c O Q OB = P R P OAQ QABR AC = MON P R AB + BA = AA 0 0 a

Transkript

1 "!$#&%')(+*+,-'#&,!.')%0/1*243"67+2"819)7+2;:$9<7 =&>? *FEG8H> I&8477 MO PQ:DR6=7 =":$2"?+9)7 X/)C0"Y Z243"67+2"819)7+2[:$9\2"]$7 =$? =&8 =S:)^:\:)+T<812&K = A B AB AB +TU=`_ =&a-7cb0=&> =&6@19)>+:)@46d;e+#&!$#&/1fhg"iOWjk*+(+4lFiOWjk*+(+", 7 = AB AB L+2_ 9$mn7oRc9p> =&847 =7 =`_+^qk+a-7 =&@"= 7 =2"@1:$9)?6=&@"= Esrt+:4Hu4vuwMO AB AB xy*+!&zp{1zg{ =fl+24:$2"6=&@"= :$92"L>+9O_ 9OK+R2"@F].342"> =f7 =f6> =&}7@19-@12"?6Ǹ ~;6=&]$8 =F:$94q?+9`L+2"7R602" )= 9` w_ 243"7+2 EG 9 :$9; 349O_++Tƒ8`@12"8 = L+2 d6^:)7+2 Mt_ = > =&]$ DK 9)a_ ="Tt9p (+ &W '$A&*A"')/1,#DC * < p wk 9)8`819)6@12"> 9H81:$9)6 7 =-602"}@12\7 =$? =wk 2"@12\6> =&R@\T< c:)^ 84L =w_ =&@&q7 =&L>+Tt9)> AA Jr<=&].3"> =\:$94 L+2"7R@4+9)@12\L+24:$2"6=ˆ7+99 2"L>+9O_ 9OK 9)7+2ˆ]w=7 wk 9)84@19n819)6@12"> Z@"=&6="L+24:$2"6=&@"=Jn_+^qK+a-7 =&@"=H:.=Q8 =&a-7nb0=&> =&6@19)>+:)@46`7 =_ =w_ 9)7 819)6@12">Š=y]w="349OK 9)a-+TJ?+9n_+^qK+a-7 =&@"=\:$9 2"L>+9O_ 9OK+RcK =ˆ:)S9nL>+24:)@12ˆ_+^qK+a-7 =&@"=f7 =-2"@1:$9)?6=&@"= QŒn24:$2"6=&@"= AB Tt2.à 9S_ =ƒ3"^q_ 9S2"L>+9O_ 9OK 9)7 = L+2ƒ> =&])K+?7k7 =ƒ7 =& DK 9O_ 9)7 6=&6@12L> 2"L>+9O_ 9OK+R7+9)@127 =F_+^qK+a-7 =&@"=Œn2Fd 7 =w_ 2&K+ 9ˆ:$9ˆ 349O_++TJ?+9f:_ =w_ 9)7 819)6@12">ƒL+2 9$:)@19$:)@4819)7\7 =$?7F:.= :)84^ >]w=&7\> =&])K+?7\?+:OK 2"84ˆb0=&> =&6@19)>+:)@46ƒEG?+:OK =1MOt 602"@12 L+2ˆ:)812"}+:)@48 =&@"=\:)cl>k+? =&@\7 =`_+^qk+a-7 =&@"=f 9n7 =&>? ="Tt9`:)6=wK =&>QŒn2ˆ@12"?7+2!$/1.W+ Cƒ9n81:DR602?+:OK 2kEG602"7+:)@"=&7@"=ZK+ L>+24Tt9)7K+8 =1MO;60249)@12ƒ_ =&8 =ƒ7r6=&68 = ( ' &4A&*+!$*0 n b0=&> =&6@19)>+:)@46=Z7 = 819)6@12"> =Hdq6=&6@12p:)L+24Tt9)7 =wbtt9447 =&L>+Tt9)>_+^qK+a-7 =&@"= 7 =H819)6@12"> =p9y:)6=wk =&> 'HA&! +/1A"' W+*+%0*+( R@12n2"L+:)8 =;7R6=&6812n:)812"}+:)@4812p7 =;819)6@12">@19J9y:)6=wK =&>t =&L>+Tt9)>-]w=J_ = 2"L>+9O_ 9OK+RTt9QL+24:$2"6@19 7 =š819)6@12">@19 9_ 24:)@"=&@4^?7+2 _ =œmn60:)> ="Tt9ƒ7R6=&68 = 24: 8 > =&8477 =&@"=ž =Z_ =œ2"l>+9o_ 9OK++T l L+24:$2"6=&@"=š7 = AB L>+9)6=&>8 ="Tt9hL>+9)]7 =$? =wk 2"@12 7 = A AB 24: 1:)L+2">+9O_+7 =ZB9O_ 2"L+24:$2"?7 =œ: l l ~;?+9)84_+7+2ˆL+24:$2"6=&@"=ˆ7 = AB :$9n2"L>+9O_ 9OK+R9O_+7+2"]$7 =$?7+2\2"@ž#DC * ')(+,*"C0"(+*0 ^ $^qk Esrt+:4Hu4 Ÿ1MO ϕ xy*+!&zp{1zv F 7 =rt+:4ju4 Ÿ ( 'n'p:)6=wk =&>0]w=& 2"@12]w=&84+:)F7+9 ϕ AB = :)^ 2@"=&6= 2"@S24:)@"= c =&+:)@47 =S=&602> =&]$ DK 9O_ ="Tt9\_+> 0 )=24:47 0 l m Esrt+:4ƒu4 1M dp> =&].3"> =:$948@19)]$ƒK 9)6oZ:)@"=&7_ =&>@47+22"@4?@"="Tt9\^ $^qk =cl>+2"@48s? =":$2"847602"8 =&@"= :)@4>+9OK+6= :)?@"=&7+2f2"@f24:)@"=6^Tœ:)^ 2"@4819)@47Rc819)6@12"> u

2 CD xy*+!&zp{1z AB CD :.=ž 1!$e+#DCq')g&(+*ZEGL+2"7R602" )=\:$9n6=&]$8 = /1#OW+*+( ' C (+*MO+=&602ˆL> =&84@19 :.= 1:)L+2">+9O_+7ZEsrt+:4vu4 0MO AB xy*+!&zp{1z Œp>h@12"8 K 9)a=&@89O_+7 =Fƒ:)^ -=FL+2&K+ 0> =&8477 =c2"@47+24:)7+2cl> =&8 =&@"= B D ="Tt94?+9 :.= ')g&(+#&e+#&!$#&%0(+*šesrt+:4nu4 0MOJP L>+2"@4819)7[:OK+ 0? =&}@19 +TU=&@f> =&])K+?7 AB CD L+24:$2"6cK+:.= C0w &(+#&e+#&!$#&%0(+*šesrt+:4hu4 1MO xy*+!&zp{1z«ª =&]$8 ="Tt94+?+9 AB CD :.= C04A&(+* +=&602ˆ@19n:.=-9O_+7+2"L+24:$2"?7 +TU=&@f9O_+7 =&684c_+^qK+a-7 s c n9)6= 9_ =w_ AB 7+2.à 9$:)@4812"@12 2"@p81:)?6819)6@12">4602"@12 a :.=> =&847c7 = AB :$9n7 =&>? =±!.A&#&²&#&g"')([A"')/1,#DCq ³ K 9O_+8 =f_ =\2"@1349OK 9)a-+TJ?+9`81:DR6=\@12"?6= 2"@\> =&8477 =&@"=FK+[L>+24:)@4> =&7+:)@4812"@1297 =$? =wk 2"@12 7 = 9O_+7+:)@4819)7ƒ819)6@12"> 602"}@129> =&819)7ƒ7 = AB ˆ 2"} :$97 =&>? = e"cq')g&!$,4a&*+,-' Wƒ7 = EG7 = rt+:4u4 µ a AB 9 ].243"> =&].9)7 7 =@4^T<7+2L+2&K 9wMO =&6@"=?+9 AB 9 L>+9O_ :)@"=&84@19OKS7 = 9 ]w=&l+:)8 ="Tt9 a :p7r602492"@-> =&819)7+:)@48 =&@"= AB = a K+ a = AB Ÿ

3 xy*+!&zp{1zg ~ 3"> =&@47+281:$9)6 L>+9O_ :)@"=&84@19OK¹7 =š7r602"} :) $_ 9)7º819)6@12"> 2"L>+9O_ 9OK+Rº9O_+7+2"]$7 =$?7+2 or0k 2"@12 x T<7+2.à 9$:)@4812 ³ $_+7R@819)6@12">\+TU= L+24:$2"6=0602.R@12`:$9 2"L>+9O_ 9OK+R\2"@L+24:$2"6=&@"=`7 =n81:$9)6\7+9).2"8 x L>+9O_ :)@"=&84@19OKºEG6=&6@12[L+24:$2"6=&@"=[7 =7+9$:) $_+7R K+ 6=&6@12[ 9\6=&]$8 ="Tt9FL+2&d»7 =w_ 2&K+ ¼!.A"iDC+ &"( 819)6@12"> M_+^qK+a-7 = 602.R@12 9Q> =&847 =p7 =;_+^qk+a-7 =&@"=n7 = 81:$9)6ˆ7+9).2"8 L>+9O_ :)@"=&84@19OK U½;602 AB = =&@"=;7 = 92"]$7 =$? =&8 ="Tt9Q: E«_ =H7 =&L+24T<7+TJ4?+9 a a a = AB &6^q_ 9)@12 AB 9L>+2"]$812&K 9)7 L>+9O_ :)@"=&84@19OK[7 = MO a =cl>^ 8SL+2" DK 9O_š_ 9)R@"=]w=:) $_ 9)7Z819)6@12">š9f]$6 1:)@4819)7 =t7+9).9$24tt9)@4>?7 =SZ2"L>+9O_ 9OK 9) )> =&o+2"7 =wk+7 =f =&+:)@47 = :) $_+7R@819)6@12">ƒ9oR0K 2¾ n(+#wjb')!$,-a&#f2"@ 819)6@12"> ˆ =F >1_ 9h24:)7+2"8 =&@19OK+7 _ 9)R@"=Z]w=Z:) $_+7B819)6@12">BL+2"]$812&K+R8 = _ =Z:$9 2"L>+9O_ 9OK+R@À.W ÁO')²DC *+%0(+* #&e 'ÂC0"Ã+*+*º8 T<7+2.à 9$:)@4812"@12 2"@ 81:)?6 Es:) $_+7 M 819)6@12"> 60249)@ :)7+2"8 =&@"= _ 9)RB7 = =&7 =wk+@4?7 =&@"=Ž.9$24Tt9)@4>R žœn2šdf_ 2&K+ k 9ƒ:$9h 349O_++TJn?+9Z=&7 =wk+@4?7 =&@"=.9$24Tt9)@4>Rh_ =&8 =Tt2" 9)7š=&L =&> =&@S]w= ].:OK 9O_+8 =&7+9\7 =.9$24Tt9)@4>?7ZL>+243&K 9$T<[~;@c_+> 0 )=:)@4> =&7 = _ 9)R@"= ]w=ƒ:) $_ 9)7k819)6@12">kTt2.à 9S_ = :$9.9$24Tt9)@4>]$> = K 9$:)7+2 L+24:)>+9O_ :)@48124T > =&]$ DK 9)a_ =&7+97 = 602"} _ =-9pL>+9O_ :)@"=&84@19OK[7 =-:)^ `RF819)6@12">Œn2dU7 =w_ 2&K+ F 9p].243"> =&]$R8 ="Tt9:) $_+7@19819)6@12"> L+24:)>+9O_ :)@48124T@19Ob7 L>+9O_ :)@"=&84@19OK+cEG6=&6@12 9t7 =&L> =&819)7+2H7 =Hrt+:4yu4 Ä4MO š@4^ } 6=&@12;9U349)]$> =&])K+? "}ƒ@12"?7+2L>+9O_ :)@"=&84@19OKš7 = 9-> =&]$ DK 9)a_ ="Tt94t 9-].3"> ="Tt9f@19)]$602"@12S:.=F7 =&}d» w_ 243"7 ]w= a 602"76>+9)@47@197 o+9ok+ 7+9)6= a b :.=;_+8 =p:) $_+7f819)6@12"> = 7 =prt+:4qu4 Ä :.= ].243"> =&].9)7ˆL>+9O_ :)@"=&84@19OK+ 7 = a b xy*+!&zp{1z»æ Š=`].349)>+9$Tš:OK 9O_c@12"8 =L>+2"]$812&K+7 =@12"?6= 2"@> =&8477 =&@"=-K+ L>+24:)@4> =&7+:)@4812"@12SEsrt+:4Hu4 Ç1MO O ³ 7 =$? =wk 2-@12"?6=&@"= O xy*+!&zp{1z È L+24:)@4>+2.R8 ="Tt9L>+9O_ :)@"=&84@19OK 7 = Esrt+:4Hu4 É1M OA a

4 xy*+!&zp{1z Ê :OK AB 7 = b :ˆ7 =$? =wk 2@12"?6=&@"= r<=&]$ DK 9)a_ ="Tt9F@12"?6@19 A 6=&@12\7 =$? =wk 2FS6> =&}[7 =ˆ819)6@12"> = O B OB 2"]$819)6@12">2"L>+9O_ 9OK+R[:) $_+7R[819)6@12"> EG@& 94 c 9Ìe"Cq')g&!$,4A&*+,-' WF7 = MO0602"}@12` 9;7 =&>? ="Tt9!$ & n(+ˆa"')/1,#dc *+,-' OB c a b tíî9p2"@1349ok+r]$8 ="Tt9 :)^3"> =&7+9)@12n:)^:y]$7 =&6=Ï Ï""6=&6@12 L>?+:OK "6=&6@12p9L+2"6=&]w=&7+2 + c = a+ b 7 =-rt+:4hu4vu.ð xy*+!&zp{1zg{1ñ =&6= L+2&K+ 0?+9)7R@p819)6@12"> ( ' &4A&*+!$*\2"@ ].342"> =;7 =H@12"?6=&@"= =&+:)@47 ="_ =H84].9$Tt9$TS_+> 0 )= c 7 =&L>+Tt9)> :)@"=&84@19OK P P Q 7 = :-7 =$? =wk 2 ƒ:ok 9O_ƒ@12"8 = L>+9O_ :)@"=&84@19OK a P 7 = QR b :n7 =$? =wk 2ˆ@12"?6=&@"= Esrt+:4vu4vu4uwMOŠ`2"6=&à 9)@194?+9 Q OB = P 94v?+9 :) TU=&@"=-7 =819)6@12"> 9F_ 9$mn7> =&7 =[602">+9)6@47+2 EGÒ =&7+94Óc?+9)@4>^ $^qk+7o@19 :.=[ 1:)L+2">+9O_+7o P OAQ QABR ]w=& 2"@12 P Q = OA AB = QR d`2"@h@12"8 =h:ok 9O_+8 =y?+9 :)^ 2h9 1:)L+2">+9O_+76Žœ]$7 =$? P OBR AC = MO P R xy*+!&zp{1zg{1{ ½;602c9`_ =w_ 9)7h819)6@12"> = 96=&]$8 BA 9Ôe"C #&,* A&#&e+#OW+#wjB')(h7 = AB Š= ]w="349ok 9)a-+TJ?+9J:)^ DK =":)7+2 7 =&Õ-=&@"= _ 9$mn7oRˆ:) TU=&@"= AB + BA = AA _ =&8 =p7 wk 9)84R819)6@12"> 2"}c9;L>+9O_ :)@"=&84@19OK7 =Ö!.A&#&²&#&g&(+*0 c7 wk 9)8`819)6@12">602"}@12` 9p349OK 9)a-+Tš: <½;602 0 a :) $_+7R@ 819)6@12"> 2"L>+9O_ 9OK 9)7ˆ2"@ AB "@12p: 9J2"]$7 =$? =&8 ="Tt9H:) $_+7R-819)6@12">ˆ:QL>+9O_ :)@"=&84@19OK a BA Esrt+:4 u4vu.ÿ1mouœn2-@"=&6^ 8ˆ7 =$?7 a + ( a) = 0 xy*+!&zp{1zg{& ~;@_ 9$mn7oR@"=[]w=:) TU=S7 =[819)6@12">š_+>+9)6@47+2hL+2&K+ 0? =&8 ="Tt94Q?+9 š 249$:)@ a + 0 = a a :)L> RTt2h:)^3"> 9c=&7 =wk 2" c7 =S?+:OK 2"@12 2"?+9)84_+7+2h9c=&7 =wk 2" 0 0 a

5 7 = ÏOL>+2"@4812"L+2&K 2.a-7+2"@12š?+:OK 2+Ïc7 = kíî984_++t¹l+2h7 =w_ 2&K+ 1J?+9:)^3"> =&7+9)@12 7 =819)6@12">Ž a :)^3"> =&7+9)@12n7 =;?+:OK =p:.= 7 =&L^qK+7+2p_ 9)7@4?7f2"@ =wk+.9$3">?7 =p DK 9O_+7 =p@12"?6=n2"l+9)> =&oeg@& 94<+TU=&@ 9O_+7cc:)^ `S=wK+.9$3">?7S:)812"}+:)@48 =1MO Ø Ù ;Ú 1 Û1 ³ ^3"> =&7+9)@12\7 =ˆ819)6@12">[9Ü/1#O n 1,",* A&(+#"+@1249$:)@ ]w= a + b = b + a 81:$9)6 _+8 =819)6@12"> = a b xy*+!&zp{1zg{" =&+:)@47 = 7+9)6= 9nL>+2"]$812&K+7 =ˆ@12"?6= :)@"=&84@19OK 7 = :n7 =$? =wk 2f8 S:OK 9O_ O OA a = L>+9O_ :)@"=&84@19OK AC 7 = b :y7 =$? =wk 2n8 Esrt+:4Qu4vu. 1MO ³ K 9O_ˆ@12"8 =46=&6@12n9yL+2"6=&]w=&7+2n7 =.2">7 =&@"= A >+:) 076=t84].9$TU="Tt9\L>+9O_ :)@"=&84@19OK ~ :)@"=&8 =f_ =ˆ:)^ 243"> =&]$+TJ?+9@12"?6@19 OB 7 = b :ˆ7 =$? =wk 2 :)^ 84L =w_ =&@&p 2"8 =\9 L>+9O_ :)@"=&84@19OK 7 = :f7 =$? =wk 2S8 2"@12fL>+9)] BD a T<7 =&8 = B D C B 9O_+7+:)@4819)7 =pl> =&8 =1 1:)L+2">+9O_+7 = 7 = Ý(+"g$ +!$(+#p2"@ 84^ >1b `@"=&]$-L> =&8 = +TU=p9O_+7+:)@4819)7 =p@12"?6= OA B 7 => =&].:)@12.R7+9 2"@ Esrt+:4Hu4vu. 1MOU =&6@129;].243"> =&].9)7+27 =`rt+:4hu4vu T< 0@"=&@4847R ]w=&602"7 OA B ]w=\:)^3"> =&7+9-7 =f819)6@12">l+2"7r602" )= :$97 =&>? =¾e"C04A&*0W+# (+ 1!$e+#DCq')g&(+*+/14n =&+:)@47 = 9 OACB 1:)L+2">+9O_+76S:)^: :)^:$9O_+7S:)@4> =&7 OA OB +=±g&*+4á #&(+.Wi&, OC 9nL>+9O_ :)@"=&84@19OK[7 =:) $_+7R 819)6@12"> c = a + b vþsß Û1 Û1 <rt+:) 076=\u4vu) `L+2.R:)7R8 =:)@4 =&or@"=pš]w=& $> =w_ 9)7R\68 =w_+> =&@2"@.K+R812:.=p_ =w_ 9)7 ³ ^3"> =&7+9)@127 = 819)6@12">F9;:)^ "!$#&Ã+*+",* c) = ( a + b) + c 819)6@12">@19 a b +=-2"@._+R:)7+2ˆ:.=84].9)@4c@19Ob7cL>+9O_ :)@"=&84@19OK+S8-> =&])K+?+9)7>+9O_ c a + ( b + xy*+!&zp{1zg{) à 9$:)7+2-9;_ =`:$9p:)^ 243"> =&]$?+9p>+9)]$ =&@"=:OK+ 0? =&RF9p9O_+7 :)^ F Š> 0 p7 =$?7F_ =`:$9 349O_++Tš8-> =&819)7+:)@4812"@12 a + ( b + c) = ( a + b) + c _ =&8 =-rt+:4hu4vu.

