_ujâr`a OWT_A MG ^GMTG@TA BWCB 2 OAHLGO@JGH]A_ IGTB@_ @htramuéïa Gqu`pbjghta mg ^ratgéïa @hm`v`mubc Gqu`pbjghta mg ^ratgéïa Oacgt`vb _giurbhéb oaj bs Gsobmbs Igajgtr`b Dâs`ob Cg`turb g @htgrprgtbéïa mg ^rafgtas _`stgjb Jàtr`oa Mgo`jbc Ârgb,, ^gr Ârgb ^grêjgt êjgtra ra g Uacujg Jbtgr`bc pbrb Oahstruéïa Ugfb sg Bprghmgu
@hst`tuta Wh`vgrsbc Drbs`cg`ra Bmj`h`strbéïa Cu`z Egrhbhma M`h`z Hbsa Fasà Obrcas M`h`z Hbsa @hæs M`h`z Hbsa Igrghtg Igrbc Jamgsta ^bhtbcàb M`rgéïa mg Brtg Obrcas Gmubrma ^. Hbsa Bss`stghtg Furêm`ob Tbebgcb Hbsa
M`rgtarb Igrbc @rghg Tamr`iugs Tamr`iug s mg Ac`vg`rb ]g`xg`rb ]g`xg`rb T`dg`ra Oaarmghbéïa M`mât`ob Ybcmaj`ra Tgool` Eatairbe`b Tadgrta Obrcas Bcvgs Butar Ybcmaj`ra Tgool`
^ramuéïa Jbroas ^rbma mg Obrvbcla
Gm`tarbéïa @hst`tuta Wh`vgrsbc Drbs`cg`ra
@jbighs Bcgxbhmrg Jars`ccb
@jprgssïa @WDTB @WDTB - @hmöstr`b Irâe`ob g Gm`tarb Ctmb. Tamav`b Gstbmubc Da`tuvb - @pgrÿ, kj 2,2 Objpas mg Da`tuvb - Da`tuvb - _^ OG^ 2<77>->>>
Tgv`sïa Tasgc` Bhbstâo`a _`cvb Jbro`b Jarg`rb mg Obrvbcla
]amas as m`rg`tas sïa rgsgrvbmas. Hïa à pgrj`t`mb b rgpramuéïa tatbc au pbro`bc mgstg oursa.
BWCB 2 OAHLGO@JGH]A_ IGTB@_
jgsja g pbrb as bfumbhtgs g bux`c`brgs qug gcg oahtrbtbr oahtrbtbr.. As `hstrujghtas qug oajp÷gj a G^@ hb oahstruéïa o`v`c sïa gstgs6 obpbogtg, ÿoucas, pratgtargs bur`oucbrgs, jâsobrb pbrb pratgéïa oahtrb pÿ g pramutas quêj`oas, cuvbs mg oaura g datbs au dat`hbs.
@htramuéïa A tgrja oahstruéïa o`v`c ghicadb mgsmg b obsb papucbr bas oahfuhtas rgs`mgho`b`s mg cuxa, irbhmgs gm`eêo`as rgs`mgho`b`s, oajgro`b`s g `hmustr`b`s, oaj pbrt`o`pbéïa mg ghighlg`ras, brqu`tgtas, tàoh`oas g jïa mg adrb gspgo`bc`zbmb. A pgmrg`ra à uj prae`ss`ahbc qug trbdbclb hb ârgb mg oahstruéïa o`v`c gxgoutbhma bs gtbpbs mg euhmbé÷gs, cgvbhtbjghta mg pbrgmgs, rgvgst`jghta mg p`sas g pbrgmgs g bobdbjg db jght htas as.. Gs Gssg sg pr prae ae`s `ss` s`ah ahbc bc mg mgvg vg tg tgr r oahlgo`jghta sadrg cg`turb g `htgrprgtbéïa mg pcbhtbs g oartgs mg prafgtas brqu`tgtýh`oas, dgj oaja bciujbs gspgo`e`obé÷gs, s`stgjbs mg jgm`mbs, oâcouca mg ârgbs g vacujgs qug clg pgrj`tgj vgr`e`obr a oahsuja g gjprgia mg jbtgr`b`s, egrrbjghtbs g gqu`pbjghtas tàoh`oas hb prgpbrbéïa mg bribjbssbs. ]uma `ssa sïa oahlgo`jghtas gssgho`b`s pbrb earjbéïa mg uj daj pgmrg`ra. À ju`ta `jpartbhtg qug a bcuha sb`db qug, pbrb gxgrogr b prae`ssïa, mgvg rgspg`tbr bs harjbs mg sgiurbhéb mgtgrj`hbmbs pgca J`h`stàr`a ma ]rbdbcla g Gjprgia, ut`c`zbhma, mg boarma oaj b bt`v`mbmg hb adrb, a Gqu`pbjghta mg ^ratgéïa @hm`v`mubc (G^@) g a Gqu`pbjghta mg ^ratgéïa Oacgt`vb (G^O).
Obpbogtg mg sgiurbhéb Gj qubcqugr bt`v`mbmg mghtra mg ujb adrb, à adr`ibtÿr`a a usa mg obpbogtgs mg sgiurbhéb (e`iurb (e`iurb 2). 2).
E`iurb 2 ‟ Obpbogtg ‟ Obpbogtg mg sgiurbhéb sgiurbhéb..
As obpbog obpbogtgs tgs sïa m`v`m m`v`m`mas `mas gj ocbss ocbssg gBg ocbssg ocbs sg D. As mb ocbssg B sïa pbrb pbrb usa igrbc, igrbc, gxogta og ta pb pbrb rb tr trbd bdbc bcla lass oa oaj j gh ghgr gri` i`b b gc gcàt àtr` r`ob ob;; fâ as mb ocbssg D tbjdàj sïa pbrb usa igrbc, `hocus`vg pbrb pb rb tr trbd bdbc bcla lass oa oaj j gh ghgr gri` i`b b gc gcàt àtr` r`ob ob.. A ob obso soa, a, mg uj ja jama ma ig igrb rbc,c, à eb ebdr dr`o `obm bma a oa oaj j jb jbtà tàr` r`bb-pr pr`j `jb bî dbsg mg pac`gt`cgha mg bctb mghs`mbmg.
Gqu`pbjghta mg ^ratgéïa @hm`v`mubc Rubhma a pgmrg`ra à bmj`t`ma pbrb trbdbclbr gj ujb adrb oaja euho`ahâr`a, gcg rgogdgrâ ma gjprgibmar as `hstrujghtas adr`ibtÿr`as ma G^@. Fâ a pgmrg`ra butýhaja (qug trbdbclb par oahtb prÿpr`b) mgvgrâ m`spar mg `hstrujghtas mg sgiurbhéb pbrb s` Oursa mg ^gmrg`ra - Bucb 2
As obpbogtgs mg bdb erahtbc, t`pa dahà mg fÿqug`, au mg bdb tatbc pamgj sgr tbhta ocbssg B oaja D. ?
@hst`tuta Wh`vgrsbc Drbs`cg`ra
^amgj sgr mg gspujb au s`c`oahg (e`iurb ( e`iurb =). =).
]ïa `jpartbhtg oaja a obsoa, b suspghsïa à eg`tb par olbrhg`rb g o`htas ebdr`obmas mg tgo`ma, hïa mg pcâst`oa rêi`ma. Gssb suspghsïa à b rgspahsâvg rgspahsâvgcc pgca bjartgo`jgh bjartgo`jghta, ta, gj obsa 4). mg pbhobmbs (e`iurb ( e`iurb 4).
E`iurb = ‟ ^ratgtargs ‟ ^ratgtargs bur`oucbrg bur`oucbrgs. s.
Jâsobrb mgsobrtâvgc mg pratgéïa E`iurb 4 ‟ _uspghsïa ‟ _uspghsïa hb pbrtg `htgrhb ma obpbogtg.
Gssb jâsobrb pratgig a pgmrg`ra oahtrb pag`rbs ar`uhmbs mb rbspbigj au c`xbjghta mg supgreêo`gs. ^ar sgr mgsobrtâvgc, mgvg sgr traobmb qubhma gst`vgr ghtup`mb mg sufg`rb, pgreurbmb, rbsibmb au oaj olg`ra au iasta mg 7). oahtbj`hbhtgss (e`iurb 7). oahtbj`hbhtg
ßoucas mg pratgéïa Gssgs ÿoucas sïa as rgoajghmbmas pbrb pratgéïa mas aclas, gspgo`e`obmas mg boarma oaj a r`soa oaja pag`rb, jbh`pucbéïa mg pramuta mu tass qu quêj êj`o `oas as g mg jb jbtg tgr` r`b` b`ss sÿ sÿc`m c`mas as pg pgre reur urbh bh-tgs, rbm`béïa gj sgrv`éas mg sacmb g oartg b qugh qu ghtg tg oa oaj j jb jbéb ébr` r`oa oa,, eg egrr rrbi bigh ghss gt gto. o. ( e` e`iu iurb rb ?).
E`iurb 7 ‟ Jâsobrb ‟ Jâsobrb mgsobrtâvgc mg pratgéïa. E`iurb ? ‟ ßoucas ‟ ßoucas mg pratgéïa.
Cuvbs mg pratgéïa
^ratgtargs bur`oucbrgs
Bs cuvbs mg pratgéïa à a `hstrujghta oaj jb`ar m`vgrs`mbmg mg gspgo`e`obé÷gs. Hb oahstruéïa o`v`c, bs jb`s usbmbs sïa bs mg câtgx au ^UO (e`iurb (e`iurb 5) 5) g mg rbspb mg oaura.
Gcgs pratgigj as auv`mas oahtrb ruêmas bo`jb mas c`j`tgs tacgrâvg`s, qug à mg <7 mD (mgo (m go`d `dà` à`s) s) pb pbrb rb a` a`ta ta la larb rbss mg gx gxpa pas` s`éï éïa. a. Oursa mg ^gmrg`ra - Bucb 2
=
@hst`tuta Wh`vgrsbc Drbs`cg`ra
irbhmgs gm`eêo`as, bs harjbs mg sgiurbhéb adr`ibj a pgmrg`ra b usbr o`hta mg sgiurbhéb c`ibma (btbma) b uj obda mg sgiurbh rb héb pbrb tr trbd bdb bcl cla as rg rgbc bc`z `zb bmas gj bhmb`jgs jgoäh`oas suspghsas b ujb bcturb mg 4,>> j (e`iurb ( e`iurb <). <).
E`iurb 5 ‟ Cuvbs ‟ Cuvbs mg pratgéïa mg ^UO.
Bs cuvbs mg câtgx au ^UO pratgigj oahtrb egr`jghtas g oahtbta oaj a o`jghta g pramutas quêj`oas; fâ bs mg rbspb mg oaura, oahtrb jbtgr`bc oarras`va g qug passb pravaobr oartgs oaja cbsobs mg jbmg`rb, dbrrbs mg egrra gto. À `jpartbhtg qug bs cuvbs sgfbj ma tbjbhla `mgbc pbrb ibrbht`r sgiurbhéb.
_bpbtas mg sgiurbhéb E`iurb < ‟ ^gmrg`ra ‟ ^gmrg`ra oaj o`hta mg sgiurbhéb trbdbclbhma trbdbclbhma gj uj bhmb`jg jgoäh`oa.
Bs da datb tbss au da dat`t`hb hbss mg ^U ^UO O oa oaj j sa sacb cbma ma bh bh-t`mgrrbpbhtg sïa `hm`obmbs pbrb caob`s öj`mas au `huhmbmas, au oaj pramutas quêj`oas; fâ as obcébmas oaj d`qug`rb mg béa pratgigj oahtrb jbe`iu iurb rb 9). tgr`r`b` tg b`ss pg pgsb sbma mass qu qug g pa pamg mgj j ob ob`r`r ha hass pà pàss (e`
A o`hta mg sgiurbhéb ut`c`zbma gj bt`v`mbmgs oaj r`soa mg qugmb gj bcturbs supgr`args b 4,>> j à a jamgca t`pa pbrbqugm`stb oaj 0). c`ibéïa erahtbc au marsbc (e`iurb (e`iurb 0).
E`iurb 9 ‟ _bpbtas ‟ _bpbtas mg sgiurbhéb sgiurbhéb..
Gqu`pbjghta mg ^ratgéïa Oacgt`vb Rubhta Rubht a ba bass `hs `hstru trujgh jghtas tas ut` ut`c`z c`zbma bmass pgca pgmrg`ra gj supgreêo`gs gxtgrhbs mg Oursa mg ^gmrg`ra - Bucb 2
E`iurb 0 ‟ O`hta ‟ O`hta mg sgiurbhé sgiurbhéb b t`pa pbrbqug pbrbqugm`stb. m`stb. 7
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Autra m`spas`t`va mg sgiurbhéb ut`c`zbma pbrb trbdbclas gj bcturb à b obmg`rb suspghsb, qug tbjdàj adr`ib a pgmrg`ra b usbr a o`hta mg sgiurbhéb t`pa pbrbqugm`stb c`ibma b uj obda mg sgiurbhéb olbjbma trbvb-qugmb (e`iurb ( e`iurb 2>). 2>).
• A caobc ahmg b gsobmb sgrâ pas`o`ahbmb mgvg gstbr c`jpa, `sghta mg irbxb au qubcqugr autra pramuta qug tarhg a olïa gsoarrgibm`a. • @sacg a caobc mg trbdbcla, s`hbc`zbhmaa pbrb pratgigr b o`roucbéïa mg trbhsguhtgs (autrbs pgssabs qug obj`hlbj ha caobc). • _gjprg ba sud`r au mgsogr, pas`o`ahb-sg mg erghtg b gsobmb, sgiurbhma has jahtbhtgs oaj bs mubs jïas. • Bs gsobmbs mgvgj sgr usbmbs par ujb sÿ pgssab. Mg uj jama igrbc, ujb gsobmb pamg biughtbr uj tatbc mg 22> b 2?7 ki ghtrg pgssabs, pgssabs, egrrbjghtbs egrrbjghtbs g jbtgr`b`s. Tgspg`tg a pgsa gspgo`e`obma pgca ebdr`obhtg. • A oajpr`jghta jâx`ja mas jahtbhtgs à mg 5 j, g a gspbébjghta ghtrg mgirbus, mg 47 b ?> oj. • Gv`tb-sg p`htbr bs gsobmbs pbrb qug b t`htb hïa aouctg passêvg`s rbolbmurbs, rbolbmurbs, tr`hobs au qubcqugr autra mgeg`ta qubhma ear `hspgo`ahbmb bhtgs ma sgu usa. Oaja pratgéïa, pamg-sg ghvgrh`zâ-cb. • Bs gsobmbs mgvgj tgr mgirbus bht`mgrrbpbhtgs dgj bpa`bmbs hb supgreêo`g mb dbsg. _g hgogssâ hgogssâr`a, r`a, ut`c`zg obcéas ( e`iurb 22) 22 ) .
