x{ ixwxuv tx*lo jìqëuh/ o*hvsìudqfh ìwdqw fdofxoìh vrxv xqh fhuwdlqh suredelolwì grqw
o*h{lvwhqfh gìulyh gh o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh1 Dx sdvvdjh/ rq d/ g*dsuëv +614, oh
píph w|sh gh irupxoh +dx{ glylghqghv suëv, srxu oh sul{ gx vrxv0mdfhqw =
4
W w
V
H
=
+4
.
,
Vw @
w W
+4 . u,W w
Od suredelolwì v*dsshooh od suredelolwì ulvtxh0qhxwuh = hooh q*d d sulrul ulhq ã yrlu dyhf od suredelolwì
lqlwldoh s/ tx*rq dsshooh dxvvl suredelolwì remhfwlyh rx suredelolwì klvwrultxh1 Oh fdofxo gx sul{ g*xq
surgxlw gìulyì qh idlw grqf sdv lqwhuyhqlu od yudlh suredelolwì ghv ìyìqhphqwv1 Oh
whuph gh ulvtxh0qhxwuh ylhqw gluhfwhphqw gh od wkìrulh ìfrqrpltxh = oruvtxh ohv lqwhuyhqdqwv g*xq
pdufkì q*rqw sdv g*dyhuvlrq srxu oh ulvtxh +rq glw hqfruh tx*lov vrqw ulvtxh0qhxwuhv,/ lov yrqw v*dffrughu
srxu ìydoxhu oh sul{ g*xq dfwli frpph o*hvsìudqfh dfwxdolvìh ghv x{ tx*lo jìqëuh1 Od irupxoh g*ìydoxdwlrq
rewhqxh lfl shuphw gh idluh frpph vl ohv lqyhvwlvvhxuv q*dydlhqw sdv g*dyhuvlrq srxu oh ulvtxh1 Fhod qh
vxssrvh devroxphqw sdv txh ohv lqwhuyhqdqwv vrqw ulvtxh0qhxwuhv1 Hq fodlu/ o*devhqfh g*rssruwx0
qlwì g*duelwudjh vh wudgxlw pdwkìpdwltxhphqw sdu o*dssdulwlrq g*xqh suredelolwì = fhwwh suredelolwì hvw
xq duwlfh pdwkìpdwltxh txl shuphw/ gdqv ohv fdofxov g*ìydoxdwlrq/ gh idluh frpph vl ohv lqyhvwlvvhxuv
ìwdlhqw qhxwuhv dx ulvtxh1
61616 Xwlolvdwlrq sudwltxh
Gëv oruv tx*rq frqqdñw ohv sdudpëwuhv u/ / x/ hw g/ rq vdlw fdofxohu oh sul{ gh wrxw surgxlw gìulyì +dx
prlqv gh w|sh hxursìhq/ dx vhqv gìql dx fkdslwuh suìfìghqw,1 O*lpsoìphqwdwlrq qxpìultxh qh srvh sdv
gh sureoëph sxlvtx*lo vx!w g*lpsoìphqwhu xq dueuh elqrpldo1 Od vhxoh txhvwlrq hvw gh vdyrlu frpphqw
rq ìydoxh ohv sdudpëwuhv1
617 Od frqyhujhqfh yhuv oh whpsv frqwlqx
Oh irqfwlrqqhphqw ghv pdufkìv qdqflhuv uhqg gl!floh o*lgìh tx*lo h{lvwh xq sdv gh whpsv ìoìphqwdluh hw
lqvìfdeoh1 Lo hvw gh idlw soxv orjltxh g*ìfuluh xq prgëoh hq whpsv frqwlqx sxlv gh oh glvfuìwlvhu dq gh
o*dgdswhu dx{ uìdolwìv ghv pdufkìv1 Lfl/ qrxv doorqv ìfuluh oh prgëoh suìfìghqw hq idlvdqw dssdudñwuh oh sdv
gh whpsv hw hq oh idlvdqw whqguh yhuv 31 Qrxv rewlhqgurqv dlqvl xq prgëoh hq whpsv frqwlqx txl q*hvw ulhq
g*dxwuh txh oh prgëoh gh Eodfn hw Vfkrohv1
61714 Txhotxhv qrwdwlrqv
Rq qrwh oh sdv gh whpsv hw rq vxssrvh txh ohv gdwhv frqvlgìuìhv lfl vrqw ghv pxowlsohv gh fh sdv gh
whpsv1 Srxu vlpsolhu ohv h{suhvvlrqv/ qrxv vxssrvhurqv/ vdqv shuwh gh jìqìudolwì/ txh ohv wdx{ g*lqwìuíw
hw wdx{ gh glylghqgh vrqw qxov1 Oh surfhvvxv gx vrxv0mdfhqw v*ìfulw grqf g*dsuëv +614, =
Vw @ V 3
w 4
\
n@3
K+ w n,=
rx hqfruh +sdu fkdqjhphqw g*lqglfh, =
w
Oq Vw @ Oq V3 .
[
n@4
Oq Kn=
61715 Fkdqjhphqw gh sdudpìwudjh
Ohv sdudpëwuhv x hw g rqw xqh lqwhusuìwdwlrq qdqflëuh txl q*hvw sdv ìylghqwh1 Rq lqwurgxlw grqf od qrwlrq
gh yrodwlolwì frpph o*ìfduw0w|sh frqglwlrqqho gx orjdulwkph gx sul{ gx vrxv0mdfhqw/ vrlw =
4
^Y duw +Oq Vw ,` 5
Hfroh Sro|whfkqltxh
7<
txl hvw grqf ìjdoh ã +frpph srxu wrxwh orl gh Ehuqrxlool,4 =
4
^Y du +Oq Kw ,` 5 @
Rq qrwhud grqf =
k
s
xl
= +4 ,= Oq
g
s
x
= +4 ,=Oq
g
Frpph/ sdu dloohxuv/ rq d sdu gìqlwlrq gh =
@
=x . +4 ,=g @ 4
rq hq gìgxlw ohv ìtxdwlrqv gh sdvvdjh hqwuh oh sdudpìwudjh +x> g, hw oh sdudpìwudjh +> ,/ vrlw =
;
x @
A
A
A
?
A
A
A
= g @
4
.+4,= h{s
s=+4,
4
+4,.= h{s s=+4,
61716 Frqyhujhqfh yhuv oh whpsv frqwlqx
Lo qh uhvwh soxv tx*ã gìqlu frpphqw od yrodwlolwì gìshqg gx sdv gh whpsv 1 Srxu txh oh prgëoh frqyhujh
yhuv txhotxh fkrvh / lo idxw vxssrvhu txh =
s
@ = Rq shxw doruv ìfuluh =
Oq Vw
@ Oq V3 .
S w
n@4 Oq Kn=
@ Oq V3 .
s
S w
n@4 = =%n . =Oq x . +4 ,=Oq g
dyhf =
Oq Kn= +=Oq x . +4 ,=Oq g,
s
= Fhwwh qrxyhooh yduldeoh doìdwrluh d o*dydqwdjh g*íwuh fhqwuìh hw uìgxlwh/ f*hvw0ã0gluh gh pr|hqqh qxooh hw
gh yduldqfh 4 vrxv od suredelolwì 1
%n @
S w
Od frqyhujhqfh
s gx whuph n@4 =Oq x . +4 ,=Oq g hvw wulyldoh = lo vx!w gh uhpsodfhu x hw g hq irqfwlrq
gh @ = / gh idluh xq gìyhorsshphqw olplwì dx yrlvlqqdjh gh @ 3/ hw gh frqvwdwhu tx*lo frqyhujh
yhuv =
5
=w
5
Oh whuph vwrfkdvwltxh frqyhujh sdu xq wkìruëph gh w|sh wkìruëph fhqwudo olplwh1 Fh whuph v*ìfulw =
s
= w
v
w
4[
%n
w
n@4
hw frqyhujh grqf hq orl yhuv xqh orl qrupdoh g*hvsìudqfh qxooh hw gh yduldqfh 5 =w1
Dx wrwdo/ Oq Vw frqyhujh/ txdqg oh sdv gh whpsv whqg yhuv 3/ yhuv xqh orl qrupdoh =
5
Oq Vw $ Q Oq V3 =w > 5 =w
5
$3
4 Oh frqglwlrqqhphqw glvsdudñw fdu rq d vxssrvì txh ohv yduldeohv
83
M|
ìwdlhqw lqgìshqgdqwhv1
61717 Txhotxhv uhpdutxhv
Oh vfkìpd elqrpldo gìfulw lfl frqyhujh yhuv xq prgëoh rü oh sul{ gh o*dfwlrq vrxv0mdfhqwh vxlw xqh orl
orj0qrupdoh1 Lo d idoox/ srxu fhod/ idluh xqh k|srwkëvh vxu o*ruguh gh od yrodwlolwì hq irqfwlrq gx sdv gh
whpsv/ hw vxuwrxw vxssrvhu txh od yrodwlolwì vh uìgxlvdlw ã xq vhxo sdudpëwuh frqvwdqw 1 Hq uhydqfkh/ oh
sdudpëwuh d prlqv g*lpsruwdqfh sxlvtx*lo glvsdudñw gdqv od frqyhujhqfh yhuv oh whpsv frqwlqx5 1
Oh uìvxowdw gh orj0qrupdolwì suìfìghqw shxw vh jìqìudolvhu srxu wrxw frxsoh gh gdwhv +w> W ,1 Dlqvl/ od orl
frqglwlrqqhooh gh Oq VW vdfkdqw Vw dssdudñwud frpph xqh orl qrupdoh =
5
5
Q Oq Vw =+W w, > =+W w,
5
Soxv jìqìudohphqw/ wrxw q0xsohw +Oq Vw4 > ===> Oq Vwq , vxlyud xqh orl qrupdoh1 Rq glud txh +Oq Vw ,w hvw xq
surfhvvxv jdxvvlhq1
Oruvtx*rq qh vxssrvh soxv ohv wdx{ g*lqwìuíw hw wdx{ gh glylghqgh qxov/ lo hvw idfloh gh yrlu txh fhwwh orl
jdxvvlhqqh ghylhqw6 =
5
Q Oq Vw . uh =+W w, > 5 =+W w,
5
Rq yrlw doruv frpphqw fdofxohu oh sul{ g*xq surgxlw gìulyì gdqv oh prgëoh frqwlqx1 Rq shxw vrlw sduwlu
gh od irupxoh g*ìydoxdwlrq elqrpldoh +617, hw idluh whqguh oh sdv gh whpsv yhuv 3/ vrlw sduwlu gh od irupxoh
soxv jìqìudoh =
.
Fw @ hu+W w, =Hw +VW N,
hq fdofxodqw o*hvsìudqfh dyhf od orl jdxvvlhqqh rewhqxh srxu Oq VW 1 Rq prqwuh txh fhv ghx{ yrlhv
derxwlvvhqw dx píph uìvxowdw hw fh uìvxowdw fruuhvsrqg ã od irupxoh gh Eodfn hw Vfkrohv tx*rq ìwxglhud
dx fkdslwuh vxlydqw1 Hqq/ lo hq h{lvwh xqh wurlvlëph tx*rq suìvhqwh pdlqwhqdqw1
61718 Hydoxdwlrq sdu ìtxdwlrq dx{ gìulyìhv sduwlhoohv
Rq prqwuh lfl txh oh sul{ g*xq dfwli gìulyì shxw íwuh yx frpph od vroxwlrq g*xqh ìtxdwlrq dx{ gìulyìhv
sduwlhoohv1 Srxu fhod/ rq uhsuhqg o*ìtxdwlrq gh uìfxuuhqfh +616, hq vxssrvdqw gh qrxyhdx txh wdx{ g*lqwìuíw
hw wdx{ gh glylghqgh vrqw qxov =
Fw @ Hw +Fw. ,
vrlw hqfruh =
Fw
@ F+w> Vw ,
@ =F+w . > Vw =x, . +4 , =F+w . > Vw =g,
Hq uhpdutxdqw txh =
u
x @ 4 . =
hw =
4 4 s
= 5 = 5
= . r+,
u
45
4 s
= 5 =
= . r+,
4
hw hq gìyhorssdqw od irqfwlrq F sdu udssruw ã vhv ghx{ dujxphqwv/ rq gìgxlw =
g @ 4 =
3@
CF
CF
4
C5 F
= .
=r+, . V 5 =5 = 5 =
Cw
CV
5
CV
vrlw =
3@
4
CF
C 5F
. V 5 = 5 = 5
Cw
5
CV
Z uì0dssdudñw = fhod vljqlh txh fh sdudpëwuh d xqh lq xhqfh vxu od
ylwhvvh gh frqyhujhqfh pdlv qrq vxu od ydohxu0olplwh hooh0píph1
6 Rq dgrswh lfl od frqyhqwlrq gh wdx{ g*lqwìuíw xwlolvìh jìqìudohphqw hq whpsv frqwlqx1 Fhod uhylhqw iruphoohphqw ã

vxevwlwxhu h o EA | ã
W w 1
5 Vl rq gìyhorssh dx{ ruguhv vxsìulhxuv/ oh sdudpëwuh
3
3
Hfroh Sro|whfkqltxh
Eno
84
txl ghylhqw txdqg ohv wdx{ g*lqwìuíw hw wdx{ gh glylghqgh vrqw qrq qxov =
uF @
CF
4
CF
C 5F
. uh=V=
. V 5 = 5 = 5
Cw
CV
5
CV
Dx wrwdo/ oh fdofxo gx sul{ 0 hq whpsv frqwlqx 0 g*xq surgxlw gìulyì who tx*xqh rswlrq hxursìhqqh shxw vh
idluh =
vrlw frpph oh fdofxo g*xqh hvsìudqfh ã sduwlu gh od irupxoh =
.
Fw @ hu+W w, =Hw +VW N,
hq lpsrvdqw txh od orl frqglwlrqqhooh gh VW vdfkdqw Vw hvw xqh orl orj0qrupdoh =
5
5
Oq VW +vdfkdqw Vw , # Q Oq Vw . uh =+W w, > =+W w,
5
vrlw frpph od vroxwlrq gh o*ìtxdwlrq dx{ gìulyìhv sduwlhoohv =
uF @
CF
CF
4
C 5F
. uh=V=
. V 5 = 5 = 5
Cw
CV
5
CV
dyhf od frqglwlrq dx{ eruqhv F+W> V, @ +V N,. 1
85
7
Oh prgëoh gh Eodfn hw Vfkrohv
714 Lqwurgxfwlrq
Qrxv dedqgrqqrqv lfl oh fdguh glvfuhw gx fkdslwuh suìfìghqw hw wudydloorqv gìvrupdlv hq whpsv frqwlqx1
Od gìpdufkh vxlylh gdqv oh fdguh glvfuhw vhud dgrswìh lfl gdqv oh fdv frqwlqx1 Lo idxw jdughu hq wíwh txh
ohv ìwdshv vrqw ohv píphv hw ohv lqwxlwlrqv qdqflëuhv vlplodluhv = vhxo fkdqjh oh qlyhdx gh wudlwhphqw
pdwkìpdwltxh1 Fh qlyhdx ìwdlw wulyldo txdqg ohv doìd hw oh whpsv ìwdlhqw glvfuhwv > lo o*hvw prlqv yrluh
ehdxfrxs prlqv gdqv ohv prgëohv frqwlqxv1
Srxu uìvxphu/ od gìpdufkh ìwdlw hw vhud od vxlydqwh =
41 grqqìh g*xqh g|qdpltxh +doìdwrluh, gx vrxv0mdfhqw >
51 wudgxfwlrq gh o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh vrxv od iruph gh o*h{lvwhqfh g*xqh
phvxuh glwh ulvtxh0qhxwuh >
61 vrxv fhwwh phvxuh/ ohv sul{ dfwxdolvìv vrqw pduwlqjdohv/ fh txl frqgxlw ã xqh irupxoh gh ydorulvdwlrq
vhorq odtxhooh oh sul{ dxmrxug*kxl g*xq dfwli dssdudñw frpph o*hvsìudqfh +vrxv od phvxuh ulvtxh0
qhxwuh, ghv x{ ixwxuv dfwxdolvìv1 Gh idêrq ìtxlydohqwh/ ohv sul{ ghv dfwliv shxyhqw dxvvl íwuh yxv
frpph ghv vroxwlrqv g*xqh ìtxdwlrq dx{ gìulyìhv sduwlhoohv >
71 vsìflfdwlrq vxssoìphqwdluh gx prgëoh sdu od grqqìh g*xqh irqfwlrq gh yrodwlolwì >
81 xqh irlv oh prgëoh vsìflì/ ohv fdofxov qxpìultxhv shxyhqw íwuh phqìv mxvtx*ã ohxu whuph1
Vfkìpdwltxhphqw/ ohv ìwdshv 4/ 5 hw 6 derxwlvvhqw dx fdguh jìqìudo gh o*ìydoxdwlrq sdu duelwudjh1 Fh fdguh
vhud ydodeoh dxvvl elhq srxu ohv gìulyìv dfwlrq txh srxu ohv gìulyìv gh wdx{ g*lqwìuíw/ gh wdx{ gh fkdqjh
hwf1 D sduwlu gh o*ìwdsh 7/ rq lqwurgxlw ghv prgëohv vsìfltxhv = lo | d grqf xqh lqqlwì gh idêrqv gh vruwlu
gh o*ìwdsh 6 srxu gìqlu xq prgëoh sduwlfxolhu1 Dlqvl/ dx fkdslwuh suìfìghqw/ rq d idlw o*k|srwkëvh txh od
yrodwlolwì +f*hvw0ã0gluh o*ìfduw0w|sh frqglwlrqqho, ìwdlw uhsuìvhqwdeoh sdu xq xqltxh sdudpëwuh 1 F*hvw od
idêrq od soxv vlpsoh gh uìsrqguh ã o*ìwdsh 7 pdlv ulhq q*hpsífkdlw gh idluh soxv frpsoltxì hq suhqdqw xqh
yrodwlolwì irqfwlrq gx vrxv0mdfhqw +l1h1 +Vw ,,1 Rq rewlhqw doruv xq dxwuh prgëoh rü oh vrxv0mdfhqw +Vw ,w q*d
d sulrul dxfxqh fkdqfh g*íwuh orj0qrupdo1
Oh prgëoh gh Eodfn hw Vfkrohv hvw mxvwhphqw oh prgëoh olplwh gx fkdslwuh suìfìghqw/ f*hvw0ã0gluh xq prgëoh
rü od vsìflfdwlrq gh o*ìwdsh 7 hvw od soxv vlpsoh tx*rq sxlvvh lpdjlqhu/ l1h xqh yrodwlolwì uìgxlwh ã xq
sdudpëwuh 1 F*hvw fh prgëoh txh qrxv ìwxglrqv lfl1 Srxu fhod/ qrxv qh uhyhqrqv sdv vxu ohv ìwdshv 4/ 5
hw 6 = qrxv vxssrvrqv tx*hoohv vrqw dftxlvhv1 Fhv ìwdshv vrqw wulyldohv ã gìprqwuhu gdqv xq fdguh glvfuhw
pdlv qìfhvvlwhqw ghv gìprqvwudwlrqv soxv vrsklvwltxìhv hq frqwlqx1 Qrxv qrxv sodêrqv gluhfwhphqw dsuëv
o*ìwdsh 6 hw qrxv glvsrvrqv grqf gh 6 remhwv =
xqh g|qdpltxh gx vrxv0mdfhqw vrxv od phvxuh ulvtxh0qhxwuh +w|sltxhphqw orj0qrupdoh, >
xqh irupxoh g*ìydoxdwlrq uholdqw oh sul{ g*xq dfwli gìulyì ã o*hvsìudqfh ghv x{ ixwxuv >
xqh vhfrqgh irupxoh g*ìydoxdwlrq 0 ìtxlydohqwh 0 vrxv od iruph g*xqh ìtxdwlrq dx{ gìulyìhv sduwlhoohv
grqw od vroxwlrq hvw oh sul{ gh o*dfwli gìulyì1
715 Od irupxoh gh Eodfn hw Vfkrohv
Od g|qdpltxh gx vrxv0mdfhqw hvw od g|qdpltxh0olplwh gx fkdslwuh suìfìghqw hw v*ìfulw grqf4 =
5
5
Oq VW +vdfkdqw Vw , # Q Oq Vw . u =+W w, > =+W w,
5
+714,
Fhwwh glvwulexwlrq orj0qrupdoh hvw uhodwlyhphqw vlpsoh ã pdqlhu1 Qrxv vxssrvrqv vdqv shuwh gh
jìqìudolwì txh oh wdx{ gh glylghqgh hvw qxo1 Oh fdv rü fh wdx{ q*hvw sdv qxo hvw xqh jìqìudolvdwlrq
gluhfwh ghv fdofxov txl vxlyhqw1
Fhwwh glvwulexwlrq shuphw gh fdofxohu oh sul{ Fw g*xqh rswlrq g*dfkdw hxursìhqqh ã sduwlu gh od irupxoh
g*ìydoxdwlrq jìqìudoh =
.
Fw @ hu=+W w, =Hw +VW N,
Od ghqvlwì gh od orl qrupdoh Q > 5 v*ìfulw =
#
$
4
+{ ,5
s
= h{s 5 5
5
fh txl grqqh lfl xqh ghqvlwì gh Oq VW +vdfkdqw Vw , =
3 4
5 =+W w, 5
.
u
{
Oq
V
w
5
4
F
E
s
i +{, @ s
= h{s C
D
5
5
+W
w,
5 W w
Dx wrwdo/ oh sul{ gh o*rswlrq g*dfkdw v*ìfulw =
Fw @ hu=+W w, =
vrlw =
Fw @ hu=+W w, =
]
]
.4
4
.4
pd{ +h{ N> 3, =i+{,=g{
h{ =i +{,=g{ N=
]
.4
i +{,=g{
Oq N
Oq N
Lo hvw doruv idfloh g*h{sulphu fh sul{ ã o*dlgh gh od irqfwlrq gh uìsduwlwlrq gh od orl qrupdoh fhqwuìh uìgxlwh =
] |
{5
4
Q+|, @ s =
h 5 =g{
5 4
fh txl grqqh od irupxoh gh Eodfn hw Vfkrohv =
Fw @ Vw =Q +g4 , N=hu=+W w, =Q +g5 ,
dyhf =
hw =
4
=Oq
g4 @ s
W w
4
g5 @ s
=Oq
W w
Vw
N=hu+W w,
Vw
N=hu+W w,
4 Oh idfwhxu g*dfwxdolvdwlrq hvw sulv lfl vrxv vd iruph frqwlqxh
87
e3oEA 3|
4 s
. W w
5
4 s
W w
5
hw qrq glvfuëwh

EnoW
w 1
Oh sul{ g*xqh rswlrq g*dfkdw dssdudñw grqf frpph xqh irqfwlrq qrq olqìdluh gx sul{ gx vrxv0mdfhqw Vw 1 Rq
.
shxw uhpdutxhu sdu dloohxuv txh oruvtxh w dwwhlqw od gdwh g*h{huflfh W doruv FW hvw elhq ìjdo ã +VW N, 1
O*xwlolvdwlrq gh fhwwh irupxoh vhud h{dplqìh gdqv ohv vhfwlrqv vxlydqwhv1
Lo idxw elhq frpsuhqguh txh fhwwh irupxoh grqqh xq sul{ wkìrultxh srxu o*rswlrq1 Frpph sdu dloohxuv
ohv dfwliv gìulyìv vrqw frwìv vxu ghv pdufkìv hw uìsrqghqw grqf ã xqh orjltxh ruh0ghpdqgh/ lo q*hvw
devroxphqw sdv dftxlv txh oh sul{ wkìrultxh vrlw ìjdo dx sul{ uìhoohphqw revhuyì vxu oh pdufkì/ qh vhudlw0
fh txh sdufh tx*rq v*hvw sodfì gdqv xq prgëoh sduwlfxolhu +l1h1 oh prgëoh gh Eodfn hw Vfkrohv, rü od yrodwlolwì
d ìwì vxssrvìh frqvwdqwh1 Dyhf xqh dxwuh k|srwkëvh/ qrxv dxulrqv derxwl ã xqh wrxw dxwuh irupxoh1 Qrxv
uhylhqgurqv vxu fh srlqw sdu od vxlwh1
Yrlfl xq h{hpsoh gh fdofxo = frqvlgìurqv xqh rswlrq g*dfkdw vxu xqh dfwlrq grqw od ydohxu dxmrxug*kxl hvw
433 hxurv +Vw @ 433,/ gh sul{ g*h{huflfh <3 hxurv +N @ <3,/ gh pdwxulwì 6 prlv +W w @ 3=58 dqqìh,1
Vxssrvrqv sdu dloohxuv txh oh wdx{ g*lqwìuíw dqqxho hvw gh 9 ( +u @ 9(, hw od yrodwlolwì dqqxhooh gh 48 (
+ @ 48(,1 Ohv whuphv g4 hw g5 ydohqw grqf =
s
4
433
4
s
=Oq
g4 @
. 3=48 3=58 @ 4=9748
3
=
39
3
=
58
<3=h
5
3=48 3=58
hw =
4
s
g5 @
=Oq
3=48 3=58
g*rü =
433
<3=h3=393=58
s
4
3=48 3=58 @ 4=89:6
5
Fw @ 433=Q +g4 , <3=h3=39{3=58 =Q +g5 , @ 44=83 hxurv
Vl oh prgëoh gh Eodfn hw Vfkrohv uh ëwh elhq od uìdolwì +hw vl rq qh v*hvw sdv wurpsì vxu ohv qlyhdx{ gh wdx{
g*lqwìuíw hw gh yrodwlolwì,/ doruv ohv lqyhvwlvvhxuv dxmrxug*kxl vrqw suíwv ã sd|hu 44183 hxurv srxu gìwhqlu
oh gurlw g*dfkhwhu o*dfwlrq dx sul{ gh <3 hxurv gdqv 6 prlv/ vdfkdqw tx*hooh hq ydxw 433 dxmrxug*kxl1
716 Oh irupdolvph gx whpsv frqwlqx
Frpph qrxv yhqrqv gh oh yrlu/ ohv glvwulexwlrqv gh suredelolwì gh +Vw ,w shuphwwhqw gh idluh wrxv ohv
fdofxov gh sul{1 Wrxwhirlv/ lo h{lvwh xq irupdolvph soxv v|qwkìwltxh pdlv pdwkìpdwltxhphqw soxv dugx
tx*rq suìvhqwh lfl gh idêrq wuëv lqiruphooh1 Xqh suìvhqwdwlrq soxv sursuh shxw vh wurxyhu gdqv xq frxuv gh
fdofxo vwrfkdvwltxh hw gh pdwkìpdwltxhv dssoltxìhv1 O*ìoìphqw gh edvh hvw oh prxyhphqw Eurzqlhq
71614 Oh prxyhphqw Eurzqlhq hw ohv ìtxdwlrqv glìuhqwlhoohv vwrfkdvwltxhv
Oh prxyhphqw Eurzqlhq hvw qrwì lfl +Zw ,w = Fh surfhvvxv hvw gìql frpph oh surfhvvxv vwrfkdvwltxh ã
dffurlvvhphqwv lqgìshqgdqwv hw jdxvvlhqv/ l1h1 =
Zw3 Zw lqgìshqgdqw gh +Zv ,vw srxu wrxw w ? w3
hw =
Zw3 Zw vxlw xqh orl qrupdoh g*hvsìudqfh qxooh hw gh yduldqfh w3 w
Dlqvl/ juåfh ã fhw remhw/ od glvwulexwlrq gx vrxv0mdfhqw gdqv oh prgëoh gh Eodfn hw Vfkrohv shxw doruv
v*ìfuluh vrxv od iruph frqghqvìh vxlydqwh5 =
5
+715,
Oq Vw @ Oq V3 . u =w . =Zw
5
dyhf/ sdu frqyhqwlrq/ Z3 @ 31 Oh prxyhphqw Eurzqlhq shuphw dlqvl g*ìfuluh gh idêrq v|qwkìwltxh hw
pdqldeoh o*k|srwkëvh frqfhuqdqw od glvwulexwlrq gh suredelolwì gx vrxv0mdfhqw1 Lo idxw elhq yrlu tx*ã fh
vwdgh lo v*djlw g*xqh vlpsoh uì0ìfulwxuh = ìfuluh +715, rx gluh txh ohv orlv frqglwlrqqhoohv gh Oq VW vdfkdqw
Vw vrqw orj0qrupdohv +ìtxdwlrq +714,,/ f*hvw gluh ã shx suëv ghx{ irlv od píph fkrvh1
5 Rq vxssrvh wrxmrxuv txh oh wdx{ gh glylghqgh hvw qxo1
Hfroh Sro|whfkqltxh
88
Rq shxw dxvvl gh pdqlëuh ìtxlydohqwh ìfuluh o*ìtxdwlrq +715, vrxv xqh iruph glìuhqwlhooh/ l1h1 xqh ìtxdwlrq
glìuhqwlhooh vwrfkdvwltxh rx ìtxdwlrq gh glxvlrq =
5
gOq Vw @ u =gw . =gZw
5
txl hvw dqdorjxh ã o*ìtxdwlrq gh uìfxuuhqfh vwrfkdvwltxh gx fkdslwuh suìfìghqw/ txl grqqdlw Vw hq irqfwlrq
gh Vw4 hw g*xqh shuwxuedwlrq doìdwrluh K1 Lfl/ gZ hw K mrxhqw oh píph u÷oh/ l1h1 xq fkrf qdqflhu txl
shuphw gh vruwlu gx prqgh gìwhuplqlvwh 1 Xqh ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh surorqjh od qrwlrq
g*ìtxdwlrq glìuhqwlhooh wudglwlrqqhooh +l1h1 qrq vwrfkdvwltxh, ã wudyhuv oh whuph =gZw 1 Sdu frqvìtxhqw/
ulhq q*hpsífkh gh frqvlgìuhu ghv ìtxdwlrqv soxv jìqìudohv txl v*ìfuludlhqw =
gVw @ +w> Vw ,=gw . +w> Vw , =gZw
rü +=> =, hw +=> =, vrqw ghv irqfwlrqv txhofrqtxhv +ã ghv frqglwlrqv gh uìjxodulwì suëv6 ,1 Wrxwhirlv/ ohv
ìtxdwlrqv glìuhqwlhoohv vwrfkdvwltxhv qh vh pdqlhqw sdv wrxw ã idlw gh od píph idêrq txh ohv ìtxdwlrqv
ruglqdluhv1
71615 Oh ohpph g*Lw÷
Oh ohpph g*Lw÷ gìfulw frpphqw rq rewlhqw o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh g*xqh irqfwlrq gh Vw hw
gh w/ sdu h{hpsoh k+w> Vw ,/ txdqg rq frqqdñw o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh gh +Vw ,w 1 Gdqv oh fdv
ruglqdluh/ qrq vwrfkdvwltxh/ oh sdvvdjh gh o*xqh ã o*dxwuh vh idlw sdu od gìulyìh gh k sdu xqh irupxoh gx
w|sh =
Ck
Ck
=gw .
=gV
gk+w> Vw , @
Cw
CV
Lfl/ fh q*hvw soxv yudl = soxv h{dfwhphqw xq whuph vxssoìphqwdluh dssdudñw1 Rq d doruv =
Sursrvlwlrq 6 +Ohpph g*Lw÷, Vl oh surfhvvxv +Vw ,w vxlw o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh =
doruv +k+w> Vw ,,w vxlw o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh +rü k hvw gh fodvvh F 5 , =
Ck 4
C 5k
Ck
5
. +w> Vw , = 5 =gw .
=gV
gk+w> Vw , @
Cw
5
CV
CV
vrlw hqfruh =
gk+w> Vw , @
Ck
Ck 4
C 5k
. +w> Vw ,=
. +w> Vw ,5 = 5
Cw
CV 5
CV
=gw .
Ck
= +w> Vw , =gZw
CV
5 C5k5 uhsrvh vxu o*reoljdwlrq g*doohu
O*lqwxlwlrq txl vh wurxyh ghuulëuh oh whuph vxssoìphqwdluh 45 +w> Vw , = CV
ã o*ruguh ghx{ hq V gdqv oh gìyhorsshphqw gh k+=> =,1 Od qìfhvvlwì g* doohu ã o*ruguh ghx{ ìwdlw
gìmã suìvhqwh gdqv ohv gìyhorsshphqwv olplwìv gx fkdslwuh suìfìghqw1 Rq h{sulph fh idlw hq glvdqw txh
gZ 5 hvw g*xq ruguh R+gw,1 Hq hhw/ sdu gìqlwlrq gx prxyhphqw Eurzqlhq/ rq d H+Zw3 Zw , @ 3 hw
H +Zw3 Zw ,5 @ w3 w/ tx*rq ìfulw H +gZw ,5 @ gw1
H{hpsoh 5 Rq shxw grqqhu xq suhplhu h{hpsoh hq fdofxodqw o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh gh
+Vw ,w ã sduwlu gh fhooh gh +Oq Vw ,w gdqv oh prgëoh gh Eodfn hw Vfkrohv1 Lo hvw idfloh gh yìulhu txh vl =
5
gOq Vw @ u =gw . =gZw
5
doruv =
gVw @ u=Vw =gw . =Vw =gZw
+716,
6 Frpph srxu ohv ìtxdwlrqv glìuhqwlhoohv ruglqdluhv/ o*h{lvwhqfh g*xqh vroxwlrq q*hvw sdv irufìphqw ìylghqwh hw uhtxlhuw
ghv frqglwlrqv gh uìjxodulwì vxu ohv frh!flhqwv gh o*ìtxdwlrq1
89
tx*rq ìfulw dxvvl =
gVw
@ u=gw . =gZw
Vw
w
Lo idxw wrxwhirlv íwuh suxghqw fdu/ g*dsuëv oh ohpph g*Lw÷/ gV
Vw 9@ gOq Vw $
H{hpsoh 6 Uhwurxyhu o*ìtxdwlrq dx{ gìulyìhv sduwlhoohv gh od q gx fkdslwuh suìfìghqw hvw xq ghx{lëph
h{hpsoh g*dssolfdwlrq1 Vxssrvrqv xqh g|qdpltxh soxv jìqìudoh rü od yrodwlolwì hvw txhofrqtxh =
gVw @ u=Vw =gw . +w> Vw , =gZw
Oh sul{ g*xq dfwli gìulyì gh sul{ Fw @ F+w> Vw , vdwlvidlw grqf o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh =
4 5
CF
CF
C 5F
CF
gF+w> Vw , @
. u=Vw =
. +w> Vw ,= 5 =gw .
= +w> Vw , =gZw
+717,
Cw
CV
5
CV
CV
Rq shxw doruv idluh oh udlvrqqhphqw wuëv lqirupho vxlydqw1 Frqvwuxlvrqv xq sruwhihxlooh frpsrvì gh o*rswlrq
hw g*xqh txdqwlwì gh o*dfwlrq vrxv0mdfhqwh = Yw @ F+w> Vw ,.=Vw 1 Od g|qdpltxh gh fh sruwhihxlooh v*ìfulw =
4
CF
CF
C 5F
CF
gYw @
. u=Vw =
. . 5 +w> Vw ,= 5 =gw .
. = +w> Vw , =gZw
Cw
CV
5
CV
CV
Rq vxssrvh txh qh ydulh sdv hqwuh w hw w . gw = o*lqyhvwlvvhxu q*dfkëwh sdv rx qh yhqg sdv g*dfwlrqv
vxssoìphqwdluhv +oh sruwhihxlooh hvw glw dxwr0qdqfì,1 Frpph gdqv oh fdv glvfuhw gx fkdslwuh suìfìghqw/
rq fkrlvlw gh whooh vruwh txh oh sruwhihxlooh vrlw vdqv ulvtxh1 Lfl/ fhod vljqlh txh/ gdqv o*ìtxdwlrq
glìuhqwlhooh vwrfkdvwltxh gh +Yw ,w /ohv fkrfv gZ rqw ìwì ìolplqìv 1 Lo idxw grqf suhqguh =
@ hw rq d doruv =
gYw @
CF
CV
4
CF
C 5F
. 5 +w> Vw ,= 5
Cw
5
CV
=gw
Fh sruwhihxlooh hvw orfdohphqw vdqv ulvtxh hw qh grlw grqf udssruwhu ql soxv ql prlqv txh oh wdx{ g*lqwìuíw
u/ vrlw =
gYw @ u=Yw =gw
hw grqf/ hq lghqwldqw ohv ghx{ h{suhvvlrqv =
CF
4
C5 F
CF
. u=V=
. 5 +w> Vw ,= 5 @ uF
Cw
CV
5
CV
txl hvw o*ìtxdwlrq dx{ gìulyìhv sduwlhoohv gh od q gx fkdslwuh suìfìghqw gdqv xq fdguh wrxwhirlv soxv jìqìudo
sxlvtxh od irqfwlrq gh yrodwlolwì hvw txhofrqtxh1 Oh prgëoh gh Eodfn hw Vfkrohv hvw xq fdv sduwlfxolhu1 Hqq/
vl rq lqmhfwh fh uìvxowdw gdqv o*ìtxdwlrq +715,/ rq rewlhqw doruv =
gF+w> Vw , @ u=F+w> Vw ,=gw .
CF
=+w> Vw ,=gZw
CV
H{hpsoh 7 Vl rq dssoltxh oh ohpph g*Lw÷ vxu od yduldeoh ]w @ huw =F+w> Vw ,/ rq rewlhqw =
g]w @ huw =
CF
==Vw =gZw
CV
txl v*ìfulw vrxv od iruph g*xqh lqwìjudoh vwrfkdvwltxh =
] W
CF
]W @ ]w .