6 =++TU="Tt9 xy*+!&zp{1zg{4ª ( a + b) + c = ( AB + BC) + CD = AC + CD = AD a + ( b + c) = AB + ( BC + CD) = AB + BD = AD vþs ~;@ 6024T< 0@"=&@4847Rhƒ=":$2"o =&@4819)7h]w=&602"7ƒ:OK 9O_+8 =?+9á!$ & n",c(+ e"c #&* "A&#OW ')(h²dc #&Y A"')/1,#DC *œ'\( ' &4A&*+!$*0 n[#&, Cq')g&&âne+#/1#&Y+,#c,-'\!.'\!.i&²&*"C0",`âp/1"/1,#*š#&, ÁãC 1e+*"C0"( '),#c*0 ÜA %0"!$,*+%0(+*S!$ & n* 2"8 =2"]$7 =$? =&8 =<?+9ˆ]w=F_ =c:$9ˆl+24:)@4>+2"z:) TU=&@"= 7 = 7R602&K+602S819)6@12"> =:$9ˆ7 =$?+9)>@"=&8 =7 =$? 0L+9)7 = K+7R q?@12\7 =":$2"?+9)7h2"@1:$9)?6L>+9O_ :)@"=&84R@ EG@& 94 :.=ˆL>+9O_ :)@"=&84@19OK+ƒ7 =1MH@19)]$[819)6@12"> 84].9)@4 8 e"c #&* "A&#OW ')(S>+9O_ 2" )=&8 7 =":$2"?+9)7S2"@ˆ7 =$? =wk 2"@12f6^Tš6> =&Rc7 =7 =$? 0L+9)7 >+:$9)7 =&@"=\:) TU=yrt+:4nu4vu.µ-L+2.R:)7R8 =f:)@4 =&or@"=hr9.=wk+]$> =&}@19`]w=- 0L> =&a-7+9)7+9819)6@12"> = AJ L+24:)>+9O_ :)@48124Tœ7 =$? 0L+9)7 =K+7R 602.R@12ˆ7+9 :$9n:.="Tt2"L>+9$:)? = xy*+!&zp{1zg{1 vþ«å n9)6= Œp>+31=&84RTt9\6^TB_+819)@19F:)@4> =&7 7 =c@12"8 =c> =&819)7+:)@4812S819)6@12"> = c + b = a b L+2&K+ 0? =&8 ="Tt9 t Q^ }F6=&@12 0@12 ( c + b) + ( b) = a + ( b) ( c + b) + ( b) = c + ( b + ( b)) = c + 0 = c L+2&K+ 0? =&8 ="Tt )@12 9H]w=&L+:)8 ="Tt9pL>+24:)@126=&@12 6=&6@129;L>\]$8 =&a_ =&7+9)@127 = c = a + ( b) a b?+:ok =tœn2"> =w_+f@12"8 9J7 =&>? ="Tt9ŽC0w $W+*+/1`7 =p819)6@12">@19 c = a b a b <r<=&])k+6=&@"= 7 = a b 9].243"> =&].9)7 =;7 =;rt+:4qu4vuwä0"6^q_ 9)@12 8 ]w=&@4^t<7+9)7r`68 =w_+> =&@n:.=h].243"> =&].9)7->+9O_ :)@"=&84@19OK+ 7 = a b +=L+2$_@wR0b 9nL+24:)@4>+249)7 =EGL+2`_+8 =7 =$?7 =1MQ> =&])K+6=&@"=ˆ7 = b a b µ

7 xy*+!&zp{1zg{æ ä02">7 076=QL+2"6=&]$8 =H:)^ =&.2"7 =wk7 =Q 1:)L+2">+9O_+76=4243"> =&]$ 08 a b 9p@wR0b7 =&@"=-> =&])K+6= ~;L+9)> =&or@"=f]$8 =&a_ =&7+97 =f819)6@12">l+2"]$812&k+r8 =f_ =f2"l>+9o_ 9OK++TB:) $_+7R[819)6@12"> AB =&602 :.=p2"l>+9o_ 9OK 9)7f6> =& -=&@"= T< < = _ = L+2"6=&à 9$Tƒ6=&6f:)@"=&8 =p@12"8 =1@4> R3"8 =pl^ >812p_ =pl+2.r:)7+t A B 6=&6 Tt2.à 9$T _ = ]$L+2&K+]$8 ="Tt9p819)6@12">F]w= _ =2"LÕ9$T > =&]$L+2&K 2.à 9)7+9)@12-7 =@12"?6@19 8`> =&8477 =&@"=` L>+24:)@4> =&7+:)@4812"@12 vþ Ø Š=`mn60:)> 2"@> =&8477 =&@"=-K+ L>+24:)@4> =&7+:)@4812"@12yÍÎ9pR\7 =&>? ="Tt9 O (+"%0.W+#"ŽPQ:DR6=h@12"?6= 8h@"=&6^ 8 :OK+ 0? =&} 2"L>+9O_ 9OK+R 9O_+7+:)@4819)7 :) $_ 9)7Ž819)6@12"> A a = Es: OA L>+9O_ :)@"=&84@19OK OA MO ³ $_+7R@Z819)6@12"> :$9c7 =&>? =åc0"g&*+ 1![A"')/1,#DCB7 =@12"?6=&@"= B 2"} a A 2"L>+9O_ 9OK+R\9O_+7+2"]$7 =$?7+2L+2&K 2.à 9)7+9)@127 = 8> =&8477 =&@"=K+\L>+24:)@4> =&7+:)@4812"@12y =p_ =2"@1349OK 9)a-+T A m=&6@"=?+9 9p> =w_+ 1:n819)6@12">7 = 9nLÕ9$TZL+2"7R602" )= a A A( a) Ì@"=&6=7+9)6=F:.=ˆ_ =w_ 9)7@12"?6@19 Esrt+:4fu4vu.Ç1MOǸ æ<_+7 =F2"@ 24:)7+2"847@19-]w=w_ =$?7 = A( a) B( b) 819)6@12">7+2"@12S:$T<R@"=&7+9\9_ = 2"L>+9O_ 9OK++TË819)6@12"> = AB f 249$:)@_ =F]$> =&]$+TË:) $_+7Rƒ819)6@12">7 = 602"}@12f:)84^ >]w=&7r@f819)6@12"> AB 9pL>+9O_ :)@"=&84@19OK xy*+!&zp{1zg{"è =&]$ ]w=w_ =$? =6=&6@12-:$9 84a_ =2"@.2">7 =&@"=`>+:) 076=:$9 >+9)Õ-=&8 =`349)]p 1:)K++94Ó _+> 0 $ _+ T< a + AB = b +2"@-6^q_ 9)@12-L+2&K+ 0? =&8 ="Tt94Ó OA + AB = OB ³ (fv) AB = OB OA = b a Åq ç èh ÛcéÛ"êt =&6@12\L>_ 9)}+:)@484R@"=F:?+:OK = 9$:)@19$:)@4819) Tt9$:)@12 _ =ˆLÕ9$T æu:)@19$:)@4819)7+2 a + a + a 3 a 9 :)^ 2[@"=&6=ƒ=&602]w=S7R602"}œ819)6@12"> b +TU="Tt9 b + b + b = 3 b = a U@1281Tt9$:)@12 b _ =SLÕ9$T 1 EG]w=& 2"@12 :)^ DK =":)7+2Z6=&]w=&7+2"@12 L+2&d».2">+9 MOËŒn2 :)^ -=&@"=K 2" $6= Tt2.à 9$TÌ_ =h6=&à 9$TÌ6=& b = a ]$7 =$?7 =&L>+Tt9)> du@12"8 =-9 6^q_ 9)@12 Esrt+:4yŸvuwMO 2 3 a 2 b 3 b = a 3 a xy*+!&zhzg{ p].243" 22"@[@19)]$š]w="349OK 9)a-6 9\R:)7+2[6=&6œ:$9F2"L>+9O_ 9OK+R@h7 =&L>+Tt9)>š819)6@12">@19 2 a a 18 Z a 243" `Rc:OK+ 0? =&} a6^q_ 9)@12 :.=9$:)@19$:)@4819)7?+:OK = p q p q ä02">7+2"@12h7 ="3&K+ëQ_ 9)7+9cL+2"6=&]$8 =y?+9ctt2.à 9$T _ =S.2"812">+T ]w=sl>+2"]$819o_ 9)7+9c7 =[L+2&K 2.a-@19OK+7+2 > =&o+2"7 =wk+7+2zeg@& 94ˆ@"=&602"8 =<60249)@129? =":)@47+2c7 =\_+819-o+9OK+ Mp?+:OK 2c:-819)6@12">\Œp>+2"]$819O_ 9)7+9)@127 = Ä

8 ?+:OK 2\: =ÓH=&602?+:OK α β :.=-L+2&K =&o+2"7 =wk+7 (av) (αβ) a = α(β a), ; (α + β) a = α a + β a Œn2"7R602" )=p 9yLÕ9$T[24:)819)7-@12"8 = 81Tt9$:)@12 ä09$24tt9)@4>?7+2 9y 1:)L+2">+9O_ 9)7SEG`_ 2"> aα α a α a 9O_+7+2"L+24:$2"?+9)7 M;7 = S+TU=-_+^qK+a-7 = ;ÍÎ9`2"@1349OK 9)a-+T 24:)819)7S@12"8 =?+9`9 8\:)K =fs:ok 9O_+7+2"@12 a α a _++:)@4>+3" 0@4847+2ˆL> =&84K 2Ó (cv) α( a + b) = α a + α b PQ9)>7+24:)@4@"=-7 =L> =&84K 2"@12Sì í4î<ï :$9 _ 2"6=&]$8 =K 9$:)7+2ˆ2"@frt+:4Ÿ ŸÓ xy*+!&zhzv =&+:)@47 ==&602 OA AB :.=nl>+9o_ :)@"=&84@19OK+F7 = a b EGL+2 7 =w_ 2&K+ 9JLÕ9$T OA = a AB = b OB = a+ b n9)6= :OK 9O_@12"8 = 2" )=&8 $^qk+7o@19 OA = α OA OB = α OB OAB :.=L+2$_ 243"7 ]$7 =$? ³ K 9O_ 2"8 =&@19OK+7+2 OA B AB = α A B A B = α AB 60249)@12ˆ2"]$7 =$? =&8 =?+9 < 7 =$? A B = α b OB = OA + <~;@;_+> 0 )=;:)@4> =&7 = A B = α a+α b OB = α OB = α( a+ b) Åqs ±@"=&6=HL>+2"]$819O_ 9)7+9)@12 7 =L+2&K 2.a-@19OK+7+2Z> =&o+2"7 =wk+7+2?+:ok 2ƒ:c819)6@12">ŽL>@19)a=&8 = :)812"}+:)@48 =&@"= ì ð1î<ïžì í4î<ïfeg@19ˆ:$9-7 =&>? =&@ñg&*+!$,;c *+²& 1,* A&(+*[ &"/1#&(+*MOˆà 9$:)7+2c9_ = :$9-:)^ 243"> =&]$?+9ì ð1î<ï`žì í4î<ï24:)@"=&8 =&@S819)>7ƒ =&602 _ 9$mn7> ="Tt9 < =;_ =p9y]$l^qk+7+9)7+2n@12"8 = > =&819)7+:)@4812n@4> R3"8 =n2"?+9)84_+7+2 \ )=&8 =2"@ ì ð1î<ï`l+2&k+ 0? =&8 ="Tt9 0 a = 0 0 = (1+( 1)) a = a+( 1) a ( 1) a = a ò1ó49)@12\[ 9L>+9$Tt9$T ]w=ˆ_ 9$mn7oR nœp>[@12"8 =fl+2&k 2.à 9)7+9-K 9$:)7+2FTt2.à 9$T _ =-_ 9$mn7> ="Tt9 a L>+2"]$819O_ 9)7+9ˆ7 =f819)6@12">ƒ: #&,;C *+Ã+",-' W+(+#f> =&o+2"7 =wk+7+2c?+:ok =& )="Tt9 α < 0 α a = α ( 1) a = α ( a) ä02">7@19;> =&].:)^ a_ 9)7RfL+2"6=&]$8 =&@`6=&6\:$9H2"L>+9O_ 9OK+RˆL>+2"]$819O_ 9)7+9)@12`7 =p819)6@12">\: e"c #&* "A&#OW+(+# C0"Ã+*+#&(+.W+(+#ˆ%0*+!ôW+#" Š`2"6=&à 9)@194?+9pL>c@12"8 =-9n84R>7+2:OK 9O_+7+2"@12f:)812"}+:)@4812Ó (sv) ( α) a = α( a) ; ( α)( a) = α a Œn2&dã_ 2&K+ - 9J84_++TJ?+9JL+24:)>+9O_ :)@48124T ì ð1î<ïysì í4î<ïy:$9hl+2&k+ 0? =&8 =&@@484^ >1_ 9H8 =&a-7f.9$24tt9)@4>?7 :OK 9O_ :)@484R tœp>+9o_+f@12"8 = 2431=$?+9H 9y_ 242"L>+9O_ 9OK++TƒL>+2"]$819O_ 9)7+9)@12`7 =p819)6@12">\:j?+:ok =p]w=p81:)?6 EG7+9 :.="Tt2-> =&o+2"7 =wk+7l+2&k 2.a-@19OK+7 MJ?+:OK = Åq«Å[ c Œp>+2"]$819O_ 9)7+9`7 =ˆ?+:OK 2"@12EG7+97+9)L>+9$Tt9)77+2\> =&o+2"7 =wk+7+2 M 819)6@12"> = α a b 602"}@12f9n2"L>+9O_ 9OK 9)7L+2ˆ:OK 9O_+7R 7 =$?7Ó = M =&602 b = 0 α = 0 3 M b 9p 1:)L+2">+9O_ 9)79O_+7+2"L+24:$2"?+9)7S: =&602 F_+^qK+a-7 =&@"=ft< F9p> =&847 =-7 = a α > 0 α a 8+ M b 9p 1:)L+2">+9O_ 9)7cL>+2"@4812"L+2&K 2.à 9)7[7 = =&602 F_+^qK+a-7 =&@"=ft< F9p> =&847 =7 = a α > 0 α a à 9$:)7+2f9 _ =-:$9 :)^ 243"> =&]$?+9 :)812"}+:)@48 =&@"=Sì ð1î<ïwì í4î<ï ì õ$î<ï 24:)@"=&8 =&@f8f:)k =]w=l>+2"]$819o_ 9)7R7 = 819)6@12">S:Ýe"C #&* "A&#OW+(+*ZEG7+9n7+9)L>+9$Tt9)77+2ˆ> =&o+2"7 =wk+7 MJ?+:OK = Ç

9 Š= ]w="349ok 9)a-+TJp?+9[84^ 8š81:)?6Ë:OK+ 0? =& b 9S 1:)L+2">+9O_ 9)7B7 = PyR>7+2š9S α b = α a 243"> a 0 b )=&8 =Fh:.="Tt2F@12" )=&8 =602" )=&@12:)^ 9$:)@ = ]w=\60249)@12 α b = α a =&+:)@47 =U@12" )=&8 =STt2.à 9$Ţ _ =cl+2&k 2.a-+T α = ± b [ Q^ } 6=&@12S7 wk 9)84R@[819)6@12">Z7RTU=L+24:$2"6= a Tt2.à 9$T _ 1:)L+2">+9O_ 9)7[7 =-81:$9)6819)6@12">H Î@"=&6= =&602 +@12 a 0 b )=&8 =ˆ :.="Tt2-@12" )=&8 =+602" )=&@12 b = α a ]w=7r60249 α b = 0 α = 0 Åq«Åqs n9)6= :.= _+819 C0w 9)@194U?+9f@12"?6=&@"= L>7 =w_k 9)a-š7 = A B C L> =&8 )=&8 = :.="Tt2S@12" )=&8 =<602" )=&@12:)^ 9$:)@ =9O_+7+:)@4819)7 =c_+812"}6=?+:ok = ]w= AB α β 602"@12 OC = α a + β b α + β = 1 r9)õ9)7+94ó` n9)6= 9@12"?6=c2"@ ˆ Q^ f 2"?6@19 :.= C AB A B AB 0 A, B =L> =&8 =]$7 =$? AB AC c:ok 9O_ 2"8 =&@19OK+7+2 ]w=60249)@12 λ AC = λ AB Esrt+:4 Ÿ 0MO xy*+!&zhz p].3"> ="Tt9f7 =$? =wk 2 h7+9)6= :.= > =w_+ 1:ˆ819)6@12">@19\:)^ 2"@4819)@47+27 =F@12"?6@19 O a, b c A B C ~;@ u4 + ]$7 ="9$TJQ?+9 AB = b a AC = c a ="Tt9$:)@48 ="Tt98 AC = λab œl+2&k+ 0? =&8 ="Tt9 c a = λ( b a) c = (1 ~ :)@"=&8 =_ =L+2&K 2.a-+T ]w= _ =L+2&K+ 0?+TJ?+9 λ) a + λ b α = 1 λ β = λ c = α a + β b α + β = 1 ~ 3"> =&@ )6= < 2" )=&8 = c = α a + β b α + β = 1 α = 1 β c = (1 β) a + β b = a β a + β b œœn24:ok 9O_+7+2"@12h> =&819)7+:)@4812ƒ2"]$7 =$? =&8 =Q?+9 c a = β( b a) AC = β AB y2"@h6^q_ 9)@12h:OK 9O_+8 =y?+9 AC 94t@12"?6=&@"= K 9)a-c7 =`L> =&8 =&@"= C AB Åq«Åq«Åƒö 4êt 4èZ =&6@12fL+2&d».2">+9 L+2&K+ 0? =&8 ="Tt94?+9 CB = αab yòpt<7+2.a=&8 ="Tt9:OK 9O_S@12"8 = > =&819)7+:)@4812"@12 AC = βab : > =&819)7+:)@4812"@12 α CB = αab : tœn2&k+ 0? =&8 =-:$9 β α AC β CB = 0 AC 6^T = β 0? =&}]w=-@12"?6=&@"= 6=&]$8 ="Tt94?+9 g"' W+* αcb 8f2"@47+2"Õ9)7+9 β :)?@"=&7+2\2"@ C AB α A B Åq«Åq ØSø < n9)6= :.=n@4>\@12"?6 602"@12:.= 2"L>+9O_ 9OK 9)7FL+24:)>+9O_ :)@48124T A( a) B( b) C( c) :)812"@19ˆ> =w_+ 1:Od»819)6@12">fŠ`2"6=&à 9)@194t?+9ˆTt9O_+o+9)7@4^ 7 = +TU= > =w_+ 1:Od»819)6@12"> G ABC g = 1 3 ( a + b + c) r9)õ9)7+94ó Q^ }6=&@12 :)>+9O_ =&@"= 7 = _ 9OK+ 8 2"@47+2"Õ9)7+9 1 &@12 M BC BC 1 = Esrt+:4BŸ +vuwmok½;7 =wk 2" $?7+2 _ 9OK+ 8 2"@47+2"Õ9)7+9 2 B~;@ƒ> =&819)7+:)@4812"@12 G AM 1 = α β L+2&K+ 0? =&8 ="Tt94+?+9 α = 2 3 β = 1 3 m = OM = 1 2 ( b+ c) α + β = 1 xy*+!&zhz zg{ É