E`iurb 2> ‟ Obmg`rb ‟ Obmg`rb suspghsb oaj trbvb-qugmb.
A s`stgjb mg e`xbéïa mb obmg`rb suspghsb mgvg sgr `hmgpghmghtg ma obda-iu`b ma trbvb-qugmb trbvb-qugmb..
_giurbhéb _giurb héb oaj bs Gsobmbs Gj trbdbclas oaj bcturbs, a pgmrg`ra mgvg ut`c`zbr gqu`pbjghtas mg sgiurbhéb oaja a bhmb`jg, b obmg`rb suspghsb gto. Fâ gj `jÿvg`s tàrrgas au bssadrbmbmas, ut`c`zbjas gsobmbs pbrb a trbdbcla gj ârgbs `htgrhbs g gxtgrhbs oaj ergquæho`b ergquæho`b.. Gjdarb sgfb dbstbhtg papucbr, ut`c`zbmb btà par mahbs mg obsb pbrb bcobhébr pgqughbs bcturbs mg jÿvg`s, oart`hbs gto., b gsobmb à uj `hstrujghta qug, prae`ss`ahbcjghtg, mgvg sgr bprapr`bma ba sgrv`éa au trbdbcla b sgr rgbc`zbma, rgbc`zbma, oaj rgspg`ta b bciujbs harjbs mg sgiurbhéb. Ugfbjas Ug fbjas bciuj bciujbs bs rgoaj rgoajghmb ghmbé÷gs é÷gs pbrb a usa mg gsobmbs gj trbdbclas oaj bcturbs6 Oursa mg ^gmrg`ra - Bucb 2
E`iurb 22 ‟ Gsobmb ‟ Gsobmb sadrg obcéas.
• ^brb trbdbclas gj qug lbfb prax`j`mbmg oaj `hstbcbé÷gs gcàtr`obs, sÿ à pgrj`t`ma a usa mg gsobmbs mg jbmg`rb au mg e`drbs, qug hïa sïa oahmutarbs mg gcgtr`o`mbmg. 5
@hst`tuta Wh`vgrsbc Drbs`cg`ra
• Wt`c`zg oarmb au autra jg`a pbrb cgvbhtbr g db`xbr egrrbjghtbs g autras jbtgr`b`s usbmas ha trbdbcla. • Hïa oacaobr b gsobmb sadrg qubcqugr gqu`pbjghta au jâqu`hb. • Hïa sud`r bcàj mas ma`s öct`jas mgirbus mb gsobmb (e`iurb ( e`iurb 24). 24).
Bs c`hlbs c `hlbs pamgj sgr rgtbs au ourvbs. Bs rgtbs pamgj aoupbr træs m`egrghtgs pas`é÷gs ha gspbéa6 lar`zahtbc, vgrt`obc g `hoc`hbmb; fâ bs c`hlbs ourvbs pamgj earjbr b o`rouhegræh2= g 27 27). o`b g b gc`psg (e`iurbs ( e`iurbs 2= g ).
E`iurb 2= ‟ C`hlbs ‟ C`hlbs rgtbs lar`zahtbc, vgrt`obc g `hoc`hbmb `hoc`hbmb..
E`iurb 27 ‟ C`hlbs ‟ C`hlbs ourvbs earjbhma ujb o`rouhegræho`b g ujb gc`psg. E`iurb 24 ‟ Jbhg`rb ‟ Jbhg`rb `hoarrgtb6 pgmrg`ra trbdbclbhma ha öct`ja mgirbu mb gsobmb.
C`hlb j`stb à oajpastb par pbrtgs mg c`hlbs rgtbs g ourvbs.
Igajgtr`b Dâs`ob Tgvgr bciuhs oahog`tas dâs`oas mg igajgtr`b oaja c`hlbs, ähiucas g pr`sjbs igajàtr`oas à ju`ta `jpartbhtg pbrb cg`turb g `htgrprgtbéïa mg mgsghlas g pcbhtbs db`xbs g oartgs, dgj oaja cgvbhtbjghta mg tgrrghas (pr`ho`pbc (pr`h o`pbcjghtg jghtg as mg earjb earjbtas tas `rrgiu `rrgiucbrgs cbrgs)) g caobéïa mb adrb sadrg gcgs.
_ïa oahs`mgrbmbs pbrbcgcbs mubs c`hlbs qug, par jb`s qug sg pracahiugj, huhob sg ghoahtrbj (e`iurb (e`iurb 25). 25).
C`hlbs C`hlb à mge`h`mb, igajgtr`objghtg, oaja ujb suogssïa mg pahtas (e`iurb ( e`iurb 2?). 2?). E`iurb 25 ‟ C`hlbs ‟ C`hlbs pbrbcgcbs.
_ïa pgrpghm`oucbrgs mubs c`hlbs rgtbs qug sg ghoahtrbj earjbhma uj ähiuca mg 0>¾, au sgfb, a ähiuca rgta (e`iurb ( e`iurb 29). 29).
E`iurb 2? ‟ Ma`s ‟ Ma`s t`pas mg c`hlbs. Oursa mg ^gmrg`ra - Bucb 2
9
@hst`tuta Wh`vgrsbc Drbs`cg`ra
E`iurb 29 ‟ C`hlbs ‟ C`hlbs pgrpghm`oucbrgs.
E`iurb 4> ‟ Ähiuca ‟ Ähiuca biuma.
Ähiucas
Ähiuca adtusa. À adtusa. À a qug tgj ujb bdgrturb jb`ar qug 0>¾; partbhta, à jb`ar qug a äh42). iuca rgta (e`iurb (e`iurb 42).
Olbjbj Olbj bjas as mg äh ähiu iuca cass b jg jgm` m`mb mb ma bebstbjghta ghtrg mubs c`hlbs rgtbs qug jbhtæj uj pahta oajuj olbjbma vàrt`og. Bs c`hlbs qug earjbj as ähiucas sïa olbjbmbs mg cbmas mg ähiuca (e`iurb ( e`iurb 2<). 2<).
E`iurb 42 ‟ Ähiuca ‟ Ähiuca adtusa.
A vbcar ma ähiuca hïa mgpghmg ma oajp oa jpr`j r`jght ghta a mb mbss c`h c`hlb lbss qu qug g ear earjbj jbj as cbmas, jbs ma vbcar mb bdgrturb mgcbs.
E`iurb 2< ‟ Gxgjpca ‟ Gxgjpca mg ähiuca.
Ugfb b ocbss`e`obéïa mas ähiucas6 B d`ssgtr`z ma ähiuca à b c`hlb qug m`v`mg 44). uj ähiuca gj ma`s ähiucas `iub`s (e`iurb ( e`iurb 44).
Ähiuca rgta. Or`bma rgta. Or`bma b pbrt`r mg ujb c`hlb lar`zahtbc g ujb vgrt`obc pgrpghm`oucbr, ear20). jbhma gxbtas 0>¾ ghtrg s` (e`iurb ( e`iurb 20).
E`iurb 44 ‟ D`ssgtr`z ‟ D`ssgtr`z ma ähiuca mg 5>¾.
A ähiuca rgta, pravbvgcjghtg, à a jb`s v`sta gj bpc`obé÷gs prât`obs hb oahstruéïa o`v`c6 ghoahtra mg pbrgmgs oaj a olïa, gsqubmr`bs mbs fbhgcbs gto.
E`iurb 20 ‟ Ähiuca ‟ Ähiuca rgta.
Ähiuca biuma. À biuma. À a ähiuca qug jgmg jghas qug 0>¾; partbhta, à uj ähiuca jghar qug a rgta (e`iurb (e`iurb 4>). 4>). Oursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
]r`ähiuca
Cg`turb g @htgrprgtbéïa mg ^rafgtas
À b e`iurb igajàtr`ob earjbmb par træs cbmas g træs ähiucas. Ugfb b ocbss`e`obéïa ma 4?)6 tr`ähiuca (e`iurb (e`iurb 4?)6
As mgsghlas mg uj prafgta mg brqu`tgturb oajprgghmgj uj oahfuhta mg prafgé÷gs, v`stbs g oartgs (sgoé÷gs) mg uj gm`eêo`a au pbrtg mgstg, oaj oahvghé÷gs qug ebo`c`tbj b `htgrprg `htg rprgtbéï tbéïa a ma mgs mgsghla ghla g, oah oahsgqu sgqughtg ghtg-jghtg, b gxgouéïa mb adrb.
• ]r`ähiuca rgtähiuca6 qubhma passu` uj ähiuca rgta; • ]r`ähiuca `sÿsogcgs6 qubhma ma`s cbmas tæj b jgsjb jgm`mb; • ]r`ähiuca gsobcgha6 qubhma b jgm`mb mas træs cbmas à m`egrghtg.
^cbhtb db`xb B pcb pcbhtb htb db` db`xb xb à a mgsg mgsghla hla gj pra prafgfgéïa lar`zahtbc mb prÿpr`b adrb, gj tbjbhla rgmuz`ma. Gssb rgmuéïa usbmb pbrb m`j`hu`r praparo`ahbcjghtg praparo`ahbcjgh tg bs jgm`mbs rgb`s ma pra fgta fgt a à mgha mghaj`hb j`hbmb mb gsob gsobcb. cb. ^ar gxgjpca6 b pcbhtb db`xb mg uj sbcïa oaj 5,>> x =,>> j, hb gsobcb mg 262>> (uj par ogj), sgr`b mgsghlbmb oaj 5 x = oj. @ssa s`ih`e`ob qug obmb oghtêjgtra ma mgsghla oarrgspahmg b 2,>> j hb jgm`mb rgbc.
E`iurb 4? ‟ ]r`ähiucas. ‟ ]r`ähiucas.
O`rouhegræho`b
Mgpa`s qug oahlgogr as gcgjghtas g b s`jdacai`b ut`c`zbmas hb rgprgsghtbéïa mas mgsghlas brqu`tgtýh`oas, mbrgjas bciuhs gxgjpcas mg oâcoucas s`jpcgs qug a pgmrg`ra pamgrâ ebzgr pbrb mgtgrj`hbr bs m`jghs÷gs rgb`s dbsgbmbs ha mgsghla qubhma gstbs hïa eargj gspgo`e`obmbs, au qubhma qu`sgr gcbdarbr bciuj prafgta au bprghmgr b gxgoutbr b rgmuéïa pbrb mgsghlâ-ca.
À ujb c`hlb ourvb g egolbmb s`tubmb huj jgsja pcbha. Gssb c`hlb gstâ gqu`m`stbhtg (jgsjb m`stäho`b) mg uj pahta olbjbma oghtra. B c`hlb qug pbssb pgca oghtra mb o`rouhegræho` ræ ho`b b g uh uhg g ma ma`s `s pa pahta htass mg mgstb stb olb olbjb jb-sg -sg m`äjgtra (M), g b c`hlb rgtb qug uhg a oghtra mb o`rouhegræho`b b qubcqugr pahta mgstb à a 4=. rb`a (T) ‟ e`iurb ‟ e`iurb 4=.
Hb pcbhtb db`xb, à jastrbmb b m`str`du`éïa ma mass oý oýja jama mass oa oaj j sg sgus us rgs rgspg pgot` ot`va vass hajgs (qubrta, sbcb, oaz`hlb, dbhlg`ra, ârgb mg sgrv`éa gto.) g subs m`jghs÷gs, bs qub`s sïa gspgo`e`obmbs gj jgtras au oghtêjgtras, `hocu`hma b gspgssurb mb pbrgmg, b caobc`zbéïa g m`jghs÷gs mg partbs, fbhgcbs g vïas, jbtgr`bc b sgr ut`c`zbma has p`sas, caobc`zbéïa mg p`bs, bpbrgclas sbh`târ`as, tbhqugs mg cbvbhmgr`b, brjâr`as gjdut`mas g btà jgsja jÿvg`s mgpghmghma ma obsa. ]bjdàj à jastrbmb b prafgéïa ma tgclbma qubhma gstg uctrbpbssbr as c`j`tgs mb adrb, earjbhma as dg`rb`s.
E`iurb 4= ‟ O`rouhegræho`b. ‟ O`rouhegræho`b.
A rb`a à `iubc ba m`äjgtra m`v`m`ma par 4), au a m`äjgtra à `iubc b mubs ma`s (T (T : M ù 4), T). vgzgs a rb`a (M (M : 4 x T). Oursa mg ^gmrg`ra - Bucb 2
0
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Ugfb, hb e`iurb hb e`iurb 47, 47, a mgsghla mg ujb pcbhtb db`xb.
E`iurb 49 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg pbrgmg db`xb.
^artbs g fbhgcbs sïa rgprgsghtbmbs mb sgiu`htg jbhg`rb (e`iurb ( e`iurb 4<)6 4<)6
E`iurb 47 ‟ Jamgca ‟ Jamgca mg pcbhtb db`xb.
^brb cgr g `htgrprgtbr ujb pcbhtb db`xb, à prgo`sa oahlgogr as sêjdacas irâe`oas qug rgprgsghtbj partbs, fbhgcbs, brjâr`as gjdut`mas, pgébs sbh`târ`bs, jÿvg`s gto. Bs pbrgmgs bctbs (ma p`sa btà a tgta) sïa rgprgs rgp rgsght ghtbm bmbs bs pa parr ma ma`s `s trb trbéas éas ira irassa ssass au 45). olg`as (e`iurb (e`iurb 45).
E`iurb 45 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg pbrgmgs bctbs.