==Vx =gZx
hux =
CV
w
hw txl frqgxlw ã =
F+w> Vw , @ hu=+W w, =Hw +F+W> VW ,,
hq xwlolvdqw H +gZx , @ 31 Rq uhwurxyh elhq o*lgìh txh ohv sul{ dfwxdolvìv vrqw pduwlqjdohv rx hqfruh txh
oh sul{ g*xq dfwli hvw ìjdo ã od ydohxu dfwxdolvìh hvsìuìh ghv x{ ixwxuv tx*lo jìqëuh1
Hfroh Sro|whfkqltxh
8:
71616 O*ìtxdwlrq gh Irnnhu0Sodqfn
Hqq/ qrxv dyrqv glw txh ohv ìtxdwlrqv glìuhqwlhoohv vwrfkdvwltxhv ìwdlhqw xq pr|hq g*ìfuluh gh idêrq
frqghqvìh oh w|sh gh glvwulexwlrq gh suredelolwì vxlylh sdu oh surfhvvxv1 Lo idxw grqf h{solflwhu fh olhq/
f*hvw0ã0gluh prqwuhu frpphqw/ ã sduwlu gh o*ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh/ rq uìfxsëuh ohv ghqvlwìv
gh suredelolwì1 Sduwrqv gh o*ìtxdwlrq =
hw qrwrqv iw>Vw +W > {, od ghqvlwì gh VW vdfkdqw Vw = Rq d doruv o*ìtxdwlrq glwh gh Irnnhu0Sodqfn =
k
l
5
5
4 C +W> {, =i+W > {,
C ^+W> {,=i +W > {,`
Ci
@ =
5
CW
5
C{
C{
vrxv od frqglwlrq dx{ eruqhv =
iw>Vw +w> {, @ gludf +{ Vw ,
717 Uhirupxodwlrq gh o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh
Oh gìwrxu sdu oh irupdolvph hq whpsv frqwlqx d prqwuì txh oh prgëoh gh Eodfn hw Vfkrohv q*ìwdlw tx*xq
prgëoh sdupl g*dxwuhv1 Ghx{ prgëohv dgplvvleohv glëuhqw sdu ohxu irqfwlrq gh yrodwlolwì 1 Hq uhydqfkh/
rq shxw prqwuhu/ sdu k|srwkëvh g*devhqfh g*rssruwxqlwìv g*duelwudjh/ txh txhooh txh vrlw
od irqfwlrq gh yrodwlolwì xwlolvìh/ od sduwlh hq gw +tx*rq dsshooh hqfruh oh guliw, hvw wrxmrxuv
od píph1 Soxv h{dfwhphqw/ vrxv o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh/ lo h{lvwh xqh
phvxuh gh suredelolwì ulvtxh0qhxwuh vrxv odtxhooh =
gVw @ u=Vw =gw . +w> V, =gZw
+718,
rü +Zw ,w hvw xq prxyhphqw Eurzqlhq vwdqgdug/ hw =1
gVw @ +u , =Vw =gw . +w> V, =gZw
oruvtx*lo | d ghv glylghqghv7 1
Oh vhxo ghjuì gh olehuwì frqfhuqh grqf od irqfwlrq gh yrodwlolwì1 Dlqvl/ rq shxw frqvlgìuhu ghv irqfwlrqv
+w> V,/ oh prgëoh gh Eodfn hw Vfkrohv ìwdqw oh fdv ìoìphqwdluh rü =
+w> V, @ =V
Od irupxoh glwh gh Eodfn hw Vfkrohv q*d d sulrul gh vhqv txh vrxv fhwwh k|srwkëvh1
718 Dqdo|vh hw xwlolvdwlrq gh od irupxoh gh Eodfn hw Vfkrohv
D sduwlu gh fhwwh irupxoh/ rq fdofxoh glìuhqwv udwlrv glw ohv juhftxhv txh ohv rsìudwhxuv xwlolvhqw srxu
frxyulu ohxu srvlwlrq1 O*lgìh hvw od vxlydqwh = xq dfwli gìulyì who tx*xqh rswlrq hvw ìylghpphqw xq surgxlw
ulvtxì fdu vd ydohxu gìshqg gx sul{ gx vrxv0mdfhqw1 O*dfkhwhxu rx o*ìphwwhxu g*xqh rswlrq shxw wrxw ã idlw
gìflghu g*dvvxphu fh ulvtxh hw fh srxu ghx{ udlvrqv1 Hq suhplhu olhx/ lo shxw v*djlu g*xq sxu vsìfxodwhxu
txl grqf idlw xq sdul vxu o*dyhqlu frpph xq mrxhxu gdqv xq fdvlqr1 Hq vhfrqg olhx/ fhw lqyhvwlvvhxu shxw
íwuh xqh hqwuhsulvh txl vxssruwh/ sdu dloohxuv/ xq ulvtxh rssrvì = hq dfkhwdqw rx hq ìphwwdqw xqh rswlrq/
hooh frxyuh fh ulvtxh1 Sdu h{hpsoh/ xqh hqwuhsulvh txl lpsruwh ghv pdwlëuhv suhplëuhv h{sulpìhv hq groodu
fudlqw xqh kdxvvh gx groodu > vl hooh dfkëwh ghv surgxlwv gìulyìv grqw oh sul{ hvw srvlwlyhphqw fruuìoì dx
groodu/ doruv hooh vh frxyuh joredohphqw1
7 Rq shxw hqylvdjhu ghv prgëohv hqfruh soxv jìqìudx{ rü od irqfwlrq gh yrodwlolwì hvw hooh0píph xq surfhvvxv vwrfkdvwltxh
qrq uìgxfwleoh ã xqh irqfwlrq gx whpsv hw gx vrxv0mdfhqw1
8;
Lqyhuvhphqw/ xq dfkhwhxu rx xq ìphwwhxu g*rswlrq shxw gìflghu gh qh sdv dvvxphu oh ulvtxh lqgxlw1 Sdu
h{hpsoh/ xqh edqtxh yhqg ã xq gh vhv folhqwv +sdu h{hpsoh xqh hqwuhsulvh, xq surgxlw gìulyì1 Fhwwh
edqtxh shxw vrxkdlwhu uhfhyrlu od frpplvvlrq dwwdfkìh ã fhwwh yhqwh +l1h1 oh sul{ gh vhv vhuylfhv, pdlv qh
sdv yrxorlu srxu dxwdqw vxssruwhu oh ulvtxh tx*hooh d hqjudqjì gx idlw píph gh o*rsìudwlrq1 Hooh fkhufkhud
doruv ã vh frxyulu hw oh soxv vlpsoh vhud gh frpsoìwhu vd srvlwlrq sdu ghv dfwlrqv vrxv0mdfhqwhv1
Hq sudwltxh/ rq uhsurgxlw oh udlvrqqhpphqw gx fkdslwuh suìfìghqw +fdv glvfuhw, rx ghv vhfwlrqv suìfìghqwhv
+fdv frqwlqx, = rq shxw wrxmrxuv +dx prlqv hq wkìrulh, frqvwuxluh xq sruwhihxlooh frpsrvìh g*xq dfwli gìulyì
hw g*xqh fhuwdlqh txdqwlwì gx vrxv0mdfhqw gh whooh vruwh txh fh sruwhixlooh uìvxowdqw vrlw +orfdohphqw, vdqv
ulvtxh1 Frpph rq o*d yx/ od txdqwlwì gh vrxv0mdfhqw grlw íwuh ìjdo ã CF
CV 1
Rq gìqlw grqf oh ghowd gh o*dfwli gìulyì frpph od gìulyìh gx sul{ gh fhw dfwli sdu udssruw dx vrxv0mdfhqw =
@
CF
CV
txl grqqh gdqv oh fdguh gh od irupxoh gh Eodfn hw Vfkrohv srxu xqh rswlrq g*dfkdw =
@ Q+g4 , . V=
hw =
CQ +g4 ,
CQ +g5 ,
N=hu+W w, =
@ Q +g4 ,
CV
CV
@ Q +g4 , 4 @ Q +g4 ,
srxu xqh rswlrq gh yhqwh1
Delta Call
Cours
Delta Put
Ghowd hq irqfwlrq gx frxuv gx vrxv0mdfhqw
Xq sruwhihxlooh frpsrvì g*xqh xqlwì g*dfwli gìulyì +sdu h{hpsoh xqh rswlrq, hw gh xqlwìv gh vrxv0
mdfhqw hvw grqf orfdohphqw vdqv ulvtxh/ l1h1 oh fkrf gZ hvw ìolplqì gx sruwhihxlooh1 Rq glw doruv tx*rq d
xq sruwhihxlooh ghowd0qhxwuh txl/ sdu devhqfh g*rssruwxqlwìv g*duelwudjh/ qh grlw udssruwhu txh oh wdx{
g*lqwìuíw vdqv ulvtxh u1 Soxv jìqìudohphqw/ rq shxw frqvlgìuhu xq sruwhihxlooh g*rswlrqv gh glìuhqwv w|shv
+sdu h{hpsoh ghv sul{ hw ghv gdwhv g*h{huflfh glìuhqwv, hw fdofxohu oh qrpeuh wrwdo g*dfwlrqv tx*lo idxw
dfkhwhu rx yhqguh srxu txh oh ghowd joredo vrlw qxo1 Vl rq uhsuhqg o*h{hpsoh qxpìultxh gx gìexw gx
fkdslwuh/ oh ghowd ydxw Q +g4 , @ 3=<7<:1 Vl rq ìphw 433 rswlrqv/ lo idxw grqf íwuh dfkhwhxu gh <8 dfwlrqv
hqylurq1
Hfroh Sro|whfkqltxh
8<
Wrxwhirlv/ oh udlvrqqhphqw q*d gh vhqv tx*hq whpsv frqwlqx1 Hq sudwltxh/ xqh frxyhuwxuh hq whpsv frqwlqx
q*hvw sdv uìdolvdeoh hw grqf qh shxw íwuh sduidlwh1 Srxu fhwwh udlvrq/ ohv lqyhvwlvvhxuv ìydoxhqw oh ulvtxh gh
ohxuv srvlwlrqv ã wudyhuv od gìulyìh g*ruguh ghx{ tx*rq dsshooh oh jdppd =
@
C 5F
CV 5
txl ydxw/ gdqv oh fdguh gh od irupxoh gh Eodfn hw Vfkrohv srxu xqh rswlrq g*dfkdw hw gh yhqwh =
@
g54
4
s
=h 5
V 5 +W w,
hw txl hvw grqf srvlwli1 Oh jdppd phvxuh ìjdohphqw od vhqvlelolwì gx ghowd ã ghv prxyhphqwv gx frxuv
gx vrxv0mdfhqw = lo lqgltxh grqf o*dpsolwxgh hw od iuìtxhqfh gx uhfdodjh gh od frxyhuwxuh1 Vl sdu h{hpsoh
oh jdppd hvw surfkh gh }ìur/ oh ghowd ydulh shx hw od frxyhuwxuh hq ghowd q*d sdv ehvrlq g*íwuh prglìh1
D o*lqyhuvh/ vl oh jdppd hvw ìohyì/ lo idxw vrxyhqw hw vljqlfdwlyhphqw uh0frqvlgìuhu oh qrpeuh g*dfwlrqv
hq sruwhihxlooh vl rq yhxw frqvhuyhu xqh frxyhuwxuh ghowd0qhxwuh1
Cours
Jdppd hq irqfwlrq gx frxuv gx vrxv0mdfhqw
Rq gìqlw ìjdohphqw oh wkhwd frpph od gìulyìh sduwlhooh gx sul{ gh o*dfwli sdu udssruw dx whpsv =
@
CF
Cw
txl ydxw/ gdqv oh prgëoh gh Eodfn hw Vfkrohv =
4
@ u=F u=V= =5 =V 5
5
Fhwwh irupxoh vh gìgxlw lppìgldwhphqw gh o*ìtxdwlrq dx{ gìulyìhv sduwlhoohv gh od vhfwlrq suìfìghqwh1 Oh
wkhwd phvxuh gh frpelhq oh sul{ gh o*rswlrq glplqxh sdu oh vhxo sdvvdjh gx whpsv1
93
Cours
Wkhwd g*xq fdoo hq irqfwlrq gx frxuv gx vrxv0mdfhqw
Lpdjlqrqv xq sruwhihxlooh ghowd0qhxwuh ìjdo ã Yw @ Fw =Vw 1 Hqwuh w hw w . w/ od ydohxu gx sruwhihxlooh
ydulh gh =
4
Y * =w . = +V,5
5
Vl ohv prxyhphqwv gx vrxv0mdfhqw vrqw vx!vdpphqw lpsruwdqwv +gdqv xq vhqv rx gdqv xq dxwuh,/ od ydohxu
gx sruwhihxlooh ghowd0qhxwuh dxjphqwh1 Gdqv oh fdv frqwudluh/ o*hhw gþ ã o*ìfrxohphqw gx whpsv grplqh
hw oh sruwhihxlooh yrlw vd ydohxu gìfurñwuh1
Ohv udwlrv ghowd/ jdppd hw wkhwd8 shuphwwhqw gh frqqdñwuh ohv hhwv ghv yduldwlrqv gx vrxv0mdfhqw hw gx
sdvvdjh gx whpsv1 Lo idxw uhpdutxhu wrxwhirlv txh od ydohxu gh fhv udwlrv gìshqg fuxfldohphqw gx prgëoh
fkrlvl1 Lo h{lvwh grqf xq ulvtxh gh prgëoh1 Fh ulvtxh hvw olì dx idlw txh oh prgëoh fkrlvl +lfl oh prgëoh gh
Eodfn hw Vfkrohv, q*hvw shxw0íwuh sdv oh erq prgëoh1 Xwlolvhu xq pdxydlv prgëoh hvw xq ghv ulvtxhv
pdmhxuv vxssruwìv sdu ohv rsìudwhxuv txl lqwhuylhqqhqw vxu ohv pdufkìv gh surgxlwv gìulyìv1 Hq hhw/
gdqv fh fdv/ ohv udwlrv gh frxyhuwxuh vhurqw eldlvìv hw grqf xqh srvlwlrq vxssrvìh íwuh ghowd0qhxwuh +grqf
soxv rx prlqv lqvhqvleoh dx{ prxyhphqwv gx vrxv0mdfhqw, qh oh vhud hq idlw sdv/ fh txl ìylghpphqw hvw
gdqjhuhx{1 Hq rxwuh/ rq frqêrlw dlvìphqw txh/ wdqw tx*rq qh frqqdñw sdv oh yudl prgëoh +hw f*hvw
wrxmrxuv oh fdv,/ lo hvw gl!floh gh phvxuhu o*h{srvlwlrq ã fh ulvtxh1
Gdqv oh fdv gx prgëoh gh Eodfn hw Vfkrohv/ gluh txh fh q*hvw sdv xq erq prgëoh uhylhqw ã gluh txh
o*k|srwkëvh gh yrodwlolwì frqvwdqwh q*hvw sdv fruuhfwh1 Od yrodwlolwì lpsolflwh +lpsolhg yrodwlolw|, hvw xqh
idêrq gh phwwuh hq ìylghqfh fh idlw1 Od yrodwlolwì lpsolflwh fruuhvsrqg dx sdudpëwuh tx*lo idxgudlw phwwuh
gdqv od irupxoh gh Eodfn hw Vfkrohv srxu uhwurxyhu oh sul{ gh o*rswlrq uìhoohphqw revhuyì vxu oh pdufkì1
Hooh hvw grqf fdofxoìh +srxu xqh rswlrq g*dfkdw vwdqgdug, frpph od ydohxu lps vroxwlrq gh o*ìtxdwlrq =
Fw revhuyì @ Vw =Q +g4 , N=hu=+W w, =Q +g5 ,
dyhf =
g4 @
4
s
lps W w
=Oq
Vw
N=huW w,
s
4
. lps W w
5
8 Rq gìqlw ìjdohphqw oh ukr frpph od gìulyìh sdu udssruw dx wdx{ g*lqwìuíw
Hfroh Sro|whfkqltxh
o1
94
hw =
g5 @
4
s
lps W w
=Oq
Vw
N=h uW w,
s
4
lps W w
5
Vl oh prgëoh gh Eodfn hw Vfkrohv ìwdlw oh erq prgëoh/ doruv rq ghyudlw revhuyhu txh ohv yrodwlolwìv
lpsolflwhv fdofxoìhv vxu wrxwhv ohv rswlrqv g*dfkdw g*xq píph vrxv0mdfhqw vrqw ìjdohv1 Dxwuhphqw glw/ rq
ghyudlw revhuyhu =
lps +N> W w, @ frqvwdqwh
Ru/ hpslultxhphqw/ fh q*hvw sdv oh fdv = ghx{ rswlrqv gh píph vrxv0mdfhqw/ gh pdwxulwì rx gh sul{
g*h{huflfh glìuhqwv rqw ghv yrodwlolwìv lpsolflwhv glìuhqwhv = oh vploh gh yrodwlolwì hvw od irqfwlrq txl
grqqh od yrodwlolwì lpsolflwh hq irqfwlrq gx sul{ g*h{huflfh lps +=> W w,hw rq dsshooh qdssh gh yrodwlolwì
od vxuidfh lps +=> =,1 Rq frqvwdwh gh soxv txh od qdssh gh yrodwlolwì vh gìiruph dx frxuv gx whpsv gh
idêrq doìdwrluh1
Srxu uhqguh frpswh gh fhv idlwv hpslultxhv/ lo idxw hqylvdjhu ghv prgëohv soxv vrsklvwltxìhv rü od irqf0
wlrq gh yrodwlolwì q*hvw soxv frqvwdqwh1 Gdqv fh fdv jìqìudohphqw/ wrxv ohv uìvxowdwv gh orj0qrupdolwì
glvsdudlvvhqw hw ohv txhvwlrqv g*ìydoxdwlrq hw gh frxyhuwxuh ghylhqqhqw qhwwhphqw soxv frpsoh{hv1 Ohv
rsìudwhxuv irqw xq frpsurplv hq xwlolvdqw txdqg píph oh prgëoh gh Eodfn hw Vfkrohv pdlv hq fdofxodqw
oh yhjd txl hvw od gìulyìh gx sul{ sdu udssruw dx sdudpëwuh gh yrodwlolwì =
yhjd @
CF
C
Gdqv oh fdguh vwulfw gx prgëoh gh Eodfn hw Vfkrohv/ fhod q*d sdv gh vhqv gh fdofxohu fh udwlr fdu hvw
vxssrvì íwuh xq sdudpëwuh {h/ ã o*lqyhuvh gx whpsv rx gx vrxv0mdfhqw1 Fhod hq d xq vl rq frqvlgëuh txh
fh prgëoh hvw oh erq prgëoh dx suhplhu ruguh 1
95
8
H{whqvlrqv hw dssolfdwlrqv gx prgëoh gh Eodfn hw
Vfkrohv
814 Lqwurgxfwlrq
Oh prgëoh gh Eodfn hw Vfkrohv gìvljqh/ frpph suìfìghpphqw/ oh prgëoh rü oh sul{ gx vrxv0mdfhqw vxlw/
vrxv od suredelolwì ulvtxh0qhxwuh/ xqh ìtxdwlrq glìuhqwlhooh vwrfkdvwltxh gdqv odtxhooh od yrodwlolwì hvw
frqvwdqwh1 Vrxv fhwwh vsìflfdwlrq/ rq d yx txh oh orjdulwkph gx sul{ gx vrxv0mdfhqw ìwdlw jdxvvlhq/ fh
txl shuphwwdlw gh idluh dlvìphqw ghv fdofxov gh sul{ g*rswlrq vwdqgdugv g*dfkdw rx gh yhqwh1
Lfl/ qrxv uhvwrqv gdqv oh fdguh gh fh prgëoh pdlv qrxv qrxv lqwìuhvvrqv ã ghv h{hpsohv g*rswlrqv qrq
vwdqgdugv1 Hq suhplhu olhx/ qrxv h{dplqrqv fhuwdlqhv rswlrqv srxu ohvtxhoohv rq shxw hqfruh idluh ghv
fdofxov gh sul{ h{solflwhv +rswlrqv ã eduulëuh/ rswlrqv gljlwdohv, sxlv/ gdqv xq vhfrqg/ whpsv/ ghv rswlrqv
txl q*rqw sdv gh irupxoh gh ydorulvdwlrq h{solflwh píph gdqv xq prgëoh dxvvl vlpsoh txh fhoxl gh Eodfn
hw Vfkrohv +rswlrqv dpìulfdlqhv,1
815 Sulvh hq frpswh ghv glylghqghv
Mxvtx*ã pdlqwhqdqw/ ohv glylghqghv rqw ìwì qìjoljìhv1 Vl rq uhsuhqg ohv fdofxov ghv fkdslwuhv suìfìghqwv/
ohv glylghqghv shxyhqw v*lqwurgxluh ã wudyhuv xq wdx{ gh glylghqgh who txh od g|qdpltxh gx vrxv0mdfhqw
v*ìfulyh vrxv suredelolwì ulvtxh0qhxwuh gh od idêrq vxlydqwh =
gVw @ +u , =Vw =gw . =Vw =gZw
Dlqvl/ od orl gh Oq VW frqglwlrqqhoohphqw ã Vw v*ìfulw =
4
Oq VW +frqwlrqqhoohphqw ã Vw , # Q
u 5 = +W w, > 5 = +W w,
5
Lo hvw doruv idfloh gh uhsuhqguh ohv fdofxov suìfìghqwv srxu rewhqlu oh sul{ wkìrultxh g*xq fdoo vwdqgdug =
Fw @ Vw =h +W w, =Q +g4 , N=hu=+W w, =Q+g5 ,
dyhf =
hw =
4
g4 @ s
=Oq
W w
4
g5 @ s
=Oq
W w
Vw
N=h+u ,=+W w,
Vw
N=h+u ,=+W w,
4 s
. W w
5
4 s
W w
5
Fhfl q*d gh vhqv txh vl ohv glylghqghv vrqw yhuvìhv frqwlqþphqw rx txdvl0frqwlqþphqw1 Hq sudwltxh/ fh
q*hvw sdv oh fdv srxu xqh dfwlrq/ vhv glylghqghv ìwdqw yhuvìhv ã ghv gdwhv glvfuëwhv +wulphvwuh/ vhphvwuh/
dqqìh,1
Srxu whqlu frpswh gh fh fdudfwëuh glvfuhw/ rq shxw lqwurgxluh od txdqwlwì =
Gw>x @ glyw =hu+xw, . glyw.4 =hu+xw4, . === . glyx
f*hvw0ã0gluh od vrpph fdslwdolvìh ghv glylghqghv yhuvìv hqwuh ohv gdwhv w hw x1 Rq vxssrvh txh ohv glylghqghv
ixwxuv vrqw frqqxv dxmrxug*kxl1 Lo v*djlw ìylghpphqw g*xqh k|srwkëvh iruwh = hq sudwltxh/ rq frqvlgìuhud
sdu h{hpsoh txh ohv glylghqghv ixwxuv vrqw ìjdx{ dx ghuqlhu glylghqgh frqqx pxowlsolì sdu xq wdx{ gh
surjuhvvlrq dg0krf1 Od yduldeoh VW . Gw>W uhsuìvhqwh od vrpph wrwdoh dftxlvh hq W sdu o*lqyhvwlvvhxu txl
d dfkhwì xqh dfwlrq hq w1 Dlqvl/ od yduldeoh =
\w>W @
VW . Gw>W
Vw
phvxuh oh uhqghphqw wrwdo gh o*rsìudwlrq txl frqvlvwh ã dfkhwhu o*dfwlrq hq gdwh w/ fdslwdolvhu ohv glylghqghv
uhêxv hqwuh w hw W / hw uhyhqguh o*dfwlrq hq W 1 Vl oh prqgh ìwdlw gìwhuplqlvwh/ fh uhqghphqw vhudlw ìjdo ã
hu+W w, 1 Rq shxw doruv ìwhqguh oh prgëoh gh Eodfn hw Vfkrohv hq vxssrvdqw txh =
g\w>x @ u\w>x =gx . =\w>x =gZx
Rq shxw doruv dssoltxhu od irupxoh gh Eodfn hw Vfkrohv vdqv glylghqgh hq uhpdutxdqw txh =
Fw
@
hu+W w, =Hw ^VW N`.
@
hu+W w, =Hw ^Vw =\w>W +N . Gw>W ,`.
@ Vw =Q +g4 , +N . Gw>W , =hu+W w, =Q +g5 ,
dyhf =
4
=Oq
g4@5 @ s
W w
Vw
+N . Gw>W , =huW w,
4 s
.@ W w
5
816 Wdx{ g*lqwìuíw hw yrodwlolwì gìshqgdqwhv gx whpsv
Gdqv oh vrxfl gh vlpsolhu/ qrxv dyrqv vxssrvì txh oh wdx{ g*lqwìuíw ìwdlw xqh frqvwdqwh u/ fh txl hvw
dvvh} ìorljqì gh od uìdolwì1 Ohv wdx{ g*lqwìuíw ydulhqw gh idêrq vwrfkdvwltxh frpph ohv sul{ g*dfwlrqv1 Ohv
udlvrqqhphqwv txl gìfrxohqw gx prgëoh elqrpldo grlyhqw íwuh vxevwdqwlhoohphqw prglìv gëv oruv tx*rq
dxwrulvh ohv wdx{ g*lqwìuíw ã íwuh hx{0píphv ghv yduldeohv doìdwrluhv/ vxuwrxw vl fhv yduldeohv doìdwrluhv
qh vrqw sdv lqgìshqgdqwhv ghv sul{ g*dfwlrq1 Lqwxlwlyhphqw/ rq qh shxw sdv/ gdqv fh fdv/ vruwlu oh
whuph g*dfwxdolvdwlrq +l1h1 o*ìtxlydohqw gh hu+W w, , gh o*hvsìudqfh frqglwlrqqhooh +hq wrxw fdv sdv dxvvl
idflohphqw,1
Hq uhydqfkh/ vl rq vxssrvh txh ohv wdx{ g*lqwìuíw qh vrqw sdv vwrfkdvwltxhv pdlv vrqw qìdqprlqv ghv
irqfwlrqv +gìwhuplqlvwhv, gx whpsv/ o*k|srwkëvh gh orj0qrupdolwì wlhqw wrxmrxuv wdqw txh od yrodwlolwì hvw
frqvwdqwh1 Rq shxw grqf vxssrvhu txh oh wdx{ frxuw u gìshqg gx whpsv +gh idêrq gìwhuplqlvwh, hw od
irupxoh gh Eodfn hw Vfkrohv kdelwxhooh hvw frqvhuyìh ã frqglwlrq gh uhpsodfhu oh whuph g*dfwxdolvdwlrq
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s gh idêrq
gìwhuplqlvwh
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fdoo v*ìfulw =
;
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=
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hw oh x{ frqwudluh srxu xq sxw1 Oh sul{ wkìrultxh g*xqh whooh rswlrq hvw idfloh ã fdofxohu1 Rq shxw xwlolvhu
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Fw @ hu+W w, =Hw ^4 +VW A N,`
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4 5
5
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5
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4
g5 @ s
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W w
Vw
N=h uW w,
4 s
W w
5
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w, =Q 3 +g5 ,
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3
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5
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gh N1
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g*xqh judqgh txdqwlwì gx vrxv0mdfhqw srxu jdughu xqh srvlwlrq ghowd0qhxwuh/ fh txl hvw ã od irlv frþwhx{
hw srwhqwlhoohphqw ulvtxì fdu wuëv vhqvleoh ã gh shwlwhv huuhxuv1
Hfroh Sro|whfkqltxh
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srvlwlyhv/ vhorq oh vljqh gh g4 = Oh jdppd hvw qìjdwli oruvtxh =
4 5
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hw grqf xqh srvlwlrq ghowd0qhxwuh hvw wuëv h{srvìh ã xq iruw prxyhphqw gx vrxv0mdfhqw1 Gdqv oh fdv
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gN
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glvwlqjxh dlqvl =
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grqqìh >
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gh K +hqwuh dxmrxug*kxl hw od gdwh W ,1
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.
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A W +,
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k
l
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qh suìvhqwhqw sdv gh gl!fxowì sduwlfxolëuh vl rq uhpdutxh txh/ sdu v|pìwulh gx prxyhphqw Eurzqlhq/
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4. 5u5
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4. 5u5
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4. 5u5
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4. 5u5
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4. 5u5
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4. 5u5
K
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g4
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s4
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.
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9;
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pw gìvljqhqw =
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txhoohv frqglwlrqv oh gìwhqwhxu gh o*rswlrq d xq lqwìuíw ã h{hufhu vrq rswlrq1 Qrxv qh uhqwurqv sdv gdqv
ohv gìwdlov fdu od wkìrulh vrxv0mdfhqwh hvw frpsoh{h1
Hq jurv/ oh sul{ gh o*rswlrq dpìulfdlqh +rswlrq g*dfkdw vwdqgdug, v*ìfulud =
k
.l
vxs
Hw hu+ w, = +V N,
Fwdp @
3 W
rü oh vxs hvw sulv vxu wrxv ohv whpsv g*duuíw1 Xq whpsv g*duuíw hvw xqh irqfwlrq ã ydohxu gdqv ^3> W ` whooh
txh o*ìyìqhphqw @ w hvw irqfwlrq gh +Vx ,xw 1 Hq fodlu/ rq v*lqwìuhvvh dx fdv rü oh gìwhqwhxu gh o*rswlrq
qh shxw h{hufhu tx*hq irqfwlrq ghv ydohxuv sdvvìhv hw frxudqwhv gx vrxv0mdfhqw hw sdv ghv ydohxuv ixwxuhv
+fh txl sdudñw orjltxh, ql ghv ydohxuv g*dxwuhv yduldeohv1
Oh sul{ gh o*rswlrq dpìulfdlqh hvw sdu frqvwuxfwlrq wrxmrxuv vxsìulhxu rx ìjdo dx sul{ gh o*rswlrq hxur0
sìhqqh = lo hvw wrxmrxuv soxv lqwìuhvvdqw g*dyrlu xqh rswlrq h{huêdeoh ã wrxw prphqw soxw÷w tx*xqh rswlrq
h{huêdeoh ã xqh vhxoh gdwh suìflvh1 Qìdqprlqv/ gdqv fhuwdlqv fdv/ ohv ghx{ sul{ 0 dpìulfdlq hw hxursìhq
0 fròqflghqw1 Lo | d ìjdolwì ghv ghx{ sul{ vl/ ã wrxw prphqw/ oh gìwhqwhxu gh o*rswlrq d lqwìuíw ã dwwhqguh
soxw÷w tx*ã h{hufhu =
rswlrq g*dfkdw vdqv glylghqgh1 Oh sul{ gh o*rswlrq hxursìhqqh yìulh =
k
.l
Hw hu+W w, = +VW N, Vw N
Hq hhw/ {. { hw grqf =
k
l
k
l
Hw hu+W w, = +VW N,. Hw hu+W w, =VW N=hu+W w,
Ru/ Hw hu+ w, =VW @ Vw / g*rü =
k
l
Hw hu+W w, = +VW N,. Vw N=hu+W w, A Vw N
fdu u A 31 Od ydohxu gh o*rswlrq dpìulfdlqh hvw/ ã fkdtxh lqvwdqw w/ vxsìulhxuh ã od ydohxu gh o*rswlrq
hxursìhqqh/ hooh0píph vxsìulhxuh ã od ydohxu Vw N/ f*hvw0ã0gluh ã od ydohxu uhêxh vl rq gìflgh
g*h{hufhu hq w1 Sdu frqvìtxhqw/ oh gìwhqwhxu gh o*rswlrq 0 v*lo hvw udwlrqqho 0 suìiëuh wrxmrxuv dwwhqguh
hw grqf q*h{hufh tx*hq W = rswlrqv dpìulfdlqh hw hxursìhqqh vrqw ìtxlydohqwhv1 Lo hvw idfloh gh yrlu
txh fh udlvrqqhphqw qh wlhqw soxv gëv tx*lo h{lvwh xq wdx{ gh glylghqgh qrq qxo +hw srvlwli, = oh sul{
gh o*rswlrq dpìulfdlqh hvw doruv vwulfwhphqw vxsìulhxu dx sul{ gh o*rswlrq hxursìhqqh1
:3
rswlrq gh yhqwh1 Gdqv fh fdv/ tx*lo | dlw rx qrq ghv glylghqghv/ oh sul{ gh o*rswlrq dpìulfdlqh hvw
vwulfwhphqw vxsìulhxu dx sul{ gh o*rswlrq hxursìhqqh1
Lo h{lvwh/ frpph srxu ohv rswlrqv hxursìhqqhv/ xqh ylvlrq dqdo|wltxh gx sul{ ã sduwlu g*xqh ìtxdwlrq dx{
gìulyìhv sduwlhoohv1 Oh sul{ gh o*rswlrq dpìulfdlqh hvw od vroxwlrq gx surjudpph vxlydqw +gdqv oh prgëoh
gh Eodfn hw Vfkrohv, =
CF dp
CF dp 4 C 5 F dp 5 5
.
u=F dp 3
=V
.
u=V=
+814,
Cw
5 CV 5
CV
vrxv ohv frqwudlqwhv =
F dp +w> Vw , iox{+w> Vw , srxu wrxw w W
F dp +W> VW , @ iox{+W> VW ,
rü i ox{+w> Vw , hvw oh x{ hq fdv g*h{huflfh/ grqf sdu h{hpsoh +Vw N,. srxu xqh rswlrq g*dfkdw vwdqgdug1
Gh soxv/ lo | d ìjdolwì gdqv o*ìtxdwlrq +814, vl od frqwudlqwh F dp +w> Vw , A iox{+w> Vw , hvw vwulfwh1 Hq fodlu/
wdqw txh o*h{huflfh q*hvw sdv lqwìuhvvdqw +l1h1 F dp +w> Vw , A i ox{+w> Vw ,, oh sul{ vxlw o*ìtxdwlrq dx{ gìulyìhv
sduwlhoohv kdelwxhooh1
Hq sduwlfxolhu/ rq yrlw txh gdqv oh fdv gh o*rswlrq g*dfkdw vdqv glylghqgh/ od irqfwlrq F hxu +w> Vw , grqqdqw
oh sul{ gh o*rswlrq g*dfkdw hxursìhqqh yìulh fh surjudpph hw/ sdu xqlflwì gh od vroxwlrq/ rq uhwurxyh
txh ohv ghx{ sul{ fròqflghqw1 Soxv jìqìudohphqw/ ohv ghx{ sul{ vrqw ìjdx{ gëv txh oh sul{ gh o*rswlrq
hxursìhqqh yìulh od frqwudlqwh =
F hxu +w> Vw , iox{+w> Vw , srxu wrxw w W
Hfroh Sro|whfkqltxh
:4
:5
9
Sul{ }ìur0frxsrq/ wdx{ g*lqwìuíw/ sul{ Iruzdug/ sul{
Ixwxuhv = qrwdwlrqv hw frqfhswv1
914 Jìqìudolwìv
Fh fkdslwuh d srxu exw hvvhqwlho gh suìflvhu ohv edvhv gh pdwkìpdwltxhv qdqflëuhv lqglvshqvdeohv ã
od erqqh frpsuìkhqvlrq ghv prgëohv gh vwuxfwxuh ghv wdx{ = wdx{ }ìur0frxsrq/ wdx{ ã whuph/ qrwlrq
g*dfwxdolvdwlrq/ 111
Wrxwhv ohv rsìudwlrqv qdqflëuhv shxyhqw v*dqdo|vhu frpph ghv ìfkdqjhv lqwhuwhpsruhov gh ulfkhvvh1 Xq
hpsuxqw dxmrxug*kxl hvw hq idlw xqh yhqwh gh hxurv ixwxuv > oh surgxlw gh fhwwh yhqwh frqvwlwxh oh fdslwdo
hpsuxqwì hw o*ìfkìdqflhu gh uhperxuvhphqwv gh fhw hpsuxqw uhsuìvhqwh oh elhq yhqgx1 Ohv rsìudwlrqv
gh suíw v*dqdo|vhqw gh idêrq v|pìwultxh1 O*dfkdw hw od yhqwh g*xq wlwuh g*reoljdwlrq shxyhqw gh od píph
idêrq vh uhsuìvhqwhu frpph ghv ìfkdqjhv hqwuh hxurv g*dxmrxug*kxl hw hxurv ixwxuv1
Wrxw uhylhqw grqf ã ìydoxhu oh sul{/ h{sulpì hq hxurv g*dxmrxug*kxl/ g*4 hxur olyuì ã xqh gdwh ixwxuh/ hw ã
prgìolvhu frpphqw fh sul{ yd ydulhu dx frxuv gx whpsv1 Srxu wrxwhv vruwhv gh udlvrq/ od ydohxu dxmrxug*kxl
g*4 hxur ixwxu q*hvw sdv ìjdoh ã 4 hxur =
xq lqglylgx q*dffrugh sdv od píph xwlolwì dx yhuvhphqw lppìgldw g*4 hxur rx dx yhuvhphqw ixwxu
+píph fhuwdlq, g*4 hxur1 Hq hhw xq srxyrlu g*dfkdw g*4 hxur dxmrxug*kxl shuphw gh glvsrvhu gh
elhqv sk|vltxhv gëv dxmrxug*kxl1
oh yhuvhphqw ixwxu oxl0píph shxw íwuh dvvruwl g*xq ulvtxh gdqv od phvxuh rü/ sdu h{hpsoh/ lo shxw
íwuh frqglwlrqqho ã od uìdolvdwlrq g*xq ìyìqhphqw = devhqfh gh idloolwh/ devhqfh gh yhuvhphqw uhwdugì
rx dqwlflsì111
O*remhw g*xq frxuv gh pdwkìpdwltxhv qdqflëuhv hvw gh vh vh frqfhqwuhu g*derug vxu od suhplëuh fdxvh
ìqrqfìh fl0ghvvxv/ f*hvw0ã0gluh ã o*hhw whpsv/ sxlv gh yrlu gdqv txhooh phvxuh rq shxw lqwìjuhu o*hhw ulvtxh1
Lo hvw fodlu txh fhwwh vhfrqgh fdxvh qìfhvvlwh xq fdguh soxv odujh sxlvtx*hooh lpsoltxh od prgìolvdwlrq
gx ulvtxh oxl0píph = uhfkhufkh ghv fdxvhv ìfrqrpltxhv gx ulvtxh/ prgìolvdwlrq ìfrqrpìwultxh gh od
gìidloodqfh/ gx uhperxuvhphqw dqwlflsì111
Oh exw gh fhv qrwhv hvw doruv g*lqwurgxluh wurlv sureoëphv irqgdphqwdx{ hw g*hvtxlvvhu ohv vroxwlrqv
sursrvìhv srxu ohv uìvrxguh1 Fhv 6 sureoëphv vrqw =
ìydoxhu ohv sul{ dxmrxug*kxl ghv hxurv ixwxuv1 Gh idêrq ìtxlydohqwh/ lo v*djlw gh gìgxluh ghv sul{
g*reoljdwlrqv frqvwdwìv vxu oh pdufkì xqh vwuxfwxuh ghv sul{ }ìur0frxsrq1 Fh sureoëph/ oruvtx*lo shxw
íwuh uìvrox grlw shuphwwuh gh grqqhu xq sul{ wkìrultxh ã fkdtxh reoljdwlrq >
frpsuhqguh od g|qdpltxh ghv sul{ }ìur0frxsrq/ uhfkhufkhu ohv idfwhxuv frppxqv dx{ yduldwlrqv gh
sul{ dq g*ìoderuhu ghv vwudwìjlhv gh frxyhuwxuh >
hqq lo hvw qìfhvvdluh gh srxyrlu grqqhu xq sul{ ã ghv dfwliv frpsoh{hv whov ohv rswlrqv/ ohv frqwudwv
ã whuph111 srxu ohvtxhov ohv x{ jìqìuìv vrqw doìdwrluhv hw gìshqghqw gh od vwuxfwxuh ghv wdx{ +hw vhv
ìyroxwlrqv,1
915 Rsìudwlrqv dx frpswdqw
Qrxv qrxv lqwìuhvvrqv lfl dx{ rsìudwlrqv dx frpswdqw/ f*hvw0ã0gluh fhoohv txl vrqw dvvrflìhv ã xqh olyudlvrq
lppìgldwh gh wlwuhv1 Qrxv glvwlqjxhurqv ohv qrwdwlrqv gx whpsv glvfuhw hw gx whpsv frqwlqx oruvtx*hoohv
glëuhqw o*xqh gh o*dxwuh1 Elhq txh ohv prgëohv g*ìydoxdwlrq g*dfwliv qdqflhuv vh sodfhqw g*hpeoìh gdqv
xq fdguh frqwlqx/ lo hvw vrxkdlwdeoh wrxwhirlv gh elhq frpsuhqguh ohv glìuhqwv frqfhswv oruvtxh oh whpsv
hvw vxssrvì glvfuhw1 Rq frpphqfh sdu lqwrgxluh ohv wlwuhv qrppìv wlwuhv }ìur0frxsrq1
91514 Gìqlwlrqv
Gìqlwlrq 6 Xq }ìur0frxsrq g*ìfkìdqfh W hvw xq wlwuh txl grqqh gurlw ã 4 hxur ã od gdwh W=
Fhv dfwliv dssdudlvvhqw qdwxuhoohphqw ã od ohfwxuh gh o*lqwurgxfwlrq gh fh fkdslwuh rü ohv rsìudwlrqv
qdqflëuhv qh vrqw txh ghv ìfkdqjhv lqwhuwhpsruhov gh ulfkhvvh1 Ohv }ìur0frxsrqv frqvwlwxhqw hq txhotxh
vruwh xqh edvh fdqrqltxh +dx vhqv dojìeultxh, gh o*hvsdfh ghv dfwliv ã uhyhqxv {hv +l1h = ohv dfwliv whov txh
ohv reoljdwlrqv/ grqw ohv x{ ixwxuv vrqw qrq doìdwrluhv,1 Pdokhxuhxvhphqw/ píph vl g*xq srlqw gh yxh
frqfhswxho lov sdudñwurqw ylwh lqglvshqvdeohv/ hq sudwltxh lo hq h{lvwh uduhphqw1 Fhshqgdqw ghsxlv pdl
4<<4/ oh Wuìvru iudqêdlv d dxwrulvì oh gìphpeuhphqw ghv Reoljdwlrqv Dvvlplodeohv gx Wuìvru +R1D1W,14
Oh sulqflsh gx gìphpeuhphqw hvw gh fuìhu/ ã sduwlu g*xqh reoljdwlrq fodvvltxh ã frxsrq/ dxwdqw gh
wlwuhv tx*lo | d gh frxsrqv +soxv xq wlwuh fruuhvsrqgdqw dx yhuvhphqw lq0qh gx fdslwdo,/ fhv wlwuhv ìwdqw
doruv lghqwldeohv gluhfwhphqw ã ghv wlwuhv }ìur0frxsrq1 Wrxwhirlv ã fh mrxu/ oh pdufkì q*hvw sdv hqfruh
vx!vdpphqw oltxlgh srxu dyrlu sohlqh frqdqfh gdqv fhv sul{ }ìur0frxsrq1
Gìqlwlrq 7 Oh idfwhxu g*dfwxdolvdwlrq ã od gdwh w srxu od gdwh W hvw oh sul{ g*xq }ìur0frxsrq gh od gdwh
w g*ìfkìdqfh W 1 Lo vhud qrwì E+w> W ,1 Od frxueh g*dfwxdolvdwlrq hvw od frxueh =
W w $ E+w> W ,
Lo hvw lpsruwdqw gh vh {hu xqh erqqh irlv srxu wrxwh ghv qrwdwlrqv/ vdfkdqw txh ohv glìuhqwv duwlfohv
sduxv gdqv od olwwìudwxuh rqw lqwurgxlw ghv qrwdwlrqv dvvh} lqfrkìuhqwhv hqwuh hoohv1 Lfl w hw W vrqw ghv
gdwhv > hq sduwlfxolhu W hvw xqh gdwh g*ìfkìdqfh hw qrq xqh pdwxulwì +l1h = xqh gxuìh,1
Lo uhvwh ã prqwuhu txh fhv sul{ }ìur0frxsrq shxyhqw íwuh gìqlv vdqv dpeljxòwì gdqv od phvxuh rü fh qh
vrqw sdv ghv sul{ gh wlwuhv h{lvwdqw vxu oh pdufkì1 Fhwwh gìprqvwudwlrq uhsrvh vxu ohv ghx{ qrwlrqv gh
edvh gh od qdqfh = frpsoìwxgh ghv pdufkìv hw devhqfh g*rssruwxqlwì g*duelwudjh1
91515 Mxvwlfdwlrqv wkìrultxhv gdqv oh fdv glvfuhw
Rq vh sodfh vxu xq pdufkì gh wlwuhv ã uhyhqxv {hv vxu ohv gdwhv 3> 4===W / hw rq vxssrvh glvsrvhu g*xq
hqvhpeoh gh q wlwuhv jìqìudqw ohv x{ vxlydqwv =
+n,
+n ,
d4 > = = = > dW
Udsshorqv txh fhv x{ vrqw qrq doìdwrluhv hw frqqxv ã od gdwh 31 Qrxv doorqv h{srvhu ohv frqglwlrqv txl
shuphwwhqw gh gìqlu vdqv dpeljxòwì ohv glìuhqwv sul{ =
E+3> w,> w @ 4> ===> W
Qrwrqv hqq P +glwh pdwulfh gh pdufkì, grqw ohv frorqqhv grqqhqw ohv x{ jìqìuìv sdu ohv glìuhqwv
wlwuhv h{lvwdqw =
5
+4, = = = d+q, 6
d4
4 :
9 1
11 :
1
P @9
1 8
7 1
+4,
+
q,
= = = dW
dW
4 Lqlwldohphqw/ lo v*djlvvdlw gh o*R1D1W ;/8 ( 534<1 Soxv uìfhpphqw/ ohv RDW 533; hw 5356 rqw idlw o*re mhw g*xq gìphp0
euhphqw1
:7
hw s oh yhfwhxu ghv sul{ gh pdufkìv ã od gdwh 3 =
5
s@7
s+4,
s+q,
6
8=
Od frqglwlrq ìqrqfìh fl0dsuëv lpsrvh o*devhqfh g*rssruwxqlwì g*duelwudjh/ f*hvw0ã0gluh o*lpsrvvlelolwì gh
frqvwlwxhu ghv sruwhihxloohv gh frþw qìjdwli rx qxo/ jìqìudqw ghv x{ +wrxmrxuv, srvlwliv rx qxov1
K|srwkëvh = Devhqfh g*rssruwxqlwì g*duelwudjh +D1R1D vwdwltxh,
3 @, s3 = 3
41 ; 5 ?q > P=
q
3> < n3 > +P=,n A 3 @, s3 = A 3
51 ; 5 ? > P=
3
rü od qrwlrq P=
3 vljqlh txh wrxwhv ohv frpsrvdqwhv gx yhfwhxu P= vrqw srvlwlyhv rx qxoohv hw rü
od nlhph frpsrvdqwh gh fh yhfwhxu hvw qrwìh sdu dloohxuv +P=,n 1
Fhv ghx{ frqglwlrqv lpsrvhqw grqf tx*xq sruwhihxlooh grqw ohv x{ ixwxuv vrqw srvlwliv +uhvs1 srvlwliv hw
grqw o*xq dx prlqv hvw vwulfwhphqw srvlwli, hvw gh sul{ srvlwli +uhvs1 vwulfwhphqw srvlwli,1 Ohv frpsrvdqwv
gh vrqw srvlwlyhv rx qìjdwlyhv vhorq txh ohv glìuhqwv wlwuhv vrqw dfkhwìv rx yhqgxv1 Rq frqylhqgud txh
n 3 +uhvs1 n 3, lpsoltxh txh oh nlhph wlwuh hvw dfkhwì +uhvs1 yhqgx,1 Gh idlw/ rq dxwrulvh grqf ohv
yhqwhv ã gìfrxyhuw vdqv frqwudlqwhv g*hqghwwhphqw1 Pr|hqqdqw wrxwhv fhv k|srwkëvhv/ oh ohpph gh Idundv
shuphw gh frqfoxuh hw gh grqqhu oh sul{ g*xq }ìur0frxsrq vdqv dpeljxòwì =
Sursrvlwlrq 8
Sq
Sq
+n ,
+n ,
n@4 dw n 3 @, n@4 s n 3
p
SW
+n ,
< 4 > ===> W srvlwliv/ ; n @ 4> ===> N> s+n, @ w@4 w dw
; 4 > ===> q > ; w @ 4> ===> W>
Od suhplëuh sursrvlwlrq hvw h{dfwhphqw od suhplëuh frqglwlrq gh o*k|srwkëvh g*devhqfh g*duelwudjh hw ohv
4 ===W vrqw grqf ohv idfwhxuv g*dfwxdolvdwlrq = w @ E+3> w,1
SW
+n,
Od irupxoh s+n, @ w@4 w dw hvw hq idlw od irupxoh g*dfwxdolvdwlrq/ f*hvw0ã0gluh o*ìjdolwì txl grqqh oh
sul{ gx wlwuh n dxmrxug*kxl frpph od vrpph ghv x{ ixwxuv jìqìuìv sdu oh wlwuh srqgìuìv sdu ohv idfwhxuv
g*dfwxdolvdwlrq =
s+n , @
W
[
+n ,
E+3> w,=dw
w@4
O*h{lvwhqfh ìwdqw dftxlvh/ o*xqlflwì uìvxowh gh od frqglwlrq gh frpsoìwxgh gx pdufkì =
K|srwkëvh = Frpsoìwxgh +vwdwltxh,
Oh udqj gh od pdwulfh P hvw ìjdo ã W = ujP @ W 1
Fhwwh k|srwkëvh vljqlh tx*lo h{lvwh dxwdqw g*dfwliv lqgìshqgdqwv txh gh gdwhv1 Fhfl lpsoltxh grqf txh
ohv q frqglwlrqv =
W
[
+n,
dw E+3> W ,
w@4
gìqlvvhqw xq xqltxh W 0xsohw +E+3> 4,> ===> E+3> W ,,/ hw tx*rq shxw uhfrqvwlwxhu ohv }ìur0frxsrqv frpph
frpelqdlvrq olqìdluh ghv wlwuhv h{lvwdqw vxu oh pdufkì1 Hq fh vhqv/ +E+3> 4,> ===> E+3> W ,, hvw xqh edvh gh
o*hvsdfh ghv wlwuhv1 Frpph xq }ìur0frxsrq g*ìfkìdqfh w hvw uhfrqvwlwxdeoh/ lo ghylhqw xq sruwhihxlooh
sduwlfxolhu/ dxtxho rq shxw dssoltxhu od vhfrqgh frqglwlrq gh o*k|srwkëvh g*DRD srxu hq gìgxluh =
s+n, @
E+3> w, A 3> ; w @ 4===W
Fhv txhotxhv mxvwlfdwlrqv wkìrultxhv shuphwwurqw gh gìqlu gh idêrq qrq dpeljxh od frxueh g*dfwxdol0
vdwlrq ã sduwlu ghv k|srwkëvhv g*DRD hw gh frpsoìwxgh1
Hfroh Sro|whfkqltxh
:8
Rq uhwurxyhud fh w|sh gh udlvrqqhphqw irqgì vxu o*devhqfh g*rssruwxqlwì g*duelwudjh hw od frpsoìwxgh
txdqg lo v*djlud gh gìqlu qrq soxv oh sul{ gx whpsv +lh = oh sul{ g*xq wlwuh }ìur0frxsrq, pdlv oh sul{ gx
ulvtxh +lh = oh sul{ g*xq wlwuh txl gìolyuh 4 hxur vl xq fhuwdlq ìyhqhphqw vh uìdolvh,1
91516 Ohv wdx{ g*lqwìuíw
Ohv mxvwlfdwlrqv wkìrultxhv txl suìfëghqw xwlolvhqw dvvh} qdwxuhoohphqw ohv sul{ }ìur0frxsrq1 Rq shxw
ã sduwlu gh fhv sul{ gìqlu q*lpsruwh txhooh yduldeoh gìulyìh1 Od sudwltxh frqgxlw ã lqwurgxluh ghv wdx{
g*lqwìuíw uhsuìvhqwdwliv gh fhv sul{1
Whpsv glvfuhw
Gìqlwlrq 8 Oh wdx{ dfwxdulho ã od gdwh w srxu od gdwh W / u+w> W ,/ hvw gìql sdu =
E+w> W , @
4
W w
^4 . u+w> W ,`
Oh wdx{ lq0qh/ ul +w> W ,/ hvw gìql sdu =
E+w> W , @
4
4 . +W w,ul +w> W ,
Oh wdx{ suì0frpswì us +w> W , hvw gìql sdu =
E+w> W , @ 4 +W w,us +w> W ,
Rq srxuudlw gìqlu xqh pxowlwxgh g*dxwuhv wdx{ > o*lpsruwdqw hvw gh yrlu txh o*xqlfdwlrq gh fhv glìuhqwhv
qrwlrqv sdvvhqw sdu ohv sul{ }ìur0frxsrq1 Hq jìqìudo/ oh wdx{ xwlolvì uìvxowh g*xqh sudwltxh gh pdufkì1
Vhorq oh pdufkì vxu ohtxho rq wudydlooh/ rq udlvrqqhud soxw÷w hq whuph gh wdx{ lq0qh/ wdx{ suìfrpswì
rx hqfruh wdx{ dfwxdulho1 Hqq/ ohv gìqlwlrqv suìfìghqwhv vrqw hoohv0píph frpsoltxìhv sdu o*xqlwì gh
whpsv xwlolvìh/ f*hvw0ã0gluh vxlydqw tx*rq frpswh hq mrxu rx qrq hw vhorq oh qrpeuh gh mrxuv dgrswì
frqyhqwlrqqhoohphqw srxu o*dqqìh1 Oruvtxh qrxv dxurqv ã xwlolvhu xqh qrwlrq gh wdx{ g*lqwìuíw/ qrxv
xwlolvhurqv v|vwìpdwltxhphqw ohv wdx{ dfwxdulhov1
Whpsv frqwlqx
Rq gìqlw o*ìtxlydohqw ghv wdx{ dfwxdulhov u+w> W , sdu =
Gìqlwlrq 9 Oh wdx{ frpswdqw frqwlqx srxu od sìulrgh ^w> W ` hvw gìql hw qrwì sdu =
\ +w> W , @ rx =
4
Oq E+w> W ,
W w
E+w> W , @ h{s +W w,\ +w> W ,
Rq gìqlw ìjdohphqw oh wdx{ lqvwdqwdqì =
Gìqlwlrq : Oh wdx{ frxuw lqvwdqwdqì hvw ìjdo ã =
uw @ olp \ +w> \ , @ W )w
gOq E+w> W , gW
W )w
Hqq/ txh oh whpsv vrlw glvfuhw rx frqwlqx/ rq sduohud gh frxueh ghv wdx{ =
Gìqlwlrq ; Od frxueh ghv wdx{ }ìur0frxsrq hvw od frxueh =
W w $ u+w> W , rx \ +w> W ,
:9
916 Rsìudwlrqv ã whuph
Ohv rsìudwlrqv ã whuph qìfhvvlwhqw gh frpsoìwhu ohv qrwdwlrqv suìfìghqwhv1 Hq hhw/ E+w> W , gìvljqdlw oh
sul{ ã od gdwh w tx*lo idxw sd|hu srxu rewhqlu xq }ìur0frxsrq g*ìfkìdqfh W 1 Ohv rsìudwlrqv ã whuph h{ljhqw
o*lqwurgxfwlrq g*xqh wurlvlëph gdwh1 Rq shxw gìqlu dlqvl oh sul{ gìflgì dxmrxug*kxl +gdwh 3,/ ã sd|hu hq
w srxu dfkhwhu xq elhq grqqì +lfl xq }ìur0frxsrq,1
91614 Gìqlwlrqv
Gìqlwlrq < Rq dsshooh sul{ ã whuph hq 3 srxu od sìulrgh ^w> W ` oh sul{ gìflgì hq gdwh 3 hw sd|ì hq w
g*xq }ìur0frxsrq olyuì ã od gdwh w g*ìfkìdqfh W 1 Lo vhud qrwì =
E i +3> w> W ,
+i srxu iruzdug,1
Sduwdqw gh fhwwh gìqlwlrq/ rq shxw lqwurgxluh wrxv ohv wdx{ lpdjlqdeohv vxu oh prgëoh gx sdudjudskh
suìfìghqw1 Lfl hqfruh od qrwlrq gh wdx{ dfwxdulho vhud sulyloìjlìh1 Oruvtxh oh whpsv hvw frqwlqx/ rq srvh =
Gìqlwlrq 43 Oh wdx{ ã whuph frqwlqx hq 3 srxu od sìulrgh ^w> W ` hvw gìql hw qrwì sdu =
\ i +3> w> W , @ rx
4
Oq E i +3> w> W ,
W w
E i +3> w> W , @ h{s +W w, \ i +3> w> W ,
Hqq rq shxw gìqlu ohv wdx{ ã whuph lqvwdqwdqìv =
Gìqlwlrq 44 Oh wdx{ ã whuph lqvwdqwdqì ã od gdwh 3 hvw gìql sdu =
i+3> w, @ olp \ i +w> \ ,
W )w
vrlw hqfruh =
E+3> w, @ h{s ] w
3
i +3> v,gv
Oh wdx{ ã whuph lqvwdqwdqì i +3> w, hvw o*dqdorjxh gx wdx{ frxuw lqvwdqwdqì dx vhqv rü/ vl od g|qdpltxh
ghv wdx{ ìwdlw gìwhuplqlvwh +l1h = wrxv ohv wdx{ ixwxuv vrqw frqqxv gëv dxmrxug*kxl,/ doruv rq dxudlw
qìfhvvdluhphqw sdu devhqfh g*rssruwxqlwìv g*duelwudjh i+3> w, @ uw 1
91615 Duelwudjh frpswdqw0whuph
Lo hvw idfloh gh yrlu txh vxu xq sodq wkìrultxh ohv pdufkìv ã whuph vrqw uhgrqgdqwv dx vhqv rü wrxwh
rsìudwlrq ã whuph shxw íwuh uìsoltxìh sdu ghv rsìudwlrqv dx frpswdqw1 Sdu h{hpsoh/ vl rq yhxw uhfrqvwlwxhu
dxmrxug*kxl +gdwh 3, xqh rsìudwlrq ã whuph srxu od sìulrgh ^w> W `/ rq frqvwuxlw oh sruwhihxlooh frpsrvì
g*xq }ìur0frxsrq g*ìfkìdqfh W hw gh }ìur0frxsrqv g*ìfkìdqfh w > fh sruwhihxlooh d xq sul{ =
E+3> W , E+3> w,
hw jìqëuh ohv x{ =
hq w
4 hq W
>W ,
Suhqdqw @ EE+3
+3>w, / rq frqvwlwxh dxmrxug*kxl xqh rsìudwlrq gdqv odtxhooh rq rewlhqw xq sdlhphqw g*4
>W ,
hxur hq gdwh W dx sul{ sd|ì hq w gh EE+3
+3>w, 1 Dlqvl/ v*lo h{lvwh xq pdufkì dx frpswdqw hw xq pdufkì ã
whuph/ od frqglwlrq g*devhqfh g*rssruwxqlwì g*duelwudjh lpsoltxh =5
5 Ohv pdfur0ìfrqrplvwhv sduohqw gh Sdulwì Frxyhuwh ghv Wdx{ g*Lqwìuíw1
Hfroh Sro|whfkqltxh
::
E i +3> w> W , @
E+3> W ,
E+3> w,
txl/ hq whuph gh wdx{ +glvfuhwv,/ v*h{sulph vrxv xqh iruph soxv frqqxh =
W w ^4 . u+3> W ,`W
4 . ui +3> w> W ,
@
^4 . u+3> w,`w
rx hqfruh =
W w
W
w ^4 . u+3> W ,` @ ^4 . u+3> w,` = 4 . ui +3> w> W ,
Hq sudwltxh lo h{lvwh xq pdufkì dx frpswdqw hw xq pdufkì ã whuph srxu ohv udlvrqv vxlydqwhv =
devhqfh gh }ìur0frxsrqv srxu wrxwhv ohv gdwhv g*ìfkìdqfh uhqgdqw lpsrvvleoh od uìsolfdwlrq gh fhu0
wdlqhv rsìudwlrqv ã whuph1
suìvhqfh gh frþw gh wudqvdfwlrq hpsífkdqw od uìsolfdwlrq sduidlwh ghv rsìudwlrqv ã whuph ã sduwlu ghv
rsìudwlrqv dx frpswdqw1
Gdqv fh fdv/ ghx{ qrwlrqv gh sul{ ã whuph fr0h{lvwhqw =
oh sul{ ã whuph lpsolflwh who tx*lo vh gìjdjh gh od irupxoh =
E i +3> w> W , @
E+3> W ,
E+3> w,
oh sul{ ã whuph frqvwdwì vxu oh pdufkì1
Vl dxfxq duelwudjh q*hvw wroìuì/ fhv ghx{ sul{ grlyhqw íwuh ìjdx{/ dx prlqv gdqv ohv fdv rü o*rsìudwlrq ã
whuph hvw sduidlwhphqw uìsolfdeoh sdu ghx{ rsìudwlrqv dx frpswdqw vxlydqw oh pìfdqlvph gìfulw suìfì0
ghpphqw1 Fh w|sh g*duelwudjh oruvtx*lo h{lvwh v*dsshooh xq duelwudjh frpswdqw0whuph +rx fdvk dqg fduu|,1
Gdqv oh fdv qrupdo rü lo q*d sdv g*rssruwxqlwìv g*duelwudjh/ ohv wdx{ ã whuph lqvwdqwdqìv i+3> w, rqw xqh
h{suhvvlrq vlpsoh hq irqfwlrq gh od frxueh ghv wdx{ frpswdqw =
i +3> w, @
@
4
Oq E i +3> w> W ,
olp W )w W w
4
Ôq E+3> W , Oq E+3> w,`
olp W )w W w
@ gOq E+3> w,
gw
tx*rq shxw hqfruh ìfuluh =
E+3> w, @ h{s ] w
3
i +3> v,gv
Lo idxw doruv uhpdutxhu txh fh uìvxowdw lpsoltxh txh od qrwlrq gh frxueh ghv wdx{ ã od gdwh w shxw íwuh
gìqlh gh idêrq ìtxlydohqwh sdu od grqqìh gh o*xqh ghv wurlv frxuehv vxlydqwhv =
;
E+w> W ,> W w> E+w> w, @ 4
A
A
A
A
?