10 ³ 9) ŸvunL+2&K+ 0? =&8 ="Tt94+?+9 g = OG = β a + α m = 1 3 a + 2 b+ c 3 2 = 1 3 ( a + b + c) Åq«ÅqvÞ p].243" 2`7+9)6=`:.=n_ =w_ 9)7F@12"?6@ "@12`7+9;K 9)a=&@7 =`9O_+7 = L> =&8 = A( a) B( b) C( c) 2" )=&8 =S@19F2"L>+9O_ 9OK+R@9O_+7+:)@4819)7 =S> =&8477 )=&8 = )=&8 =602" )=&@12:)^ 9$:)@ =&@@"=&68 =F?+:OK = cš`2"6=&à 9)@194?+9\@12"?6=&@"= 9\2"@ τ = (A, B, C) D( d) τ?+9 α, β γ d = α a + β b + γ c α + β + γ = 1 Åq«ÅqvÞs S2.à 9$TF_ =J2"L+:)8 ="Tt9U> =&]$L+2&K 2.à 9)7+9)@12 7 =y@12"?6@19u8j> =&8477 =&@"=JK+ L>+24:)@4> =&7+:)@4812"@12 L+24:)>+9O_ :)@48124Tñ!$*0 '),;C *+%0(+* 84>^ ]$6hEG>+9OK =&o MyTt9)a_+ ˆ@19Ob7@19 > =w_+ 1:;d<819)6@12">U 9 L+2"7R602" )= :.=yl+2qd+ w_ 243"7 ]w=q>+9)õ-=&8 =&7+9)@12J7 =Q> =&])K+?7L>+243&K 9$T< =&6=J7 =&L>+Tt9)>"Tt2.à 9$T _ =Jm 2">+T< wk+> ="Tt9 ="349OK 9)a-6=\Ÿ Ÿ ŸL+2-:OK 9O_+7R 7 =$?7Ó Q>f> =&])K+?7f@12"?6 K 9)a=&@`7 =n9o_+7 =pl> =&8 =n@12" )=&8 = )=&8 = 602" )=&@12 A( a) B( b) C( c) :)^ 9$:)@ =&@ˆ@4>c> =&])K+?7S2"@ˆ7 wk =`?+:OK = ]w=`602"@12 α β γ (sl) α a + β b + γ c = 0; α + β + γ = 0 r9ok =&or@"=ì õwù»ïp2"]$7 =$? =&8 =f81:)^ > =&])_ 9OK+R@F:)@4> =&7@19 A B C BC CA AB 8ˆ2"@47+2"Õ9)7R:)^ 2"@4819)@47+2 γ α βš`2"6=&à >1_ 9OK+bTt9ˆ24:)7+2"847@19áW+*+( ')Y+(+*ƒ#&e 'ÂC0"Ã+*+* :819)6@12">hd;:)^3"> =&7+9-7 =F819)6@12">h β γ α L>+2"]$819O_ 9)7+9c7 =819)6@12"> :\?+:OK 2 9FL+2"]$812&K+R8 =&@_ =[2"LÕ9$TÎ=wK+.9$3">?7+281:)?6œ:) $_+7 819)6@12">ƒ87R602.Rh> =&8477 =K+ƒ8cL>+24:)@4> =&7+:)@4812"@12t60249)@12c81:)^ `7+24:)@SL>+9O_ :)@"=&8"K+R8 =24:)7+2"847 =&@"= _ 9)R[7 =F=&7 =wk+@4?7 =&@"=.9$24Tt9)@4>R Ǹ Œp>+9O_+[_ =\7 =&L> = 9` 0@12"?7+T Š=\7 =&L+24T<7+TJ?+9`_+8 =F819)6@12"> = a b :.=\602&K+7+9.=&>7 =&602:.=n 1:)L+2">+9O_+7 K+F=&602 7R602"}\2"@@wR0bˆ9J7 wk 9)8+ Š> 0 H7 =$?7f_ =p6=&à 9$Tƒ@12"8 = 941?+9 I&ò1ón84].9$Tt9$T L>+9O_ :)@"=&84@19OK+ 7 = a b :-243" 2F7 =$? =wk 'Ojk",ž(+')g&(+ e"c04a&zeg]w="349ok 9)à 9)@194? a-7+2;_ =H> =&]$ DK 9)a_ ="Tt9J7 wk 9)84@19y819)6@12">-2"@._ 9OK+7+2 MOtÍÎ9Q2"@1349OK 9)a-+T[7 =&6> =&R 4?+9@12"8 = :)812"}+:)@4812 =7 =$? =wk+7 =&@"=ˆ@12"?6= ½;7 =wk 2" $?7+2[ 9\6=&]$8 ="Tt94?+9\819)6@12">@19 a b :.=û/1#o ne0w+"(+ C (+* U=&602:.= 1:)L+2">+9O_+7š7 = c 9O_+7 =` :)^ -=> =&8477 =Š> 0 n7 =$?7F_ =6=&à 9$Tœ@12"8 =`94Ó a b :.=6024T<LK =&7 =&>7 c L>+9O_ :)@"=&84@19OK+Rf7 = a b 4602"@12 +TU=&@243" 2n7 =$? =wk 2nK 9)a=&@8 9O_+7 =p> =&8477 =y~;@n@4 06ˆ7+9)]w="31=&847+2 c L+2&K+ 0? =&8 ="Tt94?+9-=&602\7R602"}h2"@F819)6@12">@19 a b 9`7 wk =&7 =&>7Ǹ =&+:)@47 = c =&602`7 :)@"=&84@19OK+@19n7 = c = 0 a b L>+9)]p7 =$? =wk 2"@12-2"L>+9O_ 9OK+R@-> =&8477 =8`602.R@12 7+9)L>+9$Tt9)77+2 :$9:)^q_+^ > a=ƒ@12"?6=&@"=h602.r@12 9cL>+9O_ :)@"=&84@19OKk7 = ±@4 06œ 92"@1349OK 9)a-+TJJ?+9 a :)812"}+:)@4812"@12ˆ6024T<LK =&7 =`7 =$? =wk 2"@12 ½;7 =wk 2" $?7+2F7 =FŸ Ÿ +vutt2.à 9$T _ =f7 =&LÕ9$T :)+Tt9)@4>?7h 1:OK 2"84R q602"@12 :.=f7+9$243$b 2$_++T<h _ 24:)@"=&@4^?7c]w=6024T<LK+7 =&>7+24:)@4@"=-7 EG@& 94t6024T<LK =&7 =&>7+24:)@4@"=-7 =&L>+Tt9)> 7 = A B C D 819)6@12">@19 AB AC AD K+c7 =81:$9)6c@4> 819)6@12"> =ˆ:p6> =& -=@19)]$c@12"?6 MO Åq«ÅqvÞ«ÅËø êt é ü4 êt Š=w_ 9)7Ž:.=S@12"?6@19 A( a) B( b) C( c) D( d) 60249)@12c7602"ƒ@4> 2"@c@wR0b7+9K 9)a=&@7 = 9O_+7 =FL> =&8 =\ 9-K 9)a=&@89O_+7 =F> =&8477 =c@12" )=&8 = ƒ:.="tt2 )=&8 =+602" )=&@12f:)^ 9$:)@ =&@f?+9)@4>c> =&])K+?7S2"@-7 wk =?+:OK = ]w=`602"@12 α β γ δ (c4) α a + β b + γ c + δ d = 0 ; α + β + γ + δ = 0 =&+:)@47 =Q=&602 :.=6024T<LK =&7 =&>7Q@12SK+ <K+ :$9)?+9 8[7R602.R A B C D AB CD AB Q7 =&L>+Tt9)> ƒpìl^ >84Rš:OK+ 0? =&}œ:)^ 9$:)@ = t]w=s60249)@12 P k b a = k( d c) ]w=& 2"@12 U 7 =$? b a 0 a + b + k c kd = 0 ½;602 p@12 K 9)a-B7 = 9O_+7+2"84>+9$Tt9)77+2Ü~;@ŽŸ Ÿvu]$7 ="9$TJp?+9 AB CD = {P } P AB CD :)^ 9$:)@ =&@ƒ?+:ok = =&7 =wk 2" $?7+2 y]w=[602"@12 λ µ 1 λ 0 µ ˆ 7 =$? p = OP = λoa + (1 λ) λ a + (1 λ) b p = µ c + (1 µ) d λ a + (1 λ) b = µ c + (1 µ) d _ =&8 = 60249)@12 λ a + (1 λ) b µ c t 249$:)@ˆ]w= (1 µ) d = 0 +TU="Tt9 α = λ β = 1 λ γ = µ δ = µ 1 α a + β b + γ c + δd = 0 α + β + γ + δ = 0 u.ð k 0 OB =

11 ~ 3"> =;_ 2"L 1:)7+9$TJ?+9 ì í Þ ïy9q]$l^qk+7+9)7+2t½;602 )=&8 =p )=&8 = α+β = 0 γ +δ = 0 ì í Þ ïh:$9;]w=&l+:)8 =6=&@12 :.=f 1:)L+2">+9O_+7½;602 α( a b) + γ( c q@12\@12" )=&8 =F 60249)@122"]$7 =$? =&8 =?+9 d) = 0 α BA = γ ³ DC K 9O_@12"8 =F2"@hì í FL> =&84@19 Þ ï CD α + β 0 γ + δ = (α + β) 0 L+2&K+ 0? =&8 ="Tt94+?+9 α a + β b = (γ c + δ d) ]$7 =$? AB ( a.) 2"8 =2"]$7 =$? =&8 9O_+7 => =&8477 = P α a + β b α + β = γ c + δ d γ + δ = p 0]w= 602.R@12 OP = p 9p243" -=]w= L> =&84@19 Š=]w="349OK 9)a-+TJ+L+2-:)^ `R 7 =$?7L+2&K+ 0? =&8 ="Tt94+?+9 =&602 β + γ 0@& 94<@19;K 9)a=&@-8 AB CD ( b.) 2"8 =F2"]$7 =$? =&8 =?+9`@12"?6=&@"= =&602 γ + α Q β b + γ g β + γ = α a + δ d α + δ = q q]w=\602.r@12 OQ = q 9-243" -=f]w=fl> =&84@19 BC AD ½;7 =wk 2" $?7+2 ( c.) L+2&K+ 0? =&8 ="Tt94+?+9 ]w=`602.r@12 R Œp^qK+7R@;?+9)@4>^ $^qk "@12ˆ $c:)84^ >]$8 =&@[EG+Tt9)77+2 γ c + α a γ + α = β b + δ d β + δ = r OR = r 9pL>+9$:$9)?7 =-@12"?6=7 =L> =&84@19 AC BD Esrt+:4<Ÿ Ÿ + Ÿ1Mq+TU=Q?+9)@4> 84^ >1b0=JnÕ9$:)@H:)@4> =&7 d+l> =&84@194 ABCD MO AB DC AD BC AC BD xy*+!&zjzvz zv Q>@19F_+812"}6Ž:)>+9) ` 0L+2&K 2.a-7Ž:)@4> =&7 :$9 L>+9$:)? =&@h8h@4>@19ýg&*+4á #&(+.W+(+*Ž,#&%0/1* P Q R 2"L>+9O_ 9OK 9)7 : I MþdFÿ4 MOž 9)]$ > =&819)7+:)@48 = L+2"6=&]$8 =&@k6=&6 81:DR6=œ_+ =&.2"7 =wk+7 =&])_ 9OK+R :)@4> =&7@194+7 =`602"@12-K 9)a- ½;602 7 =&L>+Tt9)>ˆL>+2$_+^qK+à 9)7+9)@12 7 = L>+9$:)? = 8 "L+2"6=&à 9)@194?+9y@12"?6@19 _ 9OK+R@ QR AB S S P [84^ 7Õ`7+2F8 2"@47+2"Õ9)7R EG@12"8 =F:.=\2"@47+2"Õ9)7R@"= ± β AS AP K+ α SB P B AS = β αsb AP = β MOt602"@12:.=> =[:)@12"}7+24:)@& =c_ =c_ 2"6=&à =U> =&]$ DK 9)a_ ="Tt9 αp B ì Åq«ÅqvÞ«Å ïˆ ì Åq«ÅqvÞ«Å í ïw +TU="Tt94Ó α a + δ d = (α + δ) q ; β b + δ d = (β + δ) r ƒ]$8 =&a_ ="Tt9\@19)]$ƒ_+819f> =&819)7+:)@48 =SŒn2&K+ 0? =&8 =:$9 ³ K 9O_Z6=&@12 α a β b = (α + δ) q (β + δ) r > =&])_ 9OK++TœL+24:OK 9O_+7+2"@12ˆ> =&819)7+:)@4812ˆ7 = :$9nL+2&K+ 0? =&8 = α β u4u

12 α a β b α β = (α + δ) q (β + δ) r α β 6^q_ 9)@12 9 > =w_+ 7 = ZŠ=[]w="349OK 9)a-+TJJ?+9c2"@hL+24:OK 9O_+7+2"@12ƒ819)6@12">7+2ƒ> =&819)7+:)@4812 s S :OK 9O_+8 =4?+9 9L>+9$:$9)?7 = 4I;2"@ Ÿ Ÿ Ÿ;:$9yL+2&K+ 0? =&8 ="?+9 _ 9OK+ 8p2"@47+2"Õ9)7+9 S AB QR S AB t =&]$8 ="Tt94+?+9n@12"?6@19 :.= C0 n#&(+*+%0(+#-!$e"cq'$á (+",*S2"@47+24:)7+2 β α S P AB 9þ n",-'âc *+.W+(+h,#&%0/1 :ctu=":.= Åq«Åq Ë 4èh Û1 1 ½;602: 7R602.Rš@12"?6= š 9 LÕ9$T 9c:)84^ >]w=&7+2?+:ok 2 6=&]$8 ="Tt94Q?+9 M n9)6= µ :.=S_+819 M µ (M, µ) (M 1, µ 1 ) (M 2, µ 2 ) TU=&@19)> =wk+7[@12"?6; =&]$8 ="Tt94?+9 9 o+9)7@4^ >S7 =ˆ@19)à 9$:)@4@"=f7 =f:)+:)@19$tu=&@"= C {(M 1, µ 1 ); (M 2, µ 2 )} =&602 µ 1 CM 1 + µ 2 CM 2 = 0 = s xy*+!&zjzvz«ª zg{ p7@4 0@4847+2\TU=wK+6=&@"=FTU=":.=-7 = :$9\ÏO6024T<L+9)7+:)> =Ï2"@f> =&].:)@12.R7+9)@12 H =&]$S DK 9O_+7 = M 1 CM = 6=&@12 ÏOL> =&84K 2"@127 =fk 24:)@"=Ï-ŽEG> =&].3"> =c:$9wmp:$9-l+2"@484^ > a_ =&8 =cl>+9)6> =":)7+22"@ 9)à 9O_+7+9)847 =&@"=L> =&6@46=UŒn2pd 7 =w_ 2&K+ - 9J> =&]$ DK 9)a_ ="Tt9;o+9)7@4^ > = 7 =p@19)à 9$:)@4@"= 6=&@12TU=&@19)> =wk+7 TU=`7 =-TU=":)@19n7 = M 1 M ½;7 =wk 2" $?7+2o+9)7@4^ > 7 9$:)@4@"=ZEGo+9)7@4^ >Z7 =ctu=":)@194o+9)7@4>+2"_qm 2 7 =c:)+:)@19$tu=&@"=2"@@12"?6 9n7 =&>? ="Tt9`TU=&@19)> =wk+7 =&@"=ˆ@12"?6= : TU=":.=cE {(M 1, µ 1 ); (M 2, µ 2 ); ; (M n, µ n )} (C, µ) µ = µ 1 + MO]w=`602.R@12 µ µ n (cw) µ 1 CM 1 + µ 2 CM µ n CM n = n k=1 µ k CM k = 0 S2.à 9$Tƒ_ =n7 ="T<> ="Tt9 o+9)7@4^ > =7 =n@19)à 9$:)@4@"=`7 =&L>+Tt9)>\6=&6@129HL+2"6=&]w=&7+2`7 =:OK 9O_+7 =&@"=>+:) 076=Ó xy*+!&zjzvz«ª zv p].3"> ="Tt9ˆL>+2"]$812&K+7 =c@12"?6= ³ K 9O_Z@12"8 =c]w=&l+:)8 ="Tt9\> =&819)7+:)@4812"@12 h> =&]$ DK 9)a_ ="Tt9f> =w_+ 1:ˆ819)6@12">@19ˆ7 =F@12"?6@19 O M 1 M 2 :ˆ> =w_+ 1:\819)6@12">Ó C µ 1 CM 1 + µ 2 CM 2 = 0 µ 1 ( OC) + µ 2 ( OM 2 f]$7 =$? OC) = 0 µ 1 OM 1 + µ 2 OM 2 (µ 1 + µ 2 ) <½;602 OC = 0 µ = µ 1 + µ 2 9O_+7+2"@12-> =&819)7+:)@4812ˆ2"L>+9O_ 9OK+Rc9O_ 2"]$7 =$?7+2 OC =`]$7 =$?cf@12"?6=&@"= Ó C OC = µ 1 µ 1 +µ 2 µ 2 r<=&].3"> =[:$94<81:)?602S@12"8 =9ˆ84^ ].Tt2.a-7+2:.="Tt2[=&602 ½;602S243" -=&@"=STU=":.= 7 = µ 1 +µ 2 OM 2 µ 0 µ :)+:)@19$TU=&@"=n9y7 wk ="@12ÎA&! +/1,#&%0/1\Es2"@p> =&8477 =&@"=pk+l>+24:)@4> =&7+:)@4812"@12 Mt9yo+9)7@4^ >-7 =;@19)à 9$:)@4@"= u.ÿ OM 1 OM 1 +

13 =F_ =F].349) d;7 =w_ 2&K+ wk 9)8 =Ί=ƒ]w="349OK 9)a-+T 7 =&6> =&R?+9 TU=":$2"84R@co+9)7@4^ >ƒ7 =f_+819-@12"?6k 9)a-h7 =\L> =&8 =&@"=<2"L>+9O_ 9OK 9)7 $[_ 9OK+hL+2 :OK 9O_+7R 7 =$?7Ó CM 1 Åq«Åq qsbö 4êt 4èZ Ž~;@h6=&]w=&7+2"@12h_ 2@4 06 :$9F84a_ =Q?+9 L+2L>7oLœ_ 2"L 1:)6="Tt9c@12"?6 :2"@4>o =&@19OK+7BTU=":)B 2"8 = 9c@"=&6=ƒ 8ƒL>K 2.à 9)7R@"=Z7 =h819)6@12">7+2"@12z:$t<r@"=&7+9d`7 =&L>+Tt9)> Ï)2"@4>o =&@19OK+7 =ÏQTU=":.=;Tt2.à 9_ =;:$9L+2.R84"602" )=&@12p:$9y2"L+:)8 =&@n:)k+@19ytt9)a_+ 9OK 9)6@4>?+9$:)6-]w=&> R0_+ :p> =&])K+?7]$7 =&ol>c@19ok =-:p> =&])K+?7S2"@47+24:)@19OK+7S@19) DK =c@&t7 Š=H> =&]$ DK 9O_ ="Tt9Q:$9) )=;:)+:)@19$T<-:UL+2"819)?+9y2"@H_+819yTU=&@19)> =wk+7@12"?6œp>+9$t<7 =&8 ="Tt9Q6^Tc> =w_+ 1: 819)6@12">8ì í Fï; L+2&K+ 0? =&8 ="Tt9 (cwrv) CM 2 = µ 2 µ 1 µ 1 ( OM 1 OC) + µ2 ( OM 2 OC) + + µn ( OM n OC) = 0; OC = 1 µ (µ 1OM 1 + µ 2OM µ nom n ) = n k=1 µ k OM k µ r<=w_+ =po+9)7@4^ > = 7 =p@19)à 9$:)@4@"=7 =p@12"?6@19 7+9H:$9HL>+24Tt9)7R =&602 ]w="tt9)7+t {(M k, µ k )} TU=":)@19 :`L>+2"L+2">o+2"7 =wk+7ƒ7 =f@wr0btu=":) q6^q_ 9)@12F> =&].3"> =F:$9 =&+:)@47 =@12" )=&8 = µ k cµ k c 0 243" -=&@"=TU=":.=:)@"=&8 = 602"7+:)@"=&7@"=&@"= :$9\:)^ 6> =& -=&8 =8> =&819)7+:)@4812"@12šì í -î1ïwœn2"> cµ c L>?7 =`87R602" ]w=w_ =$?c]w=`2"l>+24:)@wr8 =&7+9n7 =]$?+:OK 9)7R@"=ˆ243" -=&@"=TU=":.=`:$9p:$T<R@"=`]w=> =&847 =`7 = 9O_+7o = PJ=&a-7+2Ž:)812"}+:)@ =Zo+9)7@4^ 7 =š@19)à 9$:)@4@"= 7 =œ:)+:)@19$tu= TU=&@19)> =&@"= 78 =&> =&7@47+24:)@ :)L> RTt2[L+2$_ :)+:)@19$T<ÓF7+9)6=[8:)+:)@19$TU=&@"= ]w="tt9)7+tº7r602" M = {(M k, µ k )} 2"@@12"?6@19\:-@19Ob7Rho+9)7@4^ >ƒ7 9$:)@4@"= U7+9)6=c7 =&L>+Tt9)> n k=1 n 2"8 =F2"]$7 =$? =&8 =?+9 9ˆo+9)7@4^ =c:)+:)@19$tu=&@"= (D, µ ) {(M 1, µ 1 ); (M 2, µ 2 ); (M 3, µ 3 )} µ = µ 1 + µ 2 + µ 3 OD = OC ³ K 9O_@12"8 = > =&]$ DK 9)a_ ="Tt9`:)+:)@19$TU=&@"=ˆTU=&@19)> =wk+7s@12"?6 M = {(D, µ ); (M 4, µ 4 ); ; (M n, µ n )} RZ+TU=]w=So+9)7@4^ >š7 =S@19)à 9$:)@4@"=[@12"?6=&@"= U6^q_ 9)@12 {(C, µ )} µ = µ + µ µ n = µ OC = 1 µ (µ OD +µ 4OM 4 + +µ nom n ) = OC 4]w=& 2"@12 µ OD = µ 1OM 1 +µ 2OM 2 +µ 9)]$cL>+9$:$T<R@"=&7RL+2"6=&]$8 =&@&?+9n:)+:)@19$T<@19 +TU=&@f243" o+9)7@4^ >c7 =-TU=":)@194 3 M M Åq«Åq q«å 2"8 =ˆ9n243" 2-L> =&84K 2Ót7+9)6=-@12"?6@19 2"@ :.=> =&])_ 9OK 9)77 =_+819n $> 0LÓ M OM 3 :ntu=":.= M = {(M 1, µ 1 ); (M 2, µ 2 ); ; (M p, µ p )} µ = µ 1 + µ µ p M = {(M p+1, µ p+1 ); (M p+2, µ p+2 ); ; (M n, µ n )} ktu=":.= µ = µ p+1 :)^ 84L =w_ =&@& + µ p+2 + =&+:)@47 = =&602 n9)7@4>+2"819)@19 7 = TU=":)@197 = :)+:)@19$T<@19 + µ n 9nL>+2"]$812&K+7 =-@12"?6=+@12 C D M {(C, µ ); (C, µ )} O \]w= _ =:$9 349O_++TJ0?+9 > =&8472"@Sì í FïwÓ OD = OD = µ µ + µ OC + µ µ + µ OC OC 24:)@"=&8 = _ =]w="tt9$:)@4+tz> =w_+ 1: 819)6@12">@19 OC = 1 µ (µ 1OM 1 + µ 2OM µ pom p ) u. OC OC