^brgmgs db`xbs sïa rgprgsghtbmbs par mubs c`hlbs mg gspgssurb jàm`b, hïa olg`b, oaj `hm`obéïa mb jgm`mb mg bcturb (e`iurb ( e`iurb 49). 49). Oursa mg ^gmrg`ra - Bucb 2
E`iurb 4< ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg partbs g fbhgcbs. 2>
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Adsgrvg qug, pbrb rgprgsghtbr partbs, b pcbhtb 2 s`ih`e`ob qug b partb sg ghoahtrb ghtrg oýjamas oýjamas oaj p`sas ma jgsja hêvgc g jbtgr`bc (tboas, ogräj`ob ogräj`ob gto.). Hb pcbhtb 4, a p`sa hïa gstâ ha jgsja hêvgc; partbhta, tgj sacg`rb (mgirbu) au gstâ ghtrg oýjamas oaj p`sas p`s as m`e m`egrg grghtg htgss (uj tgj p`s p`sa a ogr ogräj` äj`oa; oa; autra, tboas). Rubhta îs m`jghs÷gs mbs partbs g fbhgcbs, gcbs pamgj sgr `hm`obmbs mb sgiu`htg jbhg`rb6
jgra bhtgs ma s`hbc mg juct`pc`obéïa ha hujgrbmar rgprgsghtb b cbriurb; a höjgra mgpa`s mgssg s`hbc, ha hujgrbmar, rgprgsghtb b bcturb; g a höjgra ha mghaj`hbmar rgprgsghtb ?2). a pg`tar`c (e`iurb (e`iurb ?2).
• ^ar uj trbéa mg erbéïa gj qug a hujgrbmar rgprgsghtb b cbriurb g a mghaj`hbmar, b bcturb mb partb au mb fbhgcb (e`iurb ( e`iurb 40). 40). E`iurb ?2 ‟ M`jghsïa ‟ M`jghsïa mg partbs g fbhgcbs par uj s`hbc mg juct`pc`obéïa g trbéa mg erbéïa.
^g`tar`c à b m`stäho`b gj qug b fbhg fb hgcb cb sg s` s`tu tubb bo bo`j `jbb ma p` p`sa sa.. Bs oatbs qug qug `hm`obj as as m`egrghtgs m`egrghtgs hêvg`s mg p`sas sïa rgprgsghtbmbs mb sgiu`htg jb?4). hg`rb (e`iurb (e`iurb ?4).
E`iurb 40 ‟ M`jghsïa ‟ M`jghsïa mg partbs g fbhgcbs par uj trbéa mg erbéïa.
• Bs jgm`mbs tbjdàj pamgj sgr `hm`obmbs par uj s`hbc mg juct`pc`obéïa (x), qug s`ih`e`e`ob ob –p –par ar‚, ‚, gj qu qug g a pr pr`j `jg` g`ra ra hö höjg jgra ra rg rgpr prgs gsgh ghtb tb ?>). b cbriurb g a sgiuhma, b bcturb (e`iurb ( e`iurb ?>).
E`iurb ?4 ‟ Oatbs ‟ Oatbs `hm`obmarbs mg m`egrghtgs hêvg`s mg p`sa.
Ba s sê Bas sêjd jdac acas as bp bprg rgsg sght htbm bmas as hb öc öc-t`jb t`j b e` e`iu iurb rb,, pa pamg mgja jass bo borg rgso sogh ghtb tbr r b`hmb bs cgtrbs HB g HA ghtrg pbræhtgsgs6 tgs gs6 HB `hm `hm`ob `ob –H` –H`vgc vgc Bobdbma Bobdbma‚‚ g HA, –Hêvgc ha Assa‚ (oahtrbp`sa pbrb rgogdgr a bobdbjghta).
E`iurb ?> ‟ M`jghsïa ‟ M`jghsïa mg partbs g fbhgcbs par uj s`hbc mg juct`pc`obéïa.
• Autrb jbhg`rb à par jg`a mg uj s`hbc mg juct`pc`obéïa g uj trbéa mg erbéïa6 a höOursa mg ^gmrg`ra - Bucb 2
22
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Oartgs
Hb e`iurb ??, Hb e`iurb ??, tgjas b rgprgsghtbéïa mg m`vgrsas `tghs mg ut`c`zbéïa oaja dbo`bs sbh`târ`bs, dbhlg`rb, tbhqug ghtrg autras.
A mgsghla mg ujb pcbhtb db`xb hïa pgrj`tg oahlgogr tambs bs m`jghs÷gs g mgtbclgs mg ujb oahstruéïa. ^ar `ssa, à hgogssâr`a `hocu`r mgsghlas gj oartg. Oartg à a mgsghla gj prafgéïa vgrt`obc tbjdàj gxgoutbma gj gsobcb pbrb jastrbr b bcturb mbs pbrgmgs, oajpahghtgs ma tgclbma, bcturb qug vb` ma p`sa prahta btà a tgta (pà-m`rg`ta), bcturb mg partbs, pg`tar`s g fbhgcbs, bobdbjghta mg pbrgmgs (bzucgfa, pbst`clbs gto.), ghe`j, mgtbclgs hb vgrt`obc qug hïa sïa v`stas hb pcbhtb db`xb. A oartg, qug à rgprgsghtbma hb pcbhtb par ujb c`hlb g sgtbs `hm`obt`vbs mb m`rgéïa ?=), mgvg gj qug sg gstâ aclbhma (e`iurb ( e`iurb ?=), sgjprg pbssbr par dbhlg`ras g oaz`hlbs.
E`iurb ?= ‟ C`hlb ‟ C`hlb `hm`obt`vb mg oartg.
Adsgrvg hb pcbhtb db`xb (e`iurb 47) qug b c`hlb mg oartg BB rgprgsghtb a oartg cahi`tum`hbc g b c`hlb mg oartg DD, a oartg trbhsvgrsbc. e`iurb ?7, ?7, qug Ugfb a mgsgh mgsghla la ma oartg oartg BB hb hb e`iurb rgprgsghtb bs pbrgmgs g mgtbclgs mb vgrt`obc mb pcbhtb db`xb.
E`iurb ?? ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg m`vgrsas `tghs mg ut`c`zbéïa.
E`iurb ?7 ‟ Oartg ‟ Oartg BB. Oursa mg ^gmrg`ra - Bucb 2
24
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Ugfb, hbs e`iurbs hbs e`iurbs ?5 g ?9 ?9,, oaja rgprgsghtbjas partbs g fbhgcbs mg pbrgmgs has mgsghlas gj oartgs.
E`iurb ?5 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg partbs gj mgsghlas mg oartgs.
E`iurb ?9 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg fbhgcbs gj mgsghlas mg oartgs.
Has mgsghlas gj oartg, à oajuj bpbrgogr b rgprgsghtbéïa mg vïas sgj partbs, oaja ea` jastrbma hb e`iurb ?5, jgsja pbrb rgprgsghtbr bdgrturbs oaj partbs. Bs m` m`jg jghs hs÷g ÷gss mg pa part rtbs bs g fb fbhg hgcb cbss mb mbss pb pbrg rg-mgss ha mg hass mg mgsg sghl hlas asgj gjoa oart rtg, g, au augj gjsu sub b bp bpbr bræh æho` o`b, b, e`iu iurb rb ?< ?<.. sïa sï a eg eg`t`tbs bs oa oahe hear arjg jg ja jast strb rbja jass hb e`
e`iurb ?0, ?0, a oartg trbhs Biarb, vgfb, hb hb e`iurb vgrsbc mb pcbhtb db`xb (c`hlb mg oartg DD)6
E`iurb ?< ‟ M`jghsïa ‟ M`jghsïa mg partbs g fbhgcbs gj oartg.
E`iurb ?0 ‟ Oartg DD.
Oursa mg ^gmrg`ra - Bucb 2
2?
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Gsobcbs
]gjas ujb pbrgmg oaj gspgssurb mg uj t`faca jbo`éa (47 oj) g 4,>> j mg oajpr`jghta g qugrgjas mgsghlâ-cb hb gsobcb 267>.
Oaja m`ssgjas, as mgsghlas sïa gxgoutbmas gj gsobcbs. Bs jb`s usbmbs gj rgprgsghtbé÷gs mg prafgtas brqu`tgtýh`oas sïa gstbs6
Bhtgs mg bpc`obr b rgirb, pbrb s`jpc`e`obr, pamgjas trbhsearjbr b jgm`mb rgbc mg jgtras pbrb oghtêjgtras6
• 262>> (cæ-sg –uj par ogj); • 267> (cæ-sg –uj par o`hqughtb).
2,>> j : 2>> oj 2>> oj : 2.>>> jj
Gx`stgj autrbs gsobcbs, oaja 264>, 2697, 264>>, qug sïa tbjdàj dgj usbmbs.
^artbhta, 4,>> ^artbhta, 4,>> j à `iubc b 4>> oj. oj. B gsobcb sÿ tgrâ geg`ta pbrb a trbébma ha mgsghla. Rubhma oacaobjas bs oatbs (jgm`mbs), `hsgr`jas bs jgm`mbs rgb`s (e`iurb ( e`iurb =>). =>).
Bpc`ob hma b rgirb, tgjas qug trbhse Bpc`obhma trbhseararjbr b jgm`mb rgbc hb jgm`mb ma mgsghla. Ghtïa, m`v`m`jas b jgm`b rgbc ma adfgta (4>> oj g 47 oj) pgca vbcar mb gsobcb (qug hïa sgfb 2). Hgssg gxgjpca, ut`c`zbjas, oaja vbcar mg gsobcb, 7>6 4>> (vbcar rgbc) ù 7> (vbcar mb gsobcb) : = : = oj 47 (vbcar rgbc) ù 7> (vbcar mb gsobcb) : >,7 oj : 7 jj Mgssb jbhg`rb, oahocuêjas qug ujb pbrgmg oaj oajpr`jghta mg 4,>> j (4>> oj) g gspgssurb mg 47 oj, hb gsobcb 267>, sgrâ mgsghlbmb bpghbs oaj = oj mg oajpr`jghta g =2). 7 jj mg gspgssurb ( e`iurb =2).
E`iurb => ‟ Mgsghla ‟ Mgsghla eg`ta gj gsobcb 262>>.
Bs jgm`mbs bhiucbrgs hïa saergj trbhsearjbé÷gs gj gsobcbs. Bss`j, uj ähiuca mg 5>¾ pgrjbhgogrâ mgssg fg`ta, sgfb qubc ear b gsobcb gjprgibm gjprgibmb. b.
]rbhsearjbéïa mbs jgm`mbs Jgm`mb rgbc gj jgm`mb ma mgsghla ^brb trbhsearjbr b jgm`mb rgbc pbrb b jgm`mb ma mgsghla, m`v`m`jas a vbcar mb jgm`mb rgbc pgca vbcar hujàr`oa mb gsobcb, vbcar gssg qug hïa sgfb a 2.
E`iurb =2 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg ujb pbrgmg hb gsobcb 267>.
Jgm`mb ma mgsghla gj jgm`mb rgbc
Gxgjpca6 Oursa mg ^gmrg`ra - Bucb 2
^brb trbhsearjbr b jgm`mb ma mgsghla 2=
@hst`tuta Wh`vgrsbc Drbs`cg`ra
gj jgm`mb rgbc, juct`pc`objas a vbcar mb jgm`mb ma mgsghla pgca vbcar hujàr`oa mb gsobcb.
• Lgotýjgtra (lj) ‟ 2>> j; • Mgoäjgtra (mbj) ‟ 2> j. As sudjöct`pcas ma jgtra, au jgm`mbs jghargs, sïa gstgs6
Gxgjpca6 Ba jg jgm` m` rj rjas as oa oaj j b rà ràiu iub b uj ujb b fb hgcb mg ujb pcbhtb db`xb, ghoahtrbjas 2,7 oj. Biarb, qugrgjas sbdgr b jgm`mb rgbc mb cbriurb mgssb fbhgcb, sghma b gsobcb 262>>. ^brb `ssa, juct`pc`objas b jgm`mb ma mgsghla (2,7 oj) pgca vbcar mb gsobcb (2>>). Caia, tgrgjas6
• Mgoêjgtra (mj) ‟ gqu`vbcg î mào`jb pbrtg ma jgtra, au sgfb, >,2 j; • Oghtêjgtra (oj) ‟ oghtàs`jb pbrtg ma jgtra (>,>2 j); • J`cêjgtra (jj) ‟ j`càs`jb pbrtg ma jgtra (>,>>2 j). ^amgjas b`hmb m`zgr qug a mgoêjgtra à `iubc b 2> oj g a oghtêjgtra, b 2> jj.
2,7 x 2>> : 27> : 27> oj ^artbhta, b cbriurb mb fbhgcb à mg 27> oj =4). (2,7> j) (e`iurb ( e`iurb =4).
Ugfb, b sgiu`r, uj qubmra oaj a rgsuja tbdg dgcb cb 2)6 mass jö ma jöctct`p `pca cass g su sudj djöc öct`t`pc pcas as ma jg jgtr tra a (tb
Jöct`p `pccas
E`iurb =4 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg ujb fbhgcb mgsghlbmb hb gsobcb 262>>.
kj qu`cýjgtra
mj mgoêjgtra
lj lgotýjgtra
oj oghtêjgtra
mbj mgoäjgtra
jj j`cêjgtra
]bdgcb ]bd gcb 2 ‟ Jöct`pcas g sudjöct`pcas ma jgtra.
_`stgjb Jàtr`oa Mgo`jbc
^amgjas b`hmb adtgr uh`mbmgs jghargs qug a j`cêjgtra mb sgiu`htg jbhg`rb6
A s`stgjb jàtr`oa mgo`jbc à a bmatbma ha Drbs`c g tgj, oaja uh`mbmg, a jgtra (j).
• M`v`m`hma a j`cêjgtra par 2>, adtgjas a mào` mà o`ja ja mg j` j`cêcêjg jgtr tra, a, qu qug g oa oarr rrgs gspa pahm hmg g b >, >,2 2 jj jj;; • M`v`m`hma a mào`ja mg j`cêjgtra par 2>, tgjas a oghtàs`ja mg j`cêjgtra, qug oarrgspahmg b >,>2 jj; • M` M`v` v`m` m`hm hma a a og oght htàs às`j `ja a mg j` j`cêj cêjgt gtra ra pa parr 2> 2>,, adtgjas a j`càs`ja mg j`cêjgtra, tbjdàj oahlgo`ma oaja j`orýjgtra au jêorah.