i+w> W ,> W w>
A
A
A
A
=
\ +w> W ,> W w
91616 Glvwlqfwlrq Iruzdug2Ixwxuh
Lo dssdudñw qìfhvvdluh gh idluh fhwwh glvwlqfwlrq gdqv od phvxuh rü ohv rsìudwlrqv ã whuph whoohv tx*hoohv
vrqw sudwltxìhv vxu xq pdufkì ã whuph +who txh od PDWLI, qh vrqw sdv hq jìqìudo gh fhoohv gìfulwhv gdqv
oh sdudjudskh suìfìghqw1 Rq glvwlqjxh hq idlw ohv ghx{ w|shv g*rsìudwlrqv vxlydqwhv =
:;
Rsìudwlrqv Iruzdug = Hoohv vrqw frqiruphv dx{ rsìudwlrqv gìfulwhv dx sdudjudskh suìfìghqw1 Vfkì0
pdwltxhphqw/ 5 sduwlhv D hw E frqfoxhqw xq frqwudw iruzdug oruvtxh o*xqh ghv sduwlhv +sdu h{hpsoh
D, dvvxph o*reoljdwlrq g*dfkhwhu xqh txdqwlwì {ìh gh wlwuhv ã xq sul{ frqyhqx ã o*dydqfh +glw sul{
gx frqwudw, hw ã xqh gdwh ìjdohphqw {ìh +glwh ìfkìdqfh gx frqwudw,1 Rq glw doruv txh D +uhvs1 E, d
xqh srvlwlrq orqjxh +uhvs1 frxuwh,1 Dxfxq sdlhphqw ql olyudlvrq gh wlwuhv q*rqw olhx hqwuh od gdwh gh
frqfoxvlrq hw od gdwh g*ìfkìdqfh gx frqwudw1 Oruvtxh oh wlwuh ã olyuhu hvw xq }ìur0frxsrq g*ìfkìdqfh
W hw txh od gdwh g*ìfkìdqfh gx frqwudw hvw w/ doruv oh sul{ gx frqwudw/ f*hvw0ã0gluh oh sul{ frqyhqx ã
o*dydqfh srxu oh wlwuh ã olyuhu hq w/ q*hvw ulhq g*dxwuh txh E i +3> w> W ,1 D od gdwh lqlwldoh/ oh sul{ g*xq
frqwudw ã whuph hvw/ sdu frqvwuxfwlrq/ qxo1 Hq uhydqfkh/ sdu od vxlwh/ o*hqjdjhphqw g*dfkhwhu +rx gh
yhqguh, hq w oh wlwuh }ìur0frxsrq g*ìfkìdqfh W dx sul{ E i +3> w> W , q*d soxv hq jìqìudo xqh ydohxu
qxooh1
Rsìudwlrqv Ixwxuhv = Hoohv vrqw hq sduwlh vhpeodeohv dx{ rsìudwlrqv Iruzdug gdqv od phvxuh rü
vrqw {ìhv/ ã od frqfoxvlrq gx frqwudw/ xqh gdwh g*ìfkìdqfh gx frqwudw/ xqh txdqwlwì {ìh gh wlwuhv
hw xq sul{ frqyhqx srxu fhv wlwuhv ã od gdwh g*ìfkìdqfh1 Hq uhydqfkh/ dssdudñw hqwuh ohv 5 sduwlhv D
hw E xqh fkdpeuh gh frpshqvdwlrq sdu odtxhooh yrqw wudqvlwhu ghv x{ hqwuh D hw E shqgdqw wrxwh
od sìulrgh vìsdudqw od gdwh gh plvh hq sodfh gx frqwudw hw od gdwh g*ìfkìdqfh1 Fhv x{ fruuhvsrqghqw
dx{ yduldwlrqv gx sul{ gx frqwudw/ f*hvw0ã0gluh gx sul{ ã whuph ghv wlwuhv ã olyuhu ã od gdwh g*ìfkìdqfh1
Gh soxv/ ã wrxw prphqw/ oh frqwudw shxw íwuh gìqrxì sdu xqh srvlwlrq lqyhuvh1
Frqvlgìurqv ã wlwuh g*h{hpsoh oh fdv g*xq wlwuh }ìur0frxsrq g*ìfkìdqfh W ã olyuhu hq w +l1h = gdwh g*ìfkìdqfh
gx frqwudw,1 O*dfkdw g*xq frqwudw ixwxuh sruwdqw vxu fh wlwuh/ ã od gdwh 3/ dx sul{ qrwì E I +3> w> W , +dsshoì
sul{ gx frqwudw ã od gdwh 3,/ suryrtxh ohv x{ vxlydqwv =
gìs÷w gh jdudqwlh = E I +3> w> W ,
dsshov gh pdujh =
E I +4> w> W , E I +3> w> W ,
11
1
E I +> w> W , E I + 4> w> W ,
Ohv dsshov gh pdujh shuphwwhqw ã wrxw prphqw g*íwuh hq skdvh dyhf oh pdufkì +l1h = od srvlwlrq hvw
glwh pdunhg wr pdunhw, hw gh olplwhu dlqvl ohv ulvtxhv gh gìidxw oh mrxu gh od olyudlvrq1 Hq sudwltxh/ ohv
frqwudwv qh yrqw sdv irufìphqw ã ohxu whuph frqyhqx +lfl w,/ pdlv vrqw udfkhwìv rx uhyhqgxv dydqw w1 Fhfl
vljqlh vlpsohphqw txh o*rq idlw oh vrogh gh o*rsìudwlrq hq vrppdqw ohv yhuvhphqwv/ vrlw ã od gdwh =
E I +> w> W , E I +3> w> W ,=
Vl o*rsìudwlrq hvw phqìh ã vrq whuph/ o*dfkhwhxu gx frqwudw uhêrlw =
od vrpph ghv dsshov gh pdujh =
E I +w> w> W , E I +3> w> W ,
od olyudlvrq g*xq }ìur0frxsrq g*ìfkìdqfh W dx sul{ E I +w> w> W , dyhf/ ìylghpphqw =
E+w> W , @ E I +w> w> W ,
vrlw dx wrwdo = E I +3> w> W , f*hvw0ã0gluh od olyudlvrq g*xq wlwuh }ìur0frxsrq g*ìfkìdqfh W dx sul{
E I +3> w> W ,1
Hq fh vhqv/ xqh rsìudwlrq Ixwxuh hvw wuëv vhpeodeoh ã xqh rsìudwlrq Iruzdug ã od glìuhqfh suëv txh ohv
yhuvhphqwv rqw olhx shqgdqw wrxwh od sìulrgh ^3> w` hw qrq xqltxhphqw hq w/ hw f*hvw od udlvrq srxu odtxhooh
rq d hq jìqìudo =
E I +3> w> W , 9@ E i +3> w> W ,
Hfroh Sro|whfkqltxh
:<
917 Dqqh{h 4 = Glìuhqfh hqwuh sul{ iruzdug hw sul{ ixwxuh
+Fr{/ Lqjhuvroo hw Urvv +4<;4,/ Gx!h +4<;<,,
Qrxv looxvwurqv lfl od glìuhqfh txl h{lvwh hqwuh sul{ iruzdug hw sul{ ixwxuh hw prqwurqv txh / gdqv xq
frqwh{wh gh wdx{ gìwhuplqlvwhv/ ohv ghx{ sul{ vrqw ìjdx{1 Fh uìvxowdw hvw oh iuxlw ghv wurlv sursrvlwlrqv
vxlydqwhv +vhxoh od ghuqlëuh gìshqg gh o*k|srwkëvh gh wdx{ gìwhuplqlvwhv,1
91714 Uìvxowdw 4
Rq shxw wurxyhu xqh vwudwìjlh whooh tx*hq lqyhvwlvvdqw E i +3> w> W , ã od gdwh 3/ rq sxlvvh
+w>W ,
uhfhyrlu = E
E+3>w, ã od gdwh w1
Suhxyh1 Rq hhfwxh vlpxowdqìphqw ohv ghx{ rsìudwlrqv vxlydqwhv hq 3 =
Dfkdw gh wlwuhv }ìur0frxsrq g*ìfkìdqfh w srxu xqh ydohxu gh E i +3> w> W ,1 O*rsìudwlrq frqgxlw dx
i >w>W ,
yhuvhphqw gh EE+3
+3>w, ã od gdwh w1
i >w>W ,
Dfkdw gh E+34 >w, frqwudw iruzdug dx sul{ E i +3> w> W ,1 F*hvw grqf o*hqjdjhphqw gh sd|hu EE+3
+3>w, hq w
srxu uhfhyrlu +hq w, E+34 >w, xqlwì g*xq }ìur0frxsrq g*ìfkìdqfh W 1
Oh wrwdo gh fhv ghx{ rsìudwlrqv +hw od uhyhqwh ghv E+34 >w, wlwuhv }ìur0frxsrq g*ìfkìdqfh W , derxwlw ã =
E i +3> w> W , E i +3> w> W , E+w> W ,
.
E+3> w,
E+3> w,
E+3> w,
vrlw =
E+w> W ,
E+3> w,
91715 Uìvxowdw 5
Rq shxw wurxyhu xqh vwudwìjlh whooh tx*hq lqyhvwlvvdqw E I +3> w> W , ã od gdwh 3 rq sxlvvh uhfhyrlu
ã od gdwh w =
E+w> W ,=+3> w,
rü =
+3> w, @
# w4
\
n@3
$4
E+n> n . 4,
Suhxyh1 Rq hhfwxh ohv rsìudwlrqv vxlydqwhv =
Sodfhphqw gh E I +3> w> W , hq }ìur0frxsrq g*ìfkìdqfh 4/ sxlv uhsodfhphqw hq }ìur0frxsrq g*ìfkìdqfh
5/ hwf1 Oh surgxlw ã od gdwh w hvw ìjdo ã =
E I +3> w> W ,=
vrlw =
4
4
4
=
===
E+3> 4, E+4> 5, E+w 4> w,
E I +3> w> W ,=+3> w,
Dfkdw gh E+34>4, frqwudw Ixwxuh ã od gdwh 3 hw uhyhqwh ã od gdwh 4/ vrlw xq surw +srvlwli rx qìjdwli,
gh =
I
4
E +4> w> W , E I +3> 4> W ,
E+3> 4,
uìlqyhvwlw ã fkdtxh sìulrgh hq }ìur0frxsrq gh pdwxulwì 4 +frpph gdqv o*rsìudwlrq suìfìghqwh,/ vrlw
hq w =
I
E +4> w> W , E I +3> w> W , =
4
4
===
E+3> 4, E+w 4> w,
Dfkdw gh E+3>4,4E+4>5, frqwudw ã od gdwh 4 hw uhyhqwh ã od gdwh 5/ vrlw xq surw =
I
4
E +5> w> W , E I +4> w> W ,
E+3> 4,E+4> 5,
;3
uìlqyhvwlw frpph suìfìghpphqw/
hwf1
Dx wrwdo gh fhwwh rsìudwlrq/ rq rewlhqw =
I
E +w> w> W , E I +3> w> W , =+3> w,
vrlw =
E+w> W , E I +3> w> W , =+3> w,
Oh wrwdo ghv 5 rsìudwlrqv grqqh hqq =
E+w> W ,=+3> w,
91716 Uìvxowdw 6
Gdqv xq frqwh{wh gh wdx{ gìwhuplqlvwhv/
E+3> w, @ ++3> w,,4
Suhxyh1 F*hvw xqh frqvìtxhqfh gh o*devhqfh g*rssruwxqlwì g*duelwudjh hqwuh ohv 5 vwudwìjlhv =
lqyhvwlvvhphqw g*4 hxur hq }ìur0frxsrq g*ìfkìdqfh w/ vrlw xq x{ gh E+34 >w, hq w1
lqyhvwlvvhphqw g*4 hxur hq }ìur0frxsrq g*ìfkìdqfh 4/ sxlv uìlqyhvwlvvhphqw hq }ìur0frxsrq g*ìfkìdqfh
5/ hwf1/ vrlw xq x{ gh +3> w, hq w1
Ohv 5 vwudwìjlhv vrqw vdqv ulvtxh sxlvtxh ohv wdx{ vrqw frqqxv ã od gdwh 3/ qìfhvvlwhqw oh píph lqyhv0
wlvvhphqw lqlwldo hw frqgxlvhqw grqf dx píph x{1 O*dgglwlrq gh fhv 6 uìvxowdwv frqgxlw grqf ã o*ìjdolwì
uhfkhufkìh1
Hfroh Sro|whfkqltxh
;4
918 Dqqh{h 5 = Frqwudw hxuleru
Ohv vsìflfdwlrqv gx frqwudw Ixwxuh hxuleru 6 prlv1
;5
:
Lqwhusrodwlrq hw dqdo|vh idfwrulhooh gh od vwuxfwxuh ghv
wdx{1
:14 Lqwurgxfwlrq
Dsuëv dyrlu gìql ohv sulqflsdohv qrwlrqv uhodwlyhv ã od vwuxfwxuh ghv wdx{/ lo ghylhqw qìfhvvdluh gh srxyrlu
gìwhuplqhu qxpìultxhphqw fhv glyhuvhv txdqwlwìv +sul{ }ìur0frxsrq/ wdx{111,1 Ru/ sudwltxhphqw/ lo q*h{lvwh
sdv rx txdvlphqw sdv gh wlwuhv }ìur0frxsrq srxu ghv pdwxulwìv vxsìulhxuhv ã xq dq +yrlu fkdslwuh L,1 Ohv
sul{ }ìur0frxsrq vrqw grqf ghv yduldeohv odwhqwhv/ lqrevhuydeohv gluhfwhphqw vxu xq pdufkì qdqflhu1 Lo
idxw grqf/ ã sduwlu ghv sul{ ghv dfwliv revhuyìv vxu oh pdufkì/ +f*hvw0ã0gluh ohv reoljdwlrqv sruwdqw xq
frxsrq/ rx g*dxwuhv dfwliv frpph ohv vzdsv, gìgxluh od vwuxfwxuh ghv wdx{ }ìur0frxsrq1 Rq qh shxw
pdokhxuhxvhphqw sdv vh sdvvhu gh fhwwh ìwdsh fdu od frqqdlvvdqfh gh fhv sul{ }ìur0frxsrq hvw/ frpph
qrxv o*dyrqv gìmã glw/ irqgdphqwdoh srxu wudlwhu o*hqvhpeoh ghv sureoëphv txl shxyhqw vh srvhu/ frpph
o*ìydoxdwlrq g*xq hqvhpeoh gh x{ ixwxuv +fhuwdlqv,1 Gh soxv/ ohv prgëohv g*ìydoxdwlrq txh qrxv ìwxglhurqv
gdqv ohv fkdslwuhv vxlydqwv vrqw frqvwuxlwv ã sduwlu gh fhv sul{ }ìur0frxsrq/ sxlvtx*ã sduwlu gh fhx{0fl rq
shxw uhfrqvwlwxhu wrxwh reoljdwlrq dyhf frxsrqv1
Qrxv dyrqv yx dx fkdslwuh L tx*dx prlqv hq wkìrulh/ ohv sul{ ghv }ìur0frxsrq grlyhqw srxyrlu íwuh uhfrqv0
wlwxìv vdqv gl!fxowì ã sduwlu ghv sul{ g*reoljdwlrqv/ sxlvtxh fhv ghuqlhuv qh vrqw txh ghv frpelqdlvrqv
olqìdluhv ghv suhplhuv1 Pdokhxuhxvhphqw od sudwltxh srvh gh qrpeuhx{ sureoëphv whfkqltxhv olìv dx wuëv
judqg qrpeuh gh sul{ }ìur0frxsrq tx*lo hvw qìfhvvdluh g*hvwlphu1 O*dssurfkh kdelwxhooh fkhufkh ã lqwhusr0
ohu fhv sul{ sdu xqh idplooh gh irqfwlrqv1 Vh srvh doruv od txhvwlrq gx fkrl{ gh od idplooh g*lqwhusrodwlrq1
Rq shxw shqvhu/ gdqv xq suhplhu whpsv/ gìyhorsshu xq prgëoh vdwlvidlvdqw gh erqqhv sursulìwìv ìfrqr0
pltxhv +frpph o*devhqfh g*rssruwxqlwìv g*duelwudjh, dq g*hq gìgxluh xqh idplooh qdwxuhooh gh irqfwlrqv
srxu ohv sul{ }ìur0frxsrq1 Qrxv yhuurqv gdqv ohv fkdslwuhv txl vxlyhqw tx*xqh whooh dssurfkh hvw srvvleoh
pdlv frqgxlw ã ghv surfìgxuhv g*hvwlpdwlrq orxughv1 Vl o*remhfwli hvw g*rewhqlu udslghphqw xqh frxueh ghv
wdx{/ rq shxw suìiìuhu xqh dssurfkh soxv sudjpdwltxh hw fkrlvlu g*hpeoìh xqh idplooh vlpsoh gh irqfwlrqv
g*lqwhusrodwlrq +frpph/ sdu h{hpsoh/ ghv sro|q÷phv,1 F*hvw fh srlqw gh yxh txh qrxv sulyloìjlhurqv gdqv
fh fkdslwuh/ pdlv lo qh idxw mdpdlv rxeolhu vrq fdudfwëuh dg0krf hw oh idlw txh od frxueh ghv wdx{ dlqvl
uhfrqvwlwxìh shxw qh sdv íwuh frpsdwleoh dyhf xq prgëoh ìfrqrpltxh frkìuhqw1
Oh ghx{lëph srlqw txh qrxv vrxoëyhurqv gdqv fh fkdslwuh hvw oxl dxvvl wuëv sudjpdwltxh1 Lo hvw uhodwli ã xqh
olwwìudwxuh grqw o*dpelwlrq hvw gh gìwhuplqhu frpelhq g*doìd jrxyhuqhqw ohv prxyhphqwv gh od vwuxfwxuh
ghv wdx{1 Hq hhw/ o*revhuydwlrq gdwh sdu gdwh ghv gìirupdwlrqv gh od vwuxfwxuh ghv wdx{ dpëqh ã shqvhu
txh ohv wdx{ gh glìuhqwhv pdwxulwìv qh ydulhqw sdv lqgìshqghpphqw ohv xqv ghv dxwuhv1 Rq gìfrpsrvh
dlqvl ohv gìirupdwlrqv gh od vwuxfwxuh ghv wdx{ hq soxvlhxuv gìirupdwlrqv ìoìphqwdluhv =
xq prxyhphqw gx qlyhdx jìqìudo ghv wdx{/
xqh prglfdwlrq gh od shqwh gh od vwuxfwxuh ghv wdx{/
xqh prglfdwlrq gh od frxuexuh gh od vwuxfwxuh ghv wdx{/
hwf1
Ohv lpsolfdwlrqv sudwltxhv gh fh jhquh gh gìfrpsrvlwlrq +vl hooh hvw mxvwlìh hpslultxhphqw, vrqw lpsru0
wdqwhv sxlvtx*hoohv grqqhqw ghv vwudwìjlhv g*lppxqlvdwlrq frqwuh fkdfxqh ghv gìirupdwlrqv0w|shv ìyrtxìhv
fl0ghvvxv1 Sdu dloohxuv/ oruvtxh ghv prgëohv wkìrultxhv dxurqw ìwì sursrvìv gdqv ohv fkdslwuhv xowìulhxuv/
lo vhud lqwìuhvvdqw gh uìlqwhusuìwhu fhv gìfrpsrvlwlrqv hq doìd ìoìphqwdluhv frppxqv ã od oxplëuh gh fhv
prgëohv1
Hq uìvxpì/ fh fkdslwuh d xq exw hvvhqwlhoohphqw ghvfulswli +hw qrq sdv h{solfdwli, dxvvl elhq vxu od uhfrqv0
wlwxwlrq gh od frxueh ghv wdx{ txh vxu ohv gìirupdwlrqv gh fhooh0fl dx frxuv gx whpsv1
:15 Ohv pìwkrghv g*lqwhusrodwlrq
Fhv pìwkrghv vrqw qrpeuhxvhv hw frqgxlvhqw ã ghv uìvxowdwv txl shxyhqw íwuh dvvh} glìuhqwv > hoohv rqw
oh srlqw frppxq g*íwuh wuëv sudjpdwltxhv gdqv od phvxuh rü hoohv qh vrqw sdv irqgìhv vxu xq prgëoh
wkìrultxh gh od vwuxfwxuh ghv wdx{1 G*xq srlqw gh yxh pdwkìpdwltxh/ lo v*djlw vlpsohphqw gh uhfrqvwlwxhu
xqh frxueh frpsoëwh ã sduwlu gh od grqqìh g*xq qrpeuh ql gh srlqwv gh fhwwh frxueh/ g*rü oh qrp
g*lqwhusrodwlrq1 Qrxv frpphqêrqv sdu udsshohu ohv qrwdwlrqv hw oh srlqw gh gìsduw gh od prgìolvdwlrq1
:1514 Qrwdwlrqv
Od frxueh ghv sul{ }ìur0frxsrq ã od gdwh w d gìmã ìwì lqwurgxlwh dx fkdslwuh 4 frpph od frxueh =
$ E+w> w . ,
+:14,
Rq d yx txh fhwwh frxueh ìwdlw sduidlwhphqw ìtxlydohqwh ã od grqqìh gh od frxueh ghv wdx{ +}ìur0frxsrq,
gìqlh sdu =
;
? \ +w> w . , hq whpsv frqwlqx
$
+:15,
=
u+w> w . , hq whpsv glvfuhw
Od ghx{lëph idêrq ìtxlydohqwh gh vh grqqhu od vwuxfwxuh ghv wdx{ frqvlvwh ã xwlolvhu ohv wdx{ ã whuph
+lpsolflwhv,1 Sdu h{hpsoh/ gdqv oh fdv glvfuhw/ qrxv dyrqv gìql oh sul{ ã whuph lpsolflwh fdofxoì hq w srxu
od sìulrgh ^w . > w . . 4` sdu =
E i +w> w . > w . . 4, @
E+w> w . . 4,
E+w> w . ,
hw oh wdx{ fruuhvsrqgdqw sdu =
E i +w> w . > w . . 4, @
4
i
4 . u +w> w . > w . . 4,
Dlqvl od frxueh g*dfwxdolvdwlrq ã od gdwh w shxw íwuh grqqìh sdu =
$ E+w> w . , @
\
4
n@3
E i +w> w . n> w . n . 4,
hw grqf hvw ìtxlydohqwh ã od frxueh =
$ E i +w> w . > w . . 4,
+:16,
Ohv iruphv +:14,/ +:15, rx +:16, vrqw grqf sduidlwhphqw ìtxlydohqwhv/ hw rq shxw fkrlvlu gh prgìolvhu
o*xqh gh fhv wurlv iruphv1 Fhshqgdqw/ ohv sul{ g*reoljdwlrqv v*ìfulyhqw olqìdluhphqw hq ohv sul{ }ìur0frxsrq
sxlvtxh/ srxu xqh reoljdwlrq gh sul{ Sw ã od gdwh w/ hw jìqìudqw ohv x{ ixwxuv +fhuwdlqv, +d ,4 / rq d =
[
Sw @
d =E+w> w . ,
+:17,
4
;7
vrlw =
Sw @
[
d
^4 . u+w> w . ,`
4
rx hqfruh/ vl \ +w> w . , gìvljqh oh wdx{ frqwlqx
[
d =h=\ +w>w.,
Sw @
4
Sudwltxhphqw/ od olqìdulwì gh od irupxoh +:17, grqqdqw oh sul{ ghv reoljdwlrqv ã sduwlu gh od frxueh
g*dfwxdolvdwlrq frqgxlw ã sulyloìjlhu o*lqwhusrodwlrq gh od frxueh ghv sul{ + :14,1
:1515 Od prgìolvdwlrq
Hq wkìrulh/ srxu shx txh o*rq glvsrvh gh vx!vdpphqw g*reoljdwlrqv/ rq grlw srxyrlu gìgxluh ohv sul{
E+w> w ., hq uìvroydqw xq v|vwëph olqìdluh frpsrvì g*ìtxdwlrqv gx w|sh +:17,1 Pdokhxuhxvhphqw/ fhuwdlqv
sureoëphv whfkqltxhv dssdudlvvhqw1 Hq hhw/ ohv reoljdwlrqv qh gìwdfkhqw sdv ohxu frxsrq dx{ píphv gdwhv
gdqv o*dqqìh hw fhfl pxowlsolh oh qrpeuh gh sul{ }ìur0frxsrq tx*lo hvw qìfhvvdluh g*hvwlphu/ uìgxlvdqw
g*dxwdqw ohxu suìflvlrq1 Gh soxv/ od pdwulfh ghv x{ jìqìuìv sdu xq sdqlhu g*reoljdwlrqv frqwlhqw doruv
ehdxfrxs gh }ìurv hw hvw vrxyhqw surfkh +gx srlqw gh yxh qxpìultxh, gh od vlqjxodulwì1 Gh idlw/ od
uìvroxwlrq g*xq v|vwëph olqìdluh frqgxlw ã ghv ydohxuv wuëv lqvwdeohv qxpìultxhphqw1 Hqq/ rq qh srxuud
mdpdlv lghqwlhu E+w> w . , vl dxfxqh reoljdwlrq qh gìwdfkh xq frxsrq hq hq w . 1
Srxu fhwwh udlvrq/ rq lpsrvh dx{ frxuehv $ E+w> w . , dgplvvleohv g*dssduwhqlu ã xq hqvhpeoh
sduwlfxolhu gh frxuehv1 Dlqvl rq ohv dssurfkhud sdu ghv idploohv gh frxuehv sdudpìwuìhv sdu xq sdudpëwuh
w who txh =
+:18,
E+w> w . , @ j +> w ,
rü j hvw xqh irqfwlrq frqqxh1
O*hvwlpdwlrq ghv E+w> w., vh udpëqh ã o*hvwlpdwlrq g*xq sdudpëwuh w gh wdlooh udlvrqqdeoh/ elhq lqiìulhxuh
ã od grqqìh gh wrxv ohv E+w> w . ,1 O*hvwlpdwlrq vh idlw doruv srxu xqh gdw hgrqqìh w vxu xq sdqlhu gh q
reoljdwlrqv1 O*ìtxdwlrq +:17, shuphw doruv g*ìfuluh =
S
; 4
4
A
4 d =j+> w ,
? Sw @
11
11
1
1S
A
= q
q
Sw @
4 d =j+> w ,
Lo hvw qdwxuho g*lqwurgxluh gdqv od prgìolvdwlrq vwdwlvwltxh o*lgìh txh od irupxoh +:17, q*hvw txh wkìrultxh1
Qrxv dyrqv hq hhw yx dx fkdslwuh 4 txh od irupxoh g*dfwxdolvdwlrq +:17, uìvxowdlw g*xqh k|srwkëvh
g*devhqfh g*rssruwxqlwì g*duelwudjh1 Ru/ oh sul{ uìho gh pdufkì shxw v*ìfduwhu gh fh sul{ wkìrultxh grqqì
sdu +:17,/ qrwdpphqw sdufh txh ohv wlwuhv qh vrqw sdv wudlwìv hq frqwlqx hw txh grqf ghv duelwudjhv shxyhqw
h{lvwhu prphqwdqìphqw1 Sdu dloohxuv/ ohv sul{ xwlolvìv fruuhvsrqghqw ã ghv pr|hqqhv gh sul{ uhohyìv oruv
gh wudqvdfwlrqv hw vrqw grqf g*dxwdqw soxv suìflv txh ohv wlwuhv fruuhvsrqgdqw vrqw soxv oltxlghv1 Srxu
wrxwhv fhv udlvrqv/ lo hvw qdwxuho g*lqwurgxluh xq doìd frqgxlvdqw dx prgëoh vwdwlvwltxh vxlydqw =
[
dl j+> w , . %lw l @ 4===q
+:19,
Swl @
4
dyhf H+%lw @Iw4 , @ 3 rü Iw4 gìvljqh o*lqirupdwlrq sdvvìh +hq o*rffxuuhqfh ohv ydohxuv sdvvìhv ghv
sul{ ghv wlwuhv g*reoljdwlrq,1 Rq d grqf ddluh ã xq prgëoh gh uìjuhvvlrq qrq olqìdluh/ srvvleohphqw
kìwìurvfìgdvwltxh1 Fhwwh kìwìurvfìgdvwlflwì srxuud íwuh uholìh dx{ fdudfwìulvwltxhv gx wlwuh l +vfdolwì
sduwlfxolëuh/ oltxlglwì111,1 Hq uhydqfkh/ rq qh vxssrvhud sdv hq jìqìudo txh ghx{ sul{ shxyhqw íwuh fruuìoìv
frqglwlrqqhoohphqw ã Iw4 > rq vxssrvhud grqf =
Fry+%lw > %mw @ Iw4 , @ 3 srxu l 9@ m
Ohv dssurfkhv h{srvìhv gdqv od olwwìudwxuh glëuhqw sdu =
sdu od idplooh gh irqfwlrqv j xwlolvìhv/
sdu od iruph gh od yduldqfh Y +%lw @ Iw4 ,/
od pìwkrgh g*hvwlpdwlrq1
Hfroh Sro|whfkqltxh
;8
:1516 Sulqflshv g*hvwlpdwlrq
Qrxv udsshorqv lfl eulëyhphqw ohv pìwkrghv gh o*ìfrqrpìwulh glvsrqleohv srxu wudlwhu o*hvwlpdwlrq gx
prgëoh +:19,1 Frpswh whqx gh od uhpdutxh suìfìghqwh/ lo hvw hq jìqìudo gl!floh gh srvwxohu xqh orl srxu
ohv shuwxuedwlrqv +%lw ,l / fh txl uhqgudlw srvvleoh o*xwlolvdwlrq gx pd{lpxp gh yudlvhpeodqfh1 Rq shxw
wrxwhirlv sursrvhu ghv hvwlpdwlrqv frqyhujhqwhv +txdqg q whqg yhuv .4, gx sdudpëwuh w vdqv dyrlu ã
srvwxohu xqh orl srxu +%lw ,l 1
Sursrvlwlrq 9 Sursrvlwlrq : +Hvwlpdwlrq irqgìh vxu oh suhplhu prphqw,1 O*hvwlpdwhxu ghv prlqguhv
fduuìv qrq olqìdluhv rewhqx sdu oh surjudpph =
Sq
l S
l
5
P lq
l@4 ^Sw 4 d j+> w ,`
+:1:,
w
hvw frqyhujhqw txdqg q whqg yhuv .41
Rq qh wlhqw grqf sdv frpswh gh o*kìwìurvfìgdvwlflwì1 Vl rq vrxkdlwh hq whqlu frpswh sdu h{hpsoh vrxv od
iruph =
H+%5lw @ Iw4 > [lw , @ k+[lw > w , hw H+%lw @Iw4 > [lw , @ 3
rü [lw hvw xq yhfwhxu gh yduldeohv h{solfdwlyhv +h{rjëqhv, gh fhwwh kìwìurvfìgdvwlflwì/ rq d doruv oh uìvxowdw
vxlydqw =
Sursrvlwlrq ; +Hvwlpdwlrq irqgìh vxu ohv 5 suhplhuv prphqwv,1 O*hvwlpdwlrq gx svhxgr0pd{lpxp gh
yudlvhpeodqfh rewhqx sdu oh surjudpph =
Sq
+S l S dl j+>w,,5
P d{ l@4 45 Oq k+[lw > w , . 45 w k+[4lw > ,
+:1;,
w
w > w
hvw frqyhujhqw hw dv|pswrwltxhphqw qrupdo txdqg q $ .41
Rq shxw grqf idluh frpph vl ohv +%lw ,l vxlydlhqw +frqglwlrqqhoohphqw ã [lw hw ã Iw4 , ghv orlv jdxvvlhqqhv
píph vl who q*hvw sdv oh fdv1 Lo idxw qrwhu tx*xq fulwëuh gh prlqguhv0fduuìv gx w|sh =
Sq ^Swl S 4 dl j+>w ,`5
P lq
l@4
k+[lw > w ,
w > w
frqgxlw ã ghv hvwlpdwhxuv qrq frqyhujhqwv1
:1517 Ohv sulqflsdohv idploohv gh irqfwlrqv g*lqwhusrodwlrq
Lo v*djlw pdlqwhqdqw gh suìflvhu ohv idploohv gh irqfwlrqv g*lqwhusrodwlrq j +=> w , sursrvìhv gdqv od olwwìud0
wxuh1 Xqh whooh idplooh grlw uhvshfwhu fhuwdlqhv frqglwlrqv1 Suhplëuhphqw hooh grlw shuphwwuh xqh judqgh
h{lelolwì uhqgdqw srvvleoh o*dmxvwhphqw ã xqh vwuxfwxuh ghv wdx{ furlvvdqwh/ sodwh rx gìfurlvvdqwh/ rx
hqfruh frqfdyh111 Ghx{lëphphqw/ lo grlw v*djlu g*xqh idplooh gh irqfwlrqv frqwlqxhv hw píph frqwlqþ0
phqw glìuhqwldeohv1 Udsshorqv txh ohv wdx{ ã whuph lqvwdqwdqìv vrqw rewhqxv vrxv iruph gh gìulyìh
orjdulwkpltxh = lo idxw grqf srxyrlu gìulyhu od frxueh g*dfwxdolvdwlrq rewhqxh1
Rq hvw fhshqgdqw wrxmrxuv jíqì srxu dssuìflhu od txdolwì gh o*dmxvwhphqw rewhqx1 Hq hhw/ od suìvhqfh
g*xq whuph g*huuhxu gdqv o*ìtxdwlrq +:19, lpsoltxh tx*lo hvw qrupdo txh o*dmxvwhphqw qh vrlw sdv sduidlw >
fhfl shxw vlpsohphqw vljqlhu tx*lo h{lvwh ghv rssruwxqlwìv g*duelwudjh hw txh fhuwdlqv wlwuhv vrqw grqf
vxu0ìydoxìv +rx vrxv0ìydoxìv, sdu oh pdufkì1 F*hvw g*dloohxuv fhw ìfduw hqwuh sul{ dmxvwì hw sul{ revhuyì
txl ghyudlw orjltxhphqw frqgxluh ã ghv gìflvlrqv g*lqyhvwlvvhphqw1 Pdlv/ sdu dloohxuv/ rq shxw vrxyhqw/
txlwwh ã ìodujlu od idplooh g*lqwhusrodwlrq/ v*dssurfkhu dxvvl suëv tx*rq oh vrxkdlwh gh o*dmxvwhphqw sduidlw/
fh txl frqgxlw ã od olplwh ã gluh txh wrxv ohv wlwuhv vrqw sduidlwhphqw ìydoxìv sdu oh pdufkì1 Lo | d
grqf xq frpsurplv ã wurxyhu hqwuh oh ghjuì g*dmxvwhphqw gìvluì hw od wdlooh gh od idplooh xwlolvìh1 Fh
sureoëph q*d sdv gh vroxwlrq wdqw tx*xq prgëoh ìfrqrpltxh q*d sdv ìwì frqvwuxlw juåfh dxtxho rq sxlvvh
rewhqlu xqh idplooh qdwxuhooh gh irqfwlrqv uhvshfwdqw o*devhqfh g*rssruwxqlwìv g*duelwudjh/ shuphwwdqw
;9
dlqvl g*lqwhusuìwhu ohv huuhxuv uìvlgxhoohv frpph oh vljqh g*xqh pdxydlvh ìydoxdwlrq ghv wlwuhv hw/ hqvxlwh/
g*hq wluhu surw1
G*xq srlqw gh yxh sudwltxh/ oruvtx*rq xwlolvh pdojuì wrxw xqh idplooh gh irqfwlrqv qrq uholìh ã xq pr0
gëoh +frpph fhoohv txh qrxv suìvhqwrqv fl0ghvvrxv,/ rq mxjh od txdolwì gh o*dmxvwhphqw sdu o*ìwxgh ghv
uìvlgxv g*hvwlpdwlrq1 Vl rq furlw tx*lov vrqw oh uh hw g*rssruwxqlwìv g*duelwudjh prphqwdqìv/ lov ghyudlhqw
orjltxhphqw íwuh gh pr|hqqh qxooh vdxi ã frqvlgìuhu txh fhuwdlqv wlwuhv vrqw v|vwìpdwltxhphqw vxu0 rx
vrxv0ìydoxìv1 Dlqvl/ rq srxuud uhjdughu vl fhuwdlqv uìvlgxv vrqw v|vwìpdwltxhphqw srvlwliv +rx qìjdwliv,
srxu fhuwdlqv wlwuhv/ v*lov vrqw fruuìoìv dyhf od pdwxulwì gx wlwuh/ v*lov vrqw dxwrfruuìoìv/ hwf1
O*dmxvwhphqw sdu ghv sro|q÷phv
Klvwrultxhphqw/ ohv sro|q÷phv frqvwlwxhqw od suhplëuh idplooh xwlolvìh/ qrwdpphqw sdufh txh fhuwdlqv
srlqwv plolwhqw srxu vrq xwlolvdwlrq1 G*xqh sduw/ g*xq srlqw gh yxh sudwltxh/ ohv pìwkrghv g*hvwlpdwlrq
vrqw vlpsohv1 G*dxwuh sduw/ g*xq srlqw gh yxh wkìrultxh/ wrxwh frxueh frqwlqxh +vxu xq frpsdfw, shxw
íwuh dssurfkìh sdu xq sro|q÷ph gh ghjuì vx!vdpphqw ìohyì1 Hq jìqìudo/ rq prqwuh hpslultxhphqw txh
o*dmxvwhphqw sdu xq sro|q÷ph q*hvw sdv vdwlvidlvdqw fdu/ gdqv fh fdv/ ohv ghx{ h{wuìplwìv gh od frxueh vrqw
pdo dmxvwìhv dx surw gh o*hvsdfh lqwhupìgldluh/ txl oxl vhpeoh plhx{ sulv hq frpswh1 F*hvw srxutxrl rq
d ìwì dphqì ã xwlolvhu 5 rx 6 sro|q÷phv glìuhqwv srxu dmxvwhu fkdtxh sduwlh gh od frxueh +oh gìexw/ oh
plolhx/ od q,1 Frpph/ wrxwhirlv/ o*hqvhpeoh grlw íwuh frqwlqx hw gìulydeoh/ lo h{lvwh ghv frqwudlqwhv
vxu ohv frh!flhqwv gh fhv glìuhqwv sro|q÷phv dx{ srlqwv gh udffrughphqw1 Hqq fhwwh pìwkrgh glwh ghv