14 OC = 1 µ (µ p+1om p+1 + µ p+2om p µ nom n ) Œn2\@"=&6^ 8F7 =$?7L>[@4^ >+:$9)7+9`7 =fo+9)7@4^ > =f7 =f@19)à 9$:)@4@"= A&! +/1#Fe+#&g$ n(+#wjb')!$,-a&#f#&,,#&%0/1* n#wjb' g&f²"i&g"'p &. ')( ')(+#-!( '$Á #"A&*0 n"!$#"â à ')(+,-idcq Åq«Åq q Ø Š=c> =&]$ DK 9O_ ="Tt9c:)+:)@19$TU=&@"=[TU=&@19)> =wk+7œ@12"?6 ³ ^ DK =":)7+2 {(A, α); (B, β); (C, γ)}.2">7rˆ@19)60:)@`@12"?6@ "@12`:.=no+9)7@4>+2"819;7 =ntu=":)@19;7 =n:)+:)@19$t<@19 A B C _ 9OK+R@c:)@4> =&7@19 7 = 8 2"@47+2"Õ9)7R γ {(C, γ), (A, α)} {(A, α), (B, β)} BC CA AB ABC β Esrt+:4Ÿ Ÿ 1MO α γ β α {(B, β), (C, γ)} p:)7+2ˆ94?+9no+9)7@4^ =@19)à 9$:)@4@"= 9n:)^ 2-@"=&6=- o+9)7@4^ >c7 =&L>+Tt9)>7 =-:)+:)@19$TU=&@"= xy*+!&zjzvz«ª z P {(A, α); (A, β + γ)}; ]$7 =$? AP ;Š`2"6=&à 9)@198F? =":)@47+24:)@&?+9=&602 9`o+9)7@4^ 7 =ˆ84L+:.=&7 =&@"= 2"6>^ P A = β+γ α P AP 6^q_ 9)@12 :.=f_+^qk+a-7@19ˆ7 =F:)@4> =&7@19-7 = nš`2"6=&à 9)@19f:)^ 2?+9ˆ=&602 P A = b+c a a b c ABC 84^ >1b 2"819)@19 7 = +TU=&@TU=":) t@12so+9)7@4^ =@19)à 9$:)@4@"= ABC ctgβctgγ ctgγctgα ctgαctgβ :)^ 84L =w_ =ˆ:n2">@12"o+9)7@4^ > =-7 = ABC Åq«Åq qvþ Š=_ 2"L +TU=&@F9O_+7 =&684TU=":)ZEG7 =&L>+Tt9)>[> =&847S7 = A B C D 9O_+7o =1MOš =&@12[> =&]$ DK 9)a_ =&@19c> =&])K+?7œL+2$_ :)+:)@19$T<Ž2"@hTU=&@19)> =wk+7@19c@12"?6 A B C D _ 2"6=&à 9)@194?+9ˆEsrt+:4yŸ Ÿ 0M =1M2"@1:$9)?6@ "@12 :)^9O_+7R8 =&@:)>+9O_+@19J7 = :)>+9) ` 0L+2&K 2.a-7@19;:)@4> =&7 L 1 L 2 M 1 M 2 1 B_+ =&.2"7 =wk+@19z:$9hl>+9$:)? =&@ 8Ž9O_+7 =œ@12"?6= 2 MO0602.R@12ˆ $c> =&]$L+2&K 2"84R8 = EGo+9)7@4^ 7 =š@19)à 9$:)@4@"=Ž7 =š?+9)@4>^ $^qk+76= P ABCD xy*+!&zjzvz«ª z 3 MyŠ`2"6=&à 1@12"?6=&@"= 9)@194?+9`Tt9O_+o+9)7@4^ 7 =ˆ@4>^ $^qk+76=:84^ >1b 2"8197R602"S@4>2"@\@12"?6@19 f?+9)@484^ >@4R\84>^qb-K 9)a=&@`7 = 9O_+7 =2"@1:$9)?6= 0L>\@12"8 = R-_ 9OK+f8`2"@47+2"Õ9)7+9 A B C D P P MJr<=&].3"> =f:$94+84^ ].Tt2.a-7+2F9n@12"?6@19 _ =-7+9pK 9)a=&@\8\9O_+7 =-> =&8477 )=&8 =ˆ@19 A B C D :.=p84^ >1b 2"819;7 =n@4>^ $^qk+7 = L> ="T<_ =tä02">7@19;> =&].:)^ a_ 9)7RfL+2"6=&]$8 =&@`8? =":)@47+24:)@&0?+9JL> =&84@ "@12 :)^9O_+7R8 =&@ 81:$9)6[84>^qb[:`Tt9O_+o+9)7@4^ > =\7 =\:)>+9) ` 0L+2&K 2.a-7 =&@"= T< :)@4> =&7 =F:$9L>+9$:)? =&@ 8 9O_+7 =@12"?6=FEGo+9)7@4^ > =`7 =@19)à 9$:)@4@"=1MO602.R@12 $\> =&])_ 9OK+RF82"@47+2"Õ9)7+9 3 :)?@"=&7+2-2"@`84^ >1b0=U =&]$ 1 u)

15 @12"?6=Z:)^ > =&]$L+2&K 2"84R8 = :)>+9) ` 0L+2&K 2.a-7@19 >^342"819n7 =L> ="T<_ =&@"= Åq«Åq q n9)6= 9t?+9)@4>^ $^qk+7 =JL> ="T<_ =Esrt+:4<Ÿ Ÿ 1MO Š`2"6=&à 9)@194"?+9U2"@1:$9)?6@194 ABCDM 602"@12:)^9O_+7R8 =&@STt9O_+o+9)7@4^ > =c7 =F@4>^ $^qk+76z2"@2"602&k+7 =c:)@19)7 = K+ƒ24:)7+2"8 =&@"=:)^:ˆ:)>+9O_ =&@"= 7 =fï):)>+9) ` 0L+2&K 2.a-7RÏ T< >^3:$9HL>+9$:)? =&@`89O_+7 =n\:)^ -=n@12"?6= Š`2"6=&à 9)@1940?+9 _ 9OK+f@19)]$ P P 2"@1:$9)?68ˆ2"@47+2"Õ9)7+9 3 t =&L>+Tt9)> 1 P 2 P G 1 = 2P P G 2 = 3 2 xy*+!&zjzvz«ª z«ª =-.2">7 =&@"=f>+:) 076= :.=ftt9o_+o+9)7@4>+2"819`7 =-@4>^ $^qk+7o@19 I G 1 G 2 BCM ABC :.=-:)>+9O_+c7 = Ï):)>+9) ` 0L+2&K 2.a-7@191Ï +Tš>+9$3"> = 1 AD DM Åq«Åq q n9)6= :.=n:)+:)@19$t<ftu=&@19)> =wk+7\@12"?6\:h9o_+7?7ftu=":) {(M k, 1)} n k=1 {( k, 1)} :.=-:)^ 2"@4819)@47+2ˆo+9)7@4>+2"819)@19 7 =-TU=":)@194Š`2"6=&à n k=1 9)@194?+9 (C, n) (D, n) M M M n n = n CD Åq«Åq q Œp> =&84L>+9)6=&> 84^ >1b 2"819)@19 7 =œ@4>^ $^qk+7 = M A B C D L> ="T<_ =JL>+9$:)? =&@;> =&8477@19t7 =J:)>+9) ` 0L+2&K 2.a-7@19y:)@19)7 8J@12"?6@19 qš`2"6=&à 9)@194 A B C D?+9`:) TU=&@"=\7 =f2"@47+2"õ9)7r@"= 8\602"@12F@19)]$S@12"?6[> =&])_ 9OK+R@ 9> =&847 =f7 = MA MB MC MD 1 Åq«Åq q Œp> =&8 =HL>+9$:)? = :)@4> =&7@19pEGK+`@19Ob7@19L>+2$_+^qK+à 9)7R M7 =J> =&8477+9)7-T<7+2".2"^ $^qk+76 8 C0w $W+*+%0(+*S@12"?6 602"@12`_ 9OK+R@ˆ:)@4> =&7@19 T< f8-2"@47+2"õ9)7r P 1 P 2... P n n R 1 R 2... R n P k R k = ]w= λ k R k P k+1 k n 1 Š`2"6=&à 9)@194?+9 P n R n = λ n R n P 1 λ 1 λ 2... λ n = ( 1) Ò =&7+94ÓJŠ`2"6=&à 9)@19-@484^ >1_ 9)7+9)@12F]w= píî9`l+2"6=&à 9$Tk6=&@12\L>+Tt9)>[6=&6hTt2.à n 9 n = 3, 4,... _ = :$9> =&].:)^ a_ =&8 = L> EG7 =&L+24T<7RTt94?+9 Ó@4^ }6=&@12 K 9)a-7 =;L> =&8 =&@"= "@12 &L>60249)@12 n = 4 R 1 P 1 P 2 r 1 = λ p 1 +µ p 2 λ + µ = 1 x = 9H> =w_+ MO<à 9$:)7+2`9H_ = :$9;:)^ 243"> =&]$ OX X :OK 9O_h@12"8 =q?+9tt2.à 9$T _ =\]w=&lõ9$t 84^ 8F84_ = =&68 =\?+:OK = λ µ λ = x 1 x 1 x 2 µ = x 2 x 1 x 2 x :)^ 9$:)@ =&@&<_ =&à 9F9O_+7+2"@12[2"@@wR0b EG7 =&L>+Tt9)> MnTt2.à 9f_ =3"^q_ 9\2"@47 =&L>+9O_š].3"> =&7+2SL+2 1 x 2 x L>+2"]$812&K 9)7S7 =$?7 :)@4 )= Eã=@12"8 =-9n@"=&6= ]w=& 2"@12-@12"?6@19 1 :.= C0w $W+*+%0(+*MOy =&+:)@47 = λ 0 R =&602[7 =&L>+Tt9)> t@12 h Ö@"=&6=yTt2.à 9$Tº_ =].349)>+9$T?+:OK =&@"= k L+2S@"=&6^ 8 x 1 = 7 x 2 = 7(1 λ) λ x 7 =$?7+?+9 k 2 (x 1 x 2 ) r 1 = x 1 p 1 x 2 p 2 (x 2 x 3 ) r 2 = x 2 p 2 x 3 p 3 (x 3 x 4 ) r 3 = x 3 p 3 x 4 p 4 (x 4 x ) r 4 = x 4 p 4 x p 1 ³ ^3"> =&819)7+:)@48 =- L+2&K+ 0? =&8 ="Tt9 (x 1 x 2 ) r 1 + (x 2 x 3 ) r 2 + (x 3 x 4 ) r 3 + (x 4 x ) r 4 = (x 1 x ) p 1 Œn24:OK 9O_+7+2"@12-> =&819)7+:)@4812ˆ2"]$7 =$? =&8 =+?+9nI&6+ó EG7 =\602"@12c:) 9po+9)7@4^ > 7 =`@19)à 9$:)@4@"=7 =`@12"?6@19 x 1 x 0 P 1 MOǸ 2"8 =F2431=$?+9ˆ97+9)84^ ].Tt2.a-7+2 {(R k, x k x k+1 )} x 1 x u.

16 ]w=& ( ' W 'Ojk*h7 =fl> =&8 =&@"=q84^ >1b 602.R@12 :.=ˆ> =&]$L+2&K 2.à 9)7ƒ@12"?6@19 n 7 =$? P 1 R n2 > =&819)7+:)@4812"@12 2"]$7 =$? =&8 =4?+9@12"?6=&@"= k x 1 x = 0 x = x 1 (x k x k+1 ) r k = x k p k x k+1 p k+1 R _ 9OK+ 8c2"@47+2"Õ9)7+9 ˆ~;602"7? =&@19OK+7+2 ]w=& 2"@12 k P k P k+1 λ k = x k+1 x k λ 1 λ 2 λ 3 λ 4 = ( 1) 4 x x 1 = ( 1) 4 x = x Œp> 1 L+2&K+ 0? =&8 ="Tt9@19$2">+9$TU=&@"=F7 =\ S9)7+9OK >1_ 9)7+9)@12Ÿ Ÿ Ç n = 3 n :$9p7 =&>? =-@19$2">+9$TU=7 =- =&>7+2 Åq«Åq q r<=&8477 =SL>+9$:)? =S:)@4> =&7@19F7 =cl>+24:)@4> =&7+:)@4819)7RšT<7+2".2"^ $^qk+76 8 P 1 P 2... P n n C0w $W+*+%0(+*š@12"?6 Œn2"6=&à 9)@194t?+9-L>+2"]$819O_ 9)7+9)@12[7 =c2"@47+2"õ9)7r@"=c7 =F_ 9OK 9)7+9\2"@ P 1 P 2... P Ÿ Ÿ Ç-9n> =& = n ( 1) Åq«Åq qs Š=f:)L>L+24T<7+TBŸ n Ÿ + Ÿ ="TŽ349)Õ9L+2"6=&]w=&7+2q?+9=&602 A( a) B( b) C( c) D( d) :.=ƒ?+9)@4>k@12"?6 8Z> =&8477 =&@"= L> 60249)@12Z7602"k@4>k2"@Z@wR0bŽ7+9K 9)a=&@š7 = 9O_+7 =ƒl> =&8 =;@12 :)^ 9$:)@ =&@ˆ?+9)@4>c> =&])K+?72"@-7 wk =`?+:OK = ]w=`602"@12ˆ9p]$l^qk+7+9)7+2-> =&819)7+:)@4812"@12 α β γ δ í Þ n9)6=h:$9) )=Jmn60:)> ="Tt9t@12"?6@19.602"@12 ( 'W 'Ojk", (+p')g&(+hl> =&8 =Q 9tL>+2"]$812&K+7 = A B C =&8477 =&@"=F7 = ³ ^ DK =":)7+2Ÿ Ÿ + Ÿc=& K 9)a-7 =\7602.Rh:)@4> =&7 =\7 = ABC M :)^ 9$:)@ =&@?+:OK =&@"= ]w=\602"@12 α, β, γ µ α + β + γ + µ = 0 α a + β b + γ c + µ m = 0 Š=n]w="349OK 9)a-+TJ?+9HL+24:OK 9O_+7+2"@12`> =&819)7+:)@48122"]$7 =$? =&8 =?+9 9;o+9)7@4^ >\7 =n@19)à 9$:)@4@"=`7 = :)+:)@19$TU=&@"= M TU=&@19)> =wk+7@12"?6 :)7 ="3$_ 9)7[:nTU=":) Esrt+:4yŸ Ÿ µ1mo {A, B, C} ( α µ, β µ, γ µ ) xy*+!&zjzvz«ª zg 7 =Q602.R@12 :.=Qo+9)7@4>+2"819)@197 =JTU=":)7 =H:)^ 2"@4819)@47@19y:)+:)@19$T<`2"@H_+819UTU=&@19)> =wk+7`@12"?6 A 1, B 1 C p+:ok =&@"= 1 4 2"L>+9O_ 9OK+R@`9O_+7+2"]$7 =$?7+2> =&]$L+2&K 2.à 9)7+9)@12`7 =n@12"?6=&@"= u = α µ v = β µ w = γ µ M :$9n7 =&>? =&@Ô²O C *+à ')(+,;C *+%0(+*S/1#&#DC g&*+(+",*s7 = yíî9 ]w=&l+:)8 ="Tt9@12"8 =-@"=&6=Ó p:)7+2 M M(u, v, w) = 2"L>+9O_ 9OK+R\:)^ -=&@"=@12"?6= 02"@ _+> 0 )=:)@4> =&7 = λ 0 (λu, λv, λw) M 81:DR6= 7+9)7 wk 9)8 =F@4>+2"}6=?+:OK = Tt2.à 9-_ =F3"^q_ 9-> =&]$ DK 9)a_ =&7 =c6=&@1231=&>o+9)7@4>?7ƒ60242">1_+7 =&@4 7 Åq«Åq qs Š`2"6=&à 9)@19F7 =&L>+Tt9)><?+9fo+9)7@4^ =2"L+:.=&7 =&@"=S2"602&K 2 2"6>^ a-7+24:)@[+tu= ABC 60242">1_+7 =&@4 6^q_ 9)@12 :.=`^ DK+@19 7 = (sin 2α, sin 2β, sin 2γ) α, β γ ABC ½;602 :.=-:)@4> =&7@19 7 = 0@12ˆ2"L>+9O_ 9OK a, b c ABC M(a, b, c) Åq«Åq qså =&>+:) 08 =&}@19S@4>^ $^qk+76 EsTt2.à 9Sœ_ =9c@4^ L+2"^ $^qk 9)7 Mˆ ].243"> =&].9)@19@12"?6@194?@12ˆ31=&>o+9)7@4>?7S60242">1_+7 =&@4[:.= (1, 1, ) (3, 0, 1) (0, 1, 2) ( 1, 2, 3) Åq Ø `Ù U Û1Û1 0 Åq Ø s Ž 1 n9)6= a b :.=F_+8 =c7+9)602&k+7+9.=&>7beg7+9) 1:)L+2">+9O_+7 M 819)6@12"> =c8 > =&8477 =&@"= EG 9c7 =&L+24T<7+TJJ?+92"@ƒ@4 06 8ƒ? =":)@47+24:)@Z:OK 9O_+8 =J?+9 y]w=& 2"@12ƒ7 wk 9)84R@ a 0 b 819)6@12">9p 1:)L+2">+9O_ 9)77 =`81:$9)6 _+> 0 &My 9p6024T<LK =&7 =&>+9)7[: c a b EG@& 94 a b :.=`6024T<LK =&7 =&>7 MO c ³ ^ 9$:)@ =-9O_+7+:)@4819)7 =_+812"}6=-?+:OK = ]w=`602.r@12 (α, β) c = α a + β b Š`2"6=&]w=&@19OK :)@4812Óy p].3"> ="Tt9p7R602.RF@12"?6= :)@4>+2"+T L>+9O_ :)@"=&84@19OK+ O OA OB OC 7 = :`243" 2f7 =$? =wk 2\@12"?6=&@"= Œn2f 1:OK 2"84+9`819)6@12">@19 c O OA OB OC K 9)a=&@F8F9O_+7 =ˆ> =&8477 = Esrt+:4Ÿ 1MO u.µ a b

17 xy*+!&zhz«ª a b 2"L>+9O_ 9OK+R@c_ :)d 602.R@12T<7 =&8 = L>+9)] ƒ9-9o_+7+2"l+24:$2"?7 =c: Ol O a Om 602.R@12[T<7 =&8 =L>+9)] 9\9O_+7+2"L+24:$2"?7 =S: O b cœp>+9)]\@12"?6=&@"= L>+9)6=&>8 ="Tt9FL> =&8 =< 1:)L+2">+9O_+7 = C 7 = ƒ RZL>+9$:)? = t]w=& 2"@ :.= 1:)L+2">+9O_+7 y_ =[2"]$7 =$?+T L>+9$:$9)?7 =&@"=[@12"?6=: l m l m C Œn2F:)^ `RS7 =$?7L>+9)] L>+9)6=&>8 ="Tt9-L> =&8 = 1:)L+2">+9O_+7 =\7 = n R[L>+9$:)? = 8 n Q^ }[6=&@12 m C m l C 0@12`:)^ 9$:)@ = l OCl 0]w= 60249)@12 a α OCl t½;7 =wk 2" $?7+22"@`@12"8 =0?+9 :OK 9O_+8 =0?+9 = α a OC m b β ]w= 60249)@12 OC y½;602-_ 2"L 1:)7+9$TJ?+9 ~;@SL> =&84K 2"@127 =c 1:)L+2">+9O_+76=S:OK 9O_+8 =<?+9 m = β b :$9n]$> =&]$R8 =-L+2-_+> 0 7 =$?7 7 =&L>+Tt9)> OC = OCl + c 7 =$? OC c = α a + β b c c = λ a + µ b ]$8 =w_++t 2"@:$9$349F:) L+2&K+ 0? =&8 ="Tt9 h~;@[@12"8 => =&819)7+:)@4812:OK 9O_+8 =t?+9 c (α λ) a + (β µ) b = $mno+9)7@4@19nL>+9O_ a b :.=7 wk+t =&+:)@47 = =&6027 2"@ (α λ) 0 a = (β µ) (α λ) b 6^q_ 9)@12-L+2&K+ 0? =&8 ="Tt9 t =&6@12-]$7 ="9$TJ@12"8 =7+9 9p84R>7+2 a b Ž@"=&6=0=&602 a b :.=H_+8 =;7+9) 1:)L+2">+9O_+7ˆ819)6@12"> =;8n9O_+7 =;> =&8477 =1@12 ]w=h81:$9)6-819)6@12"> 2"@p@"=&]$ c > =&8477 =f:)^ 9$:)@ =f9o_+7+:)@4819)7s7 ="342">S2"@ˆ_+819 :$9n7 =&>? =&@ /1#&#DC g&*+(+",*c7 = 9pLÕ9$T?+:OK = +@"=&68 =?+9 α K+ β p+:ok =&@"= c = α a + β b α β c c(α, β) c = (α, β) ~;@š@19)]$k> =&].:)^ a_ 9)7RBL+2&K+ 0? =&8 ="Tt94p?+9h=&602 :OK 9O_+8 = k 2"8 =8 =&a-7+2ƒ7 ="3&K+ëQ_ 9)7+9 93"^q_ 9c]$L+2&K+]$8 =&7+2ƒ?+9$:)@12h8 a b 7+9:.= 1:)L+2">+9O_+7p@12Z2"@š> =&819)7+:)@4812"@12 A a + B b = 0 A = B = 0 :OK 9O_+8 =& 3"67+2"819)7+2š8ZL>K 2.à 9)7R@"= :.= ]$> =&]$;8Z602"@12š 0? =":)@48 =&@Z_+819 { A B 7+9)]$819$:)@47Î > =&819)7+:)@48 =&@"= A = 0 :$9Z:ôb8 =& -=&@k6=&@12k:)+:)@19$tu=-2"@ 602.R@12k:$9Z2"@46>8 =&@ à 9OK =&7@19 7+9)]$819$:)@47yæt@12ˆ9O_+7cL>+Tt9)>Ó B = 0 Åq Ø «Å n9)6=p8p@4>^ K 9)a=&@ 7 = L> =&84@19 - $_ 9OK+R@ ABC A 1 C 1 BC AB 8\2"@47+2"Õ9)7R EG@12"8 =f2"]$7 =$? =&8 =?+9 α γ CA 1 = α CB BC1 = γ BA dqrt+:4hÿ µ1mojœp>+9$:$tt9)@47+9)@19 2"@47+2"Õ9)7R@"="8H602"@12;L>+9$:$9)?7 =&@"=H@12"?6= 7 = _ 9OK+@19)]$`2"@1:$9)?6<r9)Õ9)@19U]w=w_ =$? =&@"= P AA 1 CC L^ >812-L> 1 α = 1 2 γ = 2 3 Ò =&7+94Ó n9)6= AP = λaa1 CP = µ R3"8 =_ =`L>+9$:$Tt9)@47+9$Tœ?+:OK =&@"= λ µ xy*+!&zhzg Š=cL+2&K 2.a-+Ţ ]w=c@"=&]$zo+9ok CB = a BA = c p:)7+2s94?+9f819)6@12">@19 243"> =&]$ 08 =&@31=&]w= a c 24:)819)7=&602 9 ]$>+2$_ 9)7œEG>+9)Õ9)@19`:.="T<S]w=w_ =$? =&@"=\8f@12"]$[:OK+ 0? =&} MO ³ K 9O_[6=&@12f+TU="Tt931=&]w= ABC 9ˆ].349)>+9$TË 7 =$? =wk 2cdp@12Tt2.à 9f_ =3"^q_ 9úA&! }ƒ6=&@12s+tu="tt9fl> =&8127 = ].342"> 9p].349)>+9$Tš7 84>^qb 9n243" Ì]w= B CB BA ³ K 9O_c@12"8 =]$> =&].9)@19n819)6@12"> = BP L+2 _+8 =f7 =$?7 =Ó BP = CP CB = λcc1 a q6^q_ 9)@12 :.= BP = BA + AP = µ AA1 + c λ >+:$9)7@192"@47+2"Õ9)7R UŒn2&K+ 0? =&8 =ˆ:$9n:)+:)@19$TU=&@"= uwä