Jgm`r s`ih`e`ob oajpbrbr irbhmgzbs mg jgsjb gspào`g, tajbhma ujb mgcbs oaja pbmrïa. ^ar gxgjpca6 ujb sbcb oaj `hm`obéïa mg 7,>> j mg oajpr`jghta s`ih`e`ob qug gstb jgm`mb mg oajpr`jghta à o`hoa vgzgs jb`ar qug b uh`mbmg usbmb pbrb jgm`-cb, a jgtra.
Jöct`pcas g sudjöct`pcas ma jgtra
Adsgrvbé÷gs
As jöct`pcas, au uh`mbmgs jb`args qug a jgtra, sïa gstgs6
• As sêjdacas mbs uh`mbmgs mg jgm`mbs mgvgj sgr gsor`tas oaj cgtrb j`hösoucb, sgj pahta mg bdrgv`béïa g sgj b cgtrb –s‚
• Ru`cýjgtra (kj) ‟ gqu`vbcg b 2.>>> j; Oursa mg ^gmrg`ra - Bucb 2
_udjöct ct``pcas
27
@hst`tuta Wh`vgrsbc Drbs`cg`ra
ba mgoäjgtra g borgsoghtbjas uj zgra î m`rg`tb, hb obsb ma jgtra. @ssa s`ih`e`ob a sg = x 2> : => j. iu`htg6 = iu`htg6
obsa gstgfbj ha pcurbc. ^ar gxgjpca6 2> j g hïa 2> j. au 2> js. • Hïa mgvgjas oaheuhm`r j`orýjgtra oaj a `hstrujghta mg jgm`mb usbma gj jgtracai`b.
kj
Cæ-sg
? kj
træs qu`cýjgtras
7 lj
o`hoa lgotýjgtras
24 mbj
mazg mgoäjgtras
2j
uj jgtra
7 mj
o`hoa mgoêjgtras v`htg oghtêjgtras
45 jj
v`htg g sg`s j`cêjgtras
kj
oj
jj
lj m mbbj =
j >
mj >
oj
jj
x 2> x 2> Oahvgrtgr mgoäjgtra gj oghtêjgtra. oghtêjgtra. Borgsoghtbjass træs zgras î m`rg`tb mg = mbj. Borgsoghtbja Bss`j, tgrgjas = tgrgjas = x 2> x 2> x 2> : =.>>> oj. oj. lj m mbbj =
j >
mj >
oj >
jj
x 2> x 2> x 2> Oahvgrtgr mgoäjgt Oahvgrtgr mgoäjgtra ra gj lgotýj lgotýjgtra gtra.. Hgssg obsa, borgsoghtbjas uj zgra î gsqugrmb mg = mbj, au sgfb, hb obsb ma lgotýlj. jgtra. Caia, = Caia, = ù 2> : >,= lj.
]bdgcb 4 ‟ Gxgjpcas ‟ Gxgjpcas mg gsor`tb g mg cg`turb mbs uh`mbmgs mg jgm`mbs.
]rbhsearjbéïa mg jgm`mbs Wjb jg Wjb jgsj sjb b jg jgm` m`mb mb pa pamg mg sg sgrr `h `hm` m`ob obmb mb gj uh`mbmgs m`egrghtgs. ^ar gxgjpca6 ujb p`stb mg oarrr` oa r`m mb ma @W @WD D tg tgj j 7 kj, au 7.> .>> >> j; pbrb tr trb bhsearjbr ujb jgm`mb mbmb mg ujb uh`mbmg pbrb autrb, mgvgjas ebzgr a sgiu`htg6
kj
lj mb m bj > =
j
mj
oj
jj
ù 2> Oahvgrtgr mgoäjgtra gj qu`cýjgtra. qu`cýjgtra. Borgsoghtbjas ma`s zgras î gsqugrmb mg = kj. mbj g, bss`j, tgrgjas = tgrgjas = ù 2> ù 2> : >,>= kj.
• Mg ujb uh`mbmg pbrb autrb `jgm`btbjghtg jb`ar, m`v`m`jas par 2>; • Mg ujb uh`mbmg pbrb autrb `jgm`btbjghtg jghar, jghar, juct`pc`obja juct`pc`objass par 2>.
kj >
Ugf gfb b as gx gxgj gjpc pcas as qu qug g sg sg sgiu iugj gj gj tb tbdg dg-cbs qug ebo`c`tbj bs oahvgrs÷gs, sgjprg usbhma a höjgra =6
lj mb m bj > =
j
mj
oj
jj
ù 2> ù 2> Oahvgrtgr j`cêjgtra gj jgtra. jgtra. Hgssg obsa, vbjas trbhsearjbr = jj gj jgtra. ^brb `ssa `s sa,, ba bo borg rgso sogh ghtb tbrj rjas as tr træs æs zg zgra rass î gs gsqu qugr grmb mb ma höjg hö jgra ra =, tg tgrg rgja jass =ù2>ù2>ù2>:>,>>=j.
Oahvgrtgr mgoäjgtra gj jgtra. jgtra. ^brb oahvgrtgr = mgoäjgtras (= mbj) gj jgtra, oacaobjas a höjgra = hb obsb oarrgspahmghtg Oursa mg ^gmrg`ra - Bucb 2
mj
Oahvgrtgr mgoäjgtra gj mgoêjgtra. mgoêjgtra. Hgssg obsa, borgsoghtbjas ma`s zgras î m`mj. rg`tb mg = mbj. Caia, = Caia, = x 2> x 2> : =>> mj.
kj 4> oj
j >
x 2>
tbdg dgcb cb 4, vg B sg sgiu iu`r`r,, hb tb vgfb fb gx gxgj gjpc pcas as mg gs gs-or`tb g mg cg`turb mgssbs jgm`mbs6
Gxgjpcas
lj m mbbj =
25
@hst`tuta Wh`vgrsbc Drbs`cg`ra
kj
lj mb m bj
j
mj
oj
jj
>
>
>
=
Jöct`p Jöc t`pcas cas _êj _êjdac daca a _ud _udjöc jöct`p t`pcas cas _êj _êjdac daca a Ru`cýjgtra oöd`oa
ù 2> ù 2> ù 2>
Lgotýjgtra
Oahvgrtgr Oahvgr tgr qu` qu`cýj cýjgtr gtras as gj jgt jgtras ras.. ^brb trbhsearjbr = kj gj jgtra, borgsoghtbjas træs zgras î m`rg`tb ma höjgra =. Bss`j, = x 2> x 2> x 2> : =.>>> j. j. tgrgjas = tgrgjas
oöd`oa
kj¿
Mgoêjgtra oöd`oa
mj¿
lj¿
Oghtêjgtra oöd`oa
oj¿
Mgoäjgtra J`cêjgtra mbj¿ oöd`oa oöd`oa
jj¿
]bdgcb = ‟ Jöct`pcas ‟ Jöct`pcas g sudjöct`pcas ma jgtra oöd`oa.
kj =
lj mb m bj > >
j >
mj
oj
jj
Jgm`mb mg jbssb. jbssb. A irbjb (i) à b uh`mbmg mg jgm`mb mg jbssb, au pgsa, oufas jöct`pcas sïa a qu`cairbjb (ki), lgotairbjb (li) g a mgobirbjb (mbi). Fâ as sudjöct`pcas sïa a mgo`irbjb (mi), oght`irbjb (oi) g a j`c`irbjb (ji). Obmb uh`mbmg mg jbssb à mgz vgzgs jb`ar qug b uh`mbmg `jgm`btbjghtg 76 jghar, oaja pamgjas vgr hb tbdgcb 76
x 2> x 2> x 2> _`stgjb mgo`jbc mg jgm`mbs Ha s`stgjb mgo`jbc mg jgm`mbs, tgjas autrbs uh`mbmgs oahs`mgrbmbs euhmbjghtb`s. _ïa bs jgm`mbs mg supgreêo`g (jgtra qubmrbma), mg vacujg (jgtra oöd`oa), mg jbssb (irbjb) g mg obpbo`mbmg (c`tra).
Jöct`pcas ki
Jgm`mb mg supgreêo`g. supgreêo`g. B uh`mbmg mg supgreêo`g, au mg ârgb, à a jgtra qubmrbma (j´). Gssb Gs sb uh uh`m `mbm bmg g mg jg jgm` m`mb mb à `jp `jpar artb tbhtg htg,, pa pa`s `s oa oaj j b qubht`mbmg mg j´ mb supgreêo`g, pamgjas mgtgrj`hbr b qubht`mbmg mg jbtgr`bc qug sgrâ hgogssâr`a. As jöct`pcas g as sudjöct`pcas ma ?). jgtra qubmrbma sïa as sgiu`htgs (tbdgcb ( tbdgcb ?).
Lgotýjgtra
qubmrbma
kj´
Mgoêjgtra qubmrbma
mj´
lj´
Oghtêjgtra qubmrbma
oj´
Mgoäjgtra J`cêjgtra mbj´ qubmrbma qubmrbma
li mbi li
i
2.>> 2. >>> > i 2> 2>> > i 2> i
2i
mi
oi
ji
>,2 i >,> >,>2 2 i >,> >,>>2 >2 i
]bdgcb ]bd gcb 7 ‟ Jöct`pcas g sudjöct`pcas ma irbjb.
Jgm`mb mg obpbo`mbmg. obpbo`mbmg. A c`tra à b uh`mbmg mg jgm`mb mg obpbo`mbmg oufa sêjdaca à –c‚, g obmb c`tra oarrgspahmg b 2 mgoêjgtra oöd`oa (2 mj¿). As jöct`pcas ma c`tra sïa a qu`cac`tra (qc), a lgotac`tra (lc) g a mgobc`tra (mbc); as sudjöct`pcas sïa a mgo`c`tra (mc), a oght`c`tra (oc) g a j`c`c`tra (jc). Ugfb b tbdgcb oaj bs jgm`mbs mg obpbo`mbmg b sgiu`r (tbdgcb ( tbdgcb 5). 5).
Jöct` Jö ct`pca pcass _êj _êjdac daca a _ud _udjöc jöct`p t`pcas cas _êj _êjdac daca a Ru`cýjgtra qubmrbma
Wh`mbmg _udjöct`pcas ^r`ho`pbc
Jöct`pcas kc
jj´
lc
2.> .>>> >> c 2>> c
Wh`mbmg _udjöct`pcas ^r`ho`pbc
mbc
c
mc
2> c
2c
>,2 >, 2c
oc
jc
>,>2 >, >2 c >,> ,>> >2 c
]bdgcb ? ‟ Jöct`pcas ‟ Jöct`pcas g sudjöct`pcas ma jgtra qubmrbma.
]bdgcb 5 ‟ Jöct`pcas ‟ Jöct`pcas g sudjöct`pcas ma c`tra.
Jgm`mb mg vacujg. vacujg. B uh`mbmg mg jgm`mb mg vacujg à a jgtra oöd`oa (j¿). As jöct`pcas g as sudjöct`pcas mgssb uh`mbmg mg jgm`mb sïa gstgs (tbdgcb ( tbdgcb =)6 =)6
Ha c`tra, obmb uh`mbmg à mgz vgzgs jb`ar qug b `jgm`btbjghtg jghar, bss`j oaja a irbjb. À b uh`mbmg mg jgm`mb pbrb gjdbcbighs mg t`htbs, jbssbs, vgrh`zgs gto.
Oursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
@hstrujghtas mg jgm`mb
B tr trgh ghb b à oah oahst st`t`tuê uêmb mb mg uj ujb b e`e`tb tb mg béa, e`drb au tgo`ma qubmr`oucbma gj ujb au bjdbs bs ebogs. ^brb irbhmgs m`jghs÷gs mg oajpr`jghtas, a `mgbc à ut`c`zbr gssg `hstrujghta, pr`ho`pbcjghtg gj cgvbhtbjghtas mg tgrrghas au jbrobé÷gs (caobé÷gs) mg adrb. ^amgjas ghoahtrbr b trghb ha oajàro`a oaj oajpr`jghtas mg 2>,>> btà 7>,>> j; hgssg obsa, à prgo`sa ujb pgssab pbrb jbhtgr a zgra mb gsobcb ha pahta `h`o`bc g autrb pbrb gxgoutbr b cg`turb =7). ma pahta mgsgfbma (e`iurb ( e`iurb =7).
As `hstrujghtas mg jgm`mb c`hgbr jb`s ut`c`zbmas pgca pgmrg`ra sïa a jgtra brt`oucbma =?). g b trghb (e`iurb (e`iurb =?).
E`iurb =? ‟ Jgtra ‟ Jgtra brt`oucbma g trghb.
Bjda s as `hs Bjdas `hstru trujght jghtas as tæj b gsob gsobcb cb irbvbmb gj oghtêjgtras. G gstg, par sub vgz, gstâ sudm`v`m`ma par 2>, oarrgspahmghma ba j`cêjgtra. A jgtra brt`oucbma à ju`ta ut`c`zbma pbrb pgqughbs m`jghs÷gs g sïa ghoahtrbmas ha oajàro`a hbs vgrs÷gs mg 2,>> g 4,>> j. Jgm`r oaj gssg `hstrujghta `hstrujghta à dgj s`jpcgs6 dbstb oa`ho`m`r a zgra mb gsobcb ha pahta mg pbrt`mb ahmg sg mgsgfb jgm`r g gegtubr b cg`turb hb oa`ho`mæho`b ma höjgra mb gsobcb ha pahta btà ahmg sg mgsgfb jgm`r ==). (e`iurb ==).
E`iurb =7 ‟ Jgm`hma ‟ Jgm`hma oaj b trghb.
_g hïa lauvgr bciuàj pbrb bfumbr, mgvgjas prghmgr a gxtrgja mb trghb (gsobcb zgra) mg bciujb jbhg`rb ha caobc mg rgegræho`b.