vsolqhv sro|q÷pldx{ d o*dydqwdjh g*íwuh uhodwlyhphqw idfloh ã hvwlphu xqh irlv txh fhuwdlqv sureoëphv rqw
ìwì uìjoìv +fkrl{ gx ghjuì gx sro|q÷ph/ srlqwv gh udffrughphqw111,1 Rq grqqh lfl xq h{hpsoh rü xq vhxo
srlqw gh udffrughphqw hvw xwlolvì hw rü ohv sro|q÷phv vrqw gh ghjuì 61 Iruphoohphqw rq ìfulud grqf =
E+w> w . , @ j +> w ,
@ e3 +w, . e4 +w, . e5 +w,5 . e6 +w,6 vl P
@ f3 +w, . f4 +w, . f5 +w,5 . f6 +w, 6 vl A P
+:1<,
rü w hvw grqf oh yhfwhxu ghv sdudpëwuhv +en +w,> fn +w,,3n@3>===>6 1
Dx srlqw P/ rq lpsrvh od frqwlqxlwì hw od grxeoh gìulydelolwì/ fh txl lqgxlw ohv frqwudlqwhv vxlydqwhv =
;
? e3 +w, . e4 +w,P . e5 +w,P 5 . e6 +w,P 6 @ f3 +w, . f4 +w,P . f5 +w,P 5 . f6 +w,P 6
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@ f4 +w, . 5f5 +w,P . 6f6 +w,P 5
+:143,
=
5e5 +w, . 9e6 +w,P
@ 5f5 +w, . 9f6 +w,P
Rq d gh soxv od frqwudlqwh =
E+w> w, @ 4 @ e3 +w,
+:144,
oh qrpeuh gh frqwudlqwhv lqgìshqgdqwhv hvw grqf 7 hw oh qrpeuh gh sdudpëwuhv ã hvwlphu ;1 Oruvtx*rq
revhuyh q reoljdwlrqv gh pdwxulwì +Wl ,l@4===q / hw gh sul{ Swl / oh fulwëuh ã plqlplvhu hvw grqf =
;
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A
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Lo v*djlw grqf g*xqh plqlplvdwlrq uìdolvdeoh sdu ohv prlqguhv fduuìv ruglqdluhv vrxv frqwudlqwhv olqìdluhv1
Od ghuqlëuh ìwdsh frqvlvwh doruv ã dqdo|vhu ohv uìvlgxv g*hvwlpdwlrq/ f*hvw0ã0gluh ohv ìfduwv hqwuh sul{ gh
pdufkì hw sul{ hvwlpìv/ dyhf wrxwhirlv ohv uìvhuyhv h{sulpìhv dx gìexw gh fh sdudjudskh1
Hpslultxhphqw/ rq uhoëyh/ srxu fhuwdlqv pdwxulwìv/ ghv uìvlgxv v|vwìpdwltxhphqw qìjdwliv rx v|vwìpdwl0
txhphqw srvlwliv1 Lo shxw doruv sdudñwuh vrxkdlwdeoh g*lqwurgxluh ghv yduldeohv h{solfdwlyhv whoohv od oltxlglwì/
od vfdolwì olìh ã fkdtxh reoljdwlrq/ oh idlw g*íwuh gdqv oh jlvhphqw gx PDWLI111
Hfroh Sro|whfkqltxh
;:
Sdu dloohxuv/ lo | suìvrpswlrq g*kìwìurvfìgdvwlflwì hw rq shxw shqvhu tx*hooh hvw uholìh qìjdwlyhphqw ã od
oltxlglwì = ohv wlwuhv ohv prlqv f÷wìv vrqw fhx{ vrxplv dx{ soxv iruwhv yduldwlrqv gh frxuv1 Fhwwh kìwì0
urvfìgdvwlflwì d ìwì uholìh ìjdohphqw ã od gxuìh gh ylh gh o*hpsuxqw1 Hooh grlw dssdudñwuh vrxv iruph gh
srqgìudwlrqv gdqv oh fulwëuh +:145,1
Fhwwh pìwkrgh srvh hqq ghx{ sureoëphv gìmã ìyrtxìv =
oh fkrl{ ghv srlqwv gh udffrughphqw +tx*rq srxuudlw g*dloohxuv lqwurgxluh frpph sdudpëwuh ã rswl0
plvhu gdqv oh surjudpph +:145,/
oh fkrl{ gx ghjuì ghv sro|q÷phv = vl oh ghjuì hvw wurs idleoh/ o*dmxvwhphqw hvw pdxydlv +uìvlgxv wuëv
ìohyìv,/ v*lo hvw dx frqwudluh wurs ìohyì/ o*dmxvwhphqw hvw wurs sduidlw hw od frxueh rewhqxh ghylhqw
wuëv khxuwìh/
hqq lo vhpeoh txh gh qrpeuhx{ sureoëphv g*lqvwdelolwì qxpìultxh +iruwh vhqvlelolwì/ fkrl{ ghv srlqwv
gh udffrughphqw sdu h{hpsoh, dssdudlvvhqw1
O*dmxvwhphqw sdu ghv h{srqhqwlhov gh sro|q÷phv
Qrxv lurqv lfl soxv udslghphqw gdqv od phvxuh rü ohv sulqflshv vrqw ohv píphv txh suìfìghpphqw1 O*dmxv0
whphqw sdu ghv h{srqhqwlhov gh sro|q÷phv sduw gh o*lgìh tx*xq sul{ }ìur0frxsrq v*ìfulw +hq frqwlqx, =
E+w> w . , @ h{s^\ +w> w . ,`
hw txh grqf lo hvw qdwxuho gh v*lqwìuhvvhu ã ghv idploohv gh irqfwlrqv h{srqhqwlhoohv1 Sudwltxhphqw/ fhod
uhylhqw ã dmxvwhu xq sro|q÷ph vxu ohv wdx{ }ìur0frxsrq hw qrq soxv vxu ohv sul{1 Iruphoohphqw/ rq ìfulud =
\ +w> w . , @
N
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E+w> w . , @ h{s N
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n@4
en +w,n
+:147,
Oã dxvvl rq srxuudlw sursrvhu xqh vxffhvvlrq gh sro|q÷phv dgdswìv ã fkdtxh sruwlrq gh od frxueh ghv
wdx{ pdlv o*hvwlpdwlrq hq vhudlw frpsoltxìh1 Hq hhw oh surjudpph ã uìvrxguh hvw gx píph w|sh txh
+:145, pdlv lo uhoëyh gìvrupdlv ghv prlqguhv fduuìv qrq olqìdluhv1 Lfl ìjdohphqw vh srvh oh sureoëph gx
ghjuì gx sro|q÷ph fkrlvl +rq v*duuíwh hq jìqìudo ã xq sro|q÷ph gh ghjuì 6,1 O*dqdo|vh ghv uìvlgxv idlw
hqfruh dssdudñwuh =
ghv eldlv v|vwìpdwltxhv = ohv reoljdwlrqv gh idleoh pdwxulwì rqw ghv uìvlgxv v|vwìpdwltxhphqw qìjdwliv/
gh o*kìwìurvfìgdvwlflwì = ohv reoljdwlrqv gh pdwxulwì ìohyìh rqw ghv uìvlgxv soxv yduldeohv txh fhoohv gh
pdwxulwì idleoh1
Lo vhpeoh grqf qìfhvvdluh g*lqwurgxluh od pdwxulwì frpph yduldeoh h{solfdwlyh ã od irlv gdqv o*ìtxdwlrq gh
uìjuhvvlrq pdlv ìjdohphqw gdqv od yduldqfh1 Xq whqwdwlyh d ìwì idlwh vrxv od iruph =
gw
Y +%lw @ Iw4 , @ 5w Wl>w
rü Wl>w hvw od pdwxulwì uìvlgxhooh gx wlwuh l ã od gdwh w hw +gw > w , vrqw ghv sdudpëwuhv ã hvwlphu1 O*hvwlpdwlrq
sdu svhxgr0pd{lpxp gh yudlvhpeodqfh hvw gl!floh ã phwwuh hq rhxyuh qxpìultxhphqw/ pdlv shuphw +hq
wkìrulh, xqh hvwlpdwlrq frpsoëwh ghv sdudpëwuhv dlqvl txh od plvh hq rhxyuh gh whvwv frqfhuqdqw oh ghjuì
gx sro|q÷ph +pdlv qrq vxu oh sdudpëwuh gw ,1
O*dmxvwhphqw sdu ghv sro|q÷phv g*h{srqhqwlhoohv
O*lgìh hvw gh suhqguh xqh irqfwlrq g*dfwxdolvdwlrq gh od iruph =
E+w> w . , @
;;
N
[
n@4
ln
k
en +w, h=+w,
+:148,
Ohv sdudpëwuhv vrqw doruv +en +w,> +w,,n@4===N 1 Rq uhpdutxh txh +w, d xqh lqwhusuìwdwlrq hq whuph gh
wdx{ ã whuph lqvwdqwdqì > rq d hhw =
olp W $.4
gOq E+w> W ,
@ +w,
gW
Lo hvw idfloh gh yrlu ìjdohphqw txh w shxw íwuh yx frpph oh wdx{ orqj/ f*hvw0ã0gluh od olplwh ghv wdx{
}ìur0frxsrq txdqg od pdwxulwì whqg yhuv o*lqql1
O*hvwlpdwlrq ghv sdudpëwuhv uhoëyh sduwlhoohphqw g*xqh pìwkrgh gh prlqguhv fduuìv olqìdluhv +srxu +w,
grqqì/ lo | d olqìdulwì hq en +w,,1
O*dmxvwhphqw sdu xqh iruph pl{wh
Lo v*djlw g*xqh idplooh gh irqfwlrqv prlqv frqqxh/ vxuwrxw xwlolvìh sdu ohv ìfrqrplvwhv/ gdqv odtxhooh ohv wdx{
ã whuph lqvwdqwdqìv v*ìfulyhqw vrxv iruph g*xq surgxlw g*xq sro|q÷ph hw g*xqh h{srqhqwlhooh1 O*dydqwdjh
gh fhwwh irupxodwlrq hvw/ frpph qrxv doorqv oh yrlu/ gh srxyrlu íwuh hvwlpìh gh idêrq udslgh1 Rq srvh
grqf =
+:149,
i+w> w . , @ 3 +w, . + 4 +w, . 5 +w,+w,,h+w,
fh txl grqqh srxu ohv sul{ }ìur0frxsrq =
] E+w> w . , @ h{s i +w> w . v,gv
3
@ h{s ^ 3 +w, . + 4 +w, . 5 +w,,+
4 h+w,
, 5 +w,h+w, `
+w,
Idlvdqw whqguh yhuv 3 hw yhuv .4/ rq rewlhqw =
i+w> 4, @ 3 +w,
+:14:,
i+w> w, @ 3 +w, . 4 +w,
+:14;,
fh txl shuphw gh gìwhuplqhu ohv ydohxuv gh 3 +w, hw 4 +w, vl rq frqqdlw oh wdx{ frxuw hw oh wdx{ orqj1
Vl sdu dloohxuv lo h{lvwh xq h{wuhpxp vxu od frxueh ghv wdx{ ã whuph lqvwdqwdqìv/ f*hvw0ã0gluh xqh pdwxulwì
., m@ doruv =
whooh txh Ci +w>w
C
i +w> w . , @ 3 +w, . 5 +w,h{s^4 .
+w,
@ 4
4 +w,
5 +w,
4 +w,
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5 +w,
+:14<,
+:153,
Dlqvl/ frqqdlvvdqw 3 +w, hw 4 +w,/ rq shxw fdofxohu 5 +w, sxlv +w,/ ã frqglwlrq elhq vþu tx*rq revhuyh
o*h{wuhpxp hw od pdwxulwì dvvrflìh1 Rq shxw uhpdutxhu gh soxv txh fhw h{wuhpxp hvw xq pd{lpxp +uhvs1
plqlpxp, vl 5 +w, hvw srvlwli +uhvs1 qìjdwli,1
Od surfìgxuh suìfìghqwh q*hvw fodluhphqw sdv h!fdfh sxlvtx*hooh q*xwlolvh sdv wrxwh o*lqirupdwlrq frqwhqxh
gdqv od vwuxfwxuh ghv wdx{1 Gh soxv/ hq sudwltxh lo hvw vþuhphqw gl!floh g*dyrlu ohv ydohxuv h{dfwhv gx
wdx{ lqvwdqwdqì/ gx wdx{ orqj hw gh o*h{wuhpxp ìyhqwxho ghv wdx{ ã whuph lqvwdqwdqìv > qìdqprlqv rq
shxw hvsìuhu hq dyrlu ghv ydohxuv dssurfkìhv/ shuphwwdqw dlqvl g*rewhqlu ghv ydohxuv lqlwldohv srxu ohv
sdudpëwuhv gh edvh 3 +w,/ 4 +w,/ 5 +w, hw +w,1 Lo hvw doruv qdwxuho g*xwlolvhu xqh surfìgxuh g*hvwlpdwlrq
fodvvltxh +frpph gìfulw suìfìghpphqw, lqlwldolvìh sdu fhv ydohxuv gh suhplëuh ìwdsh1
Hq jxlvh gh frqfoxvlrq gh fhwwh vhfwlrq/ lo idxw qrwhu oh fdudfwëuh wuëv sudjpdwltxh gh od gìpdufkh
sursrvìh lfl1 O*lqwurgxfwlrq g*xq doìd +o*doìd gx vwdwlvwlflhq, hvw idlwh gh idêrq dg0krf/ gh píph txh
ohv k|srwkëvhv vwdwlvwltxhv vxu fhw doìd1 Xqh gìpdufkh soxv qdwxuhooh frqvlvwhud ã vh grqqhu xq prgëoh
wkìrultxh g*ìyroxwlrq gh od vwuxfwxuh ghv wdx{/ gh oh frqwudlqguh ã uhvshfwhu o*devhqfh g*rssruwxqlwì
Hfroh Sro|whfkqltxh
;<
g*duelwudjh sxlv hqq g*hq gìgxluh ohv iruphv irqfwlrqqhoohv lpsrvìhv sdu fh prgëoh srxu ohv sul{ }ìur0
frxsrq1 Oh sureoëph gh od uhfrqvwlwxwlrq gh od frxueh ghv wdx{ }ìur0frxsrq vh udpëqh doruv ã o*hvwlpdwlrq
ghv sdudpëwuhv g*xq prgëoh wkìrultxh g*ìyroxwlrq gh od vwuxfwxuh ghv wdx{1 Hq sudwltxh/ od uhfrqvwlwxwlrq
gh od frxueh ghv wdx{ }ìur0frxsrq idlw o*remhw gh ehdxfrxs g*hpslulvph ã od irlv gdqv vd plvh hq rhxyuh
+gdqv od phvxuh rü fkdfxq d vd pìwkrgh/ vd idêrq g*lqwurgxluh od vfdolwì/ fhuwdlqhv vsìflflwìv ghv
wlwuhv111, pdlv ìjdohphqw gdqv vrq dssolfdwlrq +ã sduwlu gh txhooh ydohxu ghv uìvlgxv mxjh0w0rq txh oh wlwuh
hvw vxu0 rx vrxv0ìydoxì111 B,
:16 O*dqdo|vh idfwrulhooh gh od vwuxfwxuh ghv wdx{
O*dssurfkh suìvhqwìh lfl vh yhxw hqfruh ghvfulswlyh1 Hooh vh irqgh qrq soxv xqltxhphqw vxu xqh frxueh ghv
wdx{ ã xqh gdwh w grqqìh frpph gdqv ohv vhfwlrqv suìfìghqwhv/ pdlv vxu o*ìyroxwlrq whpsruhooh gh fhwwh
frxueh1 Vrqw ìwxglìhv dlqvl ohv gìirupdwlrqv gh od frxueh ghv wdx{ dx frxuv gx whpsv1 Frqfuëwhphqw fhod
v*dssduhqwh ã xqh dqdo|vh ghv grqqìhv vxu vìulh whpsruhooh1 Fhwwh vìulh whpsruhooh +[w , q*hvw ulhq g*dxwuh
txh oh yhfwhxu ghv wdx{ }ìur0frxsrq/ ghv sul{ rx ghv uhqghphqwv dx{ glìuhqwhv pdwxulwìv1 Sdu h{hpsoh/
+[w , srxuud íwuh =
3
4
\ +w> w . 4,
E
F
11
[w @ C
D
1
\ +w> w . Q,
rx hqfruh =
3
4
E+w> w . 4,
E
F
11
[w @ C
D
1
E+w> w . Q ,
O*lgìh qdqflëuh yhuv odtxhooh rq vrxkdlwhudlw frqyhujhu ã o*lvvxh gh fhwwh ìwxgh uhsrvh vxu oh idlw txh +[w ,
q*hvw sdv xq hpslohphqw gh Q vìulhv lqgìshqgdqwhv/ pdlv tx*dx frqwudluh fhv Q vìulhv vrqw iruwhphqw
olìhv1 Srxu h{sulphu fhwwh lgìh/ rq srvwxoh txh ohv Q vìulhv frpsrvdqw +[w , vrqw olìhv sdu ghv idfwhxuv
frppxqv/ gh whooh vruwh txh +[w , v*ìfulw =
[w @ w . F=Iw . xw
+:154,
rü F hvw xqh pdwulfh Q s/ Iw hvw xq yhfwhxu gh s vìulhv dsshoìhv idfwhxuv/ xw xq yhfwhxu gh shuwxuedwlrqv
lqgìshqgdqwhv hqwuh hoohv/ hw lqgìshqgdqwhv gh Iw hw w xqh vìulh gìwhuplqlvwh1
Gdqv fh fdv/ od g|qdpltxh gh +[w , hvw uìgxfwleoh ã od g|qdpltxh g*xq soxv shwlw qrpeuh gh vìulhv/ hq
o*rffxuhqfh s idfwhxuv +s ? Q,1 Rxwuh o*dvshfw ghvfulswli/ fh jhquh g*ìwxgh d ghv frqvìtxhqfhv lpsruwdqwhv
srxu ohv prgëohv g*ìyroxwlrq gh od vwuxfwxuh ghv wdx{ dlqvl txh srxu od sudwltxh qdqflëuh/ qrwdpphqw
sdu oh eldlv ghv vwudwìjlhv gh frxyhuwxuh1 Oh suhplhu srlqw ihud o*remhw ghv fkdslwuhv xowìulhxuv1 Oh vhfrqg
hvw gìyhorssì gh idêrq khxulvwltxh gdqv oh sdudjudskh vxlydqw1
:1614 Prwlydwlrqv qdqflëuhv
Oh sul{ g*xq wlwuh ã x{ fhuwdlqv/ hw grqf oh sul{ g*xq sruwhihxlooh gh fhv wlwuhv/ hvw gluhfwhphqw irqfwlrq
gh od vwuxfwxuh ghv wdx{1 Sdu frqvìtxhqw/ ohv yduldwlrqv gh sul{ vrqw gluhfwhphqw olìhv dx{ prxyhphqwv gh
od vwuxfwxuh ghv wdx{1 Xq sureoëph uìfxuuhqw hq qdqfh hvw ghsxlv orqjwhpsv od uhfkhufkh gh sruwhihxloohv
sduwlhoohphqw lqvhqvleohv dx{ prxyhphqwv gh od vwuxfwxuh ghv wdx{1 Fh tx*rq dsshooh frppxqìphqw frx0
yhuwxuh hq gxudwlrq frqvlvwh ã frqvwuxluh ghv sruwhihxloohv lppxqlvìv frqwuh ghv prxyhphqwv gx qlyhdx
jìqìudo ghv wdx{1 Lo v*djlw lfl g*doohu soxv orlq gdqv fh w|sh g*dqdo|vh1 G*dsuëv od irupxoh g*dfwxdolvdwlrq
+:17,/ wrxw sruwhihxlooh frqvwuxlw hq w 4/ gh sul{ Sw4 shxw v*dqdo|vhu frpph xqh frpelqdlvrq olqìdluh
gh wlwuhv }ìur0frxsrq gh whooh vruwh txh Sw4 v*ìfulyh =
Sw4 @
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4
d =E+w 4> w 4 . ,
+:155,
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@ Sw Sw4
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@
d E+w> w . ,
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Sw
rü E+w> w . , @ E+w> w 4 . , E+w 4> w 4 . ,/ rx hqfruh =
[
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@
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4
+:156,
rü
E+w 4> w 4 . ,
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{w4> @ d
V*lo q*| dydlw sdv gh ulvtxh/ oh wdx{ gh uhqghphqw gh wrxwh rsìudwlrq vxu xqh sìulrgh vhudlw ìjdo dx wdx{ gh
4 4 qrwì u+w 4> w,1 Lo hvw grqf qdwxuho gh frqvlgìuhu o*h{fëv gh uhqghphqw/
uhqghphqw vdqv ulvtxh E+w
4>w,
f*hvw0ã0gluh od glìuhqfh gx wdx{ gh uhqghphqw g*xqh rsìudwlrq dyhf oh wdx{ gh uhqghphqw vdqv ulvtxh1 Vl
rq shxw srvwxohu xqh gìfrpsrvlwlrq idfwrulhooh srxu =
E
u+w 4> w,>
E
rq ìfulud =
s
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u+w 4> w, @ >w .
n In>w . x>w
E+w 4> w 4 . ,
n@4
rü ohv idfwhxuv hw od shuwxuedwlrq vrqw fhqwuìv1
;5>
+:157,
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s
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Sw
u+w 4> w, @ w .
fn In>w . yw
Sw4
n@4
rü
w
@
fn
@
yw
@
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5
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{w4> n
{w4> x>w
Rq shxw doruv ìolplqhu oh ulvtxh v|vwìpdwltxh +f*hvw0ã0gluh oh ulvtxh olì dx{ idfwhxuv, hq lpsrvdqw/ oruv gh
od frqvwlwxwlrq gx sruwhihxlooh hq w 4/ ohv s frqwudlqwhv =
vdqv rxeolhu od s . 4hph frqwudlqwh =
fn @ 3> n @ 4> ===> s
[
4
{w4> @ 4
Xq sruwhihxlooh txl yìulh fhv frqwudlqwhv vhud glw lppxqlvì1 Uhpdutxrqv doruv txh vl/ gdqv od iruph
+:157, lo h{lvwh xq idfwhxu +oh 4hu sdu h{hpsoh, who txh 4 vrlw sursruwlrqqho ã / doruv od frqglwlrq
g*lppxqlvdwlrq v*ìfulw =
f4 @ 3
uhgrqqh od frqglwlrq xvxhooh uhodwlyh ã od gxudwlrq suhplëuh1
Ohv frqglwlrqv g*lppxqlvdwlrq qìfhvvlwh grqf od frqqdlvvdqfh ghv srlgv n lqwhuyhqdqw gdqv od gìfrp0
srvlwlrq idfwrulhooh +:157,1 Od txhvwlrq hvw grqf gh uìvrxguh ohv sureoëphv vxlydqwv =
Hfroh Sro|whfkqltxh
<4
fkhufkhu oh qrpeuh gh idfwhxuv xwlohv s/
hvwlphu ohv srlgv +n ,n> 1
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Oh whuph gh prgëoh idfwrulho dssdudñw uìjxolëuhphqw gdqv od olwwìudwxuh pdlv hvw vrxyhqw suìvhqwìh gh
idêrq dpeljxî1 Dydqw wrxw/ lo idxw uhpdutxhu txh qrxv vrpphv gdqv xq frqwh{wh g|qdpltxh > lo hvw grqf
qìfhvvdluh gh gìqlu o*hqvhpeoh g*lqirupdwlrqv glvsrqleohv ã fkdtxh gdwh1 Vl qrxv ìwxglrqv xqh vìulh +[w ,
+gh glphqvlrq Q ,/ doruv fhw hqvhpeoh g*lqirupdwlrqv Iw ã od gdwh w vhud frqvwlwxì gh [w4 > [w5 === qrwì
[ w4 1 Rq vxssrvh doruv txh +[w , dgphw od gìfrpsrvlwlrq vxlydqwh =
Gìqlwlrq 45 Gìqlwlrq 46 +Uhsuìvhqwdwlrq idfwrulhooh,1 +[w , vxlw xqh uhsuìvhqwdwlrq idfwrulhooh
gh udqj s vl =
[w @ w . F=Iw . Xw
vrxv ohv k|srwkëvhv =
+Iw , hvw xq surfhvvxv gh glphqvlrq s >
+Xw , hvw xq yhfwhxu gh shuwxuedwlrq g*hvsìudqfh frqglwlrqqhooh qxooh/ gh yduldqfh frqglwlrqqhooh
gldjrqdoh/ hw qrq fruuìoìhv dyhf Iw >
F hvw xq hpdwulfhh uhfwdqjxodluh Q { s/ gh udqj s1
Od frqglwlrq vxu od yduldqfh frqglwlrqqhooh ghv shuwxuedwlrqv +l1h = gh iruph gldjrqdoh, hvw jìqìudohphqw
lpsrvìh pdlv hooh q*hvw sdv qìfhvvdluh1 Hooh hvw wrxwhirlv qdwxuhooh = xqh irlv hqohyìv ohv idfwhxuv Iw / ohv
whuphv uìvlgxhov q*rqw soxv gh idfwhxuv frppxqv hw vrqw grqf vxssrvìv íwuh qrq fruuìoìv1
Sudwltxhphqw/ ohv duwlfohv vh sodfhqw gdqv xq fdv wuëv sduwlfxolhu gh fhwwh irupxodwlrq jìqìudoh =
; ? Y Iw @[ w4 @ frqv wdq wh
Y Xw @[ w4 @ G pdwulfh gldjrqdoh frqvwdqwh
=
@ w
Rq yrlw hq rxwuh tx*rq shxw wrxmrxuv uhgìqlu od pdwulfh ghv srlgv F srxu rewhqlu =
Y [w @[ w4 @ F=F 3 . G
H [w @[ w4 @ +:158,
Fhwwh irupxodwlrq h{foxw hq sduwlfxolhu oh fdv rü +[w , hw +Iw , vrqw qrq vwdwlrqqdluhv1 Gh soxv ohv +[w ,w
vrqw qrq dxwrfruuìoìv1
Srxu qlu/ uhpdutxrqv xqh fkrvh lpsruwdqwh = od pdwulfh F q*hvw sdv lghqwldeoh1 Srxu wrxwh pdwulfh S
lqyhuvleoh/ rq shxw uììfuluh od gìqlwlrq suìfìghqwh vrxv od iruph =
[w @ F Iw . Xw
dyhf ohv píphv k|srwkëvhv hw rü =
F
Iw
@ FS
@ S 4 Iw
Hq idlw/ vhxo o*hvsdfh hqjhqguì sdu ohv frorqqhv gh F +txl hvw gh glphqvlrq s, hvw lghqwldeoh1 Ohv glìuhqwhv
pìwkrghv g*hvwlpdwlrq qh ihurqw tx*h{klehu o*hvwlpdwlrq g*xqh edvh sduwlfxolëuh gh fhw hvsdfh qrwì ^F`1
<5
:1616 Whvw g*k|srwkëvhv qdqflëuhv
Rq d yx suìfìghpphqw txh srxu uhwurxyhu od frxyhuwxuh xvxhooh hq gxudwlrq/ lo idoodlw txh ohv srlgv dhfwìv
dx suhplhu idfwhxu +sdu h{hpsoh, dlhqw xqh iruph sduwlfxolëuh1 Hq o*rffxuuhqfh/ oh wdx{ gh uhqghphqw gh
E+w> w . ,
u+w 4> w, grlw srxyrlu v*ìfuluh =
pdwxulwì /
E+w 4> w 4 . ,
E+w> w . ,
u+w 4> w, @ w . I4>w . ===
E+w> w . ,
Gdqv oh fdguh g*xqh uhsuìvhqwdwlrq idfwrulhooh jìqìudoh =
[w @ w . F=Iw . Xw
fhod vljqlh txh o*hvsdfh ^F` frqwlhqw oh yhfwhxu h @ +4> = = = > > = = = > Q ,3 =
s 5 ^F`
+:159,
rü ^F` gìvljqh o*hvsdfh hqjhqguì sdu ohv frorqqhv gh od pdwulfh F1 O*k|srwkëvh qdqflëuh txh o*rq yrlw
vrxyhqw gdqv od olwwìudwxuh vrxv od iruph = lo h{lvwh xq idfwhxu gh qlyhdx gdqv o*ìyroxwlrq gh od vwuxfwxuh
ghv wdx{ vh wudgxlw sdu o*dssduwhqdqfh g*xq fhuwdlq yhfwhxu gdqv o*hvsdfh ghv srlgv ^F`1 Hq sudwltxh/ lo
hvw ehdxfrxs soxv idfloh gh whvwhu xqh fdudfwìulvdwlrq ìtxlydohqwh gh fhwwh k|srwkëvh ã o*dlgh g*xqh edvh
sduwlfxolëuh gh o*hvsdfh ^F`1 Hq hhw/ vl F hvw xqh edvh sduwlfxolëuh gh ^F` doruv +:159, hvw dxvvl ìtxlydohqw
ã=
4 F 3 s @ s
F +F 3 F,
+:15:,
Rq shxw gìvluhu dxvvl whvwhu tx*xq sruwhihxlooh sduwlfxolhu hvw lppxqlvì1 Gdqv oh fdv rü +[w , gìvljqh
oh yhfwhxu ghv wdx{ gh uhqghphqwv sdu pdwxulwì/ fhfl vljqlh txh oh wdx{ gh uhqghphqw gx sruwhihxlooh
sduwlfxolhu v*ìfulw =
3 [w
txl hvw lppxqlvì vl =
3 F @ 3
vrlw hqfruh =
5 ^F`B
txl shxw v*ìfuluh vrxv od píph iruph txh gdqv +:15:, ã o*dlgh g*xqh edvh sduwlfxolëuh gh o*ruwkrjrqdo gh
^F`1
Fhfl looxvwuh oh idlw txh ohv k|srwkëvhv qdqflëuhv shuwlqhqwhv vh udpëqhqw ã ghv whvwv vxu o*hvsdfh ghv
srlgv hw qrq sdv vxu od ydohxu ghv srlgv hx{0píphv1 Frpph qrxv o*dyrqv glw soxv kdxw/ fhfl hvw gþ dx idlw
txh vhxo o*hvsdfh hvw lghqwldeoh1 Gh píph/ od ydohxu sulvh sdu ohv idfwhxuv Iw q*d txh shx g*lpsruwdqfh >
od vhxoh fkrvh txl lpsruwh hvw ohxu qrpeuh1
:1617 Hvwlpdwlrq g*xq prgëoh idfwrulho
Oh fkrl{ gh od pìwkrgh g*hvwlpdwlrq gìshqg ghv k|srwkëvhv idlwhv vxu fhwwh uhsuìvhqwdwlrq1 Qrxv wudlwrqv
oh fdv fruuhvsrqgdqw dx{ k|srwkëvhv xvxhoohv txh qrxv udsshorqv =
H [w @ [ w4 @ +:15;,
Y [w @ [ w4 @ F=F 3 . G
Gdqv fh fdv/ +[w , hvw xqh vxlwh gh yduldeohv doìdwrluhv/ qrq dxwrfruuìoìhv1 Frpph qrxv o*dyrqv glw/ hw G
vrqw lghqwldeohv pdlv sdv F1 Lo idxw grqf udmrxwhu xqh frqwudlqwh g*lghqwlfdwlrq1 Rq lpsrvh hq jìqìudo =
F 3 G4 F pdwulfh gldjrqdoh
Ohv hvwlpdwhxuv gh / F/ G rewhqxv sdu uìvroxwlrq gx surjudpph +vl K gìvljqh oh qrpeuh g*revhuydwlrqv, =
5
P d{
OK +> F> G,
7 v1f1
F 3 G4 F gldjrqdoh
Hfroh Sro|whfkqltxh
<6
rü =
K
K
4[
4
Oq mFF 3 . Gm +[w ,3 +FF 3 . G, +[w ,
5
5 w@4
vrqw frqyhujhqwv hw dv|pswrwltxhphqw qrupdx{ oruvtxh oh qrpeuh g*revhuydwlrqv K whqg yhuv .41
OK +> F> G, @ Dlqvl/ píph vl +[w , q*hvw sdv xq surfhvvxv jdxvvlhq/ od svhxgr0orj0yudlvhpeodqfh OW surgxlw ghv hvwlpd0
whxuv frqyhujhqwv hw dv|pswrwltxhphqw qrupdx{1 Hqq/ rq shxw uhpdutxhu txh o*hvwlpdwhxu gh q*hvw
a od yduldqfh hpslultxh gh +[w ,/ F hw G vrqw
dxwuh txh od pr|hqqh hpslultxh gh +[w ,/ hw txh/ vl rq qrwh doruv vroxwlrq gx surjudpph =
5
r
q
a
W u +FF 3 . G,4 P d{
K5 Oq mFF 3 . Gm K
5
9
7 v1f1
F 3 G4 F gldjrqdoh
rü W u gìvljqh o*rsìudwhxu Wudfh1
Frqfhuqdqw ohv whvwv ghv glìuhqwhv k|srwkëvhv ìyrtxìhv/ lo idxw dmrxwhu hq jìqìudo xqh k|srwkëvh gh
qrupdolwì vxu oh surfhvxv +[w ,1 OK ghylhqw od yudlh orjyudlvhpeodqfh hw ohv hvwlpdwhxuv rewhqxv vrqw
ohv hvwlpdwhxuv gx pd{lpxp gh yudlvhpeodqfh1 Ohv ìwxghv txl rqw ìwì idlwhv dx{ Hwdwv0Xqlv hw hq Iudqfh
q*xwlolvhqw sdv wrxmrxuv fhwwh pìwkrgh gx svhxgr0pd{lpxp gh yudlvhpeodqfh1 Hq uhydqfkh/ rq sudwltxh
a Hq idlw/ rq
xqh Dqdo|vh hq Frpsrvdqwhv Sulqflsdohv +DFS, vxu od pdwulfh gh yduldqfh hpslultxh 1
prqwuh +yrlu dqqh{h, txh fhfl q*hvw ydolgh txh vl rq vxssrvh txh ohv ulvtxhv lglrv|qfudwltxhv/ f*hvw0ã0gluh
ohv yduldqfhv frqglwlrqqhoohv ghv shuwxuedwlrqv uìvlgxhoohv vrqw gh píph yduldqfh1 Ru lo | d hq jìqìudo
suìvrpswlrq g*kìwìurvfìgdvwlflwì hw lo hvw grqf soxv oìjlwlph gh vxssrvhu G qrq vfdodluh/ txlwwh ã whvwhu
fhwwh k|srwkëvh hqvxlwh1 Oruvtx*rq idlw xqh DFS/ oh qrpeuh gh idfwhxuv fruuhvsrqg dx qrpeuh gh ydohxuv
a Vl od qrupdolwì shxw íwuh vxssrvìh/ rq shxw uìdolvhu ã qrxyhdx xq whvw gx
sursuhv vljqlfdwlyhv gh 1
udssruw ghv yudlvhpeodqfhv1
Frqfuëwhphqw/ rq fdofxoh od pdwulfh gh yduldqfh hpslultxh gx yhfwhxu g*revhuydwlrqv +[w ,/ vrlw =
a @ 4 +[w [,+[
3
w [,
K
hvw od pr|hqqh hpslultxh gh od vìulh1 G*dsuëv od iruph gx prgëoh idfwrulho/ rq d doruv ã od olplwh
rü [
txdqg K whqg yhuv o*lqql =
a $ FI F 3 . G
Vl rq qìjoljh oh ulvtxh lglrv|qfudwltxh G sdu udssruw ã od yduldqfh ghv idfwhxuv I / doruv lo hvw hhfwlyhphqw
a =
oìjlwlph gh gldjrqdolvhu oh qrpeuh gh ydohxuv sursuhv vljqlfdwlyhv grqqh xqh hvwlpdwlrq gx qrpeuh gh idfwhxuv +s,/
ohv yhfwhxuv sursuhv dvvrflìhv ã fhv ydohxuv sursuhv qrq qxoohv frqvwlwxhqw xqh edvh gh o*hvsdfh ^F`1
Hq sudwltxh/ ghv ìwxghv rqw ìwì uìdolvìhv vxu grqqìhv dpìulfdlqhv hw iudqêdlvhv vxu ghv yhfwhxuv [w uh0
suìvhqwdwliv ghv uhqghphqwv pdlv dxvvl gh wdx{ }ìur0frxsrq/ gh wdx{ ã whuph111 Xqh suhplëuh fulwltxh