18 { 1 λ = αµ 1 µ = γλ R +TU=>+9)Õ9)7R λ = 1 α 1 αγ µ = 1 γ 1 αγ Åq Ø Ø n9)6=p:)l> RTt2p7R602.R-31=&]w= 819)6@12">@19 +TU=&@p60242">1_+7 =&@4 { a, b} p q p( 1, 1) q(2, 1) Š`2"6=&à 9)@194?+9 7+9n:.= 1:)L+2">+9O_+7 p q Åq Ø Ø s Q^ }6=&@ :.=f 1:)L+2">+9O_+@12F@19ó43"> =&]$ 08 =&@31=&]w=Ǹ ="Tt9)>+9)@ ">1_+7 =&@4@19 p q 7 = a b 8ˆ@"=&]$31=&]w= r9)õ9)7+94óíî9n>+9)õ`+tëï):)+:)@19$tu=&@"=ï ì"!+ï { p = a + b q = 2 a b 2"@47+24:)7+2 a b o+9ok L^ >812S:)^3"> ="Tt9ˆ_+819)@19f 0> =&847+9)7R tl+2&k+ 0? =&8 =c:$9 p + q = a ]$7 =$? ³ K 9O_ 2"8 =&@19OK+7+2 +TU= 60242">1_+7 =&@4 831=&]$+:.= c =F_ = L+2&K+ 0?+T a = p + q a (1, 1) { p, q} T<7+2.a=&8 ="Tt9-L^ >812"@12\ 0> =&847+9)7+98F:)+:)@19$TU=&@"=F: b ]$7 =$? S.2\L>+31=&84RTt9`6^TŽ84@12">+2"@12 L+2&K+ 0? =&8 =\:$9 2 2 p + q = b b = (2, 1) Š=`]w="349OK 9)a-+TJ+?+9p@4 06F81:)^ `7+24:)@ˆ]$84^ >Õ`bTt9n9OK 9$Tt9)7@"=&>7SL>+9$243"> =&].2"8 =&7R:pTU=&@4>o =&@"= ( ) 1 1 w602.r@12h9u:)^:)@"=&819)7 =H2"@H60242">1_+7 =&@4@197 = ( ( ) ( P = a b Œn2&K+ 0? =&8 ="Tt9 a 1 1 p 2 1 b) = 2 1 q ( ="3&K+ëQ_ =&@19OK+7R@?@"=&@19OKº3"B@4> R3"8 =wk 2œ_ =š9]w="349ok+r]w=wk n?+9htu=&@4>o =&@"= ) Q = 243"> =&@47 =&@"=;TU=&@4>o =H7 = š@12"8 =J7+9U9U:OK+ 0? =H]w=&602"7+24:)@4@"=;7 =J:OK 9O_+7+2"@12H_ 9)}+:)@484+94Ó 2 1 P ( ) ]w=&l+:)8 ="Tt9h> =&819)7+:)@4812"@12 ì"!+ïstu=&@4>?7+2 ) p k T<7+2.a=&8 ="Tt9 2"@.K+R812 : q ( a = P b Q = P 1 ( ( Œn2&K+ 0? =&8 =S:$9 ) a p SPQ:DR6=[2"@[TU=&@4>o@19 :$9f7 =&>? =þ n",;c *+Ã+S(+[e"Cq'# #&g& b) = P P Q q EGL>+9Ob 2$_[Tt9)a_+ 31=&]$@19 MO { a, b} { p, q} Œn24:)>+9O_ :)@48124TÝTU=&@4>o@19[7 =ƒl>+9ob 2$_ =štt2.à 9$TÌ_ =ƒ]w=&l+:)8 ="Tt960242">1_+7 =&@4@19h7 =ƒ819)6@12">@19 :)L> RTt2S> =&])K+?7œ31=&]$+:) =&L>+Tt9)> =&602S60242">1_+7 =&@4@19 7 =819)6@12"> = :)L> RTt231=&]w=&@"= u { a, b} :.= J@12 ŽTt2.à 9$TÎ_ =]w=&lõ9$t¹l+24:ok 9O_+7+2"@12ƒ> =&819)7+:)@4812ZTU=&@4>?7+2Ó ( 3, 2) ) u = 3 a + 2 b u = ³ K 9O_@12"8 =]$L+2&K+]$8 ="Tt9 > =&819)7+:)@4812"@12 ) cl+2&k+ 0? =&8 ="Tt ">1_+7 =&@4@19 ( a ( 3, 2) b ( u = ( 3, 2) P ( a b) = P ( )) ( ( p 1 1 = ( 3, 2) q 2 1 ( p q )) ( ) p = (7, ) q ( ) p q 7 = :)L> RTt2ˆ31=&]w=&@"= u { p, q} Åq Ø Ø «Å Š=w_ 9)7F:.=n819)6@12">@19 a = ( 3, 1) b = (0, 2) c = (1, 2) d = (1, 1) :H60242">1_+7 =&@4@19 :)ƒ:)l> RTt231=&]w=&@"= Ǹ Š`2"6=&à 9)@194<?+9-81:$9)6_+8 = 2"@c@19)]$h819)6@12"> 243"> =&]$ 08 =&@S31=&]w=f n2"ƒ:.= { e, h} 60242">1_+7 =&@4@19`7 =`819)6@12"> = :)L> RTt2-@19)]$31=&]$ x = α e + β h ÅqvÞ$ Ž$% Û1 Û1 n9)6= a b c d :.=`?+9)@4>c819)6@12"> =-F_ =`_ 2"L 1:)7+9$TJ+?+9 a :.=7+9)602"LK =&7 =&>7œE«_ =7 =&L+24T<7+TJ+?+9n@12"8 =8ˆ? =":)@47+24:)@\2"]$7 =$? =&8 =+7602"}S2"@ˆ819)6@12">@19 b c a b 7+9 9p7 wk 9)8+ =:)^ 2-@"=&6=-7602"c_+8 =2"@ˆ@19)]$c819)6@12">c7+9 :.=` 1:)L+2">+9O_+7 MO c 1:OK 2"84R :)^ 9$:)@ =h9o_+7+:)@4819)7ž7 ="342">Ž2"@ƒ@4>œ?+:OK = y]w=[602"@12 α β γ d = α a + β b + γ c =p@"=&]$ˆo+9okˆ_ =p> =&]$ DK 9O_ ="Tt9;L>+9O_ :)@"=&84@19OK+ OA OB OC OD 7 = a b c d :J243" 2n7 =$? =wk 2 tœn2.r:)7+9)7r@"=ˆ7 =-rt+:4ÿ Ä`3"c@4> R3"8 =wk 2-_ =K+ëp:)@4>> =&@ˆ819)>7+24:)@4@"=ˆ7 =`.2">7+2"@12ˆ@484^ >1_ 9)7+94 O u.ç )

19 xy*+!&zhz»æ =Q_ =JL>+9)6=&> ="Tt9yL>+9)] > =&8477" 1:)L+2">+9O_+7:)^ D 7 => =&8477@19 <PQ:DR6=`2"@@19)]$F> =&8477FL>+9$:)? =24:)@"=2"L>+9O_ 9OK 9)7 =2"@@4>+9)@4R Olm Omn Onl 819)6@12">8F:)^ 2"@4819)@47 Ä4MO =ˆ_ =fl>6k+ë?+tk:`>+9)õ9)7+9)@12f7 =f]w=w_ =$? =&@"= 24:)@"=&8 = _ =ˆ:)^ 243"> =&]$+TJq?+9`=&7 =wk 2" =-L> =&84K 2"@12\7 =- 1:)L+2">+9O_+76=f8fL>+24:)@4> =&7+:)@4812"@12F]$ DK 9)a_ =f@"=&6=ó OA + OB + OC = OD t =&+:)@47 = OA + OB = OM OD = OM + MD = OM + a b :.=;@4>7+9)6024T<LK =&7 =&>7ˆ819)6@12"> =4@12;]w=;81:$9)6819)6@12"> c d 8pL>+24:)@4> =&7+:)@4812"@12 :)^ 9$:)@ = 9O_+7+:)@4819)7f7 = 0602"@129O_+7+2"]$7 =$?7+2`2"L>+9O_ 9OK+R@819)6@12"> = (α, β, γ) d 8 :$T<+:)^qK?+9 d = α a + β b + γ c < :$9;7 =&>? = ²Ow &ˆ8`L>+24:)@4> =&7+:)@4812"@12 { a, b, c}?+:ok = :$9\7 =&>? =ý/1#&#dc g&*+(+",*ëes:)l> RTt231=&]w=&@"= Mn7 =819)6@12"> = (α, β, γ) { a, b, c} d EG 9f6=&]$8 ="Tt9 2" 94?+9 d = α a + β b + γ c 9 C0w $W+4Á "( '`7 = d L+2-31=&]w=&@"= a b MOU =_ =2"@1349OK+R]$8 ="Tt94?+9 c (α, β, γ) :.=60242">1_+7 =&@4@197 = d L+2"7R602" )=ˆ 9pLÕ9$T d(α, β, γ) K+ d = (α, β, γ) n9)6= +TU=-60242">1_+7 =&@4 :.=H_+8 =;819)6@12"> = 9y?+:OK 2à 9$:)7+2p9_ = :$9Q:)^ 243"> p(α, β, γ) q(λ, µ, ν) qi t +TU=-60242">1_+7 =&@4 ; =&+:)@47 } p + q (α + λ, β + µ, γ + ν) t p (tα, tβ, tγ) 6=&@12 p = α a + β b + γ c q = ËÍÎ2 :$9S2"@47 =":DR _ 2hL>+2"]$819O_ 9)7+9)@12œ:c?+:OK 2H@12 :OK 9O_+8 =ƒ2"@ Q@12 λ a + µ b + ν c p + q = α a + β b + γ c + λ a + µ b + ν c = (α + λ) a + (β + µ) b + (γ + ν) c t p = t(α a + β b + γ c) = (tα) a + (tβ) b + (tγ) c ä02">7+2"@12 > =&].:)^ a_ 9)7+9yL+2"6=&]$8 =4?+9y=&602 mn60:)> ="Tt9y7R602.R`31=&]w=4T<7+2.à 9$:)@4812"@12p7 =H:) $_+7@19 819)6@12">Z8SL>+24:)@4> =&7+:)@4812"@129\9)6848 =wk 9)7@47+2[7 =þ n(+#wjb')!$,-a&#&,#c(+a&!$*+%0/1*ƒ n",;c *+Ã+*œ2"@S@4L L+2 2"@47+2"Õ9)7+9H7 =p2"l+9)> =&o@19;:)^3"> =&7+9Hˆ T<7+2.à 9)7+9;:J?+:OK 2<½;7 =wk 2" $?7+2T<7+2.à 9$:)@4812"@ =c:) $_+7@19\819)6@12"> 8> =&8477 =&@"=S9f9)6848 =wk 9)7@47+2S7 =ct<7+2.à 9$:)@4812"@12[7 =ctu=&@4>o@19f2"@@4l tp :OK 9O_+8 =& `Rc@19)60:)@ˆ 9p84_++TJ+?+9p@"=&]$84>^ ]$6=-7+9n9 :OK+ 0? =&}7 = 1 2 ÅqvÞs = Mˆ n9)6= 9 31=&]w=QŒp>+9O_K =& )="Tt97 =?@"=&@19OK+R@\_ =-:)^ 243"> =&]$?+9 819)6@12">@19 B = { a, b, c} a b +TU=&@h60242">1_+7 =&@4 c a(1, 0, 0) b(0, 1, 0) ;L+2"6=&à 9)@19S:)^ 2?+9 c(0, 0, 1) 2 a = (2, 0, 0) a b = (1, 1, 0) a + b + c = (1, 1, 1) 3 Mn n9)6=h:.=_ =w_ 9)7819)6@12">@19 <Œp>+9$:$Tt9)@47+9)@ ">1_+7 =&@4@19 p(1, 1, 0) q(2, 1, 4) r(2, 1, 1) 7 =819)6@12">@19 2 p q + 3 r p 3 q + 2 r 8+ M n24t<lk =&7 =&>7cK+:.=`819)6@12">@19 p q r r9)õ9)7+94óh~;?+9)84_ :.= 1:)L+2">+9O_+7; 7 =$?[=&602 & :.=6024T<LK =&7 =&>7q@12 p q p q r p q 243"> =&]$ 08 =&@31=&]w=p8n> =&8477 =&@"=\EG8n602.R@12nK =1M< 9Q@wR0b7 = K+7+9)}7 =p6024tt3"7 =&or Ó r r = ξ p + :OK 9O_6=&@12-L>> =&847+T :)^ 2"@4819)@47@ ">1_+7 =&@4SL+2&K+ 0? =&8 ="Tt9:)+:)@19$TU=&@"= ³ K 9O_S6=&@12fL>+9$T<7+9$T 6^Tœ60242">1_+7 =&@4[L+2&K+ 0? =&8 ="Tt9 η q r = (2, 1, 1) = ξ(1, 1, 0) + η(2, 1, 4) ξ + 2η = 2 ξ + η = 1 4η = 1 ]w=602.r@12ˆ9pk 9$:)7+2_ =:$9n84_+?+9n7RTU=`>+9)Õ9)7+94 ÅqvÞ«Å p].243" 2 7+9S9@4> w_+7+2ƒ_ =ƒ:$9s:)^ 243"> =&]$;?+9@4>Ž819)6@12"> = :.=ƒ7+9)6024t<lk =&7 )=&8 = 602" )=&@12`TU=&@4>o =&@"= 7 = 602.R@12>+9O_ 2"819)@19EGK+\:)@4^qK 342"819)@19wMt:.=p60242">1_+7 =&@4@19 7 =p819)6@12">@19 9n243"> =&@4+TU= =&@12-K+ëp:)@4> =&orc 9p> =&]$ DK 9O_ ="Tt9L>+Tt9)>SŸ +vu4y~;?+9)84_+7+2ˆ+tu="tt9 u.é

20 p a a q = b = A b r c c c9p_ 24:)@"=&@4^?7+2ˆ_ =` T<7+2.a-+Tœ.2">7+2"@12ˆ> =&819)7+:)@4812f: ]w=`_ =`L+2&K+ 0?+T 1 4 A 1 = a 1 4 p b = q c r 60249)@12 L+2"6=&]$8 =?+9819)6@12">@19 243"> =&]$ 08 =&@31=&]$+:EGh:OK 9O_ 2"8 =&@19OK+7+2S:.=\7+9)6024T<LK =&7 =&>7 MO p q r =&+:)@47 ==& ">1_+7 =&@4@19p7 =n819)6@12"> = :)L> RTt2`31=&]$+:.= :.= 0@12 :)L> RTt2`31=&]$+:.= x B x = (ξ, η, ζ) a p U 7 =$? = (ξ, η, ζ)a 1 { p, q, r} +TU="Tt9 x = (ξ, η, ζ) b c x = ( ξ 6η + 4ζ) p + ( ξ η + ζ) q + (4ξ + 4η 3ζ) r 60249)@12h2"]$7 =$? =&8 =y?+9va&!.')/1*ža"')/1,#dc 9fK+7+9)}7 =[6024Tt3"7 =&orœ7 = 9n31=&]w= 8ˆ? =":)@47+24:)@ˆ819)6@12">@192"@ˆ602"@12f:$9n:)^:)@12" :.=`7+9)6024T<LK =&7 =&>7 ³ _+> 0 $Z_+ T< { p, q, r} C = { p, q, r} C [=&@4>o@19 :$9`7 =&>? = n",;c *+Ã+F(+ e"cq'# #&g&hestt9)a_+ S31=&]$@19 MOp =&>? ="Tt9 A A 1 C B L>+9Ob 2$_k]$> =&]$R8 =&7+9)@12Z7 =h60242">1_+7 =&@4@19h7 =_ =w_ 9)7 819)6@12">k:)L> RTt2ƒ_ =w_ 9)7 = 31=&]w=hL+24:)>+9O_ :)@48124T 60242">1_+7 =&@4@19QT< p:)l> RTt2Q_+> 0 )=J31=&]w=Œn2Qd+.2">+9t84_+R0bTt9t7 =&L>+Tt9)>w?+9=&602 :.=y60242">1_+7 =&@4@19 (ξ, η, ζ) 7 = =&@4@19T< :)L> RTt2 +TU="Tt9 x B (λ, µ, ν) C K+c]w=&L+:.=&7+2f:nTU=&@4>o q r λ = ξ 6η + 4ζ µ = ξ η + ζ ν = 4ξ + 4η 3ζ ( ) ( ) 1 4 λ µ ν = ξ η ζ = ( ξ η ζ ) A ³ K 9O_6=&@12- T<7+2.a-+T 2"@._+R:)7+2f: L+2&K+ 0? =&8 ="Tt9 ( ) ( ) A ξ η ζ = λ µ ν A PZT<7+2".2p]w=w_ 7 =p31=&]$+:h9j:)@"=&7_ =&>@47 =nl>+2"o+9o_+ 0> =p-+tu= ]w= o+9ok-_ =p:)^ 6> =&@4ˆ7R602" L>+2"@wRa-7 L>+9$:$T<R@"=&7R 6=&6@12`84a_ ="Tt9p2"@`.2">7@19 L>+Tt9)> =31=&]$+:p:$9 :)819)a_ = _ 2 T<7+2.à 9)7+9`:nTU=&@4>o ÅqvÞ Ø Eã=1MQ =&LÕ9)@19n60242">1_+7 =&@4@19`7 =819)6@12">@19 a b :)L> RTt2ˆ31=&]w=&@"= c C Es3 MnP 31=&]$+:.= 9_ =w_ c =&684 :.= 60242">1_+7 =&@4@19\7 = :)L> RTt2 C u = ( 2, 1, 3) u 31=&]$+:.= ÅqvÞvÞ B& n9)6= :.=_+8 =œe«_ 2"6=&à 9)@19 ( ')/1#OW+*+( ' C (+*M 819)6@12"> = u = (2, 1, 1) v = ( 1, 1, 2) PQ:)?6F6024T<LK =&7 =&>7:;@wR0bf819)6@12">FK 9)a=&@8 Es:.= 1:)L+2">+9O_+7 7 =1MU9O_+7 => =&8477 = ³ $_+7@19 τ 819)6@12"> 8Z@"=&]$ > =&8477 =;>].3"> = :$94H:$9[2"L>+9O_ 9OK+R@š2"@ ²Ow &", 8 º 249$:)@&p=&602 9 { u, v} τ q 7R602"}B6024T<LK =&7 =&>+9)7 : ;6^q_ 9)@12 :.=ƒ60242">1_+7 =&@4@19ƒ7 = u v q = α u + β v (α, β) Ÿ4Ð