Ârgb, ^grêjgtra g Uacujg Ârgb g pgrêjgtra mg e`iurbs pcbhbs Âr gb à b jg Ârgb jgm` m`mb mb mg uj ujb b su supg pgre reêo êo`g `g g, gj igrbc, à `hm`obmb pgcb cgtrb –_‚ hbs subs eÿrjucbs. ]amb jgm`mb mg ârgb à `hm`obmb ba qubmrbma. ^ar gxgjpca6 sg b uh`mbmg gjprgibmb ear a j`cêjgtra, ghtïa sgrâ j`cêjgtra ba qubmrbma (jj´). ^grêjgtra à b sajb mas cbmas ma pacêiaha. Gcg à rgprgsghtbma pgcb cgtrb –^‚. B s g i u ` r, v g r g j a s b s e ÿ r j u c b s m g ârgbs g pgrêjgtras mas pr`ho`pb`s pacêiahas.
E`iurb == ‟ Jgm`hma ‟ Jgm`hma oaj a jgtra brt`oucbma. Oursa mg ^gmrg`ra - Bucb 2
2<
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Tgtähiuca
Ârgb6 _ Ârgb6 _ : : D + d x l 4
Ha rgtähiuca, tgjas mubs rgprgsghtb=5. é÷gs6 b cbriurb (D) g b bcturb (l) ‟ e`iurb ‟ e`iurb =5.
^grêjgtra6 ^ ^grêjgtra6 ^ : : D + l + d + m
Ârgb66 _ _ : • Ârgb : D x l • ^grêjgtra ^grêjgtra66 ^ ^ : : (4 x D) + (4 x l) au ^ : D + l + D + l
E`iurb =< ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj trbpàz`a.
]r`ähiuca Ha tr`ähiuca, g ha trbpàz`a, b cgtrb –l‚, bcàj mg rgprgsghtbr b bcturb, `hm`ob b d`ssgtr`z =0). ma ähiuca (e`iurb (e`iurb =0).
E`iurb =5 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj rgtähiuca.
Rubmrbma
Ârgb6 _ Ârgb6 _ : : D + l 4
Ha qubmrbma, as qubtra cbmas sïa `iub`s (cbriurb). ^artbhta, tgjas (e`iurb (e`iurb =9)6 =9)6
^grêjgtra66 ^ ^grêjgtra ^ : : D + m + m
Ârgb6 _ Ârgb6 _ : : D x D ^grêjgtra6 ^ ^grêjgtra6 ^ : : = x D au ^ : D + D + D + D
E`iurb =0 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj tr`ähiuca.
O`rouhegræho`b ^brb a oâcouca mb ârgb g ma pgrêjgtra mb o`rouhegræho`b, hb eÿrjucb, ghtrb a vbcar mg ?,2=25 rgprgsghtbma pgcb cgtrb irgib ώ (cæ-sg –p`‚). B cgtrb –T‚ `hm`ob a rb`a, qug à b jgtbmg ma m`äjgtra (e`iurb (e`iurb 7>). 7>).
E`iurb =9 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj qubmrbma.
]rbpàz`a Ha trbpàz`a, b dbsg jb`ar à rgprgsghtbmb pgcb cgtrb –D‚ g b dbsg jghar, pgcb cgtrb –d‚. Fâ b cgtrb –m‚ à b jgm`mb mas cbmas pbrb a oâcouca =<). ma pgrêjgtra ma ähiuca (e`iurb ( e`iurb =<). Oursa mg ^gmrg`ra - Bucb 2
Ârgb6 _ Ârgb6 _ : : ώ x T´ ^grêjgtra66 ^ ^grêjgtra ^ : : 4 x ώ x T 20
@hst`tuta Wh`vgrsbc Drbs`cg`ra
é÷gs6 –D‚ pbrb cbriurb; –O‚ pbrb oajpr`jghta; g –l‚ pbrb bcturb (e`iurb ( e`iurb 74)6 74)6 U : D x O x l
E`iurb 7> ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg ujb o`rouhegræho`b.
Broa (pgrêjgtra o`roucbr) Ha oâcouca mb ârgb g ma pgrêjgtra ma broa, ghtrb hb eÿrjucb a vbcar ma ähiuca, rgprgsghtbma pgcb cgtrb β (cæ-sg –bceb‚, qug à a 72. vbcar ma ähiuca gj irbus) ‟ e`iurb ‟ e`iurb 72.
E`iurb 74 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj pbrbcgcgpêpgma.
^`räj`mg ^brb obcoucbr a vacujg mb p`räj`mg, mgvgjas obcoucbr b ârgb (_) mb dbsg g juct`pc`obr 7?). pgcb bcturb. Mgpa`s, m`v`m`r par træs (e`iurb ( e`iurb 7?).
Ârgb66 _ Ârgb _ : : ώ x T´ x β ?5> ^grêjgtra (sgt (sgtar ar o`ro o`roucbr ucbr)6 )6 ^ :T+T+broa
U : _ x l ?
E`iurb 7? ‟ Tgprgsghtbéïa mg ujb p`räj`mg.
E`iurb 72 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj broa.
Uacujg mg sÿc`mas igajàtr`oas
O`c`hmra
As sÿc`mas igajàtr`oas sïa earjbmas par træs m`jghs÷gs6 bcturb, cbriurb g praeuhm`mbmg; partbhta, aoupbj uj cuibr ha gspbéa. Gssg gspbéa aoupbma pgca oarpa à olbjbma mg vacujg, rgprgsghtbma pgcb cgtrb –U‚.
^brb a o`c`hmra, ghtrbj a vbcar ?,2=25 rg7=). prgsghtbma pgca ώ g a rb`a (e`iurb ( e`iurb 7=). U : ώ x T´ x l
A va vacu cujg jg mg uj oa oarp rpa a à ob obco couc ucbm bma a oa oaj j bs sgiu`htgss eÿrjucbs6 sgiu`htg
^brbcgcgpêpgma au ouda B eÿrjucb pbrb obcoucbr a vacujg mg uj pbrbcgcgpêpgma tgj bs sgiu`htgs rgprgsghtbOursa mg ^gmrg`ra - Bucb 2
E`iurb 7= ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj o`c`hmra. 4>
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Oahg
]amb eÿrjucb qug bprgsghtb a höjgra 4 au ? bo`jb g î m`rg`tb mg bciuj mg sgus gcgjghtas tgj a sgiu`htg s`ih`e`obma6
^brb obcoucbr a vacujg ma oahg, mgvgjas bi`r oaja ha obsa mb p`räj`mg, au sgfb, obcoucbr b ârgb mb dbsg (ώ x T´), juct`pc`obr a vbcar ghoahtrbma pgcb bcturb g, mgpa`s, m`v`m`r par træs (e`iurb (e`iurb 77). 77).
• Höjgra 4 - Gcgvbma î sgiuhmb patæho`b au ba qubmrbma. Gxgjpca6 4´ (cæ-sg –ma`s gcgvbma î sgiuhmb patæho`b‚ au –ma`s ba qubmrbma‚).
U : ώ x T´ x l ?
• Höjgra ? - Gcgvbma î tgrog`rb patæho`b au ba ouda. Gxgjpca6 5¿ (cæ-sg –sg`s gcgvbma î tgrog`rb patæho`b‚ au –sg`s ba ouda‚). Bss`j, –T´‚ rgprgsghtb rgprgsghtb a vbcar mb jgm`mb jgm`mb ma rb`a ba qubmrbma. ]ama bcibr`sja gcgvbma ba qubmrbma oarrgspahmg b gcg juct`pc`obma par gcg jgsja. Gxgjpca6 T : 4> jj T´ : 4>´ : 4> x 4> : =>> T´ : =>> jj´ Caia, T´ Caia,
E`iurb 77 ‟ Tgprgsghtbéïa ‟ Tgprgsghtbéïa mg uj oahg.
Tgobp`tucbéïa jbtgjât`ob
_g qubcqugr `tgj mg ujb eÿrjucb ear gcgvbma vbm a ba ou ouda, da, db dbstb stb ju juct`p ct`pc`o c`oâ-c â-ca a pa parr s`j jgsja træs vgzgs6
Ugfbjas b mgc`j`tbéïa s`jpc`e`obmb mb öct`jb eÿrjucb bprgsghtbmb, pbrb oâcouca ma vacujg mg uj oahg6
Gxgjpca6
U : ώ x T´ x l ?
y : 4> jj y¿ : 4>¿ : 4> x 4> x 4> : <.>>> Caia, y¿ Caia, y¿ : <.>>> jj¿
Ugfb bs `mght`e`obé÷gs6 • B cgtrb irgib ώ irgib ώ à à ujb oahstbhtg jbtgjât`ob g, oaja fâ ea` m`ta, sgu vbcar à `iubc b bprax`jbmbjghtg bprax`jbmbj ghtg ?,2=25.
B rgsacuéïa mg uj pradcgjb, usbhma-sg b eÿrjucb ma bprgsghtbmb, adgmgog î sgiu`htg armgj6
• B cgtrb –T‚, oaja m`ssgjas, à a rb`a mb o`rouhegræho`b.
2. _ud _udst` st`tu` tu`rr as sêj sêjda dacas cas (cg (cgtrb trbs) s) pgcas höjgras oarrgspahm oarrgspahmghtgs; ghtgs; 4. Ebzgr bs apgrbé÷gs;
• B cgtrb –l‚ rgprgsghtb b bcturb.
?. _giu`r oaj b juct`pc`obéïa;
• A trbéa gjdb`xa mb eÿrjucb s`ih`e`ob m`v`sïa.
=. Ghogrrbr oaj b m`v`sïa.
• A höjgra –?‚ à a höjgra mb m`v`sïa. Oursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
Jbtgr`bc pbrb Oahstruéïa
A o`jghta O^ @-?4 hïa passu` bm`éïa mg qubcqugr autra jbtgr`bc, b hïa sgr mg igssa (?%), g à ut`c`zbma oaja rgtbrmbmar mg pgib. À a o`jghta `hm`obma pbrb oahorgtbigj gj igrbc, mgsmg qug gstg hïa e`qug gxpasta b sucebtas mg saca au mg âiubs sudtgrrähgbs. A o`jghta à gjdbcbma gj sboas mg pbpgc Krbet pbrb 7> ki, qug oahtàj mbmas sadrg a ebdr`obhtg, t`pa, ocbssg (e`iurb ( e`iurb 75). 75).
Jbtgr`bc pbrb oahstruéïa à tama g qubcqugr jbtgr`b jbt gr`bcc ut`c ut`c`zbm `zbma a hb hboah oahstru struéïa éïao`v` o`v`cc (ob (obsbs sbs,, prà pràm`as m`as,, ibcp ib cp÷g ÷gs, s, pa paht htgs gs gt gto. o.), ), ad adt`t`ma ma hb hb hbtu turg rgzb zb au oa oaj jb `htgrvghéïa ma lajgj pbrb pramuz`-ca. Gcg mgvg sbt`sebzgr bs gx`iæho`bs g oahm`é÷gs hgogssâr`bs mg bo boar arma ma oa oaj j b eu euhé héïa ïa qu qug g mg mgsg sgjp jpgh ghlb lb.. Hb gsoaclb ma jbtgr`bc, hïa à sajghtg a prgéa qug mgvgjas cgvbr gj oahtb, jbs tbjdàj b gsoaclb bmgqubmb g b dab qubc`mbmg mgstg, pa`s `ssa `hecugho`b ju`ta hb goahaj`b mg tgjpa g hb qubc`mbmg e`hbc mb adrb. As jb jbtg tgr` r`b` b`ss jb jb`s `s dâ dâs` s`oa oass qu qug g eb ebzg zgj j pb pbrt rtg g mb oahstruéïa o`v`c sïa gstgs6 o`jghta, brg`b, pgmrb, obc, t`faca, sb`dra, jbmg`rb, jgtbc, v`mra, tgclb, tgc lb, bz bzucg ucgfa, fa, cbm cbmr`c r`cla, la, jâ jârja rjarg, rg, irb irbh`t h`ta, a, brg brgh`t h`ta, a, parogcbhb,, igssa, `jpgrjgbd`c`zbhtg parogcbhb `jpgrjgbd`c`zbhtg g âiub.
O`jghta A o`jghta à uj pÿ eg`ta oaj prapr`gmbmgs bicaj bi cajgr grbh bhtgs tgsg g c`i c`ibh bhtgs tgs,, qu qug g gh ghmu murgo rgog g sa sad d b bé béïa ïa mbâi mb âiub ub.. J`s J`stur turbm bma a b au autra trass oa oajp jpah ahgh ghtg tgss qu qug g ea earrjbj jb j a oa oaho horg rgta ta,, tr trbh bhse sear arjb jb-s -sg g gj uj ujb b pg pgmr mrb b br br-t`e`o` t`e `o`bc, bc,oa oaj j bct bctb b mu murbd rbd`c` `c`mb mbmg mg g rgs rgs`st `stæh æho`b o`bb b obr obrib ibss g ba eaia. Rubhma a o`jghta gstâ b`hmb gj gstbma tb ma er ergs gsoa oa,, eb ebo` o`c`t c`tb b a tr trbd bdbc bcla la g b ja jacm cmbi bigj gj.. A o`jghta ^artcbhm à b mghaj`hbéïa tàoh`ob h`o b ut ut`c` `c`zb zbmb mb ju juhm hm`b `bcjg cjght htg g pb pbrb rb a qu qug g oa oahl hlgo goggjas oaja o`jghta, m`egrghtg mas autras t`pas oaja oa ja as mg bc bcuj ujêh êh`a `a,, as mg oâ oâco co`a `a,, as am amah ahta tacÿ cÿi` i`-oas gto. Gcg rgogdgu gssg hajg par bprgsghtbr prapr`gmbmgs mg murbd`c`mbmg g sac`mgz sgjgclbh cl bhtg tgss îs ra raol olbs bs mb `c`clb lb dr dr`t`täh äh`o `ob b mg ^a ^art rtcb cbhm hm.. A o`jghta à harjbc`zbma, g gx`stgj btà 22 t`pas ha jgrobma. Ugfb b tbdgcb b tbdgcb 96 96 O^ @ O^ @-_ O^ @@-G O^ @@-V O^ @@-E O^ @@@ O^ @U O^ U-BT@ T_ DO O^D
E`iurb 75 ‟ 75 ‟ _boa mg o`jghta.