pìwkrgrorjltxh dssdudñw gdqv od phvxuh rü ohv uhsuìvhqwdwlrqv idfwrulhoohv +txl vrqw irqgdphqwdohphqw
olqìdluhv, qh vrqw sdv d sulrul frpsdwleohv hqwuh hoohv vxlydqw tx*rq ohv srvwxoh vxu ghv wdx{ }ìur0frxsrq/
ghv uhqghphqwv/ ghv sul{ }ìur0frxsrq1 Oruvtx*hooh hvw srvwxoìh vxu ohv wdx{ }ìur0frxsrq hw tx*rq sudwltxh
xqh DFS/ rq wurxyh g*ruglqdluh 5 ã 6 idfwhxuv +s @ 5 rx 6,/ f*hvw0ã0gluh ghx{ ã wurlv ydohxuv sursuhv
a Rq wurxyh ìjdohphqw txh oh suhplhu yhfwhxu sursuh gh o*DFS
lpsruwdqwhv srxu od pdwulfh gh yduldqfh 1
d ghv frrugrqqìhv frqvwdqwhv/ doruv txh oh vhfrqg d ghv frrugrqqìhv furlvvdqw +rx gìfurlvvdqw, dssur{l0
pdwlyhphqw gh idêrq olqìdluh1 Fhfl frqgxlw ohv dxwhxuv ã lghqwlhu xq idfwhxu gh qlyhdx hw xq idfwhxu gh
shqwh/ f*hvw0ã0gluh txh ohv fkrfv vxu od vwuxfwxuh ghv wdx{ vh gìfrpsrvhqw hq fkrfv vxu oh qlyhdx hw od
shqwh gh fhwwh frxueh1
Xqh vhfrqgh fulwltxh frqfhuqh oh fkrl{ gh od pìwkrgh +Dqdo|vh Idfwrulhooh rx DFS, txl qh vrqw sdv
ìtxlydohqwhv1 Hqq/ lo idxw qrwhu txh gh qrpeuhxvhv ìwxghv ìfrqrpìwultxhv uhphwwhqw hq fdxvh fhuwdlqhv
ghv k|srwkëvhv vrxv0mdfhqwhv1 Dlqvl lo q*hvw sdv vþu txh ohv wdx{ vrlhqw vwdwlrqqdluhv/ fh txl frqgxlw ã
uhmhwhu ohv pìwkrghv g*hvwlpdwlrq suìfìghqwhv +Dqdo|vh Idfwrulhooh hw DFS, rx ã wrxw oh prlqv ã ohv
<7
pdqlhu dyhf suxghqfh1 Sdu dloohxuv/ oh idlw txh ohv yduldqfhv frqglwlrqqhoohv vrlhqw vxssrvìhv frqvwdqwhv
hvw hq jìqìudo uhmhwì sdu ohv grqqìhv/ frqgxlvdqw ohv ìfrqrpëwuhv ã sursrvhu ghv prgìolvdwlrqv gh w|sh
DxwrUhjuhvvlyh Frqglwlrqdoo| Khwhurvnhgdvwlf +DUFK,1
Fh txl pdqtxh irqgdphqwdohphqw ã fhv dssurfkhv hvw gh srxyrlu wudydloohu gdqv oh fdguh g*xq prgëoh
gdqv ohtxho rq srxuudlw mxvwlhu ohv k|srwkëvhv idlwhv vxu od olqìdulwì gh od uhsuìvhqwdwlrq idfwrulhooh/ od
g|qdpltxh vrxv0mdfhqwh ghv wdx{ g*lqwìuíw111 D fhw ìjdug/ frpph gdqv oh fdv gh o*lqwhusrodwlrq gh od
vwuxfwxuh ghv wdx{ txh qrxv dyrqv yx suìfìghpphqw/ ehdxfrxs gh srlqwv v*ìfodluhurqw ã od oxplëuh ghv
prgëohv g*ìyroxwlrq gh od vwuxfwxuh ghv wdx{ hw fhod qrxv shuphwwud gh glvfxwhu ohv dssurfkhv hpslultxhv
sursrvìhv lfl1
Hfroh Sro|whfkqltxh
<8
<9
;
Ohv prgëohv gh wdx{ hq whpsv glvfuhw1
;14 Lqwurgxfwlrq
Oh fkdslwuh suìfìghqw fkhufkdlw ã hvwlphu od vwuxfwxuh ghv wdx{ ã xqh gdwh grqqìh dlqvl txh vhv gìirupd0
wlrqv dx frxuv gx whpsv1 Qrxv dyrqv yx tx*xqh ghv gl!fxowìv uhqfrqwuìhv ìwdlw gh qh srxyrlu vh uhsrvhu
vxu xq prgëoh vrxv0mdfhqw shuphwwdqw gh mxvwlhu ohv k|srwkëvhv xwlolvìhv hw g*lqwhusuìwhu ohv uìvxowdwv
rewhqxhv1 Rq glvwlqjxh ghx{ fodvvhv gh prgëohv1 Ohv suhplhuv/ glwv prgëohv g*ìtxloleuh/ prgëolvhqw ohv
irqfwlrqv g*xwlolwì ghv djhqwv srxu hq gìgxluh ghv irqfwlrqv gh ghpdqghv g*dfwliv > sxlv/ o*ìjdolwì ruh0
ghpdqgh shuphw gh gìulyhu ohv sul{ ghv dfwliv1 Od vhfrqgh fodvvh gh prgëohv qh ylvh sdv ã prgìolvhu ohv
suìiìuhqfhv ghv djhqwv pdlv fkhufkh soxw÷w ã vh uhvwuhlqguh ã o*k|srwkëvh plqlpdoh g*devhqfh g*rssruwx0
qlwìv g*duelwudjh1
Oh soxv vlpsoh srxu dssuìkhqghu fh jhquh gh sureoëphv hvw gh vh sodfhu/ gdqv xq suhplhu whpsv/ gdqv
oh fdguh g*xq prgëoh hq whpsv glvfuhw/ hw gh gìyhorsshu/ gdqv fh prgëoh/ ohv sulqflsdohv lgìhv qdqflëuhv
hw suredelolvwhv txl uìdssdudñwurqw vrxv iruph frpsoh{h +g*xq srlqw gh yxh pdwkìpdwltxh, gdqv ohv pr0
gëohv hq whpsv frqwlqx1 Dlqvl od qrwlrq g*devhqfh g*rssruwxqlwì g*duelwudjh +D1R1D, hw vhv frqvìtxhqfhv
suredelolvwhv vhurqw wuëv fodluhv gdqv ohv prgëohv hq whpsv glvfuhw1 Fh w|sh gh prgëoh hvw oh shqgdqw gx
prgëoh elqrpldo gìyhorssì srxu ohv rswlrqv vxu dfwlrq +prgëoh glw gh Fr{/ Urvv hw Uxelqvwhlq,1 Qrxv
frpphqêrqv sdu gìfuluh od vwuxfwxuh suredelolvwh g*xq prgëoh hq whpsv glvfuhw ã doìd elqrpldo dlqvl
txh ohv frqwudlqwhv lpsrvìhv sdu o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh1 Qrxv qrxv lqwìuhvvrqv
hqvxlwh ã od uìvroxwlrq frpsoëwh gh fh prgëoh gdqv oh fdv sduwlfxolhu gx prgëoh glw gh Kr hw Ohh1 Sxlv
qrxv gìyhorssrqv ohv h{whqvlrqv qdwxuhoohv gh fh w|sh gh prgëoh
Qrxv grqqrqv hqvxlwh oh sulqflsh g*ìydoxdwlrq ghv elhqv frqwlqjhqwv vxlyl gh txhotxhv h{hpsohv g*dssol0
fdwlrq1 Hqq/ qrxv prqwurqv tx*rq shxw idluh frqyhujhu/ txdqg oh sdv gh whpsv whqg yhuv 3/ fhuwdlqv gh
fhv prgëohv yhuv ghv prgëohv hq whpsv frqwlqx txh qrxv uhyhuurqv dx{ fkdslwuhv vxlydqwv1
;15 Oh prgëoh gh edvh
;1514 Ohv k|srwkëvhv qdqflëuhv
Ohv k|srwkëvhv gh wudydlo vwdqgdugv vrqw ohv vxlydqwhv =
K|srwkëvhv
41 Devhqfh gh frþwv gh wudqvdfwlrq1 Ohv dfwliv vrqw lqqlphqw glylvleohv1
51 Oh whpsv hvw glvfuhw1
61 Lo h{lvwh ã fkdtxh gdwh ghv wlwuhv }ìur0frxsrq srxu wrxwh ohv pdwxulwìv1
71 Frqglwlrqqhoohphqw ã fkdtxh gdwh/ vhxov 5 ìwdwv gx prqgh shxyhqw vh uìdolvhu1
Gh soxv/ qrxv ihurqv frqvwdpphqw o*k|srwkëvh tx*lo q*| d sdv g*rssruwxqlwìv g*duelwudjh1
;1515 Oh prgëoh suredelolvwh
Vxssrvrqv tx*lo q*| dlw sdv/ hq idlw/ g*doìd > gdqv fh fdv/ rq d od uhodwlrq rewhqxh sdu oh udlvrqqhphqw
g*duelwudjh kdelwxho =
E+w 4> w,E+w> W , @ E+w 4> W ,
rx hqfruh =
E+w> W , @
E+w 4> W ,
E+w 4> w,
+;14,
Dxwuhphqw glw/ lo h{lvwh xqh uhodwlrq gh uìfxuuhqfh grqqdqw od vwuxfwxuh ghv wdx{ hq w/ E+w> =,> hq irqfwlrq
gh od vwuxfwxuh ghv wdx{ hq w 4/ E+w 4> =,1 O*lgìh od soxv vlpsoh srxu lqwurgxluh xq doìd gdqv fhwwh
ìyroxwlrq gh od vwuxfwxuh ghv wdx{ hvw gh frqvlgìuhu o*ìtxdwlrq gh uìfxuuhqfh gìulyìh gh +;14, =
E+w> W , @
E+w 4> W ,
=Kw +W w,
E+w 4> w,
+;15,
rü Kw +=, hvw xqh irqfwlrq doìdwrluh/ suhqdqw ghx{ ydohxuv srvlwlyhv srvvleohv =
;
? kw +=, dyhf suredelolwì sw
Kw +=, @
= kw +=, dyhf suredelolwì 4 sw
D fh vwdgh/ lo hvw lpsruwdqw gh frqvwdwhu txh ohv ydohxuv kw +=, hw kw +=, dlqvl txh od suredelolwì sw shxyhqw
gìshqguh gh o*hqvhpeoh g*lqirupdwlrqv ã od gdwh w 41 Fhw hqvhpeoh g*lqirupdwlrqv qrwì Iw4 frpsuhqg
wrxwhv ohv uìdolvdwlrqv sdvvìhv +mxvtx*hq w 4 frpsulv, gh od vwuxfwxuh ghv wdx{/ rx gh idêrq ìtxlydohqwh
wrxwhv ohv uìdolvdwlrqv sdvvìhv gh od irqfwlrq gh shuwxuedwlrq K1 Sdu frqvìtxhqw ohv irqfwlrqv kw +=, hw kw +=,
hw od suredelolwì sw shxyhqw d sulrul gìshqguh ghv ydohxuv sdvvìhv gh od vwuxfwxuh ghv wdx{1 Qìdqprlqv/
lfl fhv irqfwlrqv qh gìshqgurqw sdv gh Iw4 1 Rq vxssrvhud dlqvl txh +Kw ,w hvw xqh vxlwh gh irqfwlrqv
doìdwrluhv lqgìshqgdqwhv gh orlv gh Ehuqrxool1
Lo hvw doruv srvvleoh gh gìurxohu od uìfxuuhqfh +;15, dq g*rewhqlu od vwuxfwxuh ghv wdx{ ã od gdwh w hq
irqfwlrq gh fhooh gh od gdwh 3 =
E+w> W , @
@
E+w 4> W ,
=Kw +W w,
E+w 4> w,
w E+3> W , \ Kn +W n,
=
E+3> w, n@4 Kn +w n,
dyhf od frqyhqwlrq = Kw +3, @ 4/ rx hqfruh =
E+w> W , @ E i +3> w> W ,=
w \
Kn +W n,
n@4 Kn +w n,
+;16,
rü E i +3> w> W , hvw oh sul{ ã whuph lpsolflwh hq 3 srxu od sìulrgh ^w> W `1 Dlqvl od vwuxfwxuh ghv wdx{ ã od gdwh
w hvw ìjdoh ã fhooh gìgxlwh ghv wdx{ ã whuph lpsolflwhv ã od gdwh 3/ pxowlsolìh sdu wrxwhv ohv shuwxuedwlrqv
vxuyhqxhv hqwuh 3 hw w1 Rq yd pdlqwhqdqw v*dwwdfkhu ã wudgxluh ohv frqwudlqwhv lpsrvìhv sdu o*devhqfh
g*rssruwxqlwì g*duelwudjh +D1R1D, vxu oh surfhvxv +Kw ,w 1
<;
;1516 O*devhqfh g*rssruwxqlwì g*duelwudjh
Oh udlvrqqhphqw hvw wuëv fodvvltxh hw frqvlvwh ã frqvwuxluh xq sruwhihxlooh lppxqlvì/ f*hvw0ã0gluh lqvhqvleoh
ã o*doìd Kw hw hqvxlwh gh frqfoxuh hq glvdqw tx*xq who sruwhihxlooh grlw dyrlu oh píph uhqghphqw txh oh
uhqghphqw vdqv ulvtxh1
Srxu fhod/ frqvlgìurqv ã od gdwh w/ 5 wlwuhv }ìur0frxsrqv g*ìfkìdqfh W4 hw W5 hw frqvwuxlvrqv xq sruwhihxlooh
frpsrvì gh fhv ghx{ wlwuhv =
4 wlwuh }ìur0frxsrq gh pdwxulwì W4
{ wlwuh+v, }ìur0frxsrq gh pdwxulwì W5 1
Vrlw Yw od ydohxu gh fh sruwhihxlooh ã od gdwh w1 Rq d sdu frqvwuxfwlrq =
;
@ E +w> W4 , . {=E +w> W5 ,
? Yw
=
Yw.4
@ E +w . 4> W4 , . {=E +w . 4> W5 ,
Frqqdlvvdqw od vwuxfwxuh ghv wdx{ mxvtx*ã od gdwh w/ Yw.4 qh shxw suhqguh txh ghx{ ydohxuv =
; E+w>W ,
E+w>W5 ,
4
A
? E+w>w.4, kw.4 +W4 w 4, . {= E+w>w.4, kw.4 +W5 w 4,
Yw.4 @
A
= E+w>W4 , k +W w 4, . {= E+w>W5 , k +W w 4,
E+w>w.4, w.4 4
E+w>w.4, w.4 5
+;17,
Lo hvw idfloh gh fkrlvlu {/ qrwì {3 who txh fhv 5 ydohxuv vrlhqw ìjdohv = rq shxw grqf frqvwuxluh ã od gdwh w xq
sruwhihxlooh lppxqlvì dx vhqv rü vd ydohxu ã od gdwh w . 4 hvw lqvhqvleoh dx fkrf vxuyhqdqw hqwuh w hw w . 41
Xq who sruwhihxlooh hvw grqf vdqv ulvtxh/ hw grlw grqf dyrlu oh píph uhqghphqw txh wrxwh rsìudwlrq vdqv
ulvtxh hqwuh w hw w . 4/ sdu h{hpsoh o*rsìudwlrq txl frqvlvwh ã dfkhwhu hq w xq wlwuh }ìur0frxsrq g*ìfkìdqfh
w.4 =
4
Yw.4 +{3 ,
@
Yw +{3 ,
E+w> w . 4,
Od wudgxfwlrq gh fhwwh frqglwlrq frqgxlw ã od uhodwlrq =
4 kw.4 +x4 ,
4 kw.4 +x5 ,
@
kw.4 +x4 , kw.4 +x4 ,
kw.4 +x5 , kw.4 +x5 ,
dyhf x4 @ W4 +w . 4, hw x5 @ W5 +w . 4,1 Fhwwh uhodwlrq hvw ydolgh srxu wrxwh ydohxu gh w/ W4 hw W5 +dyhf
W4 A w hw W5 A w,1 Dlqvl frpph ulhq qh uhvwuhlqw oh fkrl{ gh W4 hw W5 / o*h{suhvvlrq =
4 kw.4 +x,
kw.4 +x4 , kw.4 +x,
hvw lqgìshqgdqwh gh x hw vhud qrwìh w.4 =
w.4 @
4 kw.4 +x,
kw.4 +x4 , kw.4 +x,
rx hqfruh =
; w> ; x > w kw +x, . +4 w , kw +x, @ 4
+;18,
Fh uìvxowdw hvw irqgdphqwdo hw rq shxw uhpdutxhu gh soxv txh w hvw xqh suredelolwì =
Sursrvlwlrq < O*devhqfh g*rssruwxqlwì g*duelwudjh hvw ìtxlydohqwh ã o*h{lvwhqfh g*xqh suredelolwì @
+ w , whooh txh =
H +Kw +=, @ Iw4 , @ 4
Gh soxv/ fhwwh suredelolwì hvw ìtxlydohqwh ã od suredelolwì s dx vhqv rü =
w @ 3 +, sw @ 3
hw
Hfroh Sro|whfkqltxh
w @ 4 +, sw @ 4
<<
Suhxyh = Od suhplëuh sduwlh gh od sursrvlwlrq uìvxowh ghv fdofxov txl suìfëghqw1 Qrxv dyrqv w 5 ^3> 4` vl
vhxohphqw vl kw +=, hw kw +=, qh vrqw sdv gx píph f÷wì ylv ã ylv gh 41 Vxssrvrqv grqf sdu h{hpsoh txh o*rq
dlw =
4 ? kw +x,> kw +x,
doruv rq shxw frqvwuxluh hq w 4 xq sruwhihxlooh gh frþw qxo ãã sduwlu g*xq wlwuh g*ìfkìdqfh W hw gh {
wlwuhv gh pdwxulwì 4/ hq suhqdqw =
E+w 4> W ,
{@
E+w 4> w,
Vd ydohxu hq w hvw =
E+w> W , . { @
E+w 4> W ,
^Kw +W w, 4`
E+w 4> w,
grqw od ydohxu hvw wrxmrxuv srvlwlyh txrltx*lo duulyh1 Fhfl hvw lpsrvvleoh sdu D1R1D1 Rq shxw ìylghpphqw
uhidluh oh píph udlvrqqhphqw gdqv o*dxwuh fdv = kw +x,> kw +x, ? 41 Sdu frqvìtxhqw/ rq shxw wrxmrxuv
vxssrvhu kw +=, A 4 hw kw +=, ? 41 Sdu dloohxuv/ w @ 3 +uhvs1 w @ 4, hvw ìtxlydohqw ã kw +=, @ 4 +uhvs1
kw +=, @ 4, > hq xwlolvdqw oh píph sruwhihxlooh txh suìfìghpphqw/ rq frqvwdwh txh fhoxl0fl d wrxmrxuv xq
frþw qxo hw xqh ydohxu hq w srvlwlyh rx qxooh hw píph vwulfwhphqw srvlwlyh vl Kw suhqg od ydohxu kw 1 Fhfl
q*hvw srvvleoh txh vl hw vhxohphqw vl sw @ 3 +uhvs1 sw @ 4,1
Sdu frqvìtxhqw/ lo dssdudñw qdwxuhoohphqw xqh qrxyhooh suredelolwì w / gìshqgdqw gh Iw4 vl kw +=, hw kw +=,
hq gìshqghqw hx{0píph/ vrxv odtxhooh Kw +=, hvw g*hvsìudqfh frqglwlrqqhooh ìjdoh ã 4 =
H +Kw +=, @ Iw4 , @ 4
Rq gìvljqhud gìvrupdlv o*hvsìudqfh fdofxoìh dyhf ohv suredelolwìv s sdu H hw fhooh fdofxoìh dyhf ohv sured0
elolwìv w sdu H 1 Uhsuhqdqw o*ìtxdwlrq gx prgëoh gh edvh +;15, rq shxw doruv ìfuluh =
H Ê+w> W , @ Iw4 ` @ E i +w 4> w> W , @
E+w 4> W ,
E+w 4> w,
hw soxv jìqìudohphqw =
%
&
w\
4
E+n> n . 4, @ I3
E+3> W , @ H E+w> W ,=
n@3
Fhwwh suredelolwì hvw dsshoìh suredelolwì ulvtxh0qhxwuh +ulvn0qhxwudo suredelolw|,1 Hooh hvw d sulrul glìuhqwh
gh od suredelolwì remhfwlyh s1 Hooh hvw glwh ulvtxh0qhxwuh fdu/ g*dsuëv fh txl suìfëgh/ rq d =
E+w> W , @ H +E+w . 4> W , @ Iw ,=E+w> w . 4,
fh txl vljqlh txh oh sul{ dxmrxug*kxl w g*xq }ìur0frxsrq g*ìfkìdqfh W hvw o*hvsìudqfh +vrxv , gx sul{ gh
fh wlwuh hq w . 4/ dfwxdolvì +oh idfwhxu g*dfwxdolvdwlrq ìwdqw E+w> w . 4,,1 Ru xqh ìfrqrplh gdqv odtxhooh ohv
sul{ dxudlhqw fh jhquh gh sursulìwì vhudlw xqh ìfrqrplh rü ohv djhqwv vhudlhqw qhxwuhv ylv0ã0ylv gx ulvtxh1
Qrwrqv txh fh q*hvw sdv oh fdv lfl sxlvtxh od suredelolwì hvw glìuhqwh gh od suredelolwì remhfwlyh s1
;1517 Sursulìwì gh pduwlqjdoh
Vrxv od suredelolwì / gh qrpeuhxvhv sursulìwìv shxyhqw íwuh rewhqxhv1 Dlqvl/ vrxv / oh surfhvvxv ghv
sul{ dfwxdolvìv hvw xqh pduwlqjdoh1 Dq gh suìflvhu fh srlqw/ qrxv gìqlvvrqv oh idfwhxu g*dffxpxodwlrq =
w @
% w4
\
&4
E+n> n . 4,
n@3
w hvw grqf od ydohxu rewhqxh hq fdslwdolvdqw xq lqyhvwlvvhphqw g*4 hxur hq 3/ dx{ wdx{ frxuwv vxffhvvliv1
Dyhf fhwwh qrwdwlrq/ rq d =
Sursrvlwlrq 43 Vrxv +,/ oh surfhvvxv E+w>W , hvw xqh pduwlqjdoh/ l1h =
w
w
E+w> W ,
E+v> W ,
H
@ Iv @
w
v
433
Suhxyh = G*dsuëv od uhodwlrq irqgdphqwdoh +;15, =
E+w> W , @
E+w 4> W ,
Kw +W w,
E+w 4> w,
vrlw =
H +E+w> W , @ Iw4 , @
4 @ w4 / rq gìgxlw =
Frpph E+w
4>w,
w
H
E+w> W ,
@ Iw4
w
E+w 4> W ,
E+w 4> w,
@
E+w 4> W ,
w4
Hq uìlwìudqw o*dujxphqw/ rq hq gìgxlw od sursulìwì fkhufkìh1 hvw grqf od suredelolwì txl uhqg pduwlqjdoh
oh surfhvvxv ghv sul{ dfwxdolvìv1 Hq sduwlfxolhu/ rq rewlhqw sxlvtxh E+W> W , @ 4 =
4
@ I3 @ E+3> W ,
;W > H
W
;1518 Fkdqjhphqw gh sdudpìwudjh
D xqh gdwh w grqqìh/ w / kw +=, hw kw +=, vrqw lqfrqqxv hw ghyurqw grqf íwuh vsìflìv1 Rq shxw gìmã uhpdutxhu
txh o*devhqfh g*rssruwxqlwìv g*duelwudjh shuphw g*ìolplqhu xq gh fhv sdudpëwuhv sdu od uhodwlrq +;18,1
O*remhw ghv sdudjudskhv vxlydqwv vhud gh udmrxwhu ghv k|srwkëvhv vxssoìphqwdluhv srxu rewhqlu od vsìfl0
fdwlrq frpsoëwh gx prgëoh1 Fhshqgdqw/ ohv irqfwlrqv kw +=, hw kw +=, q*rqw sdv g*lqwhusuìwdwlrq qdqflëuh
lppìgldwh hw lo hvw grqf soxv qdwxuho gh wurxyhu xq fkdqjhphqw gh sdudpìwudjh soxv qdqflhu1 Srxu fhod/
qrxv doorqv prqwuhu txh od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv shxw frqvwlwxhu xqh sdudpìulvdwlrq
qdwxuhooh/ txl shuphw hq rxwuh gh fh udssurfkhu gx srlqw gh yxh ghv prgëohv hq whpsv frqwlqx1
Gìqlvvrqv ohv wdx{ ã whuph lqvwdqwdqìv frpph =4
i +w> W , @ Oq
E+w> W . 4,
E+w> W ,
Lo hvw idfloh gh yrlu/ ã o*dlgh gh +;16, txh =
w
E+3> W . 4, \ Kn +W . 4 n,
E+w> W . 4,
@
E+w> W ,
E+3> W ,
n@4 Kn +W n,
vrlw =
i +w> W , @ i +3> W , .
dyhf
w
[
n@4
i+3> W , @ Oq
Oq
Kn +W . 4 n,
Kn +W n,
+;19,
E+3> W . 4,
E+3> W ,
Fhfl lpsoltxh ìjdohphqw txh =
i+w> W , i +w 4> W , @ Oq
Kw +W w,
Kw +W . 4 w,
Gìqlvvrqv sdu +w> W , od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv +vrxv od suredelolwì ulvtxh0qhxwuh , > rq
d=
s
kw +W w, kw +W . 4 w,
=
+w> W , @ w +4 w , Oq +;1:,
kw +W w, kw +W w,
4 Lo v*djlw g*xq dexv gh qrwdwlrq sdu udssruw ã fh txl d ìwì gìql dx{ fkdslwuhv suìfìghqwv > hq wrxwh uljxhxu/ lo v*djlw lfl
gh
t s E|c Ac A n tx*rq uhpsodfh grqf sdu s E|c A dq g*dooìjhu ohv qrwdwlrqv1
Hfroh Sro|whfkqltxh
434
G*dsuëv fhwwh ìtxdwlrq/ lo hvw grqf ìtxlydohqw gh vh grqqhu +w > kw +=,> kw +=,, rx +w > +w> =,,1 Hq hhw/ qrxv
dyrqv =
kw +W . 4 w,
+w> W ,
kw +W w,
Oq @s
Oq kw +W w,
kw +W . 4 w,
w +4 w ,
vrlw sdu uìfxuuhqfh =
W[
4
4
kw +W w,
s
@
+w> v,
+;1;,
kw +W w,
w +4 w , v@w
Od grqqìh gh od irqfwlrq +w> =, hvw grqf ìtxlydohqwh ã fhooh gh Oq kkw ++==,, 1 Rq shxw doruv h{sulphu fhv ghx{
w
irqfwlrqv ã o*dlgh gh +w> W , =
Oq
;
kw +W w,
A
A
A
?
@
A
A
A
= kw +W w, @
4
S
w .+4w ,h{s s +44 , Wv@w4 +w>v,
w
w
4
+4w ,.w h{s sw+44 w, S Wv@w4 +w>v,
+;1<,
Hqq/ rq shxw uhpdutxhu tx*lo h{lvwh xqh uhodwlrq hqwuh o*hvsìudqfh gh i+w> W , frqglwlrqqhoohphqw ã Iw4
hw vd yrodwlolwì +w> W ,1 Rq shxw grqf gluh txh o*devhqfh g*rssruwxqlwì g*duelwudjh lqgxlw xqh g|qdpltxh
sduwlfxolëuh +vrxv , srxu ohv wdx{ ã whuph lqvwdqwdqìv/ uhpdutxh txh o*rq uhwurxyhud gdqv ohv prgëohv
hq whpsv frqwlqx1
Srxu frqfoxuh/ lo dssdudñw txh o*devhqfh g*rssruwxqlwì g*duelwudjh lpsrvh ghv frqwudlqwhv vxu ohv prgëohv
dgplvvleohv pdlv qh shuphw sdv gh vsìflhu od iruph ghv irqfwlrqv kw +=, hw kw +=, rx/ gh idêrq ìtxlydohqwh/
od irqfwlrq +w> W ,1 Lo hvw grqf qìfhvvdluh gh idluh ghv k|srwkëvhv vxssoìphqwdluhv1
;16 Uìvroxwlrq gx prgëoh gdqv xq fdv vlpsoh = oh prgëoh gh Kr hw Ohh
Rq shxw sduyhqlu ã xqh vsìflfdwlrq frpsoëwh gx prgëoh pr|hqqdqw txhotxhv k|srwkëvhv vxssoìphq0
wdluhv1
K|srwkëvh 4 = Od suredelolwì w qh gìshqg sdv gh w hw od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv qh
gìshqg txh gh od pdwxulwì W w =
@ w
+w> W , @ +W w,
rü +=, hvw xqh irqfwlrq gìwhuplqlvwh1
Frpswh0whqx gh fh txh qrxv dyrqv glw vxu ohv olhqv hqwuh ohv ghx{ w|shv gh sdudpìwudjh/ lo hvw fodlu txh
fhwwh k|srwkëvh hvw ìtxlydohqwh ã gluh txh ohv irqfwlrqv k hw k qh gìshqghqw sdv gx whpsv1 Sdu dloohxuv/
rq idlw o*k|srwkëvh glwh g*lqgìshqgdqfh gx fkhplq vxlyl
K|srwkëvh 5 = Od ydohxu gh wrxw sruwhihxlooh ã od gdwh w qh gìshqg txh gx qrpeuh gh irlv rü K d sulv
od ydohxu k1
Soxv suìflvìphqw/ xq sruwhihxlooh frqvwlwxì hq w d od píph ydohxu hq w . 5 gdqv ohv 5 frqjxudwlrqv
vxlydqwhv =
Kw.4 k
Kw.4 k
hw +,
+,
Kw.5 k
Kw.5 k
Gdqv oh vfìqdulr +-,/ oh sul{ hq w . 5 g*xq }ìur0frxsrq g*ìfkìdqfh W hvw/ hq irqfwlrq gh o*lqirupdwlrq
glvsrqleoh hq w =
k+W w 4,
E+w> W ,
k +W w 5,=
E+w . 5> W , @
E+w> W . 5,
k+4,
435
doruv txh gdqv oh vfhqdulr +--, =
E+w . 5> W , @
k +W w 4,
E+w> W ,
k+W w 5,
E+w> W . 5,
k +4,
Hq ìjdolvdqw ohv 5 h{suhvvlrqv/ rq rewlhqw =
k+W w 4, k +4,
k+W w 5,
@ =
k +W w 5,
k +W w 4, k+4,
vrlw =
k+x 4, k+4,
k+x,
@ =
> ;x4
k +x,
k +x 4, k +4,
+;143,
H{sulpìh hq irqfwlrq ghv yrodwlolwìv ghv wdx{ ã whuph lqvwdqwdqìv/ fhwwh frqglwlrq vljqlh vlpsohphqw txh
+w> W , hvw frqvwdqwh =
Sursrvlwlrq 44 O*k|srwkëvh g*lqgìshqgdqfh gx fkhplq vxlyl hvw ìtxlydohqwh ã o*k|srwkëvh gh yrodwlolwì
frqvwdqwh +vrxv od suredelolwì ulvtxh0qhxwuh, ghv wdx{ ã whuph lqvwdqwdqìv1
Fhwwh sursrvlwlrq shuphw dlqvl gh grqqhu xq frqwhqx qdqflhu ã o*k|srwkëvh g*lqgìshqgdqfh gx fkhplq
vxlyl1 Xwlolvdqw ohv irupxohv gh sdvvdjh hqwuh ohv ghx{ sdudpìwudjhv +;1<,/ rq ghgxlw oh uìvxowdw vxlydqw =
Sursrvlwlrq 45 Vrxv ohv ghx{ k|srwkëvhv suìfìghqwhv 4 hw 5/ ohv irqfwlrqv k+=, hw k +=, rqw od iruph
h{solflwh vxlydqwh =
4
4
k+x, @
k +x, @
+;144,
x
4 . x
. +4 ,
dyhf
s k+4,
@ @ h +4,
k +4,
rü hvw od yrodwlolwì +frqvwdqwh, ghv wdx{ ã whuph lqvwdqwdqìv vrxv od suredelolwì ulvtxh0qhxwuh1
Rq yrlw grqf txh o*k|srwkëvh g*devhqfh g*rssruwxqlwì g*duelwudjh frpelqìh ã o*k|srwkëvh g*lqgìshqgdqfh
gx fkhplq vxlyl frqgxlw ã ghv irupxodwlrqv h{solflwhv srxu k hw k hq irqfwlrq gh 5 sdudpëwuhv lqfrqqxv = hw +rx , tx*lo hvw grqf qìfhvvdluh g*hvwlphu1 Xqh jìqìudolvdwlrq qdwxuhooh gh fh prgëoh hvw gh frqvlgìuhu
gìvrupdlv txh od yrodwlolwì +w> W , hvw qrq soxv frqvwdqwh pdlv irqfwlrq gh W w1
;17 Ohv h{whqvlrqv gx prgëoh gh Kr hw Ohh
;1714 Oh prgëoh ã yrodwlolwì h{srqhqwlhooh
Fh prgëoh sursrvh gh uhpsodfhu o*k|srwkëvh g*lqgìshqgdqfh gx fkhplq vxlyl +l1h gh yrodwlolwì frqvwdqwh,
sdu o*k|srwkëvh vxlydqwh =
K|srwkëvh 6 Od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv hvw gh od iruph =
+w> W , @ +W w, @ h+W w,
Fh fdguh frqwlhqw frpph fdv sduwlfxolhu oh prgëoh gh Kr hw Ohh oruvtx*rq suhqg @ 31 Xwlolvdqw ã
qrxyhdx ohv irupxohv gh sdvvdjh +;1<,/ oh prgëoh v*ìfulw grqf =
Sursrvlwlrq 46 Vrxv ohv k|srwkëvhv 4 hw 6/ ohv irqfwlrqv k hw k vrqw grqqìhv sdu =
;
4
k+x, @
A
A
.+4,h{s^ s+4, 4h4h+Ww, `
A
?
A
A
A
= k +x, @
4
4.h{s^ s+4, 4h4h+Ww, `
Oh prgëoh hvw grqf sdudpìwuì sdu / hw 1
Hfroh Sro|whfkqltxh
436
Rq yrlw grqf txh fh prgëoh lqwurgxlw xq ghjuì gh olehuwì vxssoìphqwdluh ã wudyhuv oh sdudpëwuh 1
Rq shxw wrxwhirlv vh ghpdqghu srxutxrl xqh iruph h{srqhqwlhooh d ìwì fkrlvlh1 Lo h{lvwh dx prlqv ghx{
fdudfwìulvdwlrqv ìtxlydohqwhv gh fh prgëoh txl phwwhqw hq ìylghqfh vrq lqwìuíw1
Od suhplëuh mxvwlfdwlrq srvvleoh frqvlvwh ã gluh txh oh prgëoh ã yrodwlolwì h{srqhqwlhooh hvw oh vhxo prgëoh
gh w|sh elqrpldo srxu ohtxho lo h{lvwh xqh yduldeoh g*ìwdw lghqwldeoh ã xq wdx{1 Fhfl vljqlh txh wrxv ohv
wdx{ ã od gdwh w qh gìshqghqw txh g*xqh vhxoh yduldeoh doìdwrluh tx*rq shxw lghqwlhu ã xq wdx{ sduwlfxolhu =
Sursrvlwlrq 47 Oh prgëoh hvw ã yrodwlolwì h{srqhqwlhooh vl hw vhxohphqw vl wrxv ohv wdx{ vrqw irqfwlrqv
olqìdluhv g*xqh yduldeoh g*ìwdw1
Suhxyh = Sduwrqv gh od g|qdpltxh gh i +w> W , grqqìh gdqv o*ìtxdwlrq +;19, =
i +w> W , @ i +3> W , .
w
[
n@4
Oq
Kn +W n,
Kn +W . 4 n,
i +w> W , dssdudñw frpph od vrpph g*xqh vxlwh gh shuwxuedwlrqv gh Ehuqrxool1 Lqwurgxlvrqv od vxlwh gh
yduldeohv doìdwrluhv lqgìshqgdqwhv +n , rü n suhqg ohv ghx{ ydohxuv =
; t
4 dyhf suredelolwì A
A
?
t
A
A
=
4
dyhf suredelolwì 40
O*doìd n lqgltxh grqf vl ã od gdwh n/ Kn d sulv od ydohxu k rx k > sdu dloohxuv/ vd gìqlwlrq sduwlfxolëuh
dvvxuh txh +n , hvw xqh vxlwh gh yduldeohv doìdwrluhv lqgìshqgdqwhv fhqwuìhv hw uìgxlwhv +vrxv ,1 Lo v*djlw
grqf g*xq euxlw eodqf1 Rq shxw doruv h{sulphu ohv shuwxuedwlrqv gh od g|qdpltxh gh i+w> W , hq irqfwlrq
gh +n ,1 Soxv suìflvìphqw/ rq ìfulud =
Oq
Kn +W n,
@ +W n,n . e+W n,
Kn +W . 4 n,
rü od irqfwlrq e shxw vh fdofxohu hq ìjdolvdqw ohv hvsìudqfhv/ gh whooh vruwh tx*rq rewlhqw =
i +w> W , @ i +3> W , .
w
[
n@4
+W n,n .
w
[
n@4
e+W n,
Vrxv fhwwh iruph/ rq shxw gìprqwuhu od sursrvlwlrq1 Vxssrvrqv g*derug txh ohv yrodwlolwìv vrlhqw h{sr0
qhqwlhoohv/ rq d doruv =
i +w> W , @ i +3> W , . h+W w,
w
[
n@4
h+wn, %n .
w
[
n@4
e+W n,
+;145,
Idlvdqw W @ w/ rq rewlhqw ìjdohphqw o*ìtxdwlrq gx wdx{ frxuw =
u+w, @ i +3> w, . w
[
n@4
h+wn, %n .