21 Ì =ƒ 0L> =&a-7+9)7+9[_ 9S6024T<LK =&7 =&>+9)7Ë: ¹ q q(4, 1, 7) u v 60242">1_+7 =&@4@19`7 = :)L> RTt2f31=&]w=&@"= 8 q { u, v} τ ÅqvÞ 243"> =&]$ 08 =&@31=&]w=1_ =n6=&à n9)6= 7+9H9J6024T<LK =&7 =&>+9)7 : 7 =&L>+Tt9)>f7+9)6= p 9$T 'p:)7+2`940=&602 U 2" )=&8 = 819)6@12">@19 u v p = (1, 1, 1) 9ºe"C #&* "A&#OW u v p P x(α, β, γ) 7R6=& ">1_+7 =&@4[:)L> RTt2 _ =`6=&à 9$T P (ξ, η, π) n9)6= (QC #"')/1Ã+* A = x = ( α β γ ) a b = ( ξ η π ) u v = ( ξ η π ) a A b c p c ~;@ˆ@12"8 =-:OK 9O_+8 =?+9 ( α β γ ) = ( ξ η π ) A c]$7 =$? Œp>+9$:$T<R@"="Tt9`243"> =&@47 =&@"=-7 = A Ó 9nTU=&@4>o =&@"=ˆ7 =`L>+9Ob 2$_ =ˆTt9)a_+ P B t ntu="tt9 ( ξ η π ) = ( α β γ ) A A 1 = ]w=&l+:)8 ="Tt9 L+24:OK 9O_+7+2"@12f> =&819)7+:)@4812-L ">1_+7 =&@4Ó PŽ? =":)@47+24:)@ ξ = α + β η = 2α 3β + γ π = 3α β + γ { ξ = α + β η = 2α 3β + γ Š=Q> =&]$ DK 9O_ ="Tt9:$9) )=J819)6@12"> = w@12"}:$9t7 =&>? = e"c #"')/1Ã+*0 `7 = 8 L+2Q7 =&L> =&8"K 9)7+94 x τ = ξ u+η v x τ 1:)L+2">+9O_+7+2ˆ7 = EG84a-@19 >+:) 076=&@"=)(QC #"')/1Ã+*0 MO~;@ˆ.2">7@19nL>+9$:$T<R@"=&7RL+2&K+ 0? =&8 ="Tt94?+9 p x τ = (α + β) u + ( 2α 3β + γ) v = (α + β)(2 a b + c) + ( 2α 3β + γ)( a + b + 2 c) ³ K 9O_ 2"8 =&@19OK+7+2 x τ +TU=60242">1_+7 =&@4 α τ = 4α + β γ β τ = 3α 4β + γ γ τ = 3α β + 2γ Ÿu

22 : DK 9)a_ α τ α β τ = P β γ τ γ ^q_ 9)@12 P 9nTU=&@4>o =&@"= ÍÎ9ˆ7 =&>? ="Tt9 n",;c *+Ã+(+e"C #"')/1,*"C0"( '),#\8 \ Rh+TU= > =&])K+?7 8 =&a-7ƒ.9$24tt9)@4>?7 P τ :)812"}+:)@48 =Ó ÅqvÞ Eã=1MQŒp>+2"819)>+9)@194+?+9 P 2 = P Es3 MqPQ:$9)6 819)6@12">&602"}@12;9t6024T<LK =&7 =&>+9)7: ÏO])_+^ > a=ïu7 =QL>+2"]$819O_ 9)7+9y: =&L>+Tt9)> u v P =&602 q + +L>+2"819)>+9)@194?+9 EG@4 06FL>+9O_+L+2&K =& )="Tt94? ">1_+7 =&@4@19 7 = :.= P q = q q ]w=&l+:.=&76=&@12ˆtu=&@4>o =ˆ:)@4^qK 3UŒp>+2"819)>+9)@19:)^ 2?+9 P x τ = x EG8 τ EG@1249$:)@-K 9)a-c8 MUŠ`2"6=&à 9)@194?+9`=&602ˆ]w=-7R602"}819)6@12"> MO 9 ]$L^qK+7+9)7+2 +@12 9n6024T<LK =&7 =&>+9)7: w P w = w w u v τ EG8 M n9)6= 97+9)7 wk 9)8 = TU=&@4>o ="]w=h602.r@12p9]$l^qk+7+9)7+2 ½;602 9y243"> =&@4+TU= Q (3 3) Q 2 = Q 9$Ţ _ =c T<7+2.a-+TºL+24:OK 9O_+7+2"@12[> =&819)7+:)@4812: ]w=c_ =cl+2&k+ 0?+T <6^q_ 9)@12 9 Q 1 Q = E 3 E 9O_+7?7 =&@"= TU=&@4>o =FŠ= _ 2"L 1:)7+9$Ţ :$9) )=<?+9 9f7+9$243"> =&@4+TU= Q^ } 3 (3 3) Q Q 0 > =&7 7 = 9pK+ƒu4K+SŸUÍÎ9p> =&]$ DK 9O_ ="Tt9`2"@._ 9OK+7+2-_+819)@19n84^ ].Tt2.a-7+24:)@4Ó r(q) Q EG8 &\A&!.')/1*S819)6@12"> M E MyŠ`2"6=&à 9)@194q?+9:)^ 9$:)@ =ˆ@"=&6^ 8F7+9)7 wk 9)8\819)6@12">?+9 1 r(q) = 1 EGL>+249)6oR 84^ >1b FL> =&8 = 1:)L+2">+9O_+7 =-7 = MO 9 602&K+7+9.=&>+9)7 u Q x u x u EG8 9H6024T<LK =&7 =&>+9)7: M E MŠ`2"6=&à 9)@194?+9:)^ 9$:)@ =&@ˆ_+8 =-7+9)602&K+7+9.=&>7[819)6@12"> = 2 r(q) = 2 &ˆA&!.')/1*c819)6@12"> EGL>+249)6oRF84^ >1b ˆ> =&8477 = 1:)L+2">+9O_+7 =7 = +@"=&68 =?+9 u v MO Q x u v x u v ÅqvÞ+* e i&,-a&"( '`*[/1#O ')(+, C, =_ =->+9)Õ`+Tœ.2">7@19 L+2$_+ 1:OK 2"84R[9 L+2&K 9)]$7+2-_ ]w=\602"@12 -r<=&].3"> w_ 2"8"K 9)@4812"> R8 =cl+24:ok 9O_+7+2"@12 x Q x = λ x x = 0 0> =&847+9)7+9ž &ca&! +/1# S@4^ 6+Tt2F]w=&@12"8 =f7+9`l>+9o_ :)@"=&8"K+R8 =F7@19)>+9$:4 >+:)+TŽ7+9)7 wk 9)84 λ 819)6@12"> ]w=f602"@12 L>[7R60249 npq:$9)6[@"=&6^ 8 819)6@12">h:$97 =&>? =¾!$#&²&!$,-A"')(ƒA"')/1,#DC Q x = λ x λ 7 = +=`?+:OK 2"@12 ḑ!$#&²&!$,-a"')(+#-%0*+!ôw+#.-»!$,#&y+(+#&!$,0/n7 = Q λ Q ÅqvÞ n9)6= 9FL>+2"]$812&K+7 =[68 =w_+> =&@47 =TU=&@4>o =Z 7+2.à 9$:)@4812"@12h7 =81:)?6 :$2434:)@4819)7 Q?+:OK =7 = :$9p7 =&>? =!$e ')/1,-iDC7 = Q Q ³ 2434:)@4819)7@19-?+:OK =\[819)6@12">h7 =\TU=&@4>oh:.=F9O_+72"@c24:)7+2"847@19L+2"7R@4R[8F@19$2">R@"=F7 = TU=&@4>?7+2"@12:$T<R@"=&7+94f 9ˆ:.=F8 =&a-7h]w=\l>k 2.à 9)7R@"=ZEG8Tt9Ob0=&76=&@"=.9$24Tt9)@4>R@"=c[_+> MO= :)^ 2- ]w=`@19$2">r@"=-7 =-TU=&@4>?7+2"@12f:$T<R@"=&7+94 Š=:$9 84^ >7+9$Tš6^TZ]w=w_ =$? =ˆŸ + µ EG8 MUƒEG8 MOU =&L+:)8 ="Tt9n> =&819)7+:)@4812"@12 L+2:OK 9O_+7R Q x = λ x 7 =$?7Ó 06^q_ 9)@12 9p9O_+7?7 =&@"=-TU=&@4>o =` :OK 9O_c@12"8 =L>+9Ob84^ >K+RTt9 Q x = λ x = E(λ x) = λe x E 2"@.K+R812Ó ³ ^ DK =":)7+2n6=&]w=&7+2"@12nL+2&d».2">+941@12"8 = TU=&@4>?7+2n 0> =&847+9)7+9 Q x λe x = 0 (Q λe) x = >+:)+T ( ')(+ &W '$A&#f>+9)Õ9)7+9 7 =f@12"8 =FTU=&@4>?7+2 0> =&847+9)7+94Ǹ =&602"8 = :)^ 9$:)@ =F:.="Tt2 ( x 0) =&602_ 9)@19)>+T<7 =&7@"=&@"= 7 =@"=&]$:)+:)@19$TU=ˆ9p7 wk = Q =`:.=>+9)Õ9)7R\7 ==wk+.9$3">?7+2"@12-0> =&847+9)7+9 t 2-9 2"@ Q λe = 0 :)@19)L+9)7[ =&602fTU=&@4>o =&@"= 9 K+2"@f:)@19)L+9)7[Ÿ =&602 Q 3 3 Q dytu=&@4>o =Q =&L>+Tt9)>[=&602 ) 2 Q = ( q λe = 1 λ λ = (1 λ)2 4 2"@ˆ60249)@12ˆ:OK 9O_+8 =+?+9n:$2434:)@4819)7@19?+:OK =-7 = :.= Q λ 1 = 1 λ 2 = 3 n9)6=\:$9) )= :.=f:)^ 2"@4819)@47+2F:$2434:)@4819)7[819)6@12">S[:$2434:)@4819)7 =f:)@12"}7+24:)@f7 = JÒpT<7+2.a=&8 ="Tt9 x λ Q 2"@.K+R812[> =&819)7+:)@4812"@12 Ý@"=&6= Q x : L+2&K+ 0? =&8 ="Tt9 n2 <]$7 =$? = λ x Q Q 2 x = λq x Q x = λ x Q 2 x = :.=:$2434:)@4819)7Z?+:OK =7 = còpt<7+2.a=&8 =&}6œ: L+2&K+ 0? =&8 ="Tt94t?+9 λq x = λ 2 x λ λ 2 Q Q :$2434:)@4819)7Z?+:OK =7 = :.=S:)^ =&@"=&@4^ 621e ')Ã+*+.W+(+#/1#"Á ",# ' Q 2 Q 3 λ 3 λ 4 Q n",;c *+Ã+(+e"C #"')/1,*"C0"( 'Oâ,# *F &(+"%0* â *c,fzv(qz ³ K 9O_ 2"8 =&@19OK+7+2 Q 2 = Q Q 3 = Q Q 4 = Q λ 2 λ 3 Ÿ4Ÿ

23 =\7 K+ n2 :$2434:)@4819)7@19-?+:OK =\7 = :.=f7 =&}[dht<7+2".2c 7 =$? Q ³ K 9O_š@12"8 =c7 ="T<> ="Tt9F:$2434:)@4819)7@19\819)6@12">Z7 = Q λ 2 = λ λ 3 = λ λ = 0 λ = ±1 TU=&@4>o =&@"=7 =cl>+249)6@4> =&7+9)@12 2"@S]w=w_ =$? =[Ÿ + ƒe«_ 2"6=&à 9)@194U?+9f@19f:.= MOŒp> :$9 P 0 1 λ = 1 L+2&K+ 0? =&8 = c:)^ 2"@4819)@47+2\:)+:)@19$TU=&@"=ˆ x 1 0 (P E) x = x 2 = x 3 0 (ev) 3x 1 + x 2 x 3 = 0 3x 1 x 2 + x 3 = 0 3x 1 x 2 + x 3 = 0 R 2"?+9)84_+7+2ˆ:$9n:)819)a_ =`_ 2 0> =&847+9)7+9)@12 t 2"8 = 0> =&847+9)7+9n+TU=`]w=`>+9)Õ9)7R 3x 1 + x 2 x 3 = 0 7 =&L>+Tt9)>819)6@12">@19 x = (1, 0, 3) y = (0, 1, ) l\#&/1&jb'),-'$1?+9a&! +/1 W+*+( ')Y+(+/1#O n²&*+(+"ã+*0 F(+ * :)^ 29J>+9)Õ9)7+9H7 =fì431î<ïwtœp>+2"819)>+9)@194 x y?+9 :.=F>+9)Õ9)7Rh7 =hì431î<ï`h]$7 =$?ƒ81:dr6=2"@ _+812"}6@19 9ˆ31=&]w=F8T<7+2.à 9$:)@4812"@12 u v u v x y 2"@œ819)6@12">@ T<LK =&7 =&>7º: Š`2"6=&à 9)@19 :)^ 2n?+9 A&!.')/1*,;C *šcq'6-')(+*0 Ë7 = ì431î<ï[:.= τ 6024T<LK =&7 =&>7 EGk:.=ƒ6024T<LK =&7 =&>7 : MO¹ Q^ }k6=&@12š:$2434:)@4819)7 =&@"=œ:)@12"}7+24:)@œ9zu4;@12 ]w= 81:$9)6 τ :$2434:)@4819)7F819)6@12">hEGK )@129 :)^ 2"@12 d<]w=n81:$9)6f819)6@12"> 2"@ M +TU="Tt9 EGL>+249)6@4> ="Tt9 τ x P x = x 84^ >1b MO τ Œp> L+2&K+ 0? =&8 ="Tt9 λ = 0 c:)^ 2"@4819)@47 =&@ If:)+:)@19$TU= (ev1) 4 1 x 1 0 (P E) x = x 2 = x 3 0 4x 1 + x 2 x 3 = 0 3x 1 4x 2 + x 3 = 0 3x 1 x 2 + 2x 3 = 0 =;L>+249)6@4> =&7+9)@12 2"@ Ÿ + 9y>+9)Õ9)7+9y7 =;@"=&]$ˆ:)+:)@19$TU= p = (1, 1, 1) n9) 2\L+2"819)?+94q_ 2"6=&à 9)@194?+9 A&! +/1#FgDC Á #f>+9)õ9)7+9`+tu=f84_ = EGL>+249)6@4> =&7+9)@12c9` 1:)L+2">+9O_+7+2 t p 7 = MO p ÅqvÞ s n9)6= :)^ 2œ9ƒTU=&@4>o =š7 =šl>+249)6@4> =&7+94±Œn2Zd\@12"?7+2 Q = Q _ 2"6=&à 9)@194? Œp>@12"8 =fl>+249)6@4> =&7+9)@12:$9`]$84^ >Õ`8 =\ 1:)L+2">+9O_+7+2F7 =\> =&8477 =&@"=602.R@12 T<7 =&8 = 9TU=&@4>o =F7 =fl>+249)6@4> =&7+9)@12c84^ >1b SL> =&8 = q602.r@12c9` 1:)L+2">+9O_+7 =F7 =f819)6@12"> = Q l p = (1, 1, 1) L>+9)] 7 =$? =wk 2"@12 c819)6@12">@19 O a = (1, 1, 0) b = (1, 0, 1) ÅqvÞ «Å =&6[]$ DK 9)a_ =\L>+249)6@4> =&7+9)@12 8F> =&8477 =&@"= q2"l>+9o_ 9OK 9)7 =F2"@F@12"?6=&@"= π M(1, 1, 1) 819)6@12">@19 (a) = ( 2, 1, 0) b = (1, 0, 1) )@12f9p 1:)L+2">+9O_+7+2 EãI1My7 =`819)6@12"> = (1, 0, 0) Es3 My7 =`819)6@12"> = (1, 0, 1) ÅqvÞ Ø n9)l>+9)6^:)7 =&@"=nb 24Tt2".9)77 = :)>+9O_ = 9QL+2$_K 2.à 9)7 =7 = K+7+9)}7 = _ 9$m 2">+TU=&oR L>ˆ602.R@12 31=&]$+:)7 =&@"=-L>].TU=cEG@& 94v+L>].TU=&@"=-L+2">+2$_ 9)7 =ˆ2"@ˆ31=&]$+:)7@19 819)6@12"> a b My:$9p@4> =&7+:$m 2">+T<> = c 8SL>].TU=&@"=tL+2">+2$_ 9)7 =[2"@S819)6@12">@19 (1, 0, 2)7 =&684Z:.= p = 1 2 (3, 1, 1) q = 1 2 (1, 2, 0) r = 1 2 Ÿ4

24 & DK =`_ 9$m EGÒ =-:.=7 =&L> =&8"K =&]$8 :)^ 9$m 2">+TU=&oR M 7;ó4Tt9)7@"=&>Ó<Œn2"6=&à 9)@1944?+9 243"> =&]$ 08 =&@p31=&]$+:4 àn7+9)}7 =&@"=H_ 9$m =&7+:$m 2">+T<> = p q r 81:$9)6œ819)6@12"> Eã=&602 84^ 8819)6@12"> = 6^q_ 9)@12 x :$9]w=&L =&]$8 =\SL+24:$2"6=&@"=1MO 7 =$?@4> R3"8 =ˆ_ =\:$97 ="Tt9)> R@ƒEã=&602q> =&].3"> =\:$94 9 TU=&@4>o =œ =&L> =&8"K 9)7+9)@12 :$9 ]w=&l =&]$8 =Q=&602 D x D x D x = λ x λ > 0 +TU=`@"=&68 =1MQ:$2434:)@4819)7@19n819)6@12"> 7 =TU=&@4>o =&@"=7 =L>+9Ob 2$_ =-2"@ˆ31=&]$+:.= a b 8-31=&]$+:.= c p q r ~;@J_+> 0 )=J:)@4> =&7 =&:$2434:)@4819)7@19U?+:OK =H:.=y8 =&a-7nb0=&> =&6@19)>+:)@46`7 =yk+7+9)}7 =&@"=Q_ 9$m 2">+TU=&oR ~;@4 O λ 1 = 1 2 λ 2 = 1 λ 3 = 2 p = ( 1, 1, 1) q = (0, 1, 1) r = (2, 1, 1) Åq ŽÙ U ºÛ1Û1 0 Š=cmn60:)> 2"@L>+24:)@4> =&7+:)@4812"@12œEGK+ O > =&8477 =&@"=1Mƒ7+9)6= 2"L>+9O_ 9OK+R :) $_+7Rk819)6@12"> 9f31=&]w= EG8S> =&8477 =&@"=U> =&].3"> =:$94t31=&]w=&@"=S9 MOPQ:DR6=@12"?6= { a, b, c} { a, b} :SL>+9O_ :)@"=&84@19OK ¹~ 3"> =&@ :$9)6k:) $_ 9)7B819)6@12"> A a OA 2"L>+9O_ 9OK+R\9O_+7+:)@4819)7 =&@"=`@12"?6= ]w=n602.r@12 b B OB = b < 2"?6=&@"= EG602.R@12`L+2"7R602" )= 9H7 =&>? ="Tt9 O (+"%0.W+# M 31=&]w=&@"= 9y7 =&>? ="Tt9k/1#&#DC g&*+(+",(+n!$*+!$,-' n4 Ž@"=&6=17+9)6= 9 { a, b, c} K = {O; a, b, c} 60242">1_+7 =&@47 =`:)+:)@19$TU=tŒn24:)>+9O_ :)@48124T87BTt2.à 9$Th_ =p>+9o_+ 0o> ="Tt9;81:DR6=n.9$24Tt9)@4>?7 = ]w=w_ =$? = 6^T 7 ="342"> =wk+.9$3">?7f> =&].:)^ a_ 9)7R =&+:)@47 =0> =&]$L+2&K 2.à 9)7+9)@12Es:)L> RTt297M<7 = 81:DR6=@12"?6= 8 A > =&8477 =&@"=ˆK+cL>+24:)@4> =&7+:)@4812"@12\:$9n2"L>+9O_ 9OK+R2"@ˆ> =w_+ EG8ˆ> =&8477 =&@"=\9n7 ="342">S2"@ˆ_+819?+:OK = MJ 9n7 =&>? ="Tt9Ü/1#&#DC g&*+(+",*s7 =-@12"?6=&@"= OA = (α, β, γ) y Q>+2"}6=&@"= (α, β, γ) (α, β) A c 9pLÕ9$T J~ :)7+2"847@19]w=w_ =$?@4 06c:$9 2"@47 =":DR@ˆ_ 2-]$> =&]$R8 =&7+9)@12ˆ7 =60242">1_+7 =&@4@19 A(α, β, γ) 7 =w_ 9)7c.9$24Tt9)@4>?7 1:OK 2"84R Ó Åq qs = M Š=w_ 9)7œ:.=c@12"?6@19 SŠ=S:$9fL>+9$:$Tt9)@47 =&@60242">1_+7 I&@4@19 A(0, 2, 1) B(3, 4, 1) 7 = 819)6@12"> = AB p].243" 2 _ 2"6=&à A(α A, β A, γ A ) B(α B, β B, γ B ) AB = (αb α A, β B β A, γ B γ A ) 3 MfŠ=w_ 9)7 =\9`24:)819)7S@12"8 =ˆS@12"?6=&@"= ;Š`2"6=&à 9)@194q?+9@12"?6@19 K 9)a=&@F7 = C(1, 0, 1) A B C 9O_+7 =nl> =&8 = Š`2"6=&à 2"@1:$9)?6=&@"= 9)@19p:)^ 21?+9H?+9 9J> =&]$L+2&K 2.à 9)7 =Tt9)a_+ Pš6=&68122"@47+2"Õ9)7+9H_ 9OK+ C A B C 8+ M Š=w_ 9)7Z:.=c@12"?6@19 AB& 6=&@12 A A B B OA = 3 OA OB = 2 OB 7+9 A B O K 9)a=&@f7 =9O_+7 =L> =&8 =<Š`2"6=&à 9)@194?+9pL> =&84@19 :$9nL>+9$:)? AB A B P ½;602 +TU=60242">1_+7 =&@4 7 ="Tt9)>+9)@19p60242">1_+7 =&@4@19 7 = t ="Tt9)>+9)@19 A A(2, 1, 1) B(1, 1, 1) P 2"@47+2"Õ9)7+9)@12 AP ÍÎ9p_ =w_ 9$Tœ:ôb 9$TU=-7 =`>+9)Õ9)7+9)@12SEsrt+:4Ÿ P B Ç1MOÓ xy*+!&zhz È ~;?+9)84_+7+2Tt2.à 9$T_ = > =&]$ DK 9)a_ ="Tt9;819)6@12">@19 OA OB 6=&@12 31=&]w=p8n> =&8477 =&@"= ³ L> RTt2 OAB,w &*²Ow & 819)6@12">@19 OA OB +TU=&@H60242">1_+7 =&@4:)^ 2"@4819)@47+2 &]$7 A (3, 0) B (0, 2) +TU=`:)^ DK =":)7+2-L+2$_+@12"?6=f= M60242">1_+7 =&@4 A B :)^ 2"@4819)@ ( 3, 2) AB : 7 =$? =wk 2 F6> =&} A(1, 0) <½;602H819)6@12">@19 B(0, 1) ( 1, 1) AB :.=J 1:)L+2">+9O_+7"@12H@4> R3"8 =J_ =Q+TU=J?+:OK 2 A B t ]w= 60249)@12 A B = t AB < 7 =$? < 249$:)@@4> R3"8 = _ =p9j]$l^qk+7+9)7+2 ( 3, 2) = t( 1, 1) = ( t, t) t = 3 9O_+7+2"84>+9$Tt9) )@12ˆ7+9 9p84R>7+2Q 7 =$? t = 2 AB 7+9 :.= 1:)L+2">+9O_+7+:OK 9O_ 2"8 =&@19OK+7+2 A B L> =&84@19 :$9\L>+9$:)? =&@8[7R602.RZ@12"?6= EG]w=& 2"@12SK 9)a=&@89O_+7 => =&8477 =SZ7+9F:.= A B AB P 1:)L+2">+9O_+7 MO Ÿ&