Mgvgjas Mgvgj as brj brjbzg bzghb hbrr as sb sboas oas mg o`jghta gj ârgb oadgrtb, sadrg gstrbmas mg jbmg`rb `sghtas mg uj`mbmg. Hïa à rgoajghmbma ebzgr p`clbs oaj jb`s mg mgz sboas, hgj mg`xbr gstaobmas par jb`s mg uj jæs. Ba rgogdgr a o`jghta hb adrb, a pgmrg`ra mgvg bpbcpbr g bpgrtbr a sboa pbrb vgr`e`obr sg gstg sg ghoahtrb –sacta‚ (daj gstbma) au gjpgmrbma au mura (jbu gstbma).
O`jghta ^artcbhm oajuj O`jghta ^artcbhm oajuj oaj bm`éïa rgmuz`mb mg jbtgr`bc pazacäh`oa O`jghta ^artcbhm oajpasta par gsoÿr`b O`jghta ^artcbhm oajpasta par pazacbhb O`jghta ^artcbhm oajpasta par eêcgr O`jghta ^artcbhm mg bcta-earha O`jghta ^artcbhm pazacäh`oa O`jghta ^artcbhm mg bctb rgs`stæho`b `h`o`bc O`jghta ^artcbhm rgs`stghtg b sucebta O`jghta ^artcbhm mg db`xa obcar mg l`mrbtbéïa O`jghta ^artcbhm drbhoa
]bdgcb 9 ‟ 9 ‟ ]`pas mg o`jghta gx`stghtgs ha jgrobma. Oursa mg ^gmrg`ra - Bucb 2
44
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Brg`b
Mgvgjas brjbzghbr b brg`b gj uj caobc oadgrta g sgoa. Jaclbmb, `hecugho`b hb qubht`mbmg mg âiub b sgr bm`o`ahbmb hb oahegoéïa ma oahorgta. Hb adrb, a oahtracg mg rgogd`jghta mb brg`b mgvg sgr eg`ta mb sgiu`htg jbhg`rb6 b`hmb ha obj obj`hl `hlïa ïa db dbsou soucbh cbhtg, tg, taj tajbj bjas as træ træss jg jgm`m m`mbs bs oaj ujb dbrrb mg egrra oaj b bcturb ma jahtg mg brg`b, pas`o`ahbhma-bs hbs mubs gxtrgj`mb7<). mgs ma obj`hlïa g ha oghtra (e`iurb ( e`iurb 7<).
]bj bjdà dàj j oah oahlg lgo`m o`mb b oa oaja ja bi birgi rgibm bma a j`öma, b brg`b pamg sgr ghoahtrbmb gj mubs 79). jbhg`rbs6 hbturbc g brt`e`o`bc (e`iurb (e`iurb 79). • Hbturbc6 birgibma j`öma qug sg bprgsghtb gj earjb mg irïas, gxtrbêma gj r`as, j`hbs g vârzgbs. • Brt`e`o`bc6 rgsêmua e`ha mbs pgmrg`rbs g ma pÿ mg pgmrb.
E`iurb E`i urb7< 7<-- ]r ]ræs æsjgm jgm`mb `mbss ha hajah jahtg tgmg mgbrg brg`b `bma maobj obj`hl `hlïa ïadbs dbsouc oucbhtg. bhtg. E`iurb 79 ‟ 79 ‟ Jahtg mg brg`b.
Ubj bjas as su supa parr qu qug g bs jg jgm` m`mb mbss ea earb rbj j gs gstb tbs6 s6 B brg`b à ocbss`e`obmb mg boarma oaj sub irbhucajgtr`b (jgm`mb ma irïa) gj ju`ta e`hb, e `hb, e`hb, jàm`b g irassb. Ugfb, hb tbdgcb hb tbdgcb <, <, tambs gssbs jgm`mbs6
]`pa mg brg`b Ju`ta e`hb E`hb Jàm`b Irassb
• ^r`jg`rb dbrrb (gxtrgj`mbmg prÿx`jb î obd`hb ma jatar`stb)6 >,<> j; • _giuhmb dbrrb (oghtra)6 2,4> j; • ]gro grog`r g`rb b dbr dbrrb rb (gx (gxtrg trgj`m j`mbmg bmg e`hb e`hbc)6 c)6 2,> 2,>> >j
]bjbhla haj`hbc (jj) Jêh`ja >,27 >,5 2,4 4,=
Oaj gssbs jgm`mbs, ebzgjas as sgiu`htgs oâcoucas pbrb adtgrjas b jàm`b mgssbs jgm`mbs6
Jâx`ja >,5 2,4 4,= =,<
>,<> + 2,4> + 2,>> : ?,>> j (bcturb jàm`b mb brg`b) ?,>> ù ? : 2,>> : 2,>> j (bcturb Bpÿs mgs Bpÿs mgsobr obrrgib rgibrr b brg brg`b, `b, taj tajbja bjass b jgm`mb ma oajpr`jghta g mb cbriurb mb obébjdb. Ubjas supar qug a oajpr`jghta sgfb =,>> j g b cbriurb, 4,4> j. Oaj `ssa, pamgjas sbdgr qubhtas jgtras oöd`oas mg brg`b gstbjas rgogdghma, ebzghma a sgiu`htg oâcouca6
]bdgcb < ‟ Jgm`mbs mb brg`b.
B qubc`mbmg mb brg`b à ju`ta `jpartbhtg. Gcb mgvg gstbr c`vrg mg `jpurgzbs oaja tgrrb, rbêzgs g passu`r ujb irbhucbéïa uh`earjg (hïa j`sturbmb). B brg`b mb prb`b hïa sgrvg pbrb a trbdbcla gj oahstruéïa o`v`c, pa`s sub sbc`h`mbmg oajprajgtg b sgiurbhéb mb jbssb ma oahorgta. Oursa mg ^gmrg`ra - Bucb 2
=,>> x 4,4> x 2,>> : <,<> : <,<> j¿ 4?
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Oajaà Oaja à oa oaju juj j rg rgog ogdg dgrr b ob obri rib b oa oaj j uj ujb b ta ta-cgräh cgr äho`b o`bmg mg2> 2>% % pb pbrb rbjb jb`s `sau aupb pbrb rbjgh jghas as,, tgr tgrêbj êbjas as66
As erbijghtas rgsuctbhtgs mb gxpcasïa sïa tr`turbmas par gqu`pbjghtas mghaj`hbmas dr`tbmg`rb. Hb sbêmb mb dr`tbmg`rb, bs pgmrbs sïa sgcgo`ahbmbs, pbssbhma-bs par ujb pghg`rb. Mgpa`s, rgogdgj b sgiu`htg ocbss`e`obéïa gj euhéïa ma tbjbhla6 >, 2, 4, ?, = g 7. Hb tbdgcb Hb tbdgcb 0, 0, bprgsghtbjas as mbmas ma tbjbhla mb dr`tb, rgegrghtgs î HDT 9422 (Birgibmas pbrb oahorgta) g HDT 9447 (Jbtgr`b`s mg pgmrb g birgibmas hbturb`s) mb Bssao`béïa Drbs`cg`rb mg Harjbs ]àoh`obs (BDH]).
<,<> + >,<< : 0,5< j¿ _g b oajprb eassg mg 2>,>> j¿, ebctbr`b >,?4 j¿ mg brg`b.
^gmrb ]bjd bjdàj àj oah oahlgo lgo`mb `mb oaj oaja a bir birgib gibma ma irb irböma öma,, b pg pgmrb mrbpa pamg mgsg sgrr hb hbtur turbc bc (p (pgm gmrgi rgiucl ucla) a),, sg sg`xa `xara racbm cbma, a, obso ob sobc bcla la g dr dr`tb `tb (p (pgm gmrb rb dr dr`tb `tbmb mb au pg pgmrb mrb br brt`e t`e`o` `o`bc bc). ). B dr`t dr`tb b à adt` adt`mb mb mg ujb uh`mbmg uh`mbmg `hmustr`b `hmustr`bcc j`hgrb j`h grbmar marb b oah oahlgo lgo`mb `mb oaj oaja a pgm pgmrg`r rg`rb, b, ahm ahmg, g, par jg`a mg gxpcas÷gs, aoarrg b mgs`htgirbéïa mb 70). raolb, mbhma ar`igj î dr`tb (e`iurb ( e`iurb 70).
Adsgrvbé÷gs • ^brb geg`ta mg masbigj, pamgjas usbr m`äjgtra jâx`ja mg 47 jj pbrb ujb j`sturb mg dr`tb 2 jb`s dr`tb 4. • ]bjbhla jb`ar qug 95 jj à gqu`vbcghtg î pgmrb mg jïa. • Hb oahegræho`b ma rgogd`jghta mb qubht`mbmg mg pgmrb, a praogm`jghta à a jgsja mb brg`b.
E`iurb 70 ‟ 70 ‟ Dr`tb.
]bjbhla haj`hbc ‟ jbclb mb pghg`rb (jj) Hujgrbéïa mb dr`tb
HDT 9422 / HDT 9447
Oajgro`bc
Jêh`ja
Jâx`ja
Jêh`ja
Jâx`ja
Dr`tb >
‟
‟
=,<
0,7
Dr`tb 2
=,<
24,7
0,7
20
Dr`tb 4
24,7
47
20
?<
Dr`tb ?
47
7>
?<
7>
Dr`tb =
7>
95
7>
95
Dr`tb 7
95
2>>
‟
‟
]bdgcb 0 ‟ 0 ‟ Hujgrbéï Hujgrbéïa a g tbjbhla mbs dr`tbs. Oursa mg ^gmrg`ra - Bucb 2
4=
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Obc
A t`faca jbo`éa bprgsghtb bs sgiu`htgs 52). obrbotgrêst`obs (e`iurb (e`iurb 52).
B obc v`rigj, tbjdàj oahlgo`mb oaja ÿx`ma mg oâco`a, à adt`mb par mgoajpas`éïa tàrj`ob mg obcoâr`a b 0>> ¶O. À uj pÿ ju`ta drbhoa g pura, jbs pamg oahtgr `jpurgzbs qubhma oajgro`bc`zbm oajgro`bc`zbma. a. Bhtgs mg sgr sgr j`sturbmb j`sturbmb oaj âiub, âiub, b obc à mghaj`hbmb v`vb; paràj, ba sgr pastb gj oahtbta oaj b âiub (qug`jb), gcb à mghaj`hbmb l`mrbtbmb au gxt`htb, ut`c`zbmb gj p`hturbs g gj bribjbssb bribjbssbs. s. Bob B obcc bm bmqu qu`r`r`m `mb b pb pbrb rb oa oahs hstr trué uéïa ïa,, ig igrb rbcj cjgh ghtg tg,, à boahm`o`ahbmb gj sboas mg pbpgc Krbet (e`iurb (e`iurb 5>). 5> ). Jgsja bss`j, oaj a tgjpa, gcb bdsarvg ogrtb uj`mbmg ma br, tarhbhma-sg gxt`htb pbucbt`hbjghtg; par `ssa gcb hïa mgvg sgr oahsgrvbmb parr ju pa ju`t`ta a tg tgjp jpa a ha ob obht htg` g`ra ra mg ad adrb rb,, jg jgsj sja a qu qug g gstg gs tgfb fb ju ju`ta `ta dg dgj j pr prat atgi gi`m `mb b oa oaht htrb rb b uj uj`m `mbm bmg. g.
• M`jghsï M`jghsïa a jb`s oajuj oajuj (Cbriurb, (Cbriurb, Bcturb g Oajpr`jghta - C x L x O)6 2> x 7 x 42 oj; • ^gsa6 4,7 ki; • Tgs`stæho`b6 stæho`b64> 4> kie (qu`cairbjb cairbjb-ear -earéb)/ éb)/oj´; oj´; • Rubht`mbmg par j´6 b) ^brgmg mg jg`a-t`faca6 99 t`facas; d) ^brgmg mg uj t`faca6 2=< t`facas.
E`iurb 52 ‟ 52 ‟ ]`faca jbo`éa.
^brb ob ^brb obmb mb t`f t`faca acajb jbo`é o`éa, a, bs m`j m`jgh ghs÷g s÷gss pb pb-mra rah h`z `zb bmbs pgcbBD cbBDH H] sïa mg 0 x 7,9 x 20 oj. Dcaoa ogräj`oa au t`faca eurbma ]bjdàj olbjbma, gj bciujbs rgi`÷gs, mg t`faca db`bha, à uj t`faca ogräj`oa vbzbma, jacmbma oaj brgstbs v`vbs g rgt`cêhgbs, ebdr`obma oaj ogräj`ob vgrjgclb adt`mb par jg`a mg gxtrusïa.
E`iurb 5> ‟ 5> ‟ _boa mg obc.
]`faca À uj gcgjghta mg bcvghbr`b bssghtbma oaj bribjbssb, ut`c`zbma pbrb gxgoutbr euhmbé÷gs g hb oahstruéïa mg pbrgmgs. As t`t`pa pass mg t`t`fa faca cass jb jb`s `s oa oaju juhs hs sï sïa a gs gstg tgs6 s6 jbo`éa, dcaoa ogräj`oa, dcaoa mg oahorgta g saca-o`jghta.
_ubs obrbotgrêst`obs sïa bs sgiu`htgs 54)6 (e`iurb 54)6 • M`jghsïa jb`s oajuj6 0 x 20 x 20 oj; • ^gsa6 ? ki; • Tgs`stæho`b mb pbrgmg6 =7 kie/oj´; • Tgs Tgs`st `stæho æho`b `b mb pbr pbrgmg gmg-gs -gspgc pgcla6 la6?> ?> kie kie/oj /oj´; ´; • Tgs`s Tgs`stæh tæho`b o`bmb mbpb pbrgm rgmg g mg mguj ujt`f t`faca6 aca6 2> 2>kie kie/oj /oj´. ´. • Rubht`mbmg par j´6 b) ^brgmg mg jg`a-t`faca6 44 dcaoas; d) ^brgmg mg uj t`faca6 =4 dcaoas.