w
[
n@4
e+w n,
+;146,
Holplqdqw od sduwlh doìdwrluh hqwuh ohv ghx{ ìtxdwlrqv/ rq rewlhqw =
i +w> W , @ h+W w, uw . e+W w,
Wrxv ohv wdx{ ã whuph hw wrxv ohv wdx{ }ìur0frxsrq vrqw grqf ghv irqfwlrqv +olqìdluhv, g*xqh vhxoh yduldeoh
doìdwrluh sulvh lfl ìjdoh dx wdx{ frxuw1
Lqyhuvhphqw/ fkhufkrqv txhooh iruph gh yrodwlolwì shuphw g*dyrlu xq prgëoh ã xqh yduldeoh g*ìwdw lghqwl0
deoh ã xq wdx{ +sdu h{hpsoh oh wdx{ frxuw,1 Lo idxw wudgxluh =
i +w> W , @ I +w> W> uw ,
437
rü I hvw xqh irqfwlrq gìwhuplqlvwh1 Rq prqwuh g*derug txh qìfhvvdluhphqw od irqfwlrq I hvw olqìdluh1 Hq
hhw/ i +w> W , hw uw v*ìfulyhqw vrxv od iruph vxlydqwh +dyhf ghv qrwdwlrqv ìylghqwhv, =
;
Sw
? i +w> W , @
n@4 +W n,%n . +W w,
+;147,
Sw
=
@
uw
n@4 +w n,%n . +3,
Rq shxw lqyhuvhu od uhsuìvhqwdwlrq gh uw gh whooh vruwh txh %w v*h{sulph hq irqfwlrq gh uw > uw4 > = = = > u4 1
Dlqvl i +w> W , shxw oxl0píph v*ìfuluh hq irqfwlrq gh uw > uw4 > = = = > u4 / f*hvw0ã0gluh +ohv sduwlhv gìwhuplqlvwhv
vrqw rplvhv dq gh vlpsolhu ohv ìfulwxuhv, =
i +w> W , &
w
[
n@4
+W n,un
Qìfhvvdluhphqw/ od irqfwlrq I hvw olqìdluh hq uw 1 Hqq srxu txh i +w> W , v*ìfulyh =
i +w> W , & +W w,uw
lo idxw txh/ g*dsuëv +;147, =
+W n, @ +W w,+w n,> ; n w W
Idlvdqw x @ w n/ W @ w . 4/ rq d =
+x . 4, @ +4,+x,> ; x 3
fh txl lpsoltxh txh od txdqwlwì +x, hvw h{srqhqwlhooh hq x1 Dlqvl/ vl rq vrxkdlwh dyrlu xq prgëoh gh
wdx{ gìshqgdqw g*xqh yduldeoh g*ìwdw lghqwldeoh ã xq wdx{/ doruv qìfhvvdluhphqw ohv yrodwlolwìv vrqw
h{srqhqwlhoohv1 Uhpdutxrqv ìjdohphqw txh od olqìdulwì ghv wdx{ ã whuph hq irqfwlrq gx wdx{ frxuw shuphw/
hq lqyhuvdqw od irupxoh/ g*lghqwlhu od yduldeoh g*ìwdw sduwlfxolëuh sulvh lfl +hq o*rffxuuhqfh oh wdx{ frxuw
uw , ã wrxw dxwuh wdx{ rx píph ã xqh frpelqdlvrq gh wdx{1 Oh idlw gh fkrlvlu oh wdx{ frxuw frpph yduldeoh
g*ìwdw q*hvw grqf sdv lfl uhvwulfwli1 Hqq/ fh prgëoh hvw xq gìexw gh mxvwlfdwlrq dx{ uhsuìvhqwdwlrqv
idfwrulhoohv olqìdluhv ìwxglìhv dx fkdslwuh suìfìghqw > lo hvw gh soxv ehdxfrxs soxv suìflv sxlvtxh qrxv
dyrqv prqwuì txh/ srxu íwuh frpsdwleoh dyhf o*devhqfh g*rssruwxqlwìv g*duelwudjh/ ohv vhqvlelolwìv dx{
idfwhxuv g*xqh uhsuìvhqwdwlrq idfwrulhooh olqìdluh vxu ohv wdx{ ghydlhqw dyrlu xqh iruph sduwlfxolëuh +gh
w|sh h{srqhqwlho,1
Xqh vhfrqgh fdudfwìulvdwlrq gh fh prgëoh d wudlw ã od g|qdpltxh sduwlfxolëuh ghv wdx{1 Hq hhw/ o*ìtxdwlrq
gh od g|qdpltxh gh +uw , grqqìh sdu o*ìtxdwlrq +;146,/ rq yrlw txh =
uw @ h uw4 . w . w
rü w hvw xqh irqfwlrq gìwhuplqlvwh gh w1 Rq frqvwdwh grqf txh/ vrxv od suredelolwì ulvtxh0qhxwuh/ +uw , hvw
xq surfhvvxv dxwruìjuhvvli g*ruguh 4 +DU+4,,1 Vrxv fhwwh iruph/ rq lqwhusuëwh frpph xq sdudpëwuh gh
uhwrxu ã od pr|hqqh1 Lqyhuvhphqw/ vl +uw , vxlw xq DU+4,/ doruv ohv yrodwlolwìv +W w, vrqw qìfhvvdluhphqw
gh w|sh h{srqhqwlho1 Gh soxv/ frpph wrxv ohv wdx{ vrqw irqfwlrqv olqìdluhv gh uw / lov vxlyhqw ìjdohphqw
ghv surfhvvxv DU+4,1 Gdqv oh fdv sduwlfxolhu gx prgëoh gh Kr hw Ohh + @ 3,/ +uw , hvw grqf xqh pdufkh
doìdwrluh/ grqf vdqv uhwrxu ã od pr|hqqh1
Hqq/ lo idxw uhpdutxhu txh +uw , q*d sdv od píph g|qdpltxh vrxv od yudlh suredelolwì s sxlvtxh +w , q*hvw
soxv fhqwuìh vrxv s1 Srxu uìvxphu/ rq shxw ìqrqfhu od sursrvlwlrq vxlydqwh =
Sursrvlwlrq 48 Vrxv o*k|srwkëvh 4/ oh prgëoh ã yrodwlolwì h{srqhqwlho shxw íwuh gìql sdu ohv 6 fdudfwì0
ulvdwlrqv vxlydqwhv ìtxlydohqwhv =
41 od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv hvw gh od iruph = +w> W , @ h+W w,
51 ohv wdx{ ã od gdwh w vrqw ghv irqfwlrqv olqìdluhv g*xq wdx{ sduwlfxolhu +sdu h{hpsoh/ oh wdx{ frxuw, =
i+w> W , @ h+W w, uw . e+W w,
61 oh wdx{ frxuw +dlqvl txh wrxv ohv dxwuhv wdx{, hvw xq DU+4, vrxv od suredelolwì ulvtxh0qhxwuh = uw @
w . h uw4 . w 1
Hfroh Sro|whfkqltxh
438
Rq yrlw tx*ã fh vwdgh hw ã od oxplëuh ghv glìuhqwhv fdudfwìulvdwlrqv sursrvìhv/ rq shxw hqylvdjhu gh
qrpeuhxvhv h{whqvlrqv srvvleohv =
dxjphqwhu oh qrpeuh g*doìd/
vxssrvhu txh +uw , vxlw xq DU+t, dyhf t 5/
sursrvhu g*dxwuhv irqfwlrqv gh yrodwlolwì/
uhoåfkhu o*k|srwkëvh gh edvh hq vxssrvdqw txh gìshqg gx whpsv/ +w> W , irqfwlrq gh w hw W +hw qrq
soxv gh od pdwxulwì W w,111
Qrxv gìyhorssrqv pdlqwhqdqw txhotxhv xqhv gh fhv slvwhv1
;1715 Prgëoh ã soxvlhxuv doìd
Rq ìwxglh lfl oh fdv gh soxvlhxuv doìd/ fh txl dyhf qrv qrwdwlrqv grqqh =
Oq
Kw +W w,
@ 4 +W w,%4w . 5 +W w,%5w . e+w> W ,
Kw +W . 4 w,
rü +%4w > 5w , vrqw ghx{ shuwxuedwlrqv lqgìshqgdqwhv +vrxv od suredelolwì ulvtxh0qhxwuh, gh orl gh Ehuqrxool
gh sdudpëwuhv uhvshfwliv 4 hw 5 1 Lo | d grqf 7 uìdolvdwlrqv srvvleohv gh od irqfwlrq Kw 1 Rq shxw doruv
lpsrvhu ghv irqfwlrqv gh yrodwlolwì gx w|sh h{srqhqwlho =
;
? 4 +W w, @ 4 h4 +W w,
=
5 +W w, @ 5 h5 +W w,
D wlwuh g*h{hpsoh/ Khdwk/ Mduurz hw Pruwrq +4<<3, sursrvh gh suhqguh 4 @ 3 hw 5 A 31 Rq rewlhqw
dlqvl xq prgëoh ã ghx{ idfwhxuv1 Hq vxlydqw od píph gìprqvwudwlrq txh suìfìghpphqw/ rq frqvwdwh txh
ohv wdx{ vrqw ghv irqfwlrqv olqìdluhv gh ghx{ wdx{ sduwlfxolhuv/ tx*rq shxw lghqwlhu/ sdu h{hpsoh dx wdx{
frxuw hw ã xq wdx{ orqj1
;1716 Prgëoh gh Eodfn/ Ghupdq hw Wr|
Fh prgëoh suhqg xq srlqw gh yxh xq shx glìuhqw1 Lo frqvlgëuh wrxmrxuv txh hvw frqvwdqw dx frxuv gx
whpsv pdlv od irqfwlrq gh yrodwlolwì gìshqg gh w hw gh W hw sdv vhxohphqw gh W w1 Gh soxv/ lo qh fkhufkh
sdv g*hpeoìh ã lpsrvhu xqh iruph irqfwlrqqhooh suìflvh srxu fhwwh irqfwlrq gh yrodwlolwì1 Soxv suìflvìphqw/
fh prgëoh vxssrvh wurlv fkrvhv =
od frxueh ghv wdx{ ã od gdwh lqlwldoh 3 hvw frqqxh +k|srwkëvh gìmã idlwh mxvtx*ã suìvhqw,/
od frxueh ghv yrodwlolwìv lqlwldohv hvw ìjdohphqw frqqxh +od qrwlrq gh frxueh gh yrodwlolwìv hvw suìflvìh
fl0dsuëv,/
lo | d lqgìshqgdqfh gx fkhplq vxlyl1
Wudgxlvrqv g*derug o*k|srwkëvh g*lqgìshqgdqfh gx fkhplq vxlyl gdqv fh frqwh{wh1 Hq uhpdutxdqw txh ohv
irqfwlrqv k hw k gìshqghqw gh w/ rq d =
kw4 +x . 4, kw4 +4,
kw +x,
@
> ;x 4
kw +x,
kw4 +x . 4, kw4 +4,
Hq whuph gh yrodwlolwì/ fhwwh ìtxdwlrq v*ìfulw =
+;148,
+w> W , @ +w 4> W ,
hw grqf =
+w> W , @ +3> W ,
Sdu frqvìtxhqw/ od grqqìh gh od frxueh ghv yrodwlolwìv lqlwldohv/ +i +3> W ,,W shuphw grqf gh uhfrqvwlwxhu
wrxwhv ohv yrodwlolwìv xowìulhxuhv hw grqf g*h{solflwhu ohv irqfwlrqv kw +=, hw kw +=,1
Xqh dxwuh dssurfkh frqvlvwh ã vh grqqhu od yrodwlolwì gx wdx{ frxuw +w> w . 4, srxu wrxw w1 Hq hhw/ rq
shxw wrxmrxuv gìurxohu od uìfxuuhqfh gh o*ìtxdwlrq +;148, srxu rewhqlu =
x\
4 k +4,
kw +x,
w.n
@
kw +x, n@3 kw.n +4,
439
+;149,
vrlw/ hq whuph gh yrodwlolwì =
+w> W , @ +W> W . 4,
Sdu frqvìtxhqw/ vl rq vh grqqh ã od gdwh 3 od irqfwlrq +w> w . 4, srxu wrxw w/ rq d doruv wrxwhv ohv irqfwlrqv
gh yrodwlolwì1 Fhv ghx{ dssurfkhv vrqw grqf wuëv vrxsohv sxlvtx*rq qh frqwudlqw sdv ohv irqfwlrqv gh
yrodwlolwì ã dyrlu xqh iruph sduwlfxolëuh > hq uhydqfkh/ hooh qìfhvvlwh ehdxfrxs g*lqirupdwlrq qrwdpphqw
vxu ohv yrodwlolwìv lqlwldohv1
;18 Dssolfdwlrqv ã od ydorulvdwlrq ghv elhqv frqwlqjhqwv
Mxvtx*ã suìvhqw/ qrxv qrxv vrpphv lqwìuhvvìv dx{ gìirupdwlrqv gh od vwuxfwxuh ghv wdx{ hw sulqflsdoh0
phqw dx{ frqwudlqwhv lpsrvìhv sdu o*devhqfh g*rssruwxqlwìv g*duelwudjh vxu fhv gìirupdwlrqv1 Qrxv doorqv
pdlqwhqdqw qrxv frqfhqwuhu vxu o*dssolfdwlrq ã o*ìydoxdwlrq ghv elhqv frqwlqjhqwv1 Hq hhw/ od sulqflsdoh
dssolfdwlrq ghv prgëohv gìfulwv dx{ sdudjudskhv suìfìghqwv hvw gh grqqhu xq sul{ +g*duelwudjh, dx{ elhqv
frqwlqjhqwv gìshqgdqw gh od vwuxfwxuh ghv wdx{ hw gh vhv ìyroxwlrqv1 Srxu vlpsolhu qrxv qrxv olplwrqv
dx fdv rü o*k|srwkëvh 4 hvw vdwlvidlwh1
;1814 Sulqflsh g*ìydoxdwlrq
Lo v*djlw g*xqh ìydoxdwlrq sdu duelwudjh/ hw grqf wuëv vlplodluh dx{ udlvrqqhphqwv gìmã whqxv/ qrwdpphqw
srxu idluh dssdudñwuh od suredelolwì ulvtxh0qhxwuh1 Frqvlgìurqv xq dfwli grqw oh surfhvvxv gh sul{ vhud qrwì
+Fw , who txh Fw vrlw xqh irqfwlrq frqqxh gh K4 +=,> ===> Kw +=, =
Fw @ Iw ^K4 > ===> Kw `
Dlqvl Fw hvw frqqx ã sduwlu gh od gdwh w1 Frqvwlwxrqv xq sruwhihxlooh/ ã od gdwh w frpsuhqdqw =
o*dfwli Fw +hq txdqwlwì {,
xq }ìur0frxsrq g*ìfkìdqfh W
Oh sul{ ã od gdwh w gh fh sruwhihxlooh hvw =
Yw @ {Fw . E+w> W ,
Hq w . 4/ fh sruwhihxlooh qh shxw suhqguh txh ghx{ ydohxuv =
;
E+w>W ,
A
? {Iw.4 +K4 > ===> Kw > Kw.4 @ k, . E+w>w.4, k+W w 4,
Yw.4 @
A
E+w>W ,
= {I +K > ===> K > K
w.4 4
w w.4 @ k , . E+w>w.4, k +W w 4,
Rq shxw grqf fkrlvlu ã od gdwh w/ { gh whooh vruwh txh ohv 5 ydohxuv vrlhqw ìjdohv1 Lo idxw srxu fhod frqqdñwuh
ã od gdwh w = od irqfwlrq Iw.4 +pdlv qrxv o*dyrqv vxssrvì,/ ohv irqfwlrqv k hw k hw od vwuxfwxuh ghv wdx{
E+w> =,1 Rq rewlhqw grqf xq sruwhihxlooh vdqv ulvtxh sxlvtxh vd ydohxu hq w . 4 hvw frqqxh txhotxh vrlw
o*doìd vxuyhqdqw hqwuh w hw w . 41 Sdu devhqfh g*rssruwxqlwì g*duelwudjh/ vrq uhqghphqw grlw íwuh ìjdo dx
uhqghphqw xqh sìulrgh =
4
Yw.4
@
Yw
E+w> w . 4,
fh txl frqgxlw ã =
Fw @ E+w> w . 4,= ^=Iw.4 +K4 ===Kw > Kw.4 @ k, . +4 ,Iw.4 +K4 ===Kw > Kw.4 @ k ,`
hq xwlolvdqw =
@
4 k +=,
k+=, k +=,
Sdu frqvìtxhqw/ rq d od uhodwlrq irqgdphqwdoh g*ìydoxdwlrq =
Fw @ E+w> w . 4,H +Fw.4 @ Iw ,
Hfroh Sro|whfkqltxh
43:
Oh sul{ g*xq elhq frqwlqjhqw hvw grqf vd ydohxu hvsìuìh +vrxv ,/ dfwxdolvìh1 Sdu uìfxuuhqfh =
Fv
Fw @ w H @ Iw
v
dyhf
w @
w\
4
+;14:,
4
E+n> n . 4,
n@3
Frqqdlvvdqw o*hqvhpeoh ghv sdudpëwuhv txl jrxyhuqhqw oh prgëoh/ rq shxw grqqhu xq sul{ ã wrxw dfwli
frqwlqjhqw srxu shx txh vrq sul{ hq w vrlw xqh irqfwlrq frqqxh ghv doìd vxuyhqxv mxvtx*hq w1 Dlqvl xqh
rswlrq vxu xq }ìur0frxsrq/ g*ìfkìdqfh w hvw xq dfwli txl grqqh gurlw hq v ã =
Ê+v> W , N`.
rü N hvw oh sul{ g*h{huflfh gh o*rswlrq1
Oh sul{ hq v gh o*rswlrq hvw elhq xqh irqfwlrq gìwhuplqlvwh ghv doìd K4 > ===> Kv hw grqf oh sul{ hq w gh fhwwh
rswlrq hvw =
%
&
Ê+v> W , N`.
@ Iw
F+w> v> W> N, @ w =H
v
Dydqw gh grqqhu txhotxhv dssolfdwlrqv/ lo hvw xwloh g*lqwurgxluh g*dxwuhv suredelolwìv txl shuphwwhqw
g*h{sulphu soxv vlpsohphqw ohv irupxohv g*ìydoxdwlrq1
;1815 Dxwuhv suredelolwìv 0 Suredelolwìv iruzdug0qhxwuhv
Txdqg od suredelolwì ulvtxh0qhxwuh +, d ìwì frqvwuxlwh/ qrxv dyrqv uhpdutxì tx*hooh uhqgdlw pduwlqjdoh oh
surfhvvxv ghv sul{ dfwxdolvìv +sdu oh wdx{ frxuw, > fhwwh sursulìwì lqflwh/ hq uhqyhuvdqw o*dujxphqw/ ã fkhu0
fkhu g*dxwuhv suredelolwìv uhqgdqw pduwlqjdoh g*dxwuhv surfhvvxv txh ohv sul{ dfwxdolvìv1 Fhv suredelolwìv
vrqw dsshoìv iruzdug0qhxwuhv > hoohv uhqghqw pduwlqjdohv ohv surfhvvxv =
E+w> W ,
E+w> p,
rü p hvw xqh gdwh txhofrqtxh grqqìh1 Lqwxlwlyhphqw/ fhfl uhylhqw ã fkdqjhu gh srlqw gh yxh = dx olhx
g*xwlolvhu oh qxpìudluh gh od gdwh 3 +frpph f*hvw oh fdv hq suhqdqw w ,/ rq frqvlgëuh oh qxpìudluh gh od
gdwh p1
Srxu idluh dssdudñwuh fhv qrxyhoohv suredelolwìv/ qrwìhv ip / sduwrqv gh o*ìtxdwlrq gìvluìh =
i E+v> W ,
E+w> W ,
p
@ Iw @
> ; w ? v ? p> W
H
E+v> p,
E+w> p,
hw gh od sursulìwì fruuhvsrqgdqwh vxu =
E+v> W ,
E+w> W ,
H
@ Iw @
v
w
Rq shxw o*ìfuluh =
H
E+v> W , E+v> p,
=
@ Iw
E+v> p,
v
@
E+w> W ,
w
+;14;,
Dlqvl dssdudñw qdwxuhoohphqw/ od txdqwlwì =
Hv>p @
E+v> p,
v
txl v*lqwhusuëwh frpph oh udssruw gx sul{ hq v g*4I gh od gdwh p hw oh sul{ +soxv h{dfwhphqw gh od ydohxu
dftxlvh, hq v g*4I gh od gdwh 31 F*hvw hq fh vhqv tx*rq sduoh gh fkdqjhphqw gh qxpìudluh1 Fh surfhvvxv
43;
+Hv>p ,v hvw pduwlqjdoh vrxv =
@
E+v> p,
@ Iw
v
E+w> p,
@ Hw>p
w
H +Hv>p @ Iw , @ H Rq shxw uììfuluh +;14;, vrxv od iruph =
E+v> W ,
E+w> W ,
=Hv>p @ Iw @
=Hw>p
H
E+v> p,
E+w> p,
rx =
H
E+v> W , Hv>p
=
@ Iw
E+v> p, Hw>p
E+w> W ,
E+w> p,
Gìqlvvrqv doruv o*hvsìudqfh sdu udssruw ã od suredelolwì ip sdu =
i
Hv>p
H p ^[v @ Iw ` @ H [v =
@ Iw
Hw>p
srxu wrxwh yduldeoh doìdwrluh [v / hw rq d od iruph fkhufkìh =
i
E+v> W ,
E+w> W ,
@ Iw @
H p
E+v> p,
E+w> p,
i
ã frqglwlrq txh ip sxlvvh v*lqwhusuìwhu frpph xqh suredelolwì +srxu txh H p vrlw xqh yìulwdeoh hvsì0
udqfh,1 F*hvw hhfwlyhphqw oh fdv sxlvtxh HHv>p
w>p hvw srvlwli hw txh =
i
H p +4 @ Iw , @ H Hv>p
@ Iw
Hw>p
@4
Rq d gh soxv od irupxoh gh sdvvdjh =
i
Hv>p
p
@ Iw
H ^[v @ Iw ` @ H [v
Hw>p
E+v> p, w
=
@ Iw
@ H [v =
E+w> p, v
+;14<,
Idlvdqw v @ w . 4/ rq rewlhqw =
i
H p ^[w.4 @ Iw ` @ H ^[w.4 Kw.4 +p w 4, @ Iw `
gh whooh vruwh txh/ iruphoohphqw ohv hvsìudqfhv vrxv ip v*xwlolvhqw frpph o*hvsìudqfh vrxv hq uhpsodêdqw
gdqv ohv fdofxov oh whuph sdu ip>w.4 dyhf =
ip>w.4 @ k+p w 4,
Rq uhpdutxh txh ohv suredelolwìv iruzdug0qhxwuh gìshqghqw qìfhvvdluhphqw gx whpsv w1
;19 Sdvvdjh ã od olplwh gdqv ohv prgëohv glvfuhwv
Lo hvw qdwxuho gh fkhufkhu od olplwh ghv prgëohv glvfuhwv suìvhqwìv lfl oruvtxh oh sdv gh whpsv whqg yhuv 31
Hq hhw/ vl rq suhqg o*h{hpsoh gx prgëoh elqrpldo gh Fr{/ Urvv hw Uxelqvwhlq srxu ìydoxhu ohv rswlrqv
vxu dfwlrq/ fhoxl0fl frqyhujh +srxu xq fkrl{ dgìtxdw ghv sdudpëwuhv, yhuv oh prgëoh gh Eodfn0Vfkrohv1
Qrxv yhuurqv lfl txh ohv prgëohv elqrpldx{ gh od vwuxfwxuh ghv wdx{ ìwxglìv gdqv fh fkdslwuh frqyhujhqw
yhuv fhuwdlqv prgëohv gìwdlooìv dx fkdslwuh vxlydqw1
Hfroh Sro|whfkqltxh
43<
Srxu vlpsolhu/ qrxv qrxv sodêrqv gdqv oh fdv rü od suredelolwì ulvtxh0qhxwuh qh gìshqg sdv gh w hw
rü od irqfwlrq gh yrodwlolwì qh gìshqg txh gh od pdwxulwì W w1 Srvrqv oh sdv gh whpsv ìoìphqwdluh1
Mxvtx*ã pdlqwhqdqw/ ìwdlw sulv frqyhqwlrqqhoohphqw sulv ìjdo ã 41 Qrxv doorqv grqf uììfuluh oh prgëoh
srxu txhofrqtxh1 grqqhu Srxu fhod qrxv sduwrqv ghv wdx{ iruzdug gìmã gìqlv gdqv ohv dxwuhv vhfwlrqv
hq lqwurgxlvdqw h{solflwhphqw fh sdv gh whpsv1 Qrxv vxssrvrqv sdu dloohxuv txh wrxwhv ohv frqglwlrqv gh
uìjxodulwì qìfhvvdluhv vrqw vdwlvidlwhv1 O*ìtxdwlrq gh edvh v*ìfulw gìvrupdlv =
E++q 4,> p,
Kq ++p q,,
E++q 4,> q,
E+q> p, @
+;153,
srxu q hw p ghx{ hqwlhuv whov txh q p1
Srxu w hw W ghx{ uìhov srvlwliv whov txh w W / qrxv fkrlvlurqv q hw p gh whooh vruwh txh =
;
? olp$3 q @ w
=
olp$3 p @ W
Hq gìurxodqw od uìfxuuhqfh +;153,/ rq rewlhqw =
q
E+3> +p . 4,, \ Kn ++p . 4 n,,
E+q> +p . 4,,
@
E+q> p,
E+3> p,
n@4 Kn ++p n,,
+;154,
Od gìqlwlrq qdwxuhooh gx wdx{ ã whuph lqvwdqwdqì ghylhqw =
i +q> p, @ E+q> +p . 4,,
4
Oq
E+q> p,
gh whooh vruwh txh srxu whqgdqw yhuv 3/ rq dlw =
i +q,> p, @ olp
$3
gOq E+w> W ,
gW
txl hvw elhq oh wdx{ ã whuph lqvwdqwdqì gx whpsv frqwlqx1
G*dsuëv o*ìtxdwlrq +;154,/ i+q> p, v*ìfulw =
i +q> p, @ i +3> +p . 4,, .
q
Kn ++p n,,
4 [
Oq
Kn ++p . 4 n,,
n@4
+;155,
Oh suhplhu whuph frqyhujh yhuv i+3> W , hw lo uhvwh grqf ã ìwxglhu od frqyhujhqfh gx vhfrqg1 Qrxv dgrswrqv
oh sdudpìwudjh hq irqfwlrq ghv yrodwlolwìv fdu f*hvw oh sdudpìwudjh qdwxuho ghv prgëohv hq whpsv frqwlqx1
Od yrodwlolwì gx wdx{ ã whuph lqvwdqwdqì v*ìfulw =
Y du+i+q> p, @ I+q4, , @ +4 ,
Oq
4
5
k ++p q,,
k++p q,,
Oq k++p . 4 q,,
k ++p . 4 q,,
5
@ ++p q,,5 Rq shxw doruv uhsdudpìwuhu ohv irqfwlrqv k+=, hw k +=, hq irqfwlrq gh od yrodwlolwì srxu rewhqlu frpph
suìfìghpphqw =
;
4
k+x, @
A
S x4 6@5
4
A
s
.+4
,
h{s
+v,
A
v
@3 A
+4
,
A
?
k +x, @
A
A
A
A
A
=
443
4
4 6@5+v,
+4,.h{s s+44 , S xv@3
Oh vhfrqg whuph gh o*ìtxdwlrq +;155, v*ìfulw grqf hq irqfwlrq ghv =
q
q
4 [
4 [
s
++p n,,n .
e++p n,,
n@4
n@4
rü od irqfwlrq e+=, hvw ìjdoh ã =
e+x, @ Oq
k +x,
k+x,
. +4 ,Oq k++x . 4,,
k ++x . 4,,
Rq d ã qrxyhdx ghx{ whuphv/ o*xq vwrfkdvwltxh hw o*dxwuh gìwhuplqlvwh1
Frqyhujhqfh gx whuph gìwhuplqlvwh
Hq xwlolvdqw o*h{suhvvlrq ghv irqfwlrqv k+=, hw k +=, hw hq hhfwxdqw xq gìyhorsshphqw olplwì gh od irqfwlrq
e+=, dx yrlvlqdjh gh @ 3 +dx ghx{lëph ruguh hq 6@5 ,/ rq rewlhqw =
q
p4
p[
q4
4 [
4 [
e++p n,, @ ^+
+v,,5 +
+v,,5 `5 . r+5 ,
5 v@3
v@3
n@4
txl frqyhujh grqf yhuv =
] W
] W w
4
^+
+v,gv,5 +
+v,gv,5 `
5 3
3
Frqyhujhqfh gx whuph vwrfkdvwltxh
Fh whuph v*ìfulw =
q
4 [
s
++p n,,%n
n@4
Hq xwlolvdqw oh wkìruëph Fhqwudo Olplwh/ rq gìgxlw txh fhwwh txdqwlwì frqyhujh hq orl yhuv xqh orl qrupdoh
fhqwuìh +vrxv od suredelolwì ulvtxh qhxwuh, gh yduldqfh =
] w
q
4 [
^++p n,,`5 @
+W v,5 gv
$3 n@4
3
olp
Soxv jìqìudohphqw/ hq xwlolvdqw oh wkìruëph Fhqwudo Olplwh Irqfwlrqqho/ rq dxudlw =
] w
q
4 [
s
^++p n,,`%n @,$3
+W v,gZv
n@4
3
rü +Zw , hvw xq prxyhphqw eurzqlhq vwdqgdug vrxv od suredelolwì ulvtxh qhxwuh1
Dx wrwdo/ rq d =
] W
] W w
] w
4
5
5
+v,gv, +
+v,gv, ` .
+W v,gZv
i+w> W , @ i+3> W , . ^+
5 3
3
3
Soxv jìqìudohphqw/ vl od yrodwlolwì gìshqgdqw h{solflwhphqw gx whpsv/ rq dxudlw =
i +w> W , @ i+3> W , .
] w
3
] W
] w
+v> W ,+
+v> x,gx,gv .
+v> W ,gZv
v
3
+;156,
Sdu h{hpsoh/ gdqv oh fdv gx prgëoh gh Kr hw Ohh rü +w> W , @ / rq rewlhqw =
i +w> W , @ i+3> W , . 5 +W w Hfroh Sro|whfkqltxh
w5
, . Zw =
5
444
Rq uhpdutxh txh od olplwh gx prgëoh glvfuhw gìshqg gh od irqfwlrq gh edvh +=, pdlv qh gìshqg soxv gh
od suredelolwì 1 Dlqvl oruvtx*rq fkhufkh ã dmxvwhu xq prgëoh gh Kr hw Ohh/ lo | d d sulrul ghx{ sdudpëwuhv
ã hvwlphu/ hw 1 Ru/ gdqv oh prgëoh olplwh/ vhxo lqwhuylhqw1 Fhfl vljqlh txh wrxw surfìgxuh ylvdqw ã
hvwlphu ã od irlv hw surgxlud ghv hvwlpdwhxuv wuëv lqvwdeohv sxlvtxh fhv ghx{ sdudpëwuhv qh vrqw sdv
lghqwldeohv vlpxowdqìphqw gdqv oh prgëoh olplwh1 Hq idlw/ lo vx!w gh {hu xqh ydohxu duelwudluh srxu hw g*hvwlphu vhxohphqw 1 Qìdqprlqv/ od ydohxu gh mrxh vxu od ylwhvvh gh frqyhujhqfh > hq sduwlfxolhu/ vl
rq uhjdugh oh whuph vxlydqw hq 9 gx gìyhorsshphqw gh od irqfwlrq e+=,/ rq frqvwdwh tx*lo hvw plqlpxp
srxu @ 4@51 Fhwwh ydohxu gh hvw grqf rswlpdoh1
Hqq/ qrwrqv txh s q*lqwhuylhqw sdv qrq soxv gdqv od g|qdpltxh ulvtxh0qhxwuh gh i+w> W , hw grqf qrq
soxv gdqv ohv fdofxov gh sul{ sxlvtxh ohv sul{ vrqw ghv hvsìudqfhv sulvhv vrxv od suredelolwì ulvtxh0qhxwuh1
Fhv surfhvvxv0olplwh uìdssdudñwurqw qdwxuhoohphqw dx fkdslwuh vxlydqw1
445
<
Lqwurgxfwlrq dx{ prgëohv hq whpsv frqwlqx1
<14 Lqwurgxfwlrq
Ohv sulqflsdx{ frqfhswv +devhqfh g*rssruwxqlwì g*duelwudjh/ suredelolwì ulvtxh0qhxwuh/111, rqw ìwì derugìv
gdqv oh fdguh ghv prgëohv hq whpsv glvfuhw1 Lo dssdudñw fhshqgdqw ylwh qìfhvvdluh gh wudydloohu gdqv xq
fdguh frqwlqx srxu dx prlqv ghx{ udlvrqv = ohv prgëohv hq whpsv frqwlqx grlyhqw fhuwhv íwuh glvfuìwlvìv
srxu íwuh xwlolvdeohv +ã ghv qv g*hvwlpdwlrq/ rx g*dmxvwhphqw dx{ grqqìhv,/ pdlv suhqguh srxu srlqw gh
gìsduw xq prgëoh hq whpsv glvfuhw lpsrvh gh {hu xqh xqlwì gh whpsv txl shxw qh sdv íwuh vlpxowdqìphqw
dgdswìh ã wrxwhv ohv vlwxdwlrqv gh pdufkì1 Gh soxv/ wudydloohu hq frqwlqx lpsrvh xqh wkìrulh pdwkìpdwltxh
elhq soxv frpsoh{h pdlv surgxlw ghv irupxohv g*ìydoxdwlrq vrxyhqw gluhfwhphqw h{sorlwdeohv +doruv txh/
gdqv ohv prgëohv glvfuhwv/ ghv vlpxodwlrqv vrqw qìfhvvdluhv,/ hw uhqg g*xqh fhuwdlqh idêrq/ ohv fdofxov soxv
wudfwdeohv1
Frpph gdqv wrxwhv ohv wkìrulhv qdqflëuhv/ ghx{ yrlhv rqw ìwì ìwxglìhv1 Od suhplëuh/ qdwxuhooh fkh} ohv
ìfrqrplvwhv/ hvw fhooh gh o*ìtxloleuh jìqìudo txl frqvlvwh ã prgìolvhu ohv suìiìuhqfhv ghv djhqwv +vxssrvìv
udwlrqqhov, frpswh whqx gh ohxu grwdwlrq lqlwldoh hw g*hq gìgxluh ohv sul{ gh wrxv ohv dfwliv1 Fhwwh dssurfkh
joredoh d oh gìidxw gh uhsrvhu/ srxu íwuh rsìudwlrqqhooh/ vxu ghv k|srwkëvhv gl!flohphqw yìuldeohv +frpph
sdu h{hpsoh od iruph ghv irqfwlrqv g*xwlolwì,1 Od vhfrqgh yrlh hvw soxv prghvwh fdu hooh vxssrvh grqqìv
ohv surfhvvxv gh sul{ g*xq fhuwdlq qrpeuh gh wlwuhv h{lvwdqw hw hq gìgxlw txhov grlyhqw íwuh/ srxu ghv
udlvrqv gh frkìuhqfh/ ohv sul{ g*dxwuhv dfwliv qdqflhuv1 Fhwwh uëjoh gh frkìuhqfh hvw hq idlw o*devhqfh
g*rssruwxqlwì g*duelwudjh1 Gh idlw/ rq dsshoohud fhv prgëohv ghv prgëohv g*duelwudjh1 O*dydqwdjh gh fhv
prgëohv hvw g*íwuh ehdxfrxs soxv rsìudwlrqqho hw vhurqw fhx{ ìwxglìv gdqv fh fkdslwuh1
Lo | d hq rxwuh ghx{ prwlydwlrqv gdqv od prgìolvdwlrq ghv gìirupdwlrqv gh od vwuxfwxuh ghv wdx{1 Od
suhplëuh hvw h{solfdwlyh hw frqvlvwh ã uhqguh frpswh gh od vwuxfwxuh ghv wdx{ ã wrxwh gdwh +klìudufklh ghv
wdx{/ iruph gh od frxueh111, > od vhfrqgh uìvlgh gdqv oh sureoëph gh o*ìydoxdwlrq sdu duelwudjh ghv elhqv
frqwlqjhqwv +rswlrqv/ frqwudwv ã whuph/ vzdsv111,1 Ohv suhplhuv prgëohv hq whpsv frqwlqx vrqw dssduxv
dydqw oh prgëoh glvfuhw gh Kr hw Ohh/ yhuv oh plolhx ghv dqqìhv :3/ qrwdpphqw ã od vxlwh ghv wudydx{
gh Eodfn0Vfkrohv vxu o*ìydoxdwlrq ghv sul{ g*rswlrqv vxu dfwlrqv1 Hq hhw/ frpph lo dssdudñw fodluhphqw
gdqv ohv fkdslwuhv suìfìghqwv/ ohv reoljdwlrqv/ hw soxv jìqìudohphqw ohv wlwuhv gìshqgdqw gh od vwuxfwxuh
ghv wdx{ qh shxyhqw íwuh frqvlgìuìv frpph ghv dfwlrqv/ qh vhudlw0fh txh sdufh txh ohv sul{ }ìur0frxsrq
vrqw qrq doìdwrluhv ã ohxu ìfkìdqfh1 Lo hvw doruv qìfhvvdluh gh gìyhorsshu xqh wkìrulh sursuh dx pdufkì
ghv wlwuhv gìshqgdqw gh od vwuxfwxuh ghv wdx{1
Qrxv vxlyurqv xq sodq surfkh gh fhoxl gx fkdslwuh suìfìghqw/ fruuhvsrqgdqw dx fdguh frqvwuxlw sdu
Khdwk/ Mduurz hw Pruwrq 1 Gdqv xq suhplhu whpsv/ qrxv qrxv grqqhurqv xqh vwuxfwxuh suredelolvwh srxu
ohv ìyroxwlrqv srvvleohv gh od frxueh ghv wdx{1 Sxlv qrxv wudgxlurqv o*devhqfh g*rssruwxqlwìv g*duelwudjh
srxu derxwlu dx idlw txh fhv prgëohv vrqw hqwlëuhphqw sdudpìwuìv sdu xqh irqfwlrq gh yrodwlolwì +ghv
wdx{ ã whuph lqvwdqwdqìv/ sdu h{hpsoh,1 Hqq/ ghv k|srwkëvhv dgglwlrqqhoohv vhurqw qìfhvvdluhv srxu
rewhqlu xqh vsìflfdwlrq frpsoëwh ghv prgëohv1 Fhv k|srwkëvhv dgglwlrqqhoohv frqvlvwhurqw/ sdu h{hpsoh/
ã vxssrvhu o*h{lvwhqfh gh yduldeohv g*ìwdw1 Dlqvl wrxv ohv sul{ }ìur0frxsrq ã xqh gdwh grqqìh ghylhqqhqw
irqfwlrqv gh fhv yduldeohv g*ìwdw1 Fhwwh lgìh d xqh wudgxfwlrq qdwxuhooh srxu ohv dssolfdwlrqv qdqflëuhv
sxlvtxh ohv vwudwìjlhv gh frxyhuwxuh vh uìgxlvhqw ã xqh lppxqlvdwlrq ylv ã ylv gh fhv yduldeohv g*ìwdw
vrxv0mdfhqwhv1 Gh idlw ohv qdqflhuv vrxkdlwhqw srxyrlu grqqhu ghv sul{ txl qh gìshqghqw txh g*xq shwlw
qrpeuh gh yduldeohv qdqflëuhv1
Dydqw g*hqwuhu gdqv ohv gìwdlov/ qrxv udsshorqv ohv sulqflsdohv qrwdwlrqv xwlolvìhv hq whpsv frqwlqx1 Frpph
wrxmrxuv/ qrxv qrwhurqv E+w> W , oh sul{ hq w g*4 hxur uhêx hq W dyhf/ elhq0vu E+W> W , @ 4 srxu wrxw W 1
Ohv wdx{ ã whuph lqvwdqwdqìv i+w> W , vhurqw ìjdohphqw vrxyhqw xwlolvìv > lov vrqw gìqlv sdu =
i +w> W , @ g
Oq E+w> W ,
gW
fh txl hvw ìtxlydohqw ã =
E+w> W , @ h{s ] W
w
i+w> v,gv
Oh surfhvvxv gx wdx{ frxuw +uw , hvw gìql sdu =
uw @ i +w> w,=
Od ydohxu dftxlvh hq w g*4 hxur sodfì hq 3 hw uìlqyhvwl frqwlqþphqw dx wdx{ frxuw +uv , hvw grqf =
w @ h{s
] w
3
uv gv
Rq uhpdutxh doruv txh/ gdqv oh fdv rü ohv wdx{ vrqw gìwhuplqlvwhv +l1h = qrq doìdwrluhv, hw xqltxhphqw
gdqv fh fdv/ rq d sdu devhqfh g*rssruwxqlwì g*duelwudjh =
; w W> E+w> W , @ h{s ] W
w
uv gv
<15 Oh fdguh suredelolvwh hw o*devhqfh g*rssruwxqlwìv g*duelwudjh +Khdwk/
Mduurz hw Pruwrq,
Frpph qrxv o*dyrqv h{soltxì gdqv o*lqwurgxfwlrq/ qrxv grqqrqv xqh vwuxfwxuh suredelolvwh srxu od g|0
qdpltxh gh o*hqvhpeoh gh od vwuxfwxuh ghv wdx{ sxlv qrxv gìulyrqv ohv frqwudlqwhv lpsrvìhv sdu o*devhqfh
g*rssruwxqlwìv g*duelwudjh1 Gh fh srlqw gh yxh/ ohv prgëohv h{srvìv lfl vrqw ohv surorqjhphqwv frqwlqxv
ghv prgëohv glvfuhwv gx fkdslwuh suìfìghqw1
O*devhqfh g*rssruwxqlwì g*duelwudjh hvw wudgxlwh sdu o*h{lvwhqfh g*xqh suredelolwì ulvtxh0qhxwuh1 Rq d yx/
hq hhw/ gdqv oh fdv glvfuhw/ txh ohv ghx{ k|srwkëvhv = devhqfh g*rssruwxqlwì g*duelwudjh hw h{lvwhqfh g*xqh
suredelolwì ulvtxh0qhxwuh ìwdlhqw ìtxlydohqwhv1 Gdqv oh fdv frqwlqx/ ohv fkrvhv vrqw soxv frpsoltxìhv1 Qrxv
dgphwwurqv txh o*k|srwkëvh mrlqwh g*devhqfh g*rssruwxqlwì g*duelwudjh hw gh frpsoìwxgh ghv pdufkìv hvw
ìtxlydohqwh ã o*h{lvwhqfh g*xqh xqltxh suredelolwì uhqgdqw wrxv ohv sul{ dfwxdolvìv pduwlqjdohv1
Qrxv uhwurxyhurqv hqvxlwh sdu ghv k|srwkëvhv dgglwlrqqhoohv fhuwdlqv prgëohv wudglwlrqqhov frpph fhx{
gh Ydvlfhn hw Fr{/ Lqjhuvroo hw Urvv1
Frpph srlqw gh gìsduw gx prgëoh/ lo idxw elhq vh grqqhu xqh vwuxfwxuh vwrfkdvwltxh1 Lfl/ rq prgìolvh ohv
wdx{ ã whuph lqvwdqwdqìv frpph gdqv Khdwk/ Mduurz hw Pruwrq pdlv rq srxuudlw wrxw dxvvl elhq sduwlu
ghv sul{ ghv }ìur0frxsrq frpph dx fkdslwuh suìfìghqw1 Qrxv ihurqv oh olhq hqwuh fhv ghx{ dssurfkhv
ìtxlydohqwhv1
447
<1514 Ohv prxyhphqwv gh od vwuxfwxuh ghv wdx{
Frpph qrxv yhqrqv gh oh gluh/ qrxv xwlolvrqv xqh idplooh gh surfhvvxv srxu o*hqvhpeoh gh od vwuxfwxuh ghv
wdx{1 Khdwk/ Mduurz hw Pruwrq +4<<5, fkrlvlvvhqw gh vsìflhu ohv wdx{ ã whuph lqvwdqwdqìv1 Fhwwh idplooh
hvw sulvh vrxv od iruph vxlydqwh =
] w
] w
+v> W ,gv .
+v> W ,3 =gZv
+<14,
i+w> W , @ i+3> W , .
3
3
rü +i +3> W ,,W gìfulw od vwuxfwxuh ghv wdx{ lqlwldoh hw hvw vxssrvìh grqqìh hw rü +Zw , xq prxyhphqw
Eurzqlhq gh glphqvlrq N vrxv od suredelolwì S / rü S hvw od suredelolwì remhfwlyh ghv ìyìqhphqwv1
+w> W , hvw od yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv hw +w> W ,3 vd wudqvsrvìh1
Vrxv iruph g*ìtxdwlrq gh glxvlrq/ rq shxw dxvvl ìfuluh =
gi +w> W , @ +w> W ,gw .
N
[
l@4
l +w> W ,gZlw
tx*rq ìfulud hqfruh =
gi +w> W , @ +w> W ,gw . +w> W ,3 =gZw
+<15,
Ohv irqfwlrqv +w> W , hw +w> W , vrqw d sulrul txhofrqtxhv hw shxyhqw íwuh hq sduwlfxolhu doìdwrluhv1 Hoohv
grlyhqw wrxwhirlv vdwlvidluh ghv frqglwlrqv gh uìjxodulwì dvvxudqw o*h{lvwhqfh g*xqh vroxwlrq srxu o*ìtxd0
wlrq +<14, hw od oìjlwlplwì ghv fdofxov txl yrqw vxlyuh1 G*xqh pdqlëuh jìqìudoh/ qrxv q*lqvlvwhurqv sdv vxu
fhv frqglwlrqv whfkqltxhv1 Gh soxv od irqfwlrq yrodwlolwì yìulh xqh frqglwlrq g*lghqwldelolwì/ txl dvvxuh
tx*lo q*| d sdv g*doìd uhgrqgdqwv =
K|srwkëvh Od irqfwlrq gh yrodwlolwì vdwlvidlw =
4
3
+w> W4 ,3
F
E
11
D hvw lqyhuvleoh srxu wrxw w hw wrxw +W4 > ==> WN ,
C
1
+<16,
+w> WN ,3
Oh surfhvvxv gx wdx{ frxuw hvw rewhqx wuëv vlpsohphqw ã sduwlu gh +<14, hq idlvdqw W @ w =
uw
@ i+w> w,
@ i+3> w, .
] w
3
+v> w,gv .
] w
3
+v> w,3 =gZv
+<17,
Rq gìgxlw hqq oh surfhvvxv ghv sul{ }ìur0frxsrq ã sduwlu gh od irupxoh xvxhooh =
] W
i +w> v,gv=
E+w> W , @ h{s w
Hq uhpsodêdqw i+w> v, sdu vrq h{suhvvlrq +<14,/ hw hq xwlolvdqw oh wkìruëph gh Ixelql srxu ohv lqwìjudohv
vwrfkdvwltxhv +ydolgh g*dsuëv ohv frqglwlrqv gh uìjxodulwì lpsrvìhv dx{ surfhvvxv gh gìulyh hw gh yrodwlolwì,/
rq rewlhqw =
$3
] w ] W
] w #] W
] W
i +3> ,g gv
+v> ,g +v> ,g =gZv
+<18,
Oq E+w> W , @ w
3
w
3 w
Rq shxw duudqjhu fhwwh h{suhvvlrq hq xwlolvdqw o*h{suhvvlrq gx wdx{ frxuw1 Hq hhw/ g*dsuëv o*ìtxdwlrq
+<17,/ rq d =
] w ] ] w
] w
] w ] u g @
i +3> ,g .
g
+v> ,gv .
g
+v> ,3 =gZv
3
3
@
Hfroh Sro|whfkqltxh
] w
3
3
i +3> ,g .
] w
3
3
gv
] w
v
3
+v> ,g
.
3
] w ] w
3
v
+v> ,g
3
=gZv
448
fh txl grqqh dyhf +<18, =
&
&3
] W
] w% ] W
] w%
+v> ,g gv .
+v> ,g =gZv
uv Oq E+w> W , @ Oq E+3> W , .
3
v
3
v
+<19,
O*ìtxdwlrq gh glxvlrq fruuhvsrqgdqwh v*ìfulw doruv =
&
% ]
&3
%
] W
W
+w> ,g gw . +w> ,g =gZw
gOq E+w> W , @ uw w
w
rx hqfruh/ hq xwlolvdqw oh ohpph g*Lw÷ =
gE+w> W , @ û+w, . e+w> W ,È+w> W ,gw . E+w> W ,
+w> W ,3 =gZw
rü e+w> W , hw +w> W , gìvljqhqw =
+<1:,
4
;
UW
+w> W , @ w +w> ,g
? UW
=
e+w> W , @ w +w> ,g . 45 n
+w> W ,n5
D fh vwdgh/ dxfxqh frqwudlqwh q*hvw lpsrvìh = ohv ìtxdwlrqv +<18, hw +<1:, qh vrqw txh ohv frqvìtxhqfhv gx
prgëoh jìqìudo +<14, =
Sursrvlwlrq 49 Vl od g|qdpltxh ghv wdx{ ã whuph lqvwdqwdqìv v*ìfulw =
] w
] w
+v> W ,gv .
+v> W ,3 =gZv
i+w> W , @ i +3> W , .
3
3
doruv ohv sul{ }ìur0frxsrq hw oh wdx{ frxuw lqvwdqwdqì v*ìfulyhqw =
;
l
l3
Uwk
UW
Uwk UW
A
? E+w> W , @ E+3> W ,h{s^ 3 uv v +v> ,g gv . 3 v +v> ,g =gZv `
A
=
uw
@ i +3> w, .
Uw
Uw
3
3 +v> w,gv . 3 +v> w, =gZv
Glvsrvdqw ghv surfhvvxv gh sul{/ qrxv srxyrqv pdlqwhqdqw wudgxluh ohv frqwudlqwhv lqgxlwhv sdu o*devhqfh
g*rssruwxqlwì g*duelwudjh1
<1515 Devhqfh g*rssruwxqlwìv g*duelwudjh
Oh udlvrqqhphqw khxulvwltxh kdelwxhooh frqvlvwh ã frqvwuxluh xq sruwhihxlooh orfdohphqw vdqv ulvtxh grqw
oh uhqghphqw lqvwdqwdqì grlw doruv íwuh oh píph tx*xq sodfhphqw dx wdx{ frxuw uw 1
Frqvlgìurqv xq sruwhihxlooh gh N . 4 wlwuhv }ìur0frxsrq g*ìfkìdqfh W4 ===WN .4 / hq sursruwlrq {4 ==={N .4 1
Od ydohxu gh fh sruwhihxlooh ã od gdwh w hvw grqf =
Yw @
N
.4
[
n@4
{n E +w> Wn ,
Fkrlvlvvrqv {4 ==={N .4 gh whooh vruwh txh Yw vrlw orfdohphqw vdqv ulvtxh/ hw h{sulprqv oh idlw txh Yw d doruv
oh píph uhqghphqw txh oh uhqghphqw vdqv ulvtxh > rq d doruv od frqglwlrq vxlydqwh =
N
.4
[
{n E +w> Wn , m +w> Wn , @ 3>
^
n@4
4
449
8 8 gìvljqh od qruph hxfolglhqqh gh ,g 1
N
.4
[
m @ 4===N` @, ^
{n E +w> Wn , e +w> Wn , @ 3`
n@4
G*dsuëv oh ohpph gh Idundv/ lo idxw hw lo vx!w txh =
< *4 ===*N > ; m> e +w> Wm , .