25 9c243" -=[]w=[l> 2"8 =h2"]$7 =$? =&8 =Q?+9c:)^ 08 = ]w=[60249)@12 P AB A B λ OP = OA + AP = OA + λ AB EG]w=& 2"@12 AB AP S=&7 =wk 2" $?7+2f:)^ 9$:)@ = ]w=`60249)@12 µ OP = OA + A P = OA + µ ¹ ="Tt9$:)@48 ="Tt9819)6@12">@19:S@19Ob7@19S60242">1_+7 =&@4Ë L+2&K+ 0? =&8 ="Tt9 A B n~;@\@4 06[:$9L+2&K+ 0? =&8 =f:)+:)@19$tu=&@"= (1, { 0) + λ( 1, 1) = (3, 0) + µ( 3, 2) (1 λ, λ) = (3 3µ, 2µ) 1 λ = 3 3µ q602.r@12 +TU=\>+9)Õ9)7+9 ="Tt9$:)@48 ="Tt9-7 =&L>+Tt9)>h8 λ = 4 µ = 2 OP = OA + λ = 2µ λ AB FL+2&K+ 0? =&8 ="Tt9 OP = (1, 0) + 4( 1, 1) = ( 3, 4) Š=`7 =&L+24T<7+TJ?+9p@12"8 =:.=60242">1_+7 =&@4@19 7 = :)L> RTt2ˆ31=&]w=&@"=ˆ2"@ˆ819)6@12"> P OA OP = 3 a + 4 b 6^q_ 9)@12 a = OA b = OB Š=`7 =&L+24T<7+TJ?+9p8-1:OK 2"84+9)@12-7 =`@"=&]$c]w=w_ =$? =@12"?6@19 3.R0b0=`]w=w_ =w_ 9)7cL+24:)>+9O_ :)@48124T A B 60242">1_+7 =&@4@19h:) :)L> RTt2 7R6=&68 = :)+:)@19$TU=ƒ8 L>+24:)@4> =&7+:)@4812"@12 n@wr 2"?+9)84_+7+2 +TU=h]w=h7 =$? =wk y].243"> =&].9)7 =ƒ7 =hrt+:4bÿ ÇQ7+2ƒ31=&]$+:)7@19 819)6@12">Ž:.=> =&])K+?7B 249$:)@ƒ_ =w_ 9)7+2 O 7 94y?+9 a = (2, 1, 1) b = (1, 1, 1) k ="Tt9$:)@48 ="Tt9 a b 8h]$> =&]w=]w= OP œl+2&k+ 0? =&8 ="Tt9 Œp>+9$:$Tt9)@47+9)@19@4^ >+:$9)7+2"@12f2"@47+2"Õ9)7+9 :.="Tt24:)@12.R@19OK+7+2 OP = ( 2, 7, 7) ÍÎ9 > =&]$ DK 9O_ ="Tt9n2" 9;7R602&K+602L>+Tt9)> L>K 2.à 9)7R@"=-7 =nk+7+9)}7@19 2"L+9)> =&os:n819)6@12"> Es:)^3"> =&7+9 T<7+2.à 9)7+9`:n?+:OK 2 MyL>c>+9)Õ-=&8 =&7+9)@12ˆ7 => =&])K+?7S24:)7+2"847.9$24Tt9)@4>?7]w=w_ =$? Åq q«å PQ:)^ `7+24:)@ˆ819)?+9349)Õ9nL+2"6=&]w=&7+2ˆ6=&6STt2.à 9$Tœ_ => =&]$L+2"]$7 ="9$Tœ_ =wk+c7r602.rc@12"?6=-k 9)a- 84^ >1b _ =w_ 9)7 = L> =&8 = Š=ƒ> =&]$ DK 9O_ ="Tt9hL>+243&K 9$TU=Z:SL>+Tt9)>Ó _ =w_ 9)7B:.=ƒ@12"?6=&@"= A(1, 1, 2) ¾ =œ@"=&]$ºo+9okî9h_ 243">+9ƒ_ =Ž:)º:)L+24T<7+TÖ 1:OK R3"8 =œ_ = e(1, 2, 3) w_ 2"8"K 9)@4812"> R8 =&@ 60242">1_+7 =&@4@19ˆ7 =ˆ@12"?6=&@"= q]w=-_ =f:$tt9:) $ 0>7?+9 L>7 =w_k 9)a- M(x, y, z) M 7 =L> =& R@12\9n 1:)L+2">+9O_+7 =ˆ7 = :)^q_+^ > a= y =&L+24T<7RTt94?+9 l a A x > =w_+ 1: = X x = OX p:)7+2f94+?+9 ]w=@12"8 =ˆ9n7+9$243$b 2$_++Tt2F _ 24:)@"=&@4^?7+2\819)6@12">@19 AM _ =ˆ:.=- 1:)L+2">+9O_+7J Q^ } a 6=&@12 y@12@12"8 =h9f]$l^qk+7+9)7+2ƒ@12"?7+2h602" )=&@12 :)^ 9$:)@ = e 0 Ur<=&819)7+:)@4812"@12 y]w=[60249)@12 Ž 7 =$? t AM = t e m a = t e (le) l : m = a + t e :$97 =&>? = A"')/1,#DC (+#Ze+ C0. '),;C *+%0(+#Z 0> =&847+9)7+9h7 =šl> =&8 =&@"= Ör<=&].3"> = :$94p@12 :OK+ 0a-B]w= l _ =[> =&]$L+2"]$7 =&8 ="Tt9_ =wk+ 7R602.Rœ@12"?6=hL>7 =w_k 9)a-Ž7 = K+ 7+94Ž S2.à 9$TÎ_ =[]w=&lõ9$töìôù3ïf: l 60242">1_+7 =&@4 9n:$9nL+2&K+ 0? x = 1 + t (lec) l : y = 1 + 2t z = 2 + 3t c:ok 9O_6=&@12ˆ]$L+2&K+]$8 ="Tt94+?+9 t = x 1 1 = y+1 2 = z 2 3 (cle) l : x 1 = y L+2&K+ 0? =&8 ="Tt9 0> =&847+9)7R@"= = z "@12f243"67+2"819)7+2f:$9p7 =&>? =&@ /1"(+#&(+*+%0(+*S )C04A&( ')(+*0 S7 =L> =&8 =&@"= l Åq q Ø =&LÕ9)@19ˆ819)6@12">7@19f6=&7+2"7?7 0> =&847+9)7Rh7 =FL> =&84@ "@12S:.= 2"L>+9O_ 9OK 9)7 2"@ Eã=1MH@12"?6=&@"= EG847+TU=&7+94Óp> =&]$L+2"]$7 =&}@19-6=&69`> =&]$L+2&K 2.à 9)7 = A( 1, 0, 0) e(2, 0, 1) L> =&8 =&@"= :)L> RTt2ˆ60242">1_+7 =&@47 =&@"=\:)+:)@19$TU=1MO l Es3 M@12"?6=&@"= EG]w="349OK 9)à 9)@194y?+9\> =w_+ 1: = A( 2, 1, 0) e(2, 1, 0) A :.=F> =&]$L+2&K 2.à 9)7 8cL^ >8 =&@"= 60242">1_+7 =&@47 => =&8477 =FdH]$7 =$?ƒ7+9ˆ9-7+9$243$b 2$_++Tt2c_ =FLÕ9$Tk@4> e 60242">1_+7 =&@4dUL+2-@12"]$c7 =$?7L>+9$T<7 =&8 ="Tt9 6^TËÏO> =&84777 Ïn60242">1_+7 =&@4 MO Ÿ4

26 9 Åq q Ø s c9p 1:)L+2">+9O_+7 =-7 = Eã=1M =&LÕ9)@19 81:)?6 84^ ].Tt2.a-7º 0> =&847+9)7Ŗ 7 =šl> =&8 =&@"=602.R@12 T<7 =&8 = L>+9)] A( 1, 1, 3) e(1, 1, 1) Es3 M ³ ^ 2"@12-]w= A(1, 0, 1) e(3, 2, 0) Åq qvþ r<=&].3"> =:$94U7RTU=S]$7 =$?+9)7+9F_ =wk+š 0> =&847+9)7+9)@12 ìôù3ïf:$9f> =&]$ DK 9)a_ =[8 ÏO> =&8477 =&@"=Ï K+ZÏOL>+24:)@4> =&7+:)@4812"@12+Ï"JÍÎ92"@1349OK 9)a-+TJ?+9:)^ DK =":)7+2\:)^ 84>+9$Tt9)77@19`:ôb8 =& -=&7R ÏO819)6@12">@191Ï Tt2" )=&@š_ =Z3"^q_ =&@ 9OK 9$Tt9)7@4B 7 =kï)t<7+2".24tt9)>7 Ï EGL+2"819)?+9h2"@ 1M[ÏO819)6@12">7BL>+24:)@4> =&7+:)@48 =Ï"n8 602"@12Žìôù3ïF:)^ 2h2"L+:)8 = ÏOL> =&84šK+7 =:$9 ]w=&7+tu=&8 ="Tt9: Ï)T<7+2".24Tt9)>7 ÏU7+9) -= "81:$9tL =&6@4 06 > =&]$?@"="Tt9y2"@4? =":)@4 7 = Ï):$9)@4847+2"@12+Ï2"@47+2"Õ9)7+9U7 =J?@"=&@19OK+R 6^Tš.9$24Tt9)@4>?7@19`24349)6@4ƒÓ d MO Åq q n9)6= :.=Z:)^ 2"@4819)@47+2œ@12"?6=šk819)6@12">Ë2"@š7R602.Rk> =&8477 =Ý 9 A( 1, 2) e(1, 2) 2"L>+9O_ 9OK+R@fL> =&8 = 60249)@12ˆ2"]$7 =$? =&8 =? R@12fT<7 =&8 =-L>+9)] c9n 1:)L+2">+9O_+7 =ˆ7 = ³ ^ DK =":)7+2[ìôù3 íï 0> =&847+9)7+9)@12ˆ l A e x+1 l : m = a + t e 1 = Œn24:OK 9O_+7+2"@12f> =&819)7+:)@4812\2"?+9)84_+7+2fTt2.à 9 y 2 _ =ˆ3"^q_ 9 2 ]w=&l+:.=&7+2ˆ@"=&6=ó y 2"8 =ˆ:)^ 2 l : 2x + y = 0 :$9p7 =&>? =- 0> =&847+9)7+9n7 =L> =&8 =&@"= EG6=&]w=&7+2ˆL+2dU@12"?7+2 ƒ#&²;:#- )C04A&( ')(+* ' 7 =L> =&8 =&@"= MO l l Åq q ~ 3"> =&@47+2 =& R602.R > =&8477 =-84].9$Tt9$Tš60242">1_+7 =&@47 =f:)+:)@19$tu=- > =&]$ DK 9O_ ="Tt9 =&@4 w_ 2"8"K 9)@4812"> R8 =&@F 0> =&847+9)7+9`2"@f@4L M(x, y) ax + by + c = 0 2"L+:)8 =&@ 7R6=&68 =;L> =&8 =<r<=&]$ DK 9O_ =&}@19Q7 =&>+:) 08 =&}@19yL> =&84@194Ó 6=&@12ˆ:)F7 =&>+:) 08 =&@19pL+2"7+9 _+819p> =&])K+? ">1_+7 =&@47S:)+:)@19$T< dt7 =&L>+Tt9)>c: l : x y+1 = 0 l : 2x y 2 = 0 l : 0 x y + 1 = 0 24:)@4^ >cc@4^ L ^ $^qkstt9)a_+ F60242">1_+7 =&@47@1924:) ~;@ L+24:OK 9O_+7+2"@12 0L> =&a-7+9)7+9š9 84_+7+2?+9 > =&]$L+2&K 2.à 9)7+9)@12B7 = ÁO')#O '),;C *+%0(+*0 +,ú#&²"')/1, L> =&8 =602"}@12S:$9f2"L+:)8 =c7 =&L>+Tt9)>Z2"@ 0> =&847+9)7+9)@12 ]w=&84+:)z2"@].342"> = 7 = 60242">1_+7 =&@47 = y = 1 :)+:)@19$TU= ³ K 9O_ 6=&@12h819O_+7^ a 9 mn60:)> =&7 =7R6=&68 =:)+:)@19$TU=h2431=$?+9åA&! +/1"/ A&*!.A&#&Y+!$,-A&h7 =.9$24Tt9)@4>?7@19`24349)6@4:$9 2"L+:)8 =&@ˆ7 =&L^qK+7+2f2"@-@19Ob7@19=wK+.9$3">?7S 0> =&847+9)7R Åq q Š=yL>+9O_+L+2&K 2.a-+TJ4?+9t:.=y_ =w_ 819)6@12">@19 A( 1, 1, 2) p(1, 0, 2) q(2, 1, 1) Q^ }6=&@12 7+9y:.=H 1:)L+2">+9O_+71@12 :)^ 9$:)@ = 9O_+7+:)@4819)7 = > =&8477 = "602.R@12nT<7 =&8 =;L>+9)] p q α A 9J 1:)L+2">+9O_+7 =n7 = < n9)6=n24:)819)7f@12"8 =p7ˆ9q_ =w_ 9)7 =n@12"?6=&@"= U S2.à 9$T_ =p> =&]$L+2"]$7 ="9$T p q M(x, y, z) _ =wk+ K 9)a-F8`> =&8477 =&@"= L+2`> =&])K+?7F7 =$?7UPŽŸ + Ÿ7 =&L>+Tt9)> 9;L>+9O_K 2.à 9)7+2-L> =&84K 2 M α L+24:)>+9O_ :)@48124Tœ602"}@12fTt2.à 9$Tš_ => =&]$L+2"]$7 ="9$Tš_ =wk+c@4> A"')/1,#DC0ˆK 9)a=&@ˆ8f9O_+7 => =&8477 =K+c7+94 n2@12"?6=&@"= K )=&8 =F:.="Tt2@12" )=&8 =602" )=&@12819)6@12">@19 M α p q K 9)a=&@89O_+7 = AM > =&8477 =EG84a-@19 :OK 9O_+8 =& `@19 >+:) 076 MO <H')/1,#DC *+,-' p â q * AM ( ' W 'Ojk",  ')g&(+nc04a&(+*+(+ <H')/1,#DC *+,-' p â q * AM W 'Ojk",ÖAf')g&(+pC04A&(+*+(+ Ÿ4µ