]`faca jbo`éa mg dbrra oaz`ma ]bjdàj oahlgo`ma oaja t`faca oajuj, à eg`ta mg dbrra jbo`éa, oaj brgstbs v`vbs g rgt`cêhgbs hg bs,, ja jacm cmbm bmas as gj ea earj rjbs bs g qu qug` g`jb jbma mass gj ea earrhas b ujb tgjpgrbturb ghtrg 0>> g 2>>> ¾O. Oursa mg ^gmrg`ra - Bucb 2
47
@hst`tuta Wh`vgrsbc Drbs`cg`ra
M`jghs÷gs haj`hb`s (jj)
Jgm`mbs gspgo`b` gspg o`b`ss (oj)
Cbri Cb riur urbb
Bctu Bc turb rb
Oajpr`jghta
2> x 2> x 4>
0>
0>
20>
2> x 27 x 4>
0>
2=>
20>
2> x 27 x 47
0>
2=>
4=>
24,7 x 27 x 47
227
2=>
4=>
]bdgcb 2> ‟ 2> ‟ M`jghs÷gs harjbc`zbmbs gj j`cêjgtras mas dcaoas ogräj`oas.
Dcaoa mg oahorgta _ïa gcgjgh _ïagcg jghtas tasha ha earj earjbta btamg mg pbr pbrbcg bcgcai cairbj rbja a g oa oaj j br brgst gstbs bsv`v v`vbs bs,, eb ebdr` dr`ob obma mass oa oaj j o`j o`jgh ghta, ta,brg brg`b, `b, pÿ mg pgm gmrb rb g âi âiub ub,, gj pr prap apa aré ré÷g ÷gss bm bmgq gqub ubmb mbss î rgs` rg s`st stæh æho` o`b b qu qug g sg sgmg mgsg sgfb fb.. _ï _ïa a pr pram amuz uz`m `mas asgj gjuj ujb b prghsb l`mrâuc`ob qug pgrj`tg mge`h`r uj earjbta rgiu rg iucb cbrr, oa oaj j br brgs gstbs tbsg g eb ebog ogss gj gjda daj j bo bobd bdbj bjgh ghta. ta. Bs obrbotgrês obrbotgrêst`obs t`obs mas mas dcaoas dcaoas mg oahorgta oahorgta sïa gstbs6
E`iurb 54 ‟ 54 ‟ Dcaoas ogräj`oas ogräj`oas..
As dca dcaoa oass og ogrä räj`o j`oas as sï sïa a eb ebdr` dr`ob obma mass oa oaj j eurbéïa o`cêhmr`ob au pr`sjât`ob (oaj pr`sjbs). Ha bssght bss ghtbjg bjghta hta,, b eur eurbéï béïa a e`o e`ob b pbr pbrbcg bcgcb cb î sup supgre greêo` êo`g g mg bs bssg sght htbj bjgh ghta ta,, au gj pa pas`é s`éïa ïa la lar`z r`zah ahtb tbc.c. Bs ebogs mgssgs dcaoas sïa v`tr`e`obmbs g m`e`ouctbj b bmgræho`b oaj bribjbssb mg bssghtbjghta g rgvgst`jghta; à par `ssa qug as dcaoas passugj rbhlurbs g sbc`æho`bs pbrb bujghtbr b bmgræho`b. Ugfb, hb tbdgcb hb tbdgcb 2>, 2>, bs m`jghs÷gs haj`hb`s gj j`cêjgtras mas dcaoas ogräj`oas, harjbc`zbmbs pgcb BDH]6
• Rubht`mbmg par j´6 247 dcaoas; • Tgs`stæho`b6 oahsuctbr a ebdr`obhtg; • Jb`s rbp`mgz ha bssghtbjghta; • Jghar gspgssurb hbs pbrgmgs; • Jghar pgsa. ^brb ebo`c`tbr a bssghtbjghta, bcàj mg gcgjghtas `htg`ras, tbjdàj sïa ebdr`obmas jg`a dcaoa mg oahorgta; bss`j, gv`tb-sg a oartg mg uj dcaoa. _ïa ebdr`obmas, tbjdàj, dcaoas obhbcgtbs pbrb gxgouéïa mg vgribs, oahtrbvgribs g v`ibs mg rgspbcma (e`iurb ( e`iurb 5?). 5?).
M`jghs÷gs haj`hb`s mg dcaoas mg vgmbéïa g gstruturb`s, oajuhs g gspgo`b`s ]`pa (oj) 2> x 4> x 4> 2> x 4> x 47 2> x 4> x ?> 2> x 4> x => 24,7 x 4> x 4> 24,7 x 4> x 47 24,7 x 4> x ?> 24,7 x 4> x => 27 x 4> x 4> 27 x 4> x 47 27 x 4> x ?> 27 x 4> x => 4> x 4> x 4> 4> x 4> x 47 4> x 4> x ?> 4> x 4> x =>
M`jghs÷gs haj`hb`s (jj) Cbri Cb riur urbb Bctu Bc turb rb Oajpr`jghta 0> 20> 20> 0> 20> 4=> 0> 20> 40> 0> 20> ?0> 227 20> 20> 227 20> 4=> 227 20> 40> 227 20> ?0> 2=> 20> 20> 2=> 20> 4=> 2=> 20> 40> 2=> 20> ?0> 20> 20> 20> 20> 20> 4=> 20> 20> 40> 20> 20> ?0>
Oursa mg ^gmrg`ra - Bucb 2
E`iurb 5? ‟ 5? ‟ Dcaoas mg oahorgta. 45
@hst`tuta Wh`vgrsbc Drbs`cg`ra
_b`dra
Ugfb, hb tbdgcb hb tbdgcb 22, 22, bs m`jghs÷gs haj`hb`s gj oghtêjgtras g a pgsa gqu`vbcghtg mas dcaoas mg oahorgta6
A sb`dra à ujb j`sturb mg bri`cb g brg`b irassb, usbma oaja oajpahghtg mbs bribjbssbs.
M`jghs÷gs mg ^gsa jg`a dcaoa (oj) (ki)
M`jghs÷gs mg uj dcaoa (oj)
^gsa (ki)
20 x 0 x ?0
2>
20 x 0 x 20
=,<
20 x 22 x ?0
2>,9
‟
‟
20 x 2= x ?0
2?,5
20 x 2= x 20
5,9
20 x 20 x ?0
27,7
20 x 20 x 20
<,9
Jbmg`rb À uj mas jbtgr`b`s mg oahstruéïa mg irbhmg gjprgia, oufb gsoaclb ma t`pa mgpghmg, ghtrgtbhta, mb e`hbc`mbmg mb adrb6 trbhs`tÿr`b au prav`sÿr`b (gsoarbjghtas, bhmb`jgs, tbpujgs, earjbs gto.) au mge`h`t`vb (brjbmurbs mas tgclbmas, jbroghbr`bs, bssabclas, gsqubmr`bs, v`ibs mas bssabclas, tboas, tgtas gto.). Ghtrg bs jbmg`rbs, m`st`hiugj-sg bs drbhobs, qug sïa, igrbcjghtg, gjprgibmbs hbs pbrtgs mb oahstruéïa hïa sufg`tbs b gsearéas oaja bs pgébs mg bobdbjghta, rgvgst`jghtas, arhbjghtbé÷gs gto., pa`s gssbs jbmg`rbs sïa jghas rgs`stghtgs g, partbhta, jb`s vuchgrâvg`s ba btbqug mg `hsgtas; g bs mg cg`, bs qub`s sïa murbs, rgs`ss tg s` tgh h tg tgss g mg bc bcta ta v bc bca a r oa oajg jgro ro`b `bc, c, ut`c`zbmbs pbrb oahegoéïa mg jÿvg`s, `hstrujghtas jus`ob`s g tbjdàj hb oahstruéïa o`v`c (p`sas g rgvgst`jghtas, brjâr`as gjdut`mas gto.). Ugfb as m`vgrsas t`pas mg jbmg`rb g sub bpc`obéïa bmgqubmb hb oahstruéïa o`v`c6
]bdgcb 22 ‟ 22 ‟ M`jghs÷gs haj`hb`s gj oghtêjgtras mg dcaoas mg oahorgta.
]`faca mg saca-o`jghta À uj t`faca ebdr`obma pgcb j`sturb mg saca brghasa, hb praparéïa mg 7> b <>% ma prÿpr`a tgrrgha ahmg sg gxgoutb b adrb, jb`s = b 2>% mg o`jghta g âiub, prghsbma jgobh`objghtg au jbhu jbhubcjgh bcjghtg. tg. ^brb a bssg bssghtbjg htbjghta, hta, ut`c`zbjas bribjbssb j`stb mg o`jghta, obc g brg`b oaj trbéa mg 2646<, au oacb bprapr`bmb. Bs obr obrbot botgrê grêst`o st`obs bs ma t`fa t`faca ca mg sac saca-o a-o`jg `jghta hta 5=)6 sïa gstbs (e`iurb (e`iurb 5=)6 • M`j M`jgh ghsï sïa a jb jb`s `s oa oaju juj6 j6 2> x =, =,7 7 x 4> oj oj;; • ^gsa6 2,?= ki; • Tgs`stæho`b6 ?> kie/oj´; • Rubht`mbmg par j´6 b) ^brgmg mg jg`a-t`faca6 2>> t`facas; d) ^brgmg mg uj t`faca6 4>> t`facas.
• U`ibs6 pgradb, `pæ, obhgcb, jbébrbhmudb, fbtadâ, iubr`tb gto. • Bssabclas6 pbu-jbre`j, obhgcb, pgradb, fbobrbhmâ, obdr`övb, brbruvb, tb`övb, `pæ ` pæ gto. • ]gtas6 p`hla, fgqu`t`dâ drbhoa, fgqu`t`dâ vgrjgcla, ogmra gto. • Jbroghbr`bs6 ogmra, p`hla, `jdu`b, obv`öhb, fbobrbhmâ, fgqu`t`dâ vgrjgcla gto. À `jpa `jpartbht rtbhtg g oahl oahlgogr gogr b mghs mghs`mbm `mbmg g mbs jbmg`rbs gj qu`cairbjb par jgtra oöd`oa (ki/j¿), oaja tbjdàj bs rgs`stæho`bs î oaj oajprg prgssï ssïa a gj qu qu`ca `cairb irbjb jb pa parr jgt jgtra ra qubmrbma (ki/j´) mg sgoéïa. ^brb `ssa, vgfb b tbdgcb 24 oaj 24 oaj gssgs vbcargs6
E`iurb 5= ‟ 5= ‟ ]`faca mg saca-o`jghta. Oursa mg ^gmrg`ra - Bucb 2
49
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Mghs`mbmg (ki/j¿)
Tgs`stæho`b (ki/j´)
Obhgcb
7?>
?5>
Ogmra
7?>
?7>
Obdr`övb
0<>
95>
Guobc`pta
9<>
7<>
Iubrbhtï
05>
90>
@jdu`b
57>
=7>
@pæ
05>
9=>
Fbobrbhmâ
90>
=<>
Fbtadâ
02>
94>
Fgqu`t`dâ
7?>
=2>
Jbébrbhmudb
5?>
=5>
^bu-jbre`j
<9>
5?>
^gradb
<>>
7<>
^`hla
74>
?0>
Jbmg`rbs
Autras jgtb`s oaja a bcujêh`a g a egrra euhm`ma sïa ut`c`zbmas gj gsqubmr`bs mg partbs g fbhgcbs.
U`mra A v`mra à uj jbtgr`bc rêi`ma, mg supgreêo`g pcbhb g qugdrbm`éb; gj igrbc, à trbhspbrghtg, jbs tbjdàj pamgjas ghoahtrbr v`mra obhgcbma, jbrtgcbma gto. À uj jbtgr`bc ut`c`zbma, pr`ho`pbcjghtg, gj gsqubmr`bs mg partbs g fbhgcbs, ebdr`obma gj bctbs tgjpgrbturbs mg ujb j`sturb mg s`cêo`as (brg`b) g obrdahbtas, oaj râp`mb sac`m`e`obéïa.
]gclb (2?) Bs tgclbs sïa ut`c`zbmbs gj oadgrturbs, mgvgj sgr `jpgrjgâvg`s, bcàj mg sgr `sacbhtg tàrj`oa g boöst`oa. Gcbs mgvgj tgr ujb dab murbéïa g sgr mg eâo`c rgpas`éïa, pa`s, lbvghma b qugdrb mg bciuj gcgjghta, b sudst`tu`éïa mgvg sgr a jb`s eâo`c passêvgc. Bs tg tgcl clbs bs jb jb`s `s ut ut`c`c`z `zbm bmbs bs gj gm gm`e`e`o `obbé÷gs sïa gstbs6 tgclb mg dbrra, tgclb mg e`drao`jghta, tgclb mg bcujêh`a g tgclb mg egrra z`hobma. Mghtrg gssbs, mgstbobjas bs tgclbs mg dbrra t`pa erbhogsb, par sgr b jb`ss ut` jb` ut`c`z c`zbm bmb. b. Gcb à pra pramuz muz`mb `mb par uj dbrra mgv`mbjghtg j`sturbma g lajaighg`zbma gj jâqu`hbs gspgo`b`s; mgpa`s, sïa qug`jbmbs pbrb ar`i`hbr uj gxogcghtg 55). jbtgr`bc `jpgrjgâvgc (e`iurb ( e`iurb 55).
]bdgcb 24 ‟ 24 ‟ Mghs`mb Mghs`mbmg mg g rgs`stæho`b mg jbmg`rbs.
Jgtbc A jgt gtb bc jb jb`s `sut ut`c`c`z `zbm bma a gj ad adrb rbss à a béa éa,, qug à vg vghm hm`m `ma a pa parr qu qu`c`ca a gj ea earj rjbt bta a mg db dbrr rrbs bs au ra raca cas, s, ut`c`z ut` c`zbm bma a hb oa oahst hstrué ruéïa ïa mg brj brjbmu bmurbs rbsmg mg oa oahor horgta gta e`iu iurb rb 57 57). pbrb pb rb cb cbfg fgs, s, v` v`ib ibs, s, p` p`cb cbrg rgss g vg vgri ribs bs (e` ).
E`iurb 57 ‟ 57 ‟ Brjbmurb mg béa.
B brjbmurb mg béa à oahlgo`mb pgcb s`icb OB (Oah (Oahorgt orgta a Brjb Brjbma) ma) oaj as sgiu`htgs sgiu`htgs vbcargs6 OB-47, OB-7> gto. Gssgs vbcargs `hm`obj b obrib mg trbdbcla qug b dbrrb mgvg supart pa rtbr br;; pa parr gx gxgj gjpc pca6 a6 OB OB-4 -47 7 `h `hm` m`ob ob qu qug g b brjbmurb supartb 47 ki/jj´. Oursa mg ^gmrg`ra - Bucb 2
E`iurb 55 ‟ 55 ‟ ]gclb mg dbrra erbhogsb. 4<
@hst`tuta Wh`vgrsbc Drbs`cg`ra
Tgvgst`jghta ogräj`oa As rgvgst`jghtas ogräj`oas sïa ut`c`zbmas pbrb rgvgst`r p`sas (cbmr`clas) g pbrgmgs (bzucgfas), oaj vâr`as t`pas pbrb qubcqugr bjd`ghtg oaja ârgbs oajgro`b`s, `hmustr`b`s, rgs`mgho`b`s, ebolbmbs, p`so`hbs gto. _ïa pcbobs mg jbtgr`bc ogräj`oa `jpgrjgâvgc, oaj ujb mbs ebogs c`sb g v`mrbmb g autrb röst`ob au parasb pbrb ebo`c`tbr a bssghtbjghta.
oas mg oahorgta sgj uj`mbmg, jbs à prgo`sa ibrbht`r b bmgræho`b g ujb gspgssurb `mgbc; pamg sgr bpc`obma tbjdàj gj supgreêo`gs oaj rgdaoa. B supgreêo`g b sgr rgvgst`mb oaj igssa mgvg gstbr `sghtb mg pag`rb, acgas`mbmg, uj`mbmg g autrbs `jpurgzbs. ^ar sgr bcobc`ha, a igssa prap`o`b oarrasïa gj pgébs jgtâc`obs, bs qub`s mgvgj rgogdgr uj trbtbjghta gspgo`bc au sgr pratgi`mb.
Jârjarg g irbh`ta
@jpgrjgbd`c`zbhtg
_ïa pgmrbs mg irbhmg rgs`stæho`b g mg oargs vbr`bmbs. A irbh`ta à jb`s jgsocbma, fâ b oacarbéïa ma jârjarg à jb`s uh`earjg. A irbh`ta à dgj jb`s mura, rgs`stghtg g jghas parasa qug a jârjarg, gj qug sg pgrj`tg r`soas oaj jb`s ebo`c`mbmg. _ïa rgoajghmbmas gj bcvghbr`b gj qug à gx`i`mb irbhmg `jpgrjgbd`c`mbmg, au jgsja oaja jbtgr`bc mg rgvgst`jghta pbrb p`sas g pbrgmgs. ^ar obusb mb sub murgzb, pgrj`tg uj daj pac`jghta gj sub supgreêo`g.
@jpgrjgbd`c`zbr s`ih`e`ob bpc`obr pramutas bmgqubmas pbrb pratgigr m`vgrsbs ârgbs oahtrb b béïa mg âiubs qug pamgj sgr mg oluvb, dbhlas au mg cbvbighs, gj gstbma cêqu`ma au sÿc`ma. A `jpgrjgbd`c`zbhtg à uj pramuta bm`o`ahbma gj oahorgta, bribjbssb, euhmbé÷gs, cbfgs mg p`sas g tgrrbéas, rgsgrvbtÿr`as gto. Gx`stgj ma`s t`pas mg `jpgrjgbd`c`zbéïa6 b rêi`mb, gj qug a oahorgta, au bribjbssb, sg tarhb `jpgrjgâvgc irbébs î `hocusïa mg bm`t`vas `jpgrjgbd`c`zbhtgs murbhtg a sgu prgpbra; g b ecgxêvgc, eg`tb oaj jbhtbs gcâst`obs (bseâct`obs) prà-ebdr`obmbs, au gcbstýjgras m`ssacv`mas g bpc`obmas sadrg b cbfg hb earjb mg p`hturb. B `jpgrjgbd `jpgrjgbd`c`zbéïa `c`zbéïa rêi`mb à b jb`s `hm`obmb pbrb ârgbs hïa gxpastbs ba sac g hïa su fg`tbs b tr`hoas g e`ssurbs oaja dbhlg`ras, oaz`hlbs, ârgbs mg sgrv`éa gto; fâ b ecgxêvgc à eg`tb gj cbfgs gxtgrhbs, gxpastbs b `htgjpàr`gs g sufg`tbs î oahtrbéïa g m`cbtbéïa.
Brgh`tas As brgh`tas (gj pbrt`oucbr, as s`c`o`asas) sïa dabs pgmrbs pbrb oahstruéïa, jbs, oaj a bpbrgo`jghta mas brgh`tas brt`e`o`b`s, jb`s rgs`stghtgs g mg jghar ousta, gcgs sïa pauoas praourbmas. ^ar `ssa, sïa rgsgrvbmas pbrb geg`tas mgoarbt`vas.
^arogcbhb
Âiub
À uj jbtgr`bc drbhoa, `jpgrjgâvgc g trbhscöo`ma, qug sg m`egrgho`b mg autras pramutas ogräj`oas par sub v`tr`e`obéïa, trbhspbræho`b, o` b, rg rgs` s`st stæh æho` o`b b g oa oajp jpcg cgtb tb `s `sgh ghéï éïa a mg paras`mbmg. Gcb à ut`c`zbmb gj rgvgst`jghtas g, pr`ho`pbcjghtg, hb oahegoéïa mg bpbrgclas sbh`târ`as6 vbsas, d`mæs, cbvbtÿr`as gto.
B qubc`mbmg mb âiub à ju`ta `jpartbh `jpartbhtg tg hb oahstruéïa o`v`c. Bss`j, b âiub pbrb a oahorgta g bribjbssb mgvg sgr patâvgc, g hïa sgrvgj b âiub pravgh`ghtg mg vbcbs, b sgrv`mb, b mg paéas g b âiub ma jbr, pa`s gstbs passugj `jpur `jp urgzb gzbss qu qug g beg begtbj tbj b qu qubc` bc`mb mbmg mg ma oa oaho horgt rgta. a. B qubht`mbmg mg âiub b sgr bm`o`ahbmb has mgjb`s oajpahghtgs ma oahorgta mgvg sgr r`iarasbjghtg a gspgo`e`obma pgca trbéa ma oahorgta, pa`s âiub gj gxogssa tarhb a oahorgta jb`s trbdbclâvgc g jghas rgs`stghtg; oaj jghas âiub, a oahorgta e`obrâ jb`s rgs`stghtg g jghas trbdbclâvgc.
Igssa A igssa à ju`ta ut`c`zbma ha rgvgst`jghta mg pbrgmgs g tgta. Ha bobdbjghta oaj p`hturb, gcg à bpc`obma m`rgtbjghtg gj qubcqugr supgreêo`g `htgrhb mg bcvghbr`b g mg dcaOursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
Ugfb sg Bprghmgu 2. Bss`hbcg (2) pbrb Gqu`pbjghta mg ^ratgéïa @hm`v`mubc g (4) pbrb Gqu`pbjghta mg ^ra2. Bss`hbcg tgéïa Oacgt`vb. Gj sgiu`mb, jbrqug b bctgrhbt`vb oarrgtb. ( ( ( ( (
) ßouc ßoucas as mg prat pratgéïa géïa ) _bpbt _bpbtas as mg sgiu sgiurbhé rbhéb b ) O`hta mg sgiu sgiurbhé rbhéb b ) Ob Obpb pbog ogtg tg ) Obmg Obmg`rb `rb susp suspghsb ghsb
b) ( d) ( o) ( m) (
) 4‟ 2‟ 2‟ 4- 4 ) 2‟ 2‟ 4‟ 2- 4 ) 2‟ 4‟ 4‟ 2- 2 ) 4‟ 4‟ 2‟ 2‟ 4
4 . Bss`hbcg b bctgrhbt`vb qug h ïa `hm`ob ujb mbs rgoajghmbé÷gs pbrb a usa mg gsobmbs. b) ( ) Bs gsobmbs gsobmbs mgvgj mgvgj tgr mgirbus mgirbus bht`mgrrbpbhtg bht`mgrrbpbhtgs. s. d) ( ) Hïa sud`r sud`r bcàj mas mas ma`s öct`jas öct`jas mgirb mgirbus us mb gsobmb gsobmb.. o) ( ) À rgoajghmb rgoajghmbma ma qug bs bs gsobmbs gsobmbs sgfbj sgfbj p`htbmb p`htbmbs. s. m) ( ) A oajpr`jgh oajpr`jghta ta jâx`ja jâx`ja mas jahtb jahtbhtgs htgs mb mb gsobmb gsobmb à mg 5 j. ?. Wj ähiuca oaj ujb bdgrturb jb`ar qug 0>¾ à olbjbma mg6 ?. Wj b) ( ) rg rgta ta.. d) ( ) bi bium uma. a. o) ( ) ad adtu tusa sa.. m) ( ) d`s d`ssgt sgtr`z r`z.. =. A qug à uj tr`ähiuca `sÿsogcgs1 =. A b) ( ) Rubhma Rubhma passu passu`` uj ähiuc ähiuca a rgta. rgta. d) ( ) Rubhma Rubhma b jgm`mb jgm`mb mas mas træs træs cbmas cbmas à m`egrghtg m`egrghtg.. o) ( ) Rubhma Rubhma b c`hlb c`hlb m`v`mg m`v`mg uj ähiuca ähiuca gj ma`s ma`s ähiucas ähiucas `iub`s. `iub`s. m) ( ) Rubhma Rubhma ma`s ma`s cbmas cbmas tæj b jgsjb jgsjb jgm` jgm`mb. mb. 7. Gj ujb o`rouhegræho`b, b c`hlb qug rgtb uhg a oghtra b qubcqugr pahta mgstb à olbjbmb mg6 b) ( ) rb rb`a `a.. d) ( ) m`ä m`äjgt jgtra. ra. o) ( ) rgt rgtäh ähiuc iuca. a. m) ( ) äh ähiu iuca ca.. 5. Hb cg`turb g `htgrprgtbéïa mg prafgtas, a mgsghla gj prafgéïa vgrt`obc gxgoutbma 5. Hb gxgoutbma gj gsobcbs à olbjbma mg6 b) ( ) pcbh pcbhtb tb db` db`xb xb.. d) ( ) pra prafgé fgéïa ïa.. o) ( ) oý oýja jama ma.. m) ( ) oa oart rtg. g. Oursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
9. Wj oghtêjgtra à gqu`vbcghtg b6 9. Wj b) ( ) j`càs`j j`càs`jb b pbrtg pbrtg ma jgtra. jgtra. d) ( ) 2> 2>> > j. o) ( ) oghtàs oghtàs`jb `jb pbrt pbrtg g ma jgtra jgtra.. m) ( ) 2. 2.>> >>> > j. <. Rubc à b eÿrjucb pbrb a oâcouca mb ârgb mg ujb o`rouhegræho`b1 <. Rubc b) ( ) _ : D x D d) ( ) _ : ώ x T´ o) ( ) _ : D x l m) ( ) _ : D + d x l 4 0. Bss`hbcg b bctgrhbt`vb qug `hm`ob b eÿrjucb pbrb obcoucbr a vacujg mg uj o`c`hmra. 0. Bss`hbcg b) ( ) U : ώ x T´ T´ x l d) ( ) U : D x O x l o) ( ) U : ώ x T´ T´ x β ?5> m) ( ) U : _ x l ? 2>. Rubc à a oarrgta brjbzghbjghta ma o`jghta1 2>. Rubc b) ( ) Gj ârgb oadgrtb, sadrg sadrg gstrbmas gstrbmas mg jbmg`rb jbmg`rb `sghtas mg uj`mbmg. uj`mbmg. d) ( ) Gj ârgb bdgr bdgrtb, tb, sadrg sadrg pgmr pgmrbs. bs. o) ( ) Gj ârgb ârgb oadgr oadgrtb, tb, sadrg sadrg b brg`b brg`b.. m) ( ) Gj ârgb ârgb bdgrtb, bdgrtb, sadrg sadrg t`facas t`facas jbo`é jbo`éas. as. 22. Gcgjghta oaj earjbta mg pbrbcgcairbja g brgstbs v`vbs, ebdr`obma oaj o`jghta, brg`b, 22. Gcgjghta pÿ mg pgmrb g âiub. Gstbjas ebcbhma mg qubc gcgjghta mg bcvghbr`b1 b) ( ) ]`fa ]`faca ca jbo jbo`éa `éa.. d) ( ) Dcaoa Dcaoa mg oaho oahorgta rgta.. o) ( ) ]`faca ]`faca mg saca saca-o`jg -o`jghta. hta. m) ( ) Dcao Dcaoa a ogrä ogräj`o j`oa. a. 24. Bss`hbcg (U) pbrb vgrmbmg`ra g (E) pbrb ebcsa hbs be`rjbé÷gs b sgiu`r. Mgpa`s, jbrqug 24. Bss`hbcg b bctgrhbt`vb oarrgtb. @- ( ) A jgtbc jgtbc jb`s jb`s ut`c`zbm ut`c`zbma a gj adrbs adrbs à a béa. béa. @@- ( ) B tgclb mg mg dbrra `tbc`b `tbc`bha ha à b jb`s ut`c`zb ut`c`zbmb mb gj adrbs. adrbs. @@@- ( ) @jpgrjgbd`c`zbr @jpgrjgbd`c`zbr s`ih`e`ob rgvgst`r pbrgmgs pbrgmgs g tgta oaj igssa. b) ( d) ( o) ( m) (
) ]ambs ]ambs sïa vgrmb vgrmbmg`rb mg`rbs. s. ) _ajghtg _ajghtg b pr`jg pr`jg`rb `rb à vgrmb vgrmbmg`rb mg`rb.. ) B sgiu sgiuhmb hmb g b tgrog`rb tgrog`rb sïa sïa vgrmbm vgrmbmg`rbs g`rbs.. ) ]ambs ]ambs sïa ebcsb ebcsbs. s.
Oursa mg ^gmrg`ra - Bucb 2
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@hst`tuta Wh`vgrsbc Drbs`cg`ra