N
[
n@4
*n +w> W4 ===WN .4 , n +w> Wm , @ 3
+<1;,
tx*rq shxw ìfuluh hqfruh =
+w> Wm ,
< * @ +*4 ===*N ,3 > ; m> e+w> Wm , @ *+w> W4 > = = = > WN.4 ,3 =
+<1<,
Ohv *n gìshqghqw d sulrul/ gh w> W4 > ===> WN.4 1 Fhshqgdqw frpph ohv gdwhv g*ìfkìdqfh W4 > = = = > WN .4 vrqw
txhofrqtxhv/ lov qh gìshqghqw sdv gh W4 > = = = > WN .4 1 Rq ohv qrwhud sdu od vxlwh *4 +w,> = = = > *N .4 +w,1 Lov v*lq0
whusuëwhqw frpph ghv sul{ gx ulvtxh dvvrflìv ã fkdfxq ghv surfhvvxv Eurzqlhqv1 Od frqwudlqwh g*devhqfh
g*rssruwxqlwìv g*duelwudjh lpsrvh grqf ghv uhvwulfwlrqv vxu ohv whuphv gh gìulyh hw gh yrodwlolwì gh od g|qd0
pltxh ghv sul{ }ìur0frxsrq1 Hq gìulydqw sdu udssruw ã W fhwwh frqglwlrq/ rq rewlhqw gh idêrq ìtxlydohqwh
xqh frqwudlqwh vxu ohv whuphv gh gìulyh hw gh yrodwlolwì fruuhvsrqgdqw dx{ wdx{ ã whuph lqvwdqwdqìv =
Sursrvlwlrq 4: O*devhqfh g*rssruwxqlwìv g*duelwudjh lpsoltxh od frqwudlqwh vxlydqwh vxu od g|qdpltxh
ghv sul{ hw ghv wdx{ ã whuph lqvwdqwdqìv =
+w> W ,> ; W w
< * @ +*4 ===*N ,3 who txh e+w> W , @ *+w,3 =
rx gh idêrq ìtxlydohqwh =
+w> W , . +w> W ,3 =+w> W , @ *+w,3 =+w> W ,
+<143,
+<144,
<1516 Fkdqjhphqw gh suredelolwì
Frpph gdqv ohv prgëohv hq whpsv glvfuhw/ o*devhqfh g*rssruwxqlwìv g*duelwudjh lpsoltxh o*h{lvwhqfh g*xqh
qrxyhooh phvxuh gh suredelolwì glwh ulvtxh0qhxwuh/ qrwìh T/ vrxv odtxhooh ohv surfhvvxv gh sul{ dfwxdolvìv
vrqw pduwlqjdohv1 Hq lqmhfwdqw +<143, gdqv o*ìtxdwlrq gh glxvlrq ghv sul{/ rq yrlw txh =
w
+w> W ,3 =gZ
gE +w> W , @ uw E+w> W , . E+w> W ,
dyhf
+<145,
w @ gZw *+w,gw
gZ
dyhf *+w, @ +*4 +w,> ===> *N +w,,3 =
Oh surfhvvxv
ghv sul{ dfwxdolvìv +] +w> W ,,w hvw doruv pduwlqjdoh vrxv od phvxuh gh suredelolwì txl idlw gh
w xq prxyhphqw Eurzqlhq vwdqgdug1 Fhfl uìvxowh gx wkìruëph gh Jluvdqry15
Z
Fh fkdqjhphqw gh suredelolwì v*ìfulw =
] w
]
gT
4 w
@ h{s
*+v,3 gZv n *+v, n5 gv
gS
5 3
3
] w
] w
v . 4
@ h{s
*+v,3 gZ
n *+v, n5 gv
5 3
3
Oh surfhvvxv ]+w> W , @ E+w>W , hvw grqf pduwlqjdoh vrxv T sxlvtxh/ g*dsuëv +<145, =
w
w
Ow
@
w
g]+w> W , @ +w> W ,3 =gZ
Frpph E+W> W , @ 4/ rq d doruv =
k
l
E+w> W , @ H T w @ Iw
W
k
l
UW
@ H S h{s w uv gv @ Iw
5 Oh sul{ gx ulvtxh
+<146,
) hvw vxssrvì gh fduuì lqwìjudeoh1
Hfroh Sro|whfkqltxh
44:
rx hqfruh/ hq xwlolvdqw od suredelolwì S =
%
&
] W
OW
S
E+w> W , @ H h{s uv gv=
@ Iw
Ow
w
Hq uìvxpì/ lo idxw uhwhqlu txh vrxv o*k|srwkëvh g*devhqfh g*rssruwxqlwìv g*duelwudjh/ lo h{lvwh xqh qrxyhooh
phvxuh gh suredelolwì T vrxv odtxhooh ohv sul{ rqw od g|qdpltxh grqqìh sdu o*ìtxdwlrq +<145,1 Frpph rq
oh yhuud/ wrxv ohv sureoëphv g*ìydoxdwlrq vh srvhqw vrxv fhwwh suredelolwì > rq shxw gëv oruv rxeolhu od
suredelolwì lqlwldoh S 1
Gh soxv/ d|dqw gìql fh fkdqjhphqw gh suredelolwì/ rq shxw h{sulphu wrxv ohv surfhvvxv gh glxvlrq
grqqìv mxvtx*ã suìvhqw +wdx{ ã whuph lqvwdqwdqìv/ sul{ }ìur0frxsrq/ wdx{ frxuw111, vrxv fhwwh qrxyhooh
phvxuh gh suredelolwì1
H{hpsoh 8 Xwlolvdwlrq gx wkìruëph gh Jluvdqry
Qrxv grqqrqv xq h{hpsoh wuëv vlpsoh srxu prqwuhu frpphqw oh wkìruëph gh Jluvdqry shxw v*xwlolvhu
hq sudwltxh1 Sodêrqv qrxv gdqv oh fdv N @ 4 +xq vhxo Eurzqlhq, hw *+w, @ *1 Prqwurqv sdu h{hpsoh
w vxlw/ vrxv T/ xqh orl qrupdoh g*hvsìudqfh qxooh hw gh yduldqfh w1 Srxu wrxwh
txh od yduldeoh doìdwrluh Z
irqfwlrq #/ rq d =
w , @ I3 , @ H S +#+Z
w ,+ gT ,w @ I3 ,
H T +#+Z
gS
] w
w ,h{s^*Z
w . 4 *5 w`gS +$,
#+Z
@
5
3
]
4
#+Zw *w,h{s^*Zw *5 w`gS +$,
@
5
Vrxv od suredelolwì S / Zw vxlw xqh orl qrupdoh g*hvsìudqfh qxooh hw gh yduldqfh w > rq d grqf =
] .4
4 5
4 {55w
T

#+{ *w,h*{ 5 * w s
g{
h
H +#+Zw , @ I3 , @
5w
4
] .4
4 |5
#+|, s
h 5w g|
@
5w
4
w vxlw/ vrxv T/ xqh orl qrupdoh g*hvsìudqfh qxooh hw gh
hq srvdqw | @ { *w1 Lo hvw doruv fodlu txh Z
yduldqfh w1
<16 Oh prgëoh vrxv od suredelolwì ulvtxh0qhxwuh
<1614 Ohv wdx{ ã whuph lqvwdqwdqìv
Rq rewlhqw wuëv vlpsohphqw od g|qdpltxh ghv wdx{ ã whuph lqvwdqwdqìv ã sduwlu gh o*ìtxdwlrq gh edvh =
gi +w> W , @ +w> W ,gw . +w> W ,3 =gZw
%]
&3
W
+w,
@
+w> ,g =+w> W ,gw . +w> W ,3 =gZ
w
+<147,
H{hpsoh 9 Yrodwlolwì frqvwdqwh
Qrxv qrxv lqwìuhvvrqv lfl dx fdv rü od irqfwlrq gh yrodwlolwì hvw vxssrvìh frqvwdqwh = +w> W , @ hw rü
N @ 41 Gdqv oh fkdslwuh suìfìghqw/ qrxv dyrqv yx tx*xqh ghv fdudfwìulvwltxhv gx prgëoh gh Kr hw Ohh
ìwdlw g*dyrlu mxvwhphqw ghv yrodwlolwìv ghv wdx{ ã whuph lqvwdqwdqìv frqvwdqwhv1 Gh soxv qrxv dylrqv grqqì
od olplwh hq whpsv frqwlqx gh fh prgëoh txh qrxv doorqv uhwurxyhu lfl1 Qrxv dyrqv hq hhw =
w=
gi +w> W , @ 5 +W w,gw . gZ
44;
vrlw hq lqwìjudqw =
i+w> W , @ i +3> W , . 5 +W w w5
w
, . Z
5
txl hvw elhq frqiruph ã o*h{suhvvlrq rewhqxh dx fkdslwuh suìfìghqw sdu sdvvdjh ã od olplwh1 Rq uhwurxyh
dlqvl frpph fdv sduwlfxolhu oh prgëoh gh Kr hw Ohh1 Rq srxuudlw uhwurxyhu gh píph ohv prgëohv ã yrodwlolwì
h{srqhqwlhooh hq suhqdqw +w> W , vrxv od iruph h+wv, 1
<1615 Surfhvvxv gx wdx{ frxuw
w =
O*ìtxdwlrq +<17, qrxv d grqqì od iruph gh +uw ,/ txl v*ìfulw gìvrupdlv ã o*dlgh gh Z
uw @ i +3> w, .
] w
3
^+v> w, . *+v,3 =+v> w,` gv .
] w
3
v
+v> w,3 =gZ
vrlw =
uw @ i +3> w, .
] w ] w
3
v
+v> ,g
3
=+v> w,gv .
] w
3
v
+v> w,3 =gZ
+<148,
Dlqvl/ vrxv T/ oh surfhvvxv vxlyl sdu +uw , qh gìshqg soxv gx sul{ gx ulvtxh gh píph txh ohv sul{ ghv wlwuhv
}ìur0frxsrq1
H{hpsoh : Yrodwlolwì frqvwdqwh hw yrodwlolwì h{srqhqwlhooh1 Vl ohv yrodwlolwìv vrqw frqvwdqwhv/ o*ìtxdwlrq
+<148, ghylhqw =
w5
w
uw @ i +3> w, . 5 . Z
5
rx
w
guw @ +w,gw . gZ
dyhf
+w, @ 5 w . C5 i +3> w,
Oh surfhvvxv +uw , hvw grqf xqh pdufkh doìdwrluh dyhf gìulyh1
Oruvtxh od irqfwlrq gh yrodwlolwì hvw h{srqhqwlhooh = +v> , @ h+ v, / doruv uw hvw ìjdoh ã =
]
] w
5 w +wv,
v=
h
+4 h+wv, ,gv . h+wv, gZ
uw @ i +3> w, .
3
3
Lo hvw doruv idfloh gh yrlu txh +uw , vxlw xq surfhvvxv g*Ruqvwhlq0Xkohqehfn gh od iruph =
w
guw @ ++w, uw , gw . gZ
5w
dyhf +w, @ C5 i +3> w,.i +3> w,. 5 4h5 1 Lo idxw uhpdutxhu txh oh surfhvvxv gx wdx{ frxuw hvw ìjdohphqw
xq surfhvvxv gh w|sh Ruqvwhlq0Xkohqehfn vrxv S / xqltxhphqw vl oh sul{ gx ulvtxh hvw xqh irqfwlrq qrq
doìdwrluh gx whpsv1 Gdqv fh fdv/ vhxoh od irqfwlrq +w, hvw prglìh1
<1616 Ohv sul{ }ìur0frxsrq
Vrxv od phvxuh T/ oh surfhvvxv ghv sul{ yìulh =
w
gE+w> W , @ u+w,E+w> W ,gw . E+w> W ,
+w> W ,3 =gZ
Frpph dwwhqgx/ oh wdx{ gh uhqghphqw hvsìuì +lqvwdqwdqì, hvw ìjdo/ gdqv o*xqlyhuv fruuljì gx ulvtxh +l1h =
vrxv od suredelolwì T,/ dx wdx{ frxuw vdqv ulvtxh uw 1 Qrxv uìfdslwxorqv fhv uìvxowdwv vrxv od sursrvlwlrq
vxlydqwh =
Hfroh Sro|whfkqltxh
44<
Sursrvlwlrq 4; Vrxv od suredelolwì ulvtxh0qhxwuh T/ ohv wdx{ ã whuph lqvwdqwdqìv hw ohv sul{ }ìur0frxsrq
vxlyhqw ohv ìtxdwlrqv gh glxvlrq vxlydqwhv =
%]
gi+w> W , @
W
w
&3
+w> ,g
+w,
=+w> W ,gw . +w> W ,3 =gZ
w
gE+w> W , @ u+w,E+w> W ,gw . E+w> W ,
+w> W ,3 =gZ
rü =
+w> W , @ ] W
w
+<149,
+<14:,
+w> ,g =
hw od g|qdpltxh gx wdx{ frxuw lqvwdqwdqì v*ìfulw =
uw @ i +3> w, .
] w ] w
3
v
+v> ,g
3
=+v> w,gv .
] w
3
v
+v> w,3 =gZ
+<14;,
w , hvw xq prxyhphqw Eurzqlhq vwdqgdug vrxv T1
rü +Z
Rq yrlw ã fh vwdgh tx*rq dxudlw sx vsìflhu gluhfwhphqw ohv sul{ }ìur0frxsrq hq glvdqw tx*lo h{lvwh xqh
phvxuh gh suredelolwì T whooh txh ohv sul{ }ìur0frxsrq vxlyhqw xqh glxvlrq gx w|sh +<149,1 Oh prgëoh hvw
doruv sdudpìwuì sdu od irqfwlrq +=> =, hw qrq soxv sdu +=> =,1 Od g|qdpltxh ghv wdx{ ã whuph lqvwdqwdqìv
v*hq gìgxlw sdu od irupxoh =
i+w> W , @ C5 Oq E+w> W ,
hq uhpdutxdqw txh ohv ghx{ sdudpìwudjhv +=> =, hw +=> =, vrqw ìtxlydohqwv =
;
%
&
] W
+w> W ,
? +w> W , @ C5 +w> W , @ +w> ,g +,
=
w
+w> w, @ 3 ; w
Fhwwh dssurfkh hvw fhooh gx prgëoh Ho Ndurxl/ P|qhql hw Ylvzdqdwkdq1
<1617 H{hpsoh = iruph gh od frxueh ghv wdx{ gdqv oh fdv g*xqh yrodwlolwì frqvwdqwh
Lo v*djlw g*xqh dssolfdwlrq gluhfwh ghv fdofxov vrxv od suredelolwì T =
w
T
@ Iw
E+w> W , @ H
W
vrlw =
%
E+w> W , @ H T h{s ] W
w
&
uv gv @ Iw
UW
Od orl gh +uw , hw grqf gh od yduldeoh w uv gv hvw grqqìh vrxv T sdu o*ìtxdwlrq +<14;,1 Gdqv oh fdv rü ohv
UW
irqfwlrqv gh yrodwlolwì vrqw gìwhuplqlvwhv/ +uw , hvw xq surfhvvxv jdxvvlhq vrxv T > od yduldeoh w uv gv hvw
grqf jdxvvlhqqh ìjdohphqw hw E+w> W , hvw grqqìh sdu od irupxoh xvxhooh =
Y +w> W ,
E+w> W , @ h{s P +w> W , .
5
rü
P+w> W , @ H T
%]
W
Y +w> W , @ Y duT
453
&
w
%]
uv gv @ Iw
w
W
&
uv gv @ Iw
Sduwrqv gh od g|qdpltxh gh +uw , vrxv o*k|srwkëvh gh yrodwlolwì frqvwdqwh +dyhf N @ 4, =
uw @ i +3> w, .
5 5
w
w . Z
5
Sdu frqvìtxhqw/ oh wdx{ u vxlw frqglwlrqqhoohphqw ã Iw xqh orl jdxvvlhqqh g*hvsìudqfh uw i +3> w, .
UW
5
i +3> , . 5 + 5 w5 , hw gh yduldqfh 5 + w,1 Od yduldeoh doìdwrluh \ @ w u g vxlw grqf hooh0píph xqh
orl jdxvvlhqqh =
+T,
\ @ Iw # Q +P+w> W , > Y +w> W ,,
rü
P +w> W , @ +W w,+uw i+3> w,, .
] W
w
i +3> ,g .
5
+W w,5 +W . 5w,
9
hw
] W
g @ Iw ,
Y +w> W , @ Y duT +
Z
w
] W
] v @ Iw ,
@ Y duT +
g
gZ
w
3
] W
v @ Iw ,
@ Y duT +
+W v,gZ
w
5
+W w,6
@
6
hw grqf =
E+w> W , @
5
E+3> W ,
h{s +W w,^uw i +3> w, . w+W w,`
E+3> w,
5
Oh idlw g*lpsrvhu xqh yrodwlolwì frqvwdqwh lpsoltxh grqf txh wrxwh od vwuxfwxuh ghv wdx{ hq w qh gìshqg
txh gx wdx{ frxuw uw +hw gh whuphv gìwhuplqlvwhv,1 Rq rewlhqw dlqvl xq prgëoh gh wdx{ ã xqh yduldeoh
g*ìwdw1
Rq srxuudlw gh píph fdofxohu ohv sul{ }ìur0frxsrq oruvtxh ohv yrodwlolwìv vrqw h{srqhqwlhoohv ++v> w, @
+W w,
1
h+wv, ,1 Ohv fdofxov vrqw gx píph w|sh txh suìfìghpphqw1 Oh frh!flhqw gh uw hvw doruv 4h <17 O*lqwurgxfwlrq gh yduldeohv g*ìwdw gdqv ohv prgëohv KMP
Oh prgëoh gh wdx{ gìfulw lfl hvw hqwlëuhphqw sdudpìwuì/ vrxv od suredelolwì ulvtxh0qhxwuh T/ sdu od irqfwlrq
gh yrodwlolwì +w> W ,1 D fh vwdgh/ fh prgëoh hvw hqfruh wurs jìqìudo sxlvtxh fhwwh irqfwlrq gh yrodwlolwì
hvw d sulrul txhofrqtxh1 Frpph dx fkdslwuh suìfìghqw/ lo hvw grqf qìfhvvdluh gh idluh ghv k|srwkëvhv
vxssoìphqwdluhv1 Od suhplëuh lgìh hvw gh suhqguh ghv irqfwlrqv gh yrodwlolwìv sduwlfxolëuhv +yrodwlolwì
frqvwdqwh/ h{srqhqwlhooh111,1 Qrxv dyrqv yx fhshqgdqw txh/ gdqv fhuwdlqv fdv/ fh jhquh g*k|srwkëvh ìwdlw
ìtxlydohqwh ã o*h{lvwhqfh gh yduldeohv g*ìwdw1 Qrxv doorqv v|vwìpdwlvhu fhwwh lgìh hw qrxv yhuurqv txh
fhwwh k|srwkëvh hvw gdqv fhuwdlqv fdv vx!vdpphqw frqwudljqdqwh srxu shuphwwuh xqh sdudpìwulvdwlrq
sduflprqlhxvh gh od yrodwlolwì +w> W ,1
Qrxv wudlwrqv hvvhqwlhoohphqw ghx{ fdv1 Oh suhplhu vxssrvh txh od irqfwlrq gh yrodwlolwì +w> W , hvw gìwhu0
plqlvwh > oh vhfrqg shuphw ã +w> W , g*íwuh irqfwlrq gh yduldeohv g*ìwdw1
<1714 Jìqìudolwìv vxu ohv prgëohv idfwrulhov
Qrxv suìflvrqv gdqv fh sdudjudskh ohv glìuhqwhv k|srwkëvhv gh yduldeohv g*ìwdw txl shxyhqw íwuh idlwhv1
Hoohv frqvlvwhqw ã vxssrvhu txh wrxwh od frxueh ghv wdx{ ã od gdwh w hvw irqfwlrq gh yduldeohv g*ìwdw
sduwlfxolëuhv1 Qrxv vxssrvrqv dlqvl =
Hfroh Sro|whfkqltxh
454
K|srwkëvh = ^Vwuxfwxuh idfwrulhooh olqìdluh` Lo h{lvwh N yduldeohv g*ìwdw I4w / 111/ INw whoohv txh =
; w W> i+w> W , @ e+w> W , . d4 +w> W ,I4w . = = = . dN +w> W ,INw
@ e+w> W , . d+w> W ,3 =Iw
rü ohv irqfwlrqv e+w> W , hw d+w> W , @ +d4 +w> W ,> = = = > dN +w> W ,,3 vrqw gìwhuplqlvwhv1
Dyhf od whuplqrorjlh ghv fkdslwuhv suìfìghqwv/ ohv irqfwlrqv dl +=> =, vrqw ohv vhqvlelolwìv dx{ idfwhxuv +idfwru
ordglqjv,1 Hq ìolplqdqw ohv yduldeohv Inw / rq shxw wrxmrxuv lghqwlhu ohv yduldeohv g*ìwdw ã ghv wdx{
sduwlfxolhuv +rx píph ghv frpelqdlvrqv gh wdx{, gh whooh vruwh tx*rq dxud od fdudfwìulvdwlrq ìtxlydohqwh
vxlydqwh =
; w W> i +w> W , @ e+w> W , . d4 +w> W ,i +w> w . 4 , . = = = . dN +w> W ,i+w> w . N ,
rü 4 / 111/ N vrqw N pdwxulwìv txhofrqtxhv glvwlqfwhv1 Vrxv fhwwh iruph/ ohv irqfwlrqv e+=> =, hw d+=> =,
grlyhqw vdwlvidluh ohv frqglwlrqv gh frpsdwlelolwì vxlydqwhv =
;
? e+w> w . l , @ 3
dl +w> w . l , @ 4
=
dm +w> w . l , @ 3 +m 9@ l,
<1715 Yrodwlolwì gìwhuplqlvwh hw yduldeohv g*ìwdw
Srxu vlpsolhu od suìvhqwdwlrq/ qrxv vxssrvrqv lfl txh od yrodwlolwì +w> W , hvw gìwhuplqlvwh hw krprjëqh
dx vhqv rü hooh qh gìshqg txh gh od pdwxulwì W w =
K|srwkëvh =
+w> W , @ +W w,
rü +=, hvw xqh irqfwlrq gìwhuplqlvwh1
Ohv wdx{ ã whuph lqvwdqwdqìv vrqw grqf jdxvvlhqv vrxv od suredelolwì T hw ìjdohphqw vrxv od suredelolwì
klvwrultxh S vl oh sul{ gx ulvtxh *+w, hvw gìwhuplqlvwh1 Fh prgëoh hvw dsshoì ìjdohphqw prgëoh olqìdluh
jdxvvlhq gdqv oh fdv soxv jìqìudo rü od irqfwlrq gh yrodwlolwì +w> W , gìshqg gh w hw W +hw qrq sdv vhxohphqw
gh W w,1
Vh sodêdqw pdlqwhqdqw gdqv oh fdguh gh o*k|srwkëvh idfwrulhooh olqìdluh/ qrxv doorqv wudgxluh ohv frqvì0
txhqfhv gh fhwwh k|srwkëvh hq fdofxodqw od iruph gh od yrodwlolwì +=,/ ghv vhqvlelolwìv d+=> =, hw gx whuph
e+=> =,1
Srxu fhod/ qrxv xwlolvhurqv od qrwdwlrq vxlydqwh =6
+{, @
] {
3
+|,g|
txl shuphw g*ìfuluh =
i+w> W , @ i+3> W , .
rx hqfruh =
] w
3
+W v,3 =
+W v,gv .
] w
3
v
+W v,3 =gZ
w
+W w,gw . +W w,3 =gZ
gi +w> W , @ +W v,3 =
+<14<,
Lo qrxv idxw ìjdohphqw od g|qdpltxh ghv yduldeohv g*ìwdw txh qrxv suhqrqv vrxv od iruph =
w
gIw @ Gw gw . Yw =gZ
rü ohv Gw hvw oh yhfwhxu gh gìulyh hw Yw od pdwulfh gh yrodwlolwì/ Gw ìwdqw d sulrul doìdwrluhv1
6 Lo v*djlw/ dx vljqh suëv/ gh od píph qrwdwlrq txh suìfìghpphqw1
455
+<153,
Vl rq lghqwlh ohv uhsuìvhqwdwlrqv +<14<, hw +<153,/ rq rewlhqw doruv =
;
+W w, @ d+w> W ,3 =Gw . C4 d+w> W ,=Iw . C4 e+w> W ,
? +W w,3 =
=
@ Yw3 =d+w> W ,
+W w,
+<154,
Vrxv fhwwh iruph/ rq yrlw gìmã txh Yw hvw qìfhvvdluhphqw xqh pdwulfh gìwhuplqlvwh1 Sdu dloohxuv/ olplwrqv
qrxv dx fdv d+w> W , @ d+W w, fh txl lpsoltxh txh Yw hvw xqh pdwulfh frqvwdqwh Y > rq d doruv hq uhpsodêdqw
od irqfwlrq d+W w, sdu Y 34 =+W w, gdqv od suhplëuh ìtxdwlrq gx v|vwëph +<154, =
+W w, @ +W w,3 =^Y 4 =Gw ` C+W w,3 =Y 4 =Iw . C4 e+w> W ,
+W w,3 =
Fh v|vwëph hvw yìulì txhotxh vrlw W w1 Rq shxw doruv hpslohu fhwwh ìtxdwlrq sulvh hq glìuhqwhv
ydohxuv gh W = w . 4 / 111/ w . N srxu rewhqlu xq v|vwëph gh N ìtxdwlrqv grqw ohv lqfrqqxhv vhudlhqw
ohv frpsrvdqwhv gx yhfwhxu Y 4 =Gw 1 Fh v|vwëph hvw gh soxv lqyhuvleoh sdu ohv frqglwlrqv lpsrvìhv sdu
o*k|srwkëvh lqlwldoh <15141 Od uìvroxwlrq gx v|vwëph prqwuh txh Y 4 =Gw hvw d!qh hq Iw 1 Gh soxv oh
frh!flhqw gh Iw gdqv fhwwh frpelqdlvrq d!qh grlw íwuh frqvwdqw +l1h = lqgìshqgdqw gh w,1 Rq ìfulud grqf
Gw vrxv od iruph =
Gw @ !+w, =Iw
Uhpsodêdqw Gw sdu fhwwh h{suhvvlrq gdqv od suhplëuh ìtxdwlrq gx v|vwëph +<154,/ rq rewlhqw =
+W w, @ +3 =Y 34 =+W w, Y 34 =C+W w,,3 =Iw . C4 e+w> W , . +W w,3 =Y 4 =!+w,
+W w,3 =
Sxlvtxh +=, hw +=, vrqw gìwhuplqlvwhv/ lo idxw grqf txh od sduwlh vwrfkdvwltxh v*ìolplqh > od irqfwlrq +=,
grlw grqf yìulhu =
+<155,
C+{, @ ^Y 4 ==Y `3 =+{,
grqw od vroxwlrq hvw gh od iruph =
+{, @
N
[
hl { 3l >
l@4
rü ohv sdudpëwuhv l vrqw ohv ydohxuv sursuhv gh od pdwulfh hw 3l ghv yhfwhxuv frqvwdqwv1 Gh od píph
idêrq/ od irqfwlrq d+=, vxlw oh píph w|sh g*ìtxdwlrq glìuhqwlhooh = Cd+{, @ 3 =d+{,
Qrxv srxyrqv grqf ìqrqfhu od sursrvlwlrq vxlydqwh =
Sursrvlwlrq 4< Vrxv o*k|srwkëvh gh vwuxfwxuh idfwrulhooh hw vrxv o*k|srwkëvh gh yrodwlolwì +w> W , gìwhu0
plqlvwh hw krprjëqh/ od g|qdpltxh ghv yduldeohv g*ìwdw d od iruph sduwlfxolëuh vxlydqwh =
w=
gIw @ +!+w, =Iw ,gw . Y=gZ
Gh soxv/ od yrodwlolwì hvw qìfhvvdluhphqw gh w|sh h{srqhqwlho dlqvl txh ohv vhqvlelolwìv dx{ idfwhxuv d+=,1
Fhv ghx{ irqfwlrqv vrqw vroxwlrq gh =
;
? C+{, @ ^Y 4 ==Y `3 =+{,
=
Cd+{,
@ 3 =d+{,
Lo q*hvw sdv gl!floh gh gìprqwuhu od uìflsurtxh gh fhwwh sursrvlwlrq > lo vx!w srxu fhod gh v*lqvsluhu
gx fkdslwuh suìfìghqw rü qrxv dylrqv prqwuì txh ohv yrodwlolwìv h{srqhqwlhoohv lqgxlvdlhqw ghv uhodwlrqv
+olqìdluhv, hqwuh ohv wdx{1 Od pdwulfh ìwdqw txhofrqtxh/ lo hvw fodlu txh ohv ydohxuv sursuhv shxyhqw íwuh
pxowlsohv rx frpsoh{hv1 Vl hoohv vrqw pxowlsohv/ ohv whuphv h{srqhqwlhov gdqv o*h{suhvvlrq gh od yrodwlolwì
vrqw suìfìgìv gh sro|qphv1 Oruvtx*hoohv vrqw frpsoh{hv/ od pdwulfh ìwdqw uìhooh/ od yrodwlolwì frqwlhqw
ghv whuphv hq frvlqxv hw vlqxv +ìyhqwxhoohphqw dpruwlv,1 Qrxv vxssrvhurqv hq jìqìudo txh ohv ydohxuv
sursuhv gh od pdwulfh vrqw vlpsohv hw uìhoohv1
Frpswh0whqx gh od iruph gh Gw / lo idxw uhpdutxhu txh od g|qdpltxh ghv idfwhxuv +Iw , hvw g*xq w|sh wuëv
sduwlfxolhu > rq shxw od yrlu frpph xq surfhvvxv gh w|sh Ruqvwhlq0Xkohqehfn jìqìudolvì1 Uhpdutxrqv
ìjdohphqw tx*ã fh vwdgh lo uhvwh ghv sureoëphv g*lghqwldelolwì sxlvtx*xq prgëoh idfwrulho olqìdluh hvw
wrxmrxuv gìql ã xqh wudqvirupdwlrq olqìdluh suëv ghv idfwhxuv1 Lo hvw idfloh gh yrlu txh/ srxu o*lqvwdqw/
Hfroh Sro|whfkqltxh
456
vhxohv ohv ydohxuv sursuhv gh od pdwulfh vrqw elhq lghqwldeohv1 Dlqvl/ fhv ydohxuv sursuhv vrqw lqwulqvëtxhv
dx prgëoh dx vhqv rü hoohv qh gìshqghqw sdv gh od edvh sduwlfxolëuh fkrlvlh srxu ohv idfwhxuv1 Hq uhydqfkh
od pdwulfh gh yrodwlolwì ghv idfwhxuv Y hq gìshqg ìylghpphqw1
Srxu dfkhyhu o*lghqwlfdwlrq frpsoëwh gx prgëoh/ lo vx!w g*lghqwlhu Iw ã xq yhfwhxu gh wdx{ sduwlfxolhuv
+i +w> w . 4 > ===> i +w> w . N ,,3 / fh txl uhylhqw ã lpsrvhu ohv frqglwlrqv gh frpsdwlelolwì =
;
? e+w> w . l , @ 3
@ 4
dl + l ,
=
@ 3 +m 9@ l,
dm +l ,
Fhwwh frqwudlqwh shuphw g*lpsrvhu ghv frqglwlrqv lqlwldohv srxu o*ìtxdwlrq glìuhqwlhooh vxu ohv irqfwlrqv
d+=, hw +=,1
Lo uhvwh doruv ã gìwhuplqhu od irqfwlrq e+w> W ,1 G*dsuëv oh v|vwëph +<154,/ lo ylhqw =
C4 e+w> W , @ +W w,3 =
+W w, +W w,3 =Y 4 =!+w,=
Wrxw uhylhqw grqf ã gìwhuplqhu od irqfwlrq !+w,1 Hq lqwìjudqw hw hq idlvdqw W @ w . l / rq rewlhqw =
e+3> w . l , @
] w
3
+w . l v,3 =
+w . l v,gv ] w
3
d+w . l v,3 =Y 4 =!+v,gv
Od irqfwlrq e+3> w, hvw frqqxh srxu wrxw w gëv txh od frxueh ghv wdx{ lqlwldoh hvw frqqxh +e+3> w, @
i +3> w, d+w,3 =I3 , > sdu frqvìtxhqw/ hq hpslodqw o*ìtxdwlrq suìfìghqwh srxu l @ 4> ===> N/ rq rewlhqw xq
v|vwëph gh N ìtxdwlrqv shuphwwdqw g*h{solflwhu ohv N frpsrvdqwhv gh od irqfwlrq !+=,1 Oh whuph gh gìulyh
!+w, hvw grqf irqfwlrq gh od frxueh ghv wdx{ lqlwldoh1
Uhpdutxh 4 Sxlvtx*rq d rewhqx o*h{suhvvlrq gh i +w> W ,/ rq shxw fdofxohu ohv sul{ }ìur0frxsrq sdu od
irupxoh =
] W
E+w> W , @ h{s^
i +w> v,gv`
w
Lo h{lvwh g*dxwuhv pr|hqv1 Hq hhw/ vrxv T/ E+w> W , yìulh ìjdohphqw =
E+w> W , @ H T +h{s^
] W
w
uv gv` @ Iw ,
tx*rq shxw fdofxohu gluhfwhphqw sxlvtxh +uw , hvw xq surfhvvxv jdxvvlhq1 Sdu dloohxuv/ sxlvtxh E+w> W , hvw
xqh irqfwlrq gìwhuplqlvwh E+w> W> Iw , gh w/ W hw Iw / rq shxw oxl dssoltxhu oh ohpph g*Lw÷ hw wudgxluh oh
idlw txh oh whuph gh gìulyh gdqv od g|qdpltxh gh +E+w> W ,, hvw ìjdoh ã uw E+w> W ,1 Rq rewlhqw doruv xqh
ìtxdwlrq dx{ gìulyìhv sduwlhoohv grqw E+w> W , hvw od vroxwlrq =
CE
4
C 5E
CE
.
=+!+w, I , . W u^Y 3 =
=Y ` @ uE
3
Cw
CI
5
CI CI 3
dyhf =
E+w> w> u, @ 4> ;w
H{hpsoh ; Oh prgëoh gh Ydvlfhn1 Ohv gìyhorsshphqwv suìfìghqwv shuphwwhqw gh uhfrqvlgìuhu oh prgëoh
sursrvì sdu Ydvlfhn1 Srxu fhod/ rq vh sodfh gdqv oh fdv sduwlfxolhu rü N @ 4 hw rü o*xqltxh idfwhxu Iw hvw
lghqwlì dx wdx{ frxuw uw / fh txl lpsrvh =
e+w> w, @ 3
d+3,
@ 4
Od g|qdpltxh gx wdx{ frxuw v*ìfulw grqf =
w
guw @ +!+w, uw , . gZ
457
Od irqfwlrq gh yrodwlolwì hvw ìjdoh ã =
+w> W , @ h+W w,
hw od vhqvlelolwì dx idfwhxu =
d+{, @ h{
Sxlvtxh e+w> w, @ 3/ rq d =
e+w> W , @
] W
w
d+W v,!+v,gv ] W
w
+W v,
+W v,gv
Frqqdlvvdqw od vwuxfwxuh ghv wdx{ lqlwldoh +f*hvw0ã0gluh e+3> =,,/ rq gìgxlw o*h{suhvvlrq gh od irqfwlrq !+=, =
!+w, @ C5 i+3> w, . i +3> w, .
5
+4 h5w ,
5
Rq hq gìgxlw soxv jìqìudohphqw o*h{suhvvlrq gh od irqfwlrq e+=> =, =
e+w> W , @ i +3> W , h+W w, i+3> w, .
5 +W w,
h
+4 h+W w, ,+4 h5w ,
5
Dx wrwdo/ od frxueh ghv wdx{ ã od gdwh w d od iruph vxlydqwh =
E+w> W , @ E i +3> w> W ,h{s ^+W w,+uw i+3> w,, .
rü od irqfwlrq +W w, hvw ìjdoh ã
5
+4 h5w ,+W w,5 `
7
4h+W w, 1
Uhpdutxh 5 Vl rq idlw @ 3/ rq uhwurxyh elhq od irupxoh grqqìh gdqv oh fdv gh od yrodwlolwì frqvwdqwh,1
Oh prgëoh gh Ydvlfhn sursuhphqw glw frqvlvwh ã vxssrvhu txh od irqfwlrq !+w, hvw xqh frqvwdqwh !/ fh txl
lpsoltxh xqh iruph sduwlfxolëuh gh od frxueh ghv wdx{ lqlwldoh1 Ohv sul{ }ìur0frxsrq vh vlpsolhqw srxu
grqqhu =
5
5
!
5
!
+W w,5 `
E+w> W , @ h{s^+ 5 ,+W w, +uw + 5 ,,+W w, 5
5
7
<1716 Yrodwlolwìv qrq gìwhuplqlvwhv hw yduldeohv g*ìwdw
Qrxv vxssrvrqv gìvrupdlv txh ohv yrodwlolwìv vrqw qrq gìwhuplqlvwhv > soxv suìflvìphqw od irqfwlrq +w> W ,
hvw gh od iruph =
+w> W , @ +Iw > W w,
rü Iw hvw frpph suìfìghpphqw xq yhfwhxu gh yduldeohv g*ìwdw hw +=> =, xqh irqfwlrq gìwhuplqlvwh1
Lqwurgxlvrqv ã qrxyhdx od qrwdwlrq vxlydqwh =
+Iw > {, @
] {
3
+Iw > |,g|=
Oh prgëoh olqìdluh jdxvvlhq gìfulw dx sdudjudskh suìfìghqw hvw ìylghpphqw oh fdv sduwlfxolhu rü od yrod0
wlolwì qh gìshqg sdv ghv idfwhxuv Iw 1 Qrxv qrxv sodêrqv vrxv o*k|srwkëvh gh vwuxfwxuh idfwrulhooh olqìdluh1
Od gìpdufkh dgrswìh lfl hvw wuëv vlplodluh ã fhooh gx sdudjudskh suìfìghqw1 Dlqvl/ lghqwldqw ohv sduwlhv
w / rq rewlhqw ã sduwlu gh =
gw hw gZ
i +w> W , @ d+W w,3 =Iw . e+w> W ,
gIw
w
@ Gw gw . Yw =gZ
oh v|vwëph g*ìtxdwlrqv =
Hfroh Sro|whfkqltxh
458
;
+Iw > W w, @ d+W w,3 =Gw Cd+W w,3 =Iw . C4 e+w> W ,
? +Iw > W w,3 =
=
@ Yw3 =d+W w,
+Iw > W w,
+<156,
Dq gh vlpsolhu ohv ìfulwxuhv/ rq srvh =
D+W w, @
fh txl lpsoltxh =
] W w
3
d+|,g|
+Iw > W w, @ Yw3 =D+W w,
Hq uhpsodêdqw od irqfwlrq +=> =, sdu vrq h{suhvvlrq hq irqfwlrq gh d+=,/ rq rewlhqw o*ìtxdwlrq =
d+W w,3 =Yw =Yw3 =D+W w, @ d+W w,3 =Gw Cd+W w,3 =Iw . C4 e+w> W ,
Fhwwh ìtxdwlrq hvw vdwlvidlwh srxu wrxwh ydohxu gh W w1 Rq shxw doruv xwlolvhu oh píph dujxphqw txh gdqv
oh sdudjudskh suìfìghqw1 Dlqvl/ suhqdqw vx!vdpphqw gh ydohxuv W / rq rewlhqw xq v|vwëph g*ìtxdwlrqv
g*lqfrqqxv Gw hw Yw =Yw3 tx*rq shxw lqyhuvhu1 7 Sdu frqvìtxhqw/ Gw hw Yw =Yw3 dssdudlvvhqw frpph ghv
irqfwlrqv d!qhv gh Iw 1 Rq ìfulud grqf =
5 3
6
44 =Iw . 44 +w, = = = 34p =Iw . 4p +w,
9
:
11
11
Yw =Yw3 @ 7
8
1
1
3p4 =Iw . p4 +w, = = = 3pp =Iw . pp +w,
Gw
@ !+w, =Iw
Oh fduuì gh od yrodwlolwì ghv idfwhxuv hvw grqf qìfhvvdluhphqw d!qh hq ohv idfwhxuv1 Od iruph grqqìh
fl0ghvvxv hvw hq rxwuh frqwudlqwh srxu ghv udlvrqv gh v|pìwulh sdu =
lm
lm
@ ml > ; l 9@ m
@ ml > ; l 9@ m
Vl rq uhpsodfh pdlqwhqdqw Gw hw Yw =Yw3 sdu ohxu h{suhvvlrq gdqv od suhplëuh ìtxdwlrq gx v|vwëph/ rq
yrlw txh qìfhvvdluhphqw ohv vhqvlelolwìv dx{ idfwhxuv +soxv h{dfwhphqw od irqfwlrq D+=,, grlyhqw yìulhu
o*ìtxdwlrq glìuhqwlhooh vxlydqwh +hq lqwìjudqw xqh irlv, =
N
4 [
lm Dl +{,Dm +{,,
CD+{, @ CD+3, 3 D+{, +
5 l>m @4
+<157,
txl hvw xqh ìtxdwlrq gh Ulffdwl ã N glphqvlrqv1 Rq uhwurxyh ã qrxyhdx oh prgëoh olqìdluh jdxvvlhq +hw
grqf ohv vroxwlrqv gh w|sh h{srqhqwlho, hq suhqdqw frpph fdv sduwlfxolhu lm @ 31 Rq d grqf =
Sursrvlwlrq 53 Vrxv o*k|srwkëvh gh vwuxfwxuh idfwrulhooh olqìdluh hw vl od yrodwlolwì ghv wdx{ ã whuph
lqvwdqwdqìv hvw vwrfkdvwltxh +gh od iruph +Iw > W w,,/ doruv od g|qdpltxh ghv yduldeohv g*ìwdw d od iruph
sduwlfxolëuh =
w
gIw @ +!+w, =Iw ,gw . Yw =gZ
rü Yw =Yw3 v*ìfulw =
5
6
344 =Iw . 44 +w, = = = 34p =Iw . 4p +w,
9
:
11
11
Yw =Yw3 @ 7
8
1
1
3p4 =Iw . p4 +w, = = = 3pp =Iw . pp +w,
Od irqfwlrq gh yrodwlolwì ghv wdx{ ã whuph lqvwdqwdqìv v*ìfulw =
+w> W , @ Yw3 =d+W w,
7 Hq idlw/ lo q*hvw sdv fhuwdlq tx*rq sxlvvh wrxmrxuv lqyhuvhu fh v|vwëph1
459
rü ohv vhqvlelolwìv dx{ idfwhxuv D+{, @
U{
3
d+|,g| vxlyhqw o*ìtxdwlrq gh Ulffdwl vxlydqwh =
N
4 [
lm Dl +{,Dm +{,,
CD+{, @ CD+3, 3 D+{, +
5 l>m @4
Idlvrqv txhotxhv uhpdutxhv1 Suhplëuhphqw/ frqwudluhphqw dx prgëoh olqìdluh jdxvvlhq/ rq qh shxw hq
jìqìudo uìvrxguh h{solflwhphqw fhwwh ìtxdwlrq gh Ulffdwl vdxi gdqv oh fdv g*xqh vhxoh yduldeoh g*ìwdw
N @ 41 Lo | d fhshqgdqw xq fdv rü o*ìtxdwlrq gh glphqvlrq N shxw íwuh uìvroxh1 Lo v*djlw lqwxlwlyhphqw
gx fdv rü fh v|vwëph gh Ulffdwl shxw v*ìfuluh frpph N ìtxdwlrqv gh Ulffdwl ã xqh glphqvlrq1 G*dsuëv
od iruph gh o*ìtxdwlrq +<157,/ rq yrlw txh fhfl q*hvw srvvleoh txh srxu ghv ydohxuv wuëv sduwlfxolëuhv ghv
sdudpëwuhv hw lm 1
Ghx{lëphphqw/ ulhq q*dvvxuh txh od pdwulfh Yw =Yw3 vrlw srvlwlyh > f*hvw od udlvrq srxu odtxhooh rq sursrvh
ghv iruphv sduwlfxolëuhv gh fhwwh pdwulfh txl frqvhuyhqw oh fdudfwëuh olqìdluh pdlv txl dvvxuhqw od srvlwlylwì1
Wurlvëphphqw/ frpph gdqv ohv prgëohv ã yrodwlolwì gìwhuplqlvwh/ wrxv ohv sdudpëwuhv qh vrqw sdv lghqwl0
deohv wdqw tx*rq q*d sdv suìflvì txhooh edvh gh idfwhxuv rq fkrlvlw1
Hqq/ lo uhvwh ã gìwhuplqhu od irqfwlrq e+=> =, txl vh gìgxlw ghv irqfwlrqv !+w, hw +w, =
C4 e+w> W , @
vrlw =
e+w> W , e+3> W , @
N
[
l>m @4
lm +w,dl +W w,Dm +W w, d+W w,3 =!+w,
N ] w
[
l>m @4
3
lm +v,dl +W v,Dm +W v,gv ] w
3
d+W v,3 =!+v,gv
Vl rq lghqwlh ohv idfwhxuv ã ghv wdx{ sduwlfxolhuv Iw @ +i+w> w . 4 > ===> i+w> w . N ,,3 / rq doruv sdu ohv
uhodwlrqv gh frpsdwlelolwì =
e+3> w . l , @ i +3> w . l , . d+w . l ,3 =Iw
@
] w
3
N ] w
[
l>m @4
3
lm +v,dl +w . l v,Dm +w . l v,gv
d+w . l v,3 =!+v,gv> ; l @ 4> ===> N
Rq yrlw doruv txh od frxueh ghv wdx{ lqlwldohv qh vx!w sdv srxu gìwhuplqhu ohv N +N5.4, irqfwlrqv lm +=, hw
ohv N irqfwlrqv l +=,1 Lo uhvwh grqf ghv ghjuìv gh olehuwì vxssoìphqwdluhv1 Rq shxw lpsrvhu txh fhuwdlqhv
gh fhv irqfwlrqv vrqw frqvwdqwhv/ sdu h{hpsoh ohv lm +=,1
Dq gh grqqhu ghv looxvwudwlrqv gh fh prgëoh/ qrxv uhjdugrqv ohv fdv sduwlfxolhuv sursrvìv gdqv od olwwì0
udwxuh1
<18 H{hpsohv gh prgëohv olqìdluhv ã yrodwlolwì vwrfkdvwltxh
<1814 Oh prgëoh ã xq idfwhxu +Fr{/ Lqjhuvroo hw Urvv,
Lo frqvlvwh ã vh sodfhu gdqv oh fdv N @ 41 Frpph qrxv o*dyrqv glw suìfìghpphqw/ lo hvw doruv srvvleoh
g*rewhqlu ghv irupxohv h{solflwhv srxu ohv sul{ }ìur0frxsrq1 Gdqv oh fdv N @ 4/ qrxv srxyrqv vdqv shuwh
gh jìqìudolwì/ lghqwlhu o*xqltxh idfwhxu dx wdx{ frxuw uw +vrlw 4 @ 3,1
Ohv ìtxdwlrqv suìfìghqwhv grqqhqw doruv =
Hfroh Sro|whfkqltxh
45:
; 5
Yw
A
A
A
A
A
A
A
A
? Gw
@ uw . +w,
@ !+w, uw
A
A
CD+{, @ 4 D+{, 45 D+{,5
A
A
A
A
A
A
s
=
w
guw
@ +!+w, uw ,gw . uw . +w,gZ
+<158,
Od uìvroxwlrq gh o*ìtxdwlrq gh Ulffdwl vxlylh sdu D+=, vh uìvrxg vrxv od iruph =
D+{, @
rü =
4 . n 4 h{
4 . nh{
t
;
A
@
4 . 55
A
A
?
t 5
A
4. 5 4
A
A
= n @ t 5
4. 5 .4
Rq yrlw elhq vrxv fhwwh iruph frpphqw oh prgëoh olqìdluh jdxvvlhq vh gìgxlw gx prgëoh jìqìudo1 Vl rq
suhqg @ 3/ doruv oh sdudpëwuh n hvw oxl0píph ìjdo ã 3 +hw uìflsurtxhphqw,1 Gh soxv/ hvw ìjdo ã 1 Rq
uhwurxyh elhq xqh h{suhvvlrq h{srqhqwlhooh srxu D+=, hw grqf srxu d+=,1 Lfl od vwuxfwxuh idfwrulhooh shxw
íwuh h{solflwhphqw ìfulwh =
i+w> W , @ +4 . n,5
h+W w,
uw . e+w> W ,=
+4 . nh+W w, ,5
rü od irqfwlrq e+w> W , hvw ìjdoh ã =
e+w> W , @ ] W
w
+v,d+W v,D+W w,gv .