27 Q^ 1:OK AM _ =`K 9)a-c8 = 9 M α OM OA 4@12 AM +TU=p60242">1_+7 =&@4 AM(x+1, y 1, z 2) ³ K 9O_ 2"8 =&@19OK+7+2 x + 1 y 1 z = 0 Œp>+9$:$Tt9)@47+9)@19.2">7 =&@"=`_ 9)@19)>+T<7 =&7@"=+]w=_ =`L+2&K+ 0?@19 t 2"8 => =&819)7+:)@4812 2x + 3y z + 1 = 0 :$97 =&>? = )C04A&( ')(+* '`(+C04A&(+*+(+", 2F2"]$7 =$? =&8 =?+9@12"?6=&@"= K )=&@12 α M α 60242">1_+7 =&@4@19ˆ[ w_ 2"8"K 9)@4812"> R8 =&@c 0> =&847+9)7+9)@12 7 = ]w=& 2"@12 \~;@F_+> 0 )= :)@4> =&7 =&L>+Tt9)>@12"?6=&@"= 7+9K 9)a-[8 α (2, 1, 3) 2"?+9)84_+7+2:$9 α ( 1) = 3 0 (0, 0, 3) :)^q_+^ > a=-8 U R :)^ 2`@"=&6=L>7 =w_k 9)a-c 7 =@4>+9)@"=&@"=-60242">1_+7 =&@47 =ˆ24: dt8-@12"?6=602.r@12ˆ9 7 = α > =&].:)@12.R7+9p@4>f_+^qK+a-7F7 =&@19)7c819)6@12">\8`L+2&K 2.a-@19OK+7 =`L+24:$2"6=2"@7 =$? =wk 2"@12 O =&>+:) 08 =&}@19p60242">1_+7 =&@47 =-:)+:)@19$TU=\> =&8477 =&@"= 8`7+9)R t p].243" 2` 0> =&847+9)7+9)@12`7 =n_ =w_ 9)7 = α > =&8477 =c:)^q_+^ > a=cl^qk+7 = 7+m 2">+TU=&oRh]w=F7+9)RhL+2"]$812&K+R8 =F_ =c:$9->+9)õ-=&8 =&@c> =&])K+?7 ]w=w_ =$? :)84^ >]w=&7s:p84^ L>+24:)7 =&@"=ˆ> =&8477 = Åq q s ³ ^ 2"@12 :$9[2"@47 =":DRŽŽ]w=h 0> =&847+9)7R@"= 7 =ƒl> =&84@194 =&6=ƒ7 =&L>+Tt9)>BTt2.à 9$TÌ_ = ].:OK 9O_+8 ="Tt9 6=&6S:.=> =&]$L+2&K 2.à 9)7[L> =&8 =&@"= 0> =&847+9)7+9)@12f7 =`602.R@12\9ˆì í ù3ï c> =&8477 =&@"= y = l α _ =ˆ> =&].349)>+9$T _ =wk+ :$9L>+9$:)? =&@F9 _ 24:)@"=&@4^?7+2\_ =ˆ>+9)Õ`+TŽ:)+:)@19$TU=&@"=\2"@\ 0> =&847+9)7R@"=f7 = l α l 0> =&847+9)7+9)@12ˆ7 = Ó α x 1 = y + 1 = z x + 3y z + 1 = 0 =&]$ƒ:)+:)@19$tu=\+tu=f9o_+7+:)@4819)7+2 >+9)Õ9)7+9 EG>+9)Õ9)@19R[:.="Tt24:)@12.R@19OK+7+2 MO x = 7 y = 1 z = 2"8 =-2"]$7 =$? =&8 =?+9 :$9nL>+9$:)? 9nL>+24342$_ =-7 = 16 : α l X( 7, 1, 16 ) α l Åq q «Å Œp>+2"819)>+9)@194?+9F@12"?6@19 7+9\K 9)a=&@h7 =[9O_+7 = A(1, 3, 1) B(0, 4, 1) C(1, 2, 2) L> =&8 =c7 =&LÕ9)@19n 0> =&847+9)7+9)@12ˆ7 =`> =&8477 =&@"= 602.R@12fT<7 =&8 =-L>+9)]n@wR0b β Ò =&7+94ÓUPy^ L>+24:)7 =&@"=> =&8477 =`9; 1:)L+2">+9O_+7 =`7 =FEG7+9)602&K+7+9.=&>7@19wMQ819)6@12">@19 AB AC T<7 =&8 A Åq q «Åqs n9)6= :.=J7R6=&684?+:OK =EGL+2"7R602" )=; 9< $7 =&>? ="Tt9š/1#"'>=t*+Ã+* ')(+,*M n_ = a b c d L>+9O_+L+2&K 2.a-+TJ?+9;:$Tt9;mn60:)> =wk+\7r6=&68 =60242">1_+7 =&@47 =:)+:)@19$TU=UÍÎ9H]$ 0?+Tƒ.9$24Tt9)@4>?7+2"@12 T<R:)@12 7 ]w= 60242">1_+7 =&@4@19p7 = 602"@129HL^qK+7+9)7+2 α M(x, y, z) ax + by + cz + d = 0 r<=&].3"> =:$94JÏO]$>+2$_ 9)7@191Ï\:OK+ 0? =&ƒ7+9ˆ:.= 7@19)>+9$:)7ƒdp=&60281:)?6ƒ60249$mno+9)7@4 7 =&L>+Tt9)>Z:.= 7 wk++@12ôa&!$*+%0/1*@12"?6[:.=ˆ2"@ Š> 8F:OK+ 0? =&}[9n602" )=&@12?p:)7+2\94+?+9 α a = b = c = 0 =-@12"?6c7RTU=@1249$:)@ 9pL> =&]$7+2"@12fT<7+2.à 9$:)@4812 α Åq q «Åq«Å Š=f> =&]$ DK 9O_ ="Tt9ˆ7 =&L>+Tt9)> 0> =&847+9)7+9)@12 ÍÎ9L+2"6=&à 9$TJ?+9 x + 2y + 2z 2 = 0 α > =&8477 = =f@"=&]$[o+9ok[_ =ˆ> =&]$ DK 9O_ ="Tt9-@4>S>+9)Õ9)7RS7 =f 0> =&847+9)7+9)@12 x + 2y + 2z 2 = 0 602"@12[ 9\@4^qK+6 08 ="Tt9 6=&@12Z/1#&#DC g&*+(+",*š(+[,#&%0/1* ƒ Q^ }š6=&@12[+tu="tt9 9O_+7+2[ 0> =&847+9)7+9 :\@4> 7+9)]$819$:)@47+@12ˆ9p7+2">+TU=wK+7+2-_ =:$9 2"? =&68 =?+9n 9p+TU=349)].3">+2"}ST<7+2".2->+9)Õ9)7R U º@12-7 =&+:)@47 = 2$_k@4 06Ž9c_ =[_ =w_ 9$TÌL>+2"]$812&K+7k:)@12"}7+24:)@4 7 =[_+819S2"@ 7+9)]$819$:)@47@19S _ =\2"L>+9O_ 9OK++Tk@4>+9)@"=&@"=Œp>[60249)@12=&602FZ*h,;C *+,-'`/1#"'>=t*+Ã+* ')(+, 7+9:.=\7 wk+q@12f:)@"=&7_ =&>@19)7 L+2$b8 =&@\9n_ =84_++Tœ_ =wk++tu=-@12"?6s2"@ -L+2&K+ 0? =&8 ="Tt941?+9y@12"?6=&@"= +602"@12f:.=ˆ2"@f60242">1_+7 L+2&K =& )="Tt9 α L>7 =w_k 9)a-ˆ7 = <Œn2p:)^ `R7 =$?7ˆ> =&].3"> ="Tt94 y = z = 0 A(2, 0, 0) α?+9u@12"?6@19 :)^ 2HL>7 =w_k 9)a=&@n7 = EG84a-@19>+:) B(0, 1, 0) C(0, 0, 1) 7+9yK 9)a=&@ 7 =p9o_+7 =pl> =&8 =4@19J2"L>+9O_ 9OK+R@9O_+7 =ˆEG8 :$T<+:)^qKf7 =Î')g&*+(+!$,-A"')(+ Ó d M > =&8477 =_ =`6=&à 9$T α Ÿ1Ä

28 @ l\*"c (+ˆg&"g"')(+nC04A&(+*+(+f!ôfe"C04A&*+,-'OâUe+#-/1#&*+,#, /1#&#DC C04A&(+*+(+*qz Åq q «Åq Ø Š=7 =&LÕ9$Tš 0> =&847+9)7+9)@12ˆ7 = Ó α x y z = )@12:OK 9O_\L>+9$:$T<R@"=&7+9H_ =&8 = = ]$819$:)@47 =pk+ 06=&8124:)@&07+2 x + 2y + 2z 2 = 0 A. "' '),-'H,;C *\e"c #&* "A&#OW+(+*\( ')/1#OW+*+( 'ÂC (+*ˆ>+9)Õ9)7REG7 =&L>+Tt9)>f@12"?6@19 My7 = F 9p:$9 349O_+@194?+9 0> =&847+9)7+9)@127 =.9$24Tt9)@4>?7+2"@12fT<R:)@12 A 1 (2, 1, 1) B 1 ( 4, 0, 3) ( 1, 2, 0) x + 2y + 2z 2 = 0 9`:)^ 2"@12fK+[:ôb 2$_+7+2\7 = EG@1249$:)@ 9`2"@f84_ = ]w=ˆ7r60249 MO p].243" 2 α 1 α lx + 2ly + 2lz 2l = 0 l 84].9$Tt9)@19-e"C #&* "A&#OW+(+*S 0> c9)60:)l+9)>+tt9)7@4> =&}@19`6=&6812f:$9nL+2&K+ 0? =&8 = 7+9n:.= 1:)L+2">+9O_+7 2"8 =-2"]$7 =$? =&8 =?+9n:$9nL>+9$:)? =&@ˆL+2-7R602.RcL> =&8 = Åq q Ø Eã=1My n9)6=`> =&8477 =&@"= +TU=` 0> =&847+9)7+9 Š`2"6=&à 9)@194?+9 > =&8477@19 β x y 3z = 0 α β m Es3 MQ =&LÕ9)@19n 0> =&847+9)7R@"=-7 = EGÒ =&7+94ÓU7 ="Tt9)>+9)@19n_+819p@12"?6S2"@ MO m m EG80MQ p>^:)@124:.=&7ck+:.=`l> =&84@19 Es2"@fŸ Ÿ1MQ l EG &M ³ ^ 2"@12-]w=`L> =&84@192"@fŸ SŸ vu4 m& Åq q Ø s Eã=1MŠ`2"6=&à 9)@194+?+9;> =&8477@19 α : 3x 2y + z 6 = 0 β : 9x 6y + 3z = 0 :.= 1:)L+2">+9O_+7 Es3 MQPyR>7+2K+c9n@484^ >1_ 9)7+9)@12?+9n 0> =&847+9)7+9)@12ˆ7 =ža&! +/1f> =&8477 =+602.R@12f9p 1:)L+2">+9O_+7 =-7 = α Tt2.à 9n_ =-3"^q_ 9n]w=&L+:.=&7+2-L+2ˆ:OK 9O_+7R 7 =$?7Ó ]w=7r60249 α : 3x 2y + z + u = 0 u EG80Mt p].243" 2n_ 2"6=&à Tt2.à 9ˆ_ = 3"^q_ 9ˆ]w=&L+:.=&7+2c84^ 8c84_ = 9)@194?+9J 0> =&847+9)7+9)@12 A&! +/1`> =&8477 =0602.R@12`9J 1:)L+2">+9O_+7 = 6^q_ 9)@12 7 = 9L>+2"]$812&K+7+2 α : ax + by + cz + d = 0 ax + by + cz + u = 0 u?+:ok 2 Åq q vþ ="Tt9)>+9)@ ">1_+7 =&@4@19S7 =[L>+24342$_ =7 =S81:DR6=h2"@h> =&8477@19 :F81:DR6=h2"@ α β L> =&84@192"@-@12"?6SŸ SŸ vu4 Åq q PQ:$9)6 _+819f6>^:)@124:.=&7šL> =&84 2"L>+9O_ 9OK+R@[_+812"}6=S 1:)L+2">+9O_+7Z> =&8477 m l α 81:DR6= 2"@c602"@12c:)^q_+^ > a= L+2 9O_+7 = 2"@ L> =&84@194- =F:OK 9O_+8 =& -=&@"= >+:) m α > =&8477S9p 1:)L+2">+9O_+7 =-7 = l EGK+c7 =cïo7 =&L> =&8"K+R8 =& `@191Ïn819)6@12"> MQ9O_+7+2"84>+9$Tt9)77+2ˆ m l a m a T<7 =&8 =L>+9)]\EG7R602.R MQ@12"?6=ˆ2"@f:)^ 2"@4819)@47 =&@"=ˆL> =&8 = l xu4a&(+*+(+*+,-' *!ôf 1!$e+#DCq')g&(+*c(+ * α m α l m l =&LÕ9)@19 0> =&847+9)7R@"=-7 =`> =&8477@194 2"L>+9O_ 9OK 9)7S2"@ˆL> =&84@19`Ÿ Ä0 ceg80myÿ Ä0 ceg &MO Ÿ4Ç

29 & Åq qba C0"(+!.A"'ÂC+ &.W+[7 =&84 L>+9$:)? = l m :$9ˆ7 =&>? =81:DR6=L> =&8 = t 602.R@12S $ (QC04A&", ',;C0"!.A"'ÂC+ &.W+\(+ * t m l Œp>+9)] A&! 602.R@12 7+9hK 9)a- 7 = L>+9$T<7 =&8 =Ž9O_+7+:)@4819)7 =&7+:)819)>]w=wK = A α m α =&+:)@47 ="> =&8477 =&@"= L>+9)] w602.r@12 9U 1:)L+2">+9O_+7 =H7 = l π l L a l LA :)^q_+^ > a=;l> =&8 l A =&]$c> =&8477 =-9p602">+9)6@47+2ˆ2"L>+9O_ 9OK 9)7 =+@4^ } 6=&@12-819)6@12">@19 a l LA ( 'n!ôf 1!$e+#DCq')g&(+*tEG]w=& 2 MO ½;7 =wk 2" $?7+2œ819)6@12">@19 a m MA B@12"?6=&@"= 2"L>+9O_ 9OK+R@Ž> =&8477 =&@"= p602.r@12ž:)^q_+^ > a= M L> =&8 =&@"= m A m r<=&8477@19 7+9f:.=F 1:)L+2">+9O_+7 :OK 9O_ 2"8 =&@19OK+7+2[:$9-L>+9$:)? =&@SL+2c7R602.RhL> =&8 = \ = π m π l t :OK 9O_+7 =&@"=->+:) 076=ˆ:$9p84a_ =+?+9 :)^q_+^ > a=ˆ@12"?6=&@"= L>+9$:)? =L> =&84@19 t A m l (QC04A&", e"cq')!$*+%0 &C:#&,# ' e"cq')!.')%0(+*+ã+ˆ(+ * zed;(+.w+#"á *+%0(+# e"cq')!$*+%0ˆ* z t l l π l a l t m Åq q Œn24:)>+9O_ :)@48124Tœ:)812"@19p819)6@12">7 L =&> ="Tt9)@4>?7c 0> =&847+9)7RF:.= _ =w_ 9)7 L> =&84@19 Q =&LÕ9)@19H 0> =&847+9)7R@"= 7 =p@wr0b7 =&@"=n@4> =&7+:)819)>]w=wK =0602.R@12 t(1, 0, 0) µ : q = (1, 1, 1)+τ(0, 0, 1) T<7 =&8 A(1, 2, 3) Ò =&7+94ÓyŒp> =&8 =&@"= T<7 =&8 c9n 1:)L+2">+9O_+7 =-7 =819)6@12"> = l O(0, 0, 0) a l (1, 0, 0) ³ K 9O_ 2"8 =&@19OK+7+2f 0> =&847+9)7+9)@12-7 => =&8477 =&@"= 602.R@12ˆ9n 1:)L+2">+9O_+7 =-7 =`819)6@12">@19 9 π l a l x y z = 0 K+ y]w=[> =&8477 =&@"= y602.r@12ƒ:)^q_+^ > a= 3y + 2z = 0 π 1:)L+2">+9O_+7 =-7 =819)6@12">@19 m MA = (0, 1, 2) +TU="Tt9 a m 2"@ 6^q_ 9)@12 :OK 9O_+8 =J?+9 > =&8477Ó π m x 1 y 1 z = 0 OA l : p = m EGœ]$7 =$?Ž@12"?6=&@"= M(1, 1, 1) Mˆ 9 : x 1 = 0 ³ _+> 0 $ _+ =&7+:)819)>]w=wK =&@"= t 9cL>+9$:$9)?7o =ƒ7 = t : { 3y + 2z = 0 x 1 = 0 Ÿ4É

30 Š=\]w="349OK 9)a-+TJ?+9L+24:OK = 2"L+:)8 =&@ 602" )=&@12c7R602.RL> =&8 = 9-2"L+:.=&7 =F6=&@12 t L>+9$:$9)?7o =œ7 = _+819> =&8477 :)+:)@19$TU=&@"= 2"@œ@19Ob7@19 0> =&847+9)7RË:$97 =&>? =k#&²;:#z )C04A&( ')(+*0 7 =š:)^ 2"@4819)@47 =&@"=œl> =&8 = ½;602š+:)6="Tt9_ =Z7 =&LÕ9$T 7 =&L>+Tt9)>Ë6=&7+2"7?7@19 0> =&847+9)7Rk7 = 9ƒ_ 24:)@"=&@4^?7+2 _ = 84].9$Tt9$TÖ_+819 > =&])K+?7Î@12"?6 2"@ 7+9)R Ô 249$:)@ _+819Z> =&])K+?7 >+9)Õ9)7Ŗ 7 = { :)+:)@19$TU=&@"= 3y + z = 0 2"8 = :.=h7 P (1, 0, 0) Q(1, 4, 6) x 1 = 0 a t = P Q = (0, 4, 6) 9J 1:)L+2">+9O_ 9)7f7 = 4 T<7 =&8 =nl>+9)] t 7 =$? +TU=n819)6@12">7+2 L =&> ="Tt9)@4>?7+2 t t P t 0> =&847+9)7+9 t : t = 6^q_ 9)@12 9L =&> ="Tt9)@4^ > =&L+:.=&7+2 : OP + λ at = (1, 0, 0) + λ(0, 4, 6) λ 60242">1_+7 =&@4SL+24:OK 9O_+7+2"@12ˆ]$ DK 9)a_ =ˆ@"=&6=Ó x = λ y = 0 + 4λ z = 0 + 6λ =&6@128-Ÿ ŸTt2.à 9$T _ =7 =&LÕ9$TZ6=&7+2"7?7@19n 0> =&847+9)7RF7 = :OK 9O_ 6=&@12>+9)Õ`+T.2">7@19 t 0> =&847+9)7R[:)L> RTt2\L =&> ="Tt9)@4^ > = 2"8 =f2431=$?+9`7+9`tt2.à 9`_ =f:$97 =&L> =&84[:L^ >812"@12\ 0> =&847+9)7+94 λ ]w=& 2"@12H@12J7+9U]w=&84+:)`2"@ PQ:)^ `7+24:)@ 2"@;.2">7@19 0> =&847+9)7R:$9tL+2&K+ 0? =&8 =&?+9 λ x = 1 λ = Py^ =`8`K+@19)> =&@4 0> =&@"=cegf]w=`l+2dt6> =&@4^ 6 ]w=&l+:wmy9 L>+9)@12`_ =`:$9pLÕ9 y 4 = z 6 t : x 1 0 = 6=&@12ˆ:$9nL+2$_+> =&].3"> =?+9p7+9)L>+9$Tt9)77+2 y 4 = z 6 y x = 1 Åq q s Š=y84_++T L>+9$:)? = 4 = ½;602Q7 =&L>+Tt9)> z 6 = λ :.=Q60242">1_+7 =&@4@19 t l m (x m, y m, z m ) 7 =nl>+9$:$9)?7 =&@"=`@12"?6= 7 = :$9HL+2&K+ 0? =&8 =&@`L>\7R602.R\:)@12"}7+24:)@7 = f7r602.rf:)@12"}7+24:)@ m t τ 7 = < 7 =$?ˆ@4> R3"8 = _ =p:.= ]$L^qK+7+9)7ˆ> =&819)7+:)@48 =&@"= λ x m = 1 = 1 y m = 1 = 4λ z m = 1+τ = 6λ Œp^ >812"@12`2"@@wR0bf9H@4^ a_ 9$:)@ @12">+2"@12`:$9H]$L^qK+7R8 = L> 0=n@4>+9)@12"@12L> λ = 1 4 λ = 1 4 τ = 7 =$?S243" -=&@"=@12"?6=-+TU=60242">1_+7 =&@4 1 2 (1, 1, Åq q «Å p].243" 2š@"=&6= :$97 ="T<> =œ243" -=&@"=š@12"?6=š7 = _+819L> =&84=&602n> =&].3"> =œ:$9@19h:$9 3) 2 L>+9$:)? =&@& p7 =$?+9:)+:)@19$TU=&@"=ˆ7RTU=`>+9)Õ9)7+94æt@12-7R602&K+602ˆL>+Tt9)> =Ó Eã=1M` n9)6= :.=S 0> =&847+9)7R@"=7 = l : l = ( 1, 1, 2) + t(0, 1, 2) m : m = (1, 0, 1) + τ(2, 1, 0) L> =&84@19 +TU="Tt94Ó t½;602@19p+tu=&@ˆ243" -=`@12"?6= +@12`]w=`7R602"c:)@12"}7+24:)@4c7 =`L =&> ="Tt9)@4>@19 l m M(x, y, z) t τ x = 1 = 1 + 2τ y = 1 + t = τ z = 2 + 2t = 1 :OK 9O_-60249)@12n2"@pL^ >812"@12n> =&819)7+:)@4812pL+2&K+ 0? =&8 ="Tt944?+9 42"@p84@12">+2"@12n:OK 9O_+8 =4?+9 -:OK 9O_ τ = 1 t = 2 6=&@12\]w="Tt9$:)@4+TŽ8\@4>+9)@12"@12F:$9L+2&K+ 0? =&8 = p 7 =$?[.2">7@19`> =&819)7+:)@48 =\:.=ˆL>+2"@4812">+9)?84 2 = 1 7RTU=`@"=&68 =ˆ:)@12"}7+24:)@47 =`L =&> ="Tt9)@4>@19 L> =&84@19 7+9n:$9nL>+9$:)? =&@&Q~;@-@12"8 =-:OK 9O_+8 =?+9 m 1:)L+2">+9O_+7Fn9p7+9 :.=` 1:)L+2">+9O_+7S:OK 9O_+8 =-2"@-@12"8 =?+9ˆÏO7 =&L> =&8"K+R8 =& `@191Ï 819)6@12"> 2"?+9)84_+7+2ˆ7+9n:.= 1:)L+2">+9O_+7 (2, 1, 0) (0, 1, 2) Es3 M Š=w_ 9)7š:.= L> =&84@19 l : l = OL + t a l m : m = OM còu:)@"=&7+2"819)@19f84]w=&+t<7+2"@12 + τ a +TŽ> =&]$L+2&K 2.à 9)7+9-h=&602fK 9)a=&@ 8F9O_+7 =f> =&8477 =q7 =&LÕ9)@19 0> =&847+9)7+9)@12Fp½;602F:$9L>+9$:)? =&@& m L>+9$:$Tt9)@47+9)@19-:)^ 2f@"=&6=\60242">1_+7 =&@4@19-7 =f243" 1:OK 2"84+94q?+94Ó OL = (0, 0, 3) a l = (3,, 1) OM = (1, 1, 4) a m = (1, 3, 1) OL = (1, 0, 3) a l = (3,, 1) OM = (1, 1, 4) a m = (1, 3, 1) OL = (4, 1, 3) a l = (2, 2, 1) OM = (1, 0, 3) a m = ( 4, 4, 2) OL = (1, 3, 3) a l = (1, 0, 1) OM = (2, 1, 1) a m = (1, 1, 1) OL = (4, 3, 1) a l = ( 3, 1, 1) OM = (3, 1, ) a m = (2, 3, 1) Åq q Ø n9)6=\:.=-_ =w_ 9)7[6>^:)@124:.=&7@19L> =& )6024T<LK =&7 =&>+9)7 l m p EG@1249$:)@f7 =&L> =&8"K+R8 =& `@19`819)6@12"> :.=-7+9)6024T<LK =&7 =&>7 MO S2.à 9$Tœ