] W
w
d+W v,!+v,gv
Rq yrlw doruv txh od frqqdlvvdqfh gh od frxueh ghv wdx{ lqlwldoh +f*hvw0ã0gluh hq idlw e+3> =,, q*hvw sdv
vx!vdqwh srxu gìwhuplqhu ohv ghx{ irqfwlrqv !+=, hw +=,1
Ohv sul{ }ìur0frxsrq v*rewlhqqhqw doruv sdu =
E+w> W , @ h{s ] W
w
i +w> v,gv`
@ h{s ^D+W w,uw .
dyhf =
] W
w
e+w> v,gv`
] W
]
] W
4 W
5
e+w> v,gv @ +x,D+W x, gx .
!+x,D+W x,gx
5 w
w
w
Rq qh shxw grqf sdv dyrlu xqh iruph h{solflwh hq irqfwlrq gh od frxueh ghv wdx{ lqlwldoh pdlv vhxohphqw
sdu o*lqwhupìglduh ghv irqfwlrqv !+=, hw +=,1
Oh prgëoh gh Fr{/ Lqjhuvroo hw Urvv +4<;8, sursuhphqw glw frqvlvwh ã lpsrvhu !+=, hw +=, frqvwdqwhv
dx frxuv gx whpsv/ fh txl lpsrvh ghv frqwudlqwhv vxu od frxueh ghv wdx{ lqlwldoh1 Gdqv fh fdv/ rq shxw
UW
h{solflwhu od irqfwlrq w e+w> v,gv hq irqfwlrq ghv sdudpëwuhv ! hw / fh txl grqqh dx wrwdo =
E+w> W , @ h{s ^D+W w,+uw .
dyhf =
45;
+W w,

,+W w, . 5 ! Oq 4 . nh
, . +!
`
4.n
@ ! . !
+<159,
Uhpdutxrqv txh +uw , q*hvw ìylghpphqw sdv jdxvvlhq vrxv T > rq prqwuhud dx fkdslwuh vxlydqw txh od orl
frqglwlrqqhooh gh uw vdfkdqw Iv +v ? w, hvw xqh orl gx "5 gìfhqwuìh1 Qìdqprlqv oh fdofxo suìfìghqw prqwuh
tx*rq vdlw fdofxohu =
UW
H T +h w uv gv @ Iw ,
sxlvtxh fhwwh txdqwlwì hvw h{dfwhphqw ìjdoh ã E+w> W ,1 Sdu dloohxuv/ E+w> W , frpph irqfwlrq gh w/ W hw uw
hvw vroxwlrq gh o*ìtxdwlrq dx{ gìulyìhv sduwlhoohv =
4 C 5E
CE CE
.
+! u, .
+u . , @ uE
Cw
Cu
5 Cu5
vrxv E+w> w> u, @ 4 grqw od vroxwlrq hvw grqf grqqìh sdu o*ìtxdwlrq +<159,1
<1815 Oh prgëoh gh Gx!h hw Ndq
Fh prgëoh frqvlvwh ã frqvlgìuhu xqh idplooh sduwlfxolëuh gh yrodwlolwì srxu ohv yduldeohv g*ìwdw vrxv od
iruph =
t
t
Yw @ =Gldj^ 34 =Iw . 4 +w,> ===> 3N =Iw . N +w,`
rü hvw xqh pdwulfh N N +lqyhuvleoh, hw Gldj+=> ===> =, xqh pdwulfh gldjrqdoh1 Lo v*djlw g*xq fdv sdu0
wlfxolhu gdqv od phvxuh rü wrxwh pdwulfh Y whooh txh Y=Y 3 dlhqw vhv ìoìphqwv olqìdluhv hq Iw qh shxw
hq jìqìudo sdv v*ìfuluh vrxv fhwwh iruph1 Gh idlw/ fh prgëoh frqwlhqw prlqv gh sdudpëwuhv oleuhv hw hvw
grqf fhuwdlqhphqw soxv idfloh ã fdoleuhu1 Sdu dloohxuv/ ohv sureoëphv g*h{lvwhqfh gh vroxwlrq ã o*ìtxdwlrq
glìuhqwlhooh vwrfkdvwltxh vxlylh sdu +Iw , vrqw soxv idflohv ã uìvrxguh gdqv fh fdv1
Od sduwlh idfwrulhooh gx prgëoh hvw wrxmrxuv gx píph w|sh/ ã vdyrlu xqh iruph olqìdluh hq ohv yduldeohv
g*ìwdw srxu ohv wdx{ > wrxwhirlv/ o*ìtxdwlrq gh Ulffdwl fruuhvsrqgdqwh hvw soxv vlpsoh +pdlv wrxmrxuv qrq
uìvroxeoh dqdo|wltxhphqw, hw v*ìfulw =
N
4[
l D+{,3 = l = 3l =D+{,
CD+{, @ CD+3, =D+{, 5 l@4
rü ohv yhfwhxuv l vrqw whov txh @ + 4 > ===> N ,1 Rq yrlw grqf txh ohv N iruphv txdgudwltxhv txl
dssdudlvvhqw gdqv oh v|vwëph g*ìtxdwlrqv glìuhqwlhoohv vh gldjrqdolvhqw/ gdqv fh prgëoh/ gdqv od píph
edvh ruwkrqrupìh1
<1816 Oh prgëoh txdgudwltxh jdxvvlhq +Ho Ndurxl/ P|qhql hw Ylvzdqdwkdq,
Fh prgëoh sursrvh xq pr|hq g*lqwurgxluh ghv qrq0olqìdulwìv gdqv ohv vwuxfwxuhv idfwrulhoohv/ f*hvw0ã0gluh
gh vh gìjdjhu sduwlhoohphqw gh o*k|srwkëvh gh vwuxfwxuh idfwrulhooh olqìdluh1
D wlwuh g*looxvwudwlrq/ sodêrqv qrxv gdqv oh fdv g*xqh vhxoh yduldeoh g*ìwdw/ qrwìh +]w ,1 Oh prgëoh txdgud0
wltxh jdxvvlhq frqvlvwh ã sduwlu gh iruphv idfwrulhoohv txdgudwltxhv gx w|sh =
i +w> W , @ d4 +w> W ,]w5 . d5 +w> W ,]w . e+w> W ,
+<15:,
Fh prgëoh shxw íwuh yx frpph xqh prgëoh idfwrulhooh olqìdluh dyhf frqwudlqwh =
i+w> W , @ d4 +w> W ,I4w . d5 +w> W ,I5w . e+w> W ,
dyhf =
Srvwxodqw od g|qdpltxh vxlydqwh srxu +Iw , =
I4w @ I55w > ; w
w
gIw @ Gw gw . Yw =gZ
w , hvw xqh prxyhphqw Eurzqlhq el0glphqvlrqqho/ doruv g*dsuëv ohv uìvxowdwv txl suìfëghqw/ Gw hw
rü +Z
3
Yw =Yw vrqw olqìdluhv hq Iw =
@ !+w, +w,=Iw
Gw
Yw =Yw3
Hfroh Sro|whfkqltxh
@ +lm +w,3 =Iw . lm +w,,l>m
45<
Wrxv ohv sdudpëwuhv gìshqghqw gx whpsv gdqv od phvxuh rü qrxv qh idlvrqv soxv g*k|srwkëvhv g*krpr0
jìqìlwì vxu ohv irqfwlrqv +w> W , hw d+w> W ,1
Vl rq lpsrvh doruv I4w @ I55w hw vl rq srvh =
w
gI5w @ +!5 +w, 5 +w,3 =Iw ,gw . 35w =gZ
doruv =
3
Gw @ C
5+!5 +w, 5 +w,3 =Iw ,I5w . n 5w n5
!5 +w, 35 =Iw
hw =
3
Yw =Yw3 @n 5w n5 C
7I4w
5I5w
5I5w
4
4
D
4
D
Rq yrlw grqf txh/ srxu dvvxuhu od olqìdulwì gh od pdwulfh Yw =Yw3 hq Iw / lo idxw txh n 5w n5 vrlw gìwhuplqlvwh1
Gh píph/ srxu txh Gw vrlw olqìdluh hq Iw / lo idxw txh od suhplëuh frrugrqqìh gh 5 vrlw qxooh1 Dx wrwdo/
ohv pdwulfhv hw Yw =Yw3 vrqw gh od iruph =
4
3
5+w, 5!+w,
D
@ C
3
+w,
3
Yw =Yw3
@ n +w, n5 C
7I4w
5I5w
5I5w
4
4
D
rü +w,/ !+w, hw +w, vrqw ohv sdudpëwuhv gh od g|qdpltxh gh od yduldeohv g*ìwdw ]w @ I5w =
w
g]w @ +!+w, +w,]w ,gw . +w,3 =gZ
Od g|qdpltxh gx prgëoh hvw hqwlëuhphqw uìvxpìh sdu fhooh gh +]w ,/ txl hvw gx w|sh Ruqvwhlq0Xkohqehfn1
Od sduwlh idfwrulhooh hvw grqf xq sro|qph gh ghjuì 5 hq ]w hw o*ìtxdwlrq gh Ulffdwl grqw ohv irqfwlrqv
d4 +=> =, hw d5 +=> =, vrqw vroxwlrq v*ìfulw =
;
C4 d4 +w> W , @ 5+w,d4 +w> W , . 7 n +w, n5 d4 +w> W ,D4 +w> W ,
A
A
A
A
?
C4 d5 +w> W , @ 5!+w,d4 +w> W , . +w,d5 +w> W ,
A
A
A
A
=
.5 n +w, n5 +d4 +w> W ,D5 +w> W , . D4 +w> W ,d5 +w> W ,,
tx*rq shxw uìvrxguh h{solflwhphqw sxlvtxh o*ìtxdwlrq txl grqqh od irqfwlrq d4 +=> =, hvw xql0glphqvlrqqhooh1
O*lghqwlfdwlrq frpsoëwh gx prgëoh qh shxw vh idluh hq lpsrvdqw ã ]w g*íwuh ìjdo ã xq wdx{ sduwlfxolhu8 >
rq shxw hq uhydqfkh lghqwlhu ]w5 ã xq wdx{ sduwlfxolhu1
Hqq/ rq shxw idluh ohv uhpdutxhv vxlydqwhv1 Rq frqvwdwh suhplëuhphqw txh fh prgëoh hvw ìtxlydohqw ã xq
prgëoh rü lo q*| dxudlw tx*xq vhxo Eurzqlhq hq suhqdqw n +w, n5 srxu od yrodwlolwì gh +]w ,1 Ghx{lëphphqw/
rq srxuudlw prqwuhu tx*rq qh shxw sdv frqvwuxluh xq prgëoh srxu ohtxho od vwuxfwxuh idfwrulhooh vhudlw
fxeltxh hq ]w hw soxv jìqìudohphqw xq sro|qph hq ]w gh ghjuì vxsìulhxu rx ìjdo ã 61 Wurlvlëphphqw/
rq shxw ìjdohphqw fkhufkhu wrxv ohv prgëohv idfwrulhov =
i+w> W , @ d4 +w> W ,I4w . d5 +w> W ,I5w . e+w> W ,
srxu ohvtxhov rq dxudlw xqh uhodwlrq gìwhuplqlvwh hqwuh ohv idfwhxuv gx w|sh t+I4w > I5w > w, @ 31 Rq prqwuh0
udlw tx*rq derxwlw qìfhvvdluhphqw dx fdv rü t+=> => =, hvw xqh iruph txdgudwltxh gìjìqìuìh/ f*hvw0ã0gluh oh
prgëoh txdgudwltxh jdxvvlhq1
8 Rq derxwlw doruv ã
463
@ E|c A lghqwltxhphqw qxo1
43
H{hpsohv gh surgxlwv gh wdx{ g*lqwìuíw
Rq v*lqwìuhvvh lfl ã ghv surgxlwv w|sltxhv gh wdx{ g*lqwìuíw1 Fhuwdlqv vrqw qìjrflìv gh juì ã juì hw ohv dxwuhv
vxu ghv pdufkìv rujdqlvìv1 Qrxv derughurqv od txhvwlrq gh o*ìydoxdwlrq gh fhv surgxlwv = od whfkqrorjlh
gh edvh uhsrvh vxu o*ìydoxdwlrq sdu duelwudjh whooh txh gìfulwh suìfìghpphqw ã od qxdqfh suëv txh qrxv
wlhqgurqv frpswh ghv frqyhqwlrqv gh pdufkì xwlolvìhv hq sudwltxh1 Qrxv grqqhurqv ìjdohphqw srxu fkdtxh
surgxlw ghv h{hpsohv g*xwlolvdwlrq hw ghv udlvrqv srxu ohvtxhoohv fh surgxlw shxw lqwìuhvvhu who rx who dfwhxu1
4314 Iruzdug udwh djuhhphqw +IUD,
Oh iruzdug udwh djuhhphqw hvw xq frqwudw hqwuh ghx{ sduwlhv D hw E sdu ohtxho D v*hqjdjh ã suìwhu ã
xq wdx{ g*lqwìuíw frqyhqx ã o*dydqfh1 O*hqwuìh gdqv xq IUD shuphw grqf gh {hu gëv dxmrxug*kxl ohv
frqglwlrqv gh wdx{ g*lqwìuíw srxu xqh rsìudwlrq gh suíw rx hpsuxqw1 Lo v*djlw grqf g*xq rxwlo qdwxuho
gh frxyhuwxuh frqwuh xqh kdxvvh rx xqh edlvvh ghv wdx{ g*lqwìuíw +vhorq tx*rq vrxkdlwh hpsuxqwhu rx
suíwhu ã xqh gdwh ixwxuh,1 Fh w|sh gh frqwudw shuphw sdu h{hpsoh ã xqh hqwuhsulvh txl vdlw tx*hooh ghyud
hpsuxqwhu gdqv txhotxhv prlv gh vh frxyulu frqwuh xqh kdxvvh ghv wdx{ g*lqwìuíw hqwuh dxmrxug*kxl hw od
gdwh gh o*hpsuxqw1 O*hqwuhsulvh hqwuhud gdqv fh frqwudw vrlw sdufh tx*hooh dqwlflsh xqh uhprqwìh ghv wdx{
g*lqwìuíw/ vrlw wrxw vlpsohphqw sdufh tx*hooh vrxkdlwh qh sdv suhqguh gh ulvtxh = oh pìwlhu g*xqh hqwuhsulvh
qrq qdqflëuh frqvlvwh ã suhqguh soxw÷w ghv ulvtxhv lqgxvwulhov txh ghv ulvtxhv qdqflhuv1
Soxv suìflvìphqw/ xq IUD qh grqqh sdv olhx ã od plvh hq sodfh hhfwlyh g*xqh rsìudwlrq g*hpsuxqw 2
suíw ã whuph1 Frqvlgìurqv xq IUD frqfox ã od gdwh w/ ohv ghx{ sduwlhv fkrlvlvvhqw =
xq prqwdqw glw qrwlrqqho Q vxu ohtxho vhurqw fdofxoìv ohv lqwìuíwv >
xqh gdwh gh gìsduw gh od jdudqwlh W hw od gdwh g*ìfkìdqfh W . q >
xq wdx{ gh uìiìuhqfh u+v> v . ,/ sdu h{hpsoh xq wdx{ g*lqwìuíw ã 6 prlv/ 9 prlv/ 111/ l1h1 o*hxuleru 6
prlv + @ 6 prlv,/ 9 prlv hwf1
Oh x{ hqwuh D hw E vhud doruv fdofxoì frpph od glìuhqfh hqwuh oh wdx{ ã whuph i +w> W> W ., +grqf frqqx
oruv gh od plvh hq sodfh gx IUD, hw oh wdx{ hhfwlyhphqw frqvwdwì ã od gdwh gh gìsduw gh od jdudqwlh =
Q= +u+W> W . , i +w> W> W . ,, =
q
Edvh
rü Edvh hvw od frqyhqwlrq gh mrxuv dqqxhov +sdu h{hpsoh 693 mrxuv,1 Rq yrlw tx*lo v*djlw g*xqh frqyhqwlrq
gh wdx{ olqìdluh hw qrq dfwxdulho1 Fh x{ hvw yhuvì hq W . q1 Wrxwhirlv/ vl oh yhuvhphqw d olhx ã od gdwh gh
gìsduw gh od jdudqwlh +rq sduoh gh vhwwohphqw lq dgydqfh,/ lo hvw qìfhvvdluh gh o*dfwxdolvhu1 Oh x{ ghylhqw
grqf =
q
Q= +u+W> W . , i+w> W> W . ,, = Edvh
q
4 . u+W> W . ,= Edvh
O*dfkhwhxu gx IUD hvw fhoxl txl uhêrlw fh x{ hw oh yhqghxu fhoxl txl oh yhuvh1 Oh wdx{ g*lqwìuíw gx IUD
i +w> W> W . , vxu ohtxho v*dffrughqw ohv ghx{ sduwlhv v*dsshooh oh wdx{ ã whuph rx wdx{ iruzdug1 Rq d gìmã
yx gdqv ohv fkdslwuhv suìfìghqwv txh/ sdu xq udlvrqqhphqw g*duelwudjh/ fh wdx{ ã whuph srxydlw vh fdofxohu
ã sduwlu gh od frxueh ghv wdx{ dxmrxug*kxl1 Vl rq uhsuhqg o*h{hpsoh g*xqh hqwuhsulvh hq w txl vrxkdlwh
hpsuxqwhu hq W vxu xqh sìulrgh @ q1 Srxu vh surwìjhu/ hooh dfkëwh xq IUD hq w hw grqf derxwlw dx{
x{ vxlydqwv =
gdwh w = 3
gdwh W = o*hqwuhsulvh uhêrlw oh prqwdqw gh vrq hpsuxqw Q >
=u+W> W . , hw uhêrlw
gdwh W . = o*hqwuhsulvh uhperxuvh oh qrplqdo Q gh vrq suíw/ ohv lqwìuíwv Q= Edvh
oh x{ fruuhvsrqgdqw ã vrq IUD/ vrlw =
Q= +u+W> W . , i +w> W> W . ,, =
Edvh
vrlw dx wrwdo =
Q Q=
=u+W> W .,.Q= +u+W> W . , i +w> W> W . ,, =
@ Q= 4 . i +w> W> W . ,=
Edvh
Edvh
Edvh
H{ srvw/ o*hqwuhsulvh d grqf sx hpsuxqwhu dx wdx{ ã whuph hw qrq dx wdx{ hhfwli oh mrxu gh od plvh hq
sodfh gx suíw1 Vl gdqv o*lqwhuydooh hqwuh w hw W / ohv wdx{ rqw ehdxfrxs prqwì/ o*hqwuhsulvh qh vrxuh sdv
gh fhwwh kdxvvh1 Lqyhuvhphqw/ vl vhv fudlqwhv g*xqh kdxvvh ghv wdx{ ìwdlhqw lqirqgìhv hw tx*dx frqwudluh
ohv wdx{ rqw edlvvì/ hooh qh eìqìflh sdv gh fhwwh edlvvh = hooh q*dxudlw sdv gþ uhqwuhu gdqv oh IUD1
Hqq/ vl rq dfwxdolvh hq w ohv ghx{ x{ ixwxuv +sduidlwhphqw frqqxv hq w,/ rq rewlhqw od ydohxu gh o*ìfkìdq0
flhu txl hvw qxooh hq w sxlvtx*lo q*| d dxfxq x{ hq w =
3 @ Q=E+w> W , Q= 4 . i+w> W> W . ,=
=E+w> W . ,
Edvh
vrlw =
Edvh
i +w> W> W . , @
E+w> W ,
4
E+w> W . ,
hw rq uhwurxyh elhq o*lgìh txh ohv wdx{ ã whuph shxyhqw íwuh fdofxoìv ã sduwlu gh od frxueh ghv wdx{ g*lqwìuíw
dxmrxug*kxl1
Hq sudwltxh/ o*hqwuhsulvh qh wurxyhud sdv wrxmrxuv xq IUD d|dqw h{dfwhphqw ohv fdudfwìulvwltxhv vrxkdl0
wìhv1 Hq sduwlfxolhu/ hw q shxyhqw íwuh surfkhv vdqv íwuh ìjdx{ = od frxyhuwxuh q*hvw sdv sduidlwh hw lo
uhvwh xq ulvtxh uìvlgxho +glw ulvtxh gh edvh,1
4315 Frqwudw Ixwxuh
Oh frqwudw IUD jìqëuh xq xqltxh x{/ fh x{ fruuhvsrqgdqw ã od glìuhqfh hqwuh oh wdx{ ã whuph qìjrflì
hq w hw oh wdx{ gh pdufkì hhfwlyhphqw uìdolvì ã od pdwxulwì gx IUD1 Hq sduwlfxolhu/ fh frqwudw qh jìqëuh
sdv gh x{ ã od gdwh gh frqfoxvlrq gx frqwudw +gdwh w, = oh frqwudw d xqh ydohxu gh pdufkì qxooh ã od
gdwh w1 Wrxwhirlv/ gëv od gdwh w . 4/ ohv dqwlflsdwlrqv ghv pdufkìv vh vrqw prglìhv hw lo q*| d soxv dxfxqh
udlvrq txh fh frqwudw dlw xqh ydohxu gh pdufkì qxooh1 Oh píph w|sh gh frqwudw vh qìjrflhud ã xq wdx{
i +w . 4> W> W . , suredeohphqw glìuhqw gh i +w> W> W . ,1
Gdqv oh fdv gx Ixwxuh/ fhwwh yduldwlrq gh ydohxu gh pdufkì hvw frqvwdwìh hw hquhjlvwuìh fkdtxh mrxu = rq
glw txh oh frqwudw hvw pdunhg0wr0pdunhw1 Dlqvl/ ã fkdtxh gdwh/ vl oh frqwudw Ixwxuh vljqì sdu D hw E hq w d
xqh ydohxu hq w . 4 qìjdwlyh srxu D +hw grqf srvlwlyh srxu E,/ doruv D hvw frqwudlqw gh sd|hu od glìuhqfh
gh ydorulvdwlrq +dsshoìh dssho gh pdujh, hw od sduwlh E hvw fuìglwìh gx píph prqwdqw1 G*xqh fhuwdlqh
idêrq/ od shuwh +rx oh jdlq, qdo +glìuhqfh wdx{ ã whuph hw wdx{ uìdolvì, hvw uìsduwl wrxw dx orqj gh od
ylh gx frqwudw1 Glw dxwuhphqw/ oh frqwudw Ixwxuh v*dssduhqwh ã xqh vìulh gh frqwudwv iruzdug uhqrxyhoìv
wrxv ohv mrxuv = fkdtxh mrxu/ oh frqwudw iruzdug suìfìghqw hvw gìerxfoì hw xq dxwuh vh phw hq sodfh srxu
xqh pdwxulwì g*xq mrxu1
465
Hq rxwuh/ ohv frqwudwv Ixwxuh vrqw vwdqgduglvìv hw jìuìv sdu xqh fkdpeuh gh frpshqvdwlrq/ gh whooh vruwh
txh ohv frqwuhsduwlhv D hw E qh frqwudfwhqw sdv gluhfwhphqw hqwuh hoohv1 Hqq/ ã od gdwh w gh plvh hq
sodfh gh o*rsìudwlrq/ xq gìs÷w gh jdudqwlh hvw ghpdqgì sxlv uhvwlwxì sdu od vxlwh1 Rq wurxyhud fl0dsuëv
ohv vsìflfdwlrqv gx frqwudw Hxuleru 6 prlv gx PDWLI +zzz1pdwli1iu,1 Oh frqwudw hvw frwì vrxv od iruph
433 i +w> W> W . ,1 Dlqvl/ vhorq oh vhqv gh ohxu srvlwlrq/ ohv sduwlhv D hw E uhfhyurqw rx sdlhurqw =
Q=
^+433 i +w . n> W> W . ,, +433 i+w . n 4> W> W . ,,`
Edvh
hq w . n/ hw/ vl oh frqwudw yd ã vrq whuph/ oh x{ =
Q=
û+W> W . , i+W 4> W> W . ,`
Edvh
hq W / dyhf Q @ 4333333 hxurv hw @ 6 prlv @ <3 mrxuv gdqv oh fdv gx frqwudw hxuleru 6 prlv1
Hq sudwltxh/ D +rx E, gìwlhqw xq frpswh vxu ohtxho lo grlw yhuvhu vrq gìs÷w gh jdudqwlh lqlwldo/ hw fh
frpswh hvw gìelwì rx fuìglwì ghv dsshov gh pdujh fruuhvsrqgdqw dx{ yduldwlrqv gh sul{ gx frqwudw Ixwxuh1
Vl rq idlw od vrpph ghv x{ uhêxv rx yhuvìv sdu ohv sduwlhv D hw E/ rq frqvwdwh elhq txh fhwwh vrpph qh
gìshqg hq W txh gh u+W> W . , i +w> W> W . ,/ frpph srxu oh IUD1 Wrxwhirlv/ lo idxw uhpdutxhu txh/
gdqv od phvxuh rü ghv x{ lqwhupìgldluhv rqw olhx/ fh frqwudw hvw xq surgxlw qdqflhu glìuhqw gx IUD1
Hq sduwlfxolhu/ oh wdx{ Ixwxuh i+w> W> W . , q*d sdv gh udlvrq g*íwuh ìjdo dx wdx{ iruzdug gx IUD1
G*dxwuhv frqwudwv Ixwxuhv h{lvwhqw srxu ohvtxhov oh vrxv0mdfhqw hvw xqh reoljdwlrq = D +sdu h{hpsoh, suhqg
grqf hq w o*hqjdjhphqw dyhf E +yld od fkdpeuh gh frpshqvdwlrq, g*dfkhwhu o*reoljdwlrq vrxv0mdfhqwh hq
Hfroh Sro|whfkqltxh
466
W ã xq sul{ gìflgì hq w1 Lo v*djlw g*xq frqwudw ã whuph vxu reoljdwlrq g*Hwdw gh pdwxulwì 43 dqv1 Soxv
h{dfwhphqw/ fhwwh reoljdwlrq hvw fwlyh/ g*rü oh whuph gh frqwudw qrwlrqqho1 Fhfl d vd orjltxh = oh frqwudw
grlw sruwhu ã fkdtxh lqvwdqw vxu xqh reoljdwlrq gh pdwxulwì 43 dqv1 Ru lo q*h{lvwh mdpdlv ã wrxw lqvwdqw
xqh whooh reoljdwlrq1
Vl oh frqwudw hvw gìqrxì dydqw oh whuph gx frqwudw/ D hw E v*ìfkdqjhqw ghv x{ fruuhvsrqgdqw dx{
yduldwlrqv ghv sul{ ã whuph +frpph gdqv oh frqwudw hxuleru,1 Vl oh frqwudw yd mxvtx*ã vrq whuph/ lo |
d olyudlvrq sk|vltxh1 Frpph o*reoljdwlrq vrxv0mdfhqwh q*h{lvwh sdv/ lo h{lvwh xq v|vwëph g*ìtxlydohqfh
hqwuh oh sul{ gh fhwwh reoljdwlrq fwlyh hw oh sul{ ghv reoljdwlrqv uìhoohphqw h{lvwdqwhv1 Oh jlvhphqw gx
Pdwli gìvljqh o*hqvhpeoh ghv RDW txl shxyhqw íwuh olyuìhv dx wlwuh ghv frqwudwv ã whuph qrwlrqqhov1 Oh
idfwhxu gh frqyhuvlrq hvw oh idfwhxu pdwkìpdwltxh txl shuphw gh idluh xqh fruuhvsrqgdqfh hqwuh oh sul{
gh o*reoljdwlrq fwlyh hw oh sul{ ghv reoljdwlrqv gx jlvhphqw1
Xq who frqwudw shuphw gh vh frxyulu frqwuh ohv yduldwlrqv gx wdx{ ã 43 dqv +doruv txh oh frqwudw hxuleru
6 prlv shuphw gh vh frxyulu frqwuh xqh yduldwlrq gx wdx{ ã 6 prlv,1 Ohv fdudfwìulvwltxhv ghv glìuhqwv
frqwudwv qrwlrqqhov gx Pdwli vrqw ohv vxlydqwhv +zzz1pdwli1iu, =
467
Hfroh Sro|whfkqltxh
468
Srxu qlu/ rq srxuud vh uhsruwhu ã od grfxphqwdwlrq wuëv elhq idlwh gh o*ìtxlydohqw dqjodlv gx Pdwli 0 oh
OLIIH 0 ã od q gh fh fkdslwuh1
4316 Ohv vzdsv
Xq vzds hvw xq frqwudw hqwuh ghx{ sduwlhv sdu ohtxho hoohv v*hqjdjhqw ã v*ìfkdqjhu +l1h1 wr vzds, ghv x{
ã ghv gdwhv vsìflìhv1 Gdqv oh fdv g*xq vzds vwdqgdug/ ohv x{ ìfkdqjìv vrqw ghv wdx{ g*lqwìuíw = D +sdu
h{hpsoh, yhuvh ã E xq wdx{ g*lqwìuíw {h +vsìflì lqlwldohphqw, hw E yhuvh ã D xq wdx{ gh pdufkì/ grqf
yduldeoh dx frxuv gx whpsv1 Lo q*| d sdv gh x{ ã od plvh hq sodfh gx frqwudw = oh frqwudw d xqh ydohxu
qxooh1 Od yduldeoh gh qìjrfldwlrq hvw grqf oh wdx{ {h +yhuvì lfl sdu D,1 Soxv h{dfwhphqw/ frpph srxu
ohv IUD rx ohv Ixwxuhv/ ohv x{ vrqw fdofxoìv frpph o*dssolfdwlrq ghv wdx{ g*lqwìuíw ã xq fhuwdlq fdslwdo1
O*ìfkìdqflhu gh wdx{ {h +yhuvì lfl sdu D, v*dsshooh od mdpeh {h gx vzds hw o*ìfkìdqflhu gh wdx{ gh
pdufkì od mdpeh yduldeoh1
Gh idlw/ oh wdx{ {h fulvwdoolvh ohv dqwlflsdwlrqv vxu od ydohxu ghv wdx{ g*lqwìuíw gh pdufkì wrxw dx orqj gh
od gxuìh gh ylh gx vzds1 Fhv frqwudwv vrqw wuëv xwlolvìv fdu lov shuphwwhqw gh wudqviruphu xqh ghwwh ã wdx{
gh pdufkì hq xqh ghwwh ã wdx{ {h1 Lo v*djlw grqf g*xq rxwlo hvvhqwlho xwlolvì sdu ohv hqwuhsulvhv qdqflëuhv
rx qrq qdqflëuhv srxu vh frxyulu frqwuh ohv ulvtxhv gh yduldwlrq ghv wdx{ g*lqwìuíw1
Gh qrpeuhxvhv yduldqwhv vrqw ìylghpphqw srvvleohv = oh wdx{ gh pdufkì shxw íwuh xq wdx{ ã 6 prlv/ 9
prlv hwf1/ od pdwxulwì shxw íwuh gh 4 dq/ 5 dqv/ hwf1/ ohv x{ shxyhqw íwuh yhuvìv ã ghv iuìtxhqfhv glìuhqwv
+wrxv ohv 9 prlv srxu E/ wrxv ohv dqv srxu D/ hwf1
Suhqrqv oh fdv vwdqgdug rü =
oh wdx{ g*lqwìuíw gh pdufkì hvw oh wdx{ g*lqwìuíw u+w> w . , =
E+w> w . , @
4
4 . =u+w> w . ,
ohv x{ vrqw yhuvìv dx{ píphv gdwhv n >
ohv sdlhphqwv vrqw gìwhuplqìv ã o*dydqfh = od glìuhqfh hqwuh oh wdx{ {h hw oh wdx{ yduldeoh u+w> w . ,
q*hvw yhuvìh tx*hq w . +hw qrq hq w,
Lo v*djlud sdu h{hpsoh g*xq vzds vxu o*hxuleru 9 prlv + @ 9 prlv, dyhf ìfkdqjh g*lqwìuíw wrxv ohv 9 prlv1
Od ydohxu gh pdufkì g*xq vzds oruv gh vd plvh hq sodfh +hq w, ìwdqw qxooh/ rq grlw dyrlu =
&
%
] w.n
q
[
=Q=Hw +u+w . +n 4,> w . n, U,= h{s uv gv
3@
w
n@4
rü Q hvw oh fdslwdo qrwlrqqho vxu ohtxho sruwhqw ohv fdofxov g*lqwìuíw/ U oh wdx{ {h yhuvì sdu D +sdu
h{hpsoh, hw uv oh wdx{ frxuw lqvwdqwdqì kdelwxho1 Hq uhpsodêdqw u+w . +n 4,> w . n, sdu vd gìqlwlrq/
rq rewlhqw doruv =
&
%
] w.n
q
q
[
[
4
4 h{s E+w> w . n, . Q
Hw
uv gv
3 @ =Q=U=
E+w . +n 4,> w . n,
w
n@4
n@4
hq xwlolvdqw =
%
E+w> w . n, @ Hw h{s Gh píph/ rq d =
3 @ =Q=U=
hq xwlolvdqw =
q
[
n@4
E+w> w . n, . Q
q
[
n@4
] w.n
w
&
uv gv
%
Hw +4 E+w . +n 4,> w . n,, +h{s %
E+w . +n 4,> w . n, @ Hw.+n4, h{s 469
] w.n
w.+n4,
&
uv gv
] w.+n4,
w
&
uv gv
Fh txl grqqh qdohphqw =
3 @ =Q=
q
[
n@4
U=E+w> w . n, . Q
q
[
n@4
E+w> w . +n 4,, E+w> w . n,
hq xwlolvdqw gh qrxyhdx od sursulìwì gh pduwlqjdolwì ghv sul{ dfwxdolvìv vrxv od suredelolwì ulvtxh0qhxwuh1
Rq shxw dxvvl ìfuluh fhwwh h{suhvvlrq vrxv od iruph =
&
% q
[
E+w> w . n,= î +w> w . +n 4,> w . n, U` =
3@Q
n@4
rü i +w> w . +n 4,> w . n, hvw oh wdx{ ã whuph =
4
i +w> w . +n 4,> w . n, @
E+w> w . +n 4,,
4
E+w> w . n,
Dx wrwdo/ rq rewlhqw od uhodwlrq gh ydorulvdwlrq vxlydqwh =
U@
4 E+w> w . q,
Sq
n@4 E+w> w . n,
Oh wdx{ gx vzds +tx*rq dsshooh dxvvl oh sul{ gx vzds, vh gìwhuplqh grqf/ sdu xq udlvrqqhphqw g*duel0
wudjh hw frpph srxu ohv wdx{ ã whuph/ hq irqfwlrq gh od frxueh ghv wdx{ ã od gdwh w1 Elhq ìylghpphqw/
gëv od gdwh vxlydqwh w . 4/ oh vzds q*d soxv xqh ydohxu qxooh = dqqxohu oh vzds lpsoltxh tx*xqh ghx{
frqwuhsduwlh d xqh ghwwh ylv0ã0ylv gh o*dxwuh1
Rq shxw uhpdutxhu tx*xqh srvlwlrq gh vzds hvw ìtxlydohqwh +srxu D, ã dfkhwhu xqh reoljdwlrq grqw oh
frxsrq hvw xq wdx{ yduldeoh hw ã ìphwwuh xqh reoljdwlrq ã wdx{ {h1
Ohv vzdsv vrqw sdu h{hpsoh ehdxfrxs xwlolvìv sdu ohv edqtxhv gdqv oh fdguh gh ohxu srolwltxh gh jhvwlrq
dfwli0sdvvli +DOP rx Dvvhw Oldelolw| Pdqdjhphqw,1 Lo v*djlw srxu od edqtxh gh frxyulu oh ulvtxh gh
wdx{ gh vrq elodq1 D o*dfwli gx elodq/ rq wurxyhud sdu h{hpsoh ohv suíwv +lpprelolhuv/ shuvrqqhov/ hwf1,
tx*hooh d dffrugìv ã vhv folhqwv hw dx sdvvli vhv uhvvrxufhv +uhqdqfhphqw gh od edqtxh vxu ohv pdufkìv gh
fdslwdx{,1 Oruvtxh ohv wdx{ gh pdufkì prqwhqw/ ohv uhvvrxufhv gh od edqtxh ghylhqqhqw soxv frþwhxvhv
fdu vrq uhqdqfhphqw hvw sulqflsdohphqw ã wdx{ gh pdufkì > ã o*lqyhuvh/ ohv uhyhqxv jìqìuìv sdu o*dfwli
q*dxjphqwhqw sdv fdu/ wuëv vrxyhqw/ ohv suíwv dffrugìv sdu od edqtxh vrqw ã wdx{ frqvwdqw1 Xqh edqtxh
hvw grqf jìqìudohphqw h{srvìh ã xqh kdxvvh ghv wdx{ = rq glw tx*hooh d xq h{fìghqw g*hpsorlv ã wdx{
{h1 Ohv edqtxhv phwwhqw txrwlglhqqhphqw hq sodfh ghv surjudpphv gh vzds txl ohxu shuphwwhqw gh
uì0ìtxloleuhu ohxu elodq hq wudqvirupdqw yluwxhoohphqw ghv hpsorlv ã wdx{ {h hq hpsorlv ã wdx{ yduldeoh1
Xqh frxyhuwxuh sduidlwh hvw whooh txh/ oruvtxh ohv wdx{ prqwhqw o*lpsdfw vrlw oh píph vxu oh sdvvli hw vxu
o*dfwli1 Dlqvl/ oh eìqìfh gh od edqtxh = uhyhqxv ghv hpsorlv 0 frþwv ghv uhvvrxufhv qh ydulh soxv oruvtxh
ohv wdx{ ydulhqw1
Ghuqlëuh uhpdutxh = od uhodwlrq oldqw ohv wdx{ gx vzds hw ohv wdx{ }ìur0frxsrq hvw xwlolvìh srxu uhfrqvwlwxhu
od frxueh ghv wdx{ }ìur0frxsrq1 Ohv revhuydwlrqv vrqw ohv wdx{ gh vzds U srxu ohv glìuhqwhv pdwxulwìv
hw ohv lqfrqqxhv ohv sul{ }ìur0frxsrq1
4317 Fds hw Iorru
Rq hqylvdjh ã sduwlu gh pdlqwhqdqw ghv dfwliv rswlrqqhov1 Gdqv fhv fdv/ rq qh shxw soxv rewhqlu xqh
irupxoh gh ydorulvdwlrq qh gìshqgdqw txh gh od frxueh ghv wdx{1 Gìvrupdlv od yrodwlolwì lqwhuylhqgud hw
grqf gh idlw xq prgëoh grlw íwuh vsìflì1 Ohv fds 2 rru vrqw ohv ìtxlydohqwv ghv fdoo 2 sxw1
Rq gìqlw g*derug xq fdsohw frpph xqh rswlrq txl gìolyuh xq x{ ìjdo ã od glìuhqfh 0 vl hooh hvw srvlwlyh
0 hqwuh xq wdx{ gh pdufkì hw xq wdx{ g*h{huflfh1 Hq uhsuhqdqw ohv qrwdwlrqv suìfìghqwhv/ oh sd|r v*ìfulud =
=Q= û+W> W . , I `.
Hfroh Sro|whfkqltxh
46:
Oh fds hvw doruv xqh froohfwlrq gh fdsohwv/ f*hvw0ã0gluh xq dfwli txl gìolyuh ohv x{ =
.
=Q= û+w . +n 4,> w . n, I `
dx{ gdwhv w . n1 Od ydohxu gh pdufkì gh fhw dfwli q*hvw ìylghpphqw sdv qxooh hw ydxw =
&
%
] w.n
q
[
.
Fds @ =Q=
Hw û+w . +n 4,> w . n, U` = h{s uv gv
w
n@4
tx*rq qh shxw fdofxohu h{solflwhphqw txh vl rq d vsìflì xq prgëoh sduwlfxolhu gdqv ohtxho rq frqqdlw ohv
ghqvlwìv ghv yduldeohv doìdwrluhv txl dssdudlvvhqw gdqv o*hvsìudqfh1
Rq shxw uhpdutxhu txh oh fds shxw vh frqfhyrlu frpph xqh froohfwlrq g*rswlrq vxu }ìur0frxsrq1 Hq hhw/
hq xwlolvdqw od gìqlwlrq gx wdx{ u+w. +n 4,> w. n, hw ohv píphv dvwxfhv gh fdofxo txh gdqv od vhfwlrq
suìfìghqwh/ rq rewlhqw =
&
%
.
] w.+n4,
q
[
4
E+w . +n 4,> w . n, = h{s Hw
uv gv
Fds @ +4 . U, =Q=
4 . U
w
n@4
Hqq/ rq shxw ìjdohphqw ìfuluh xqh uhodwlrq gh sdulwì fds 2 rru1 Od glìuhqfh hqwuh xq fds hw xq rru
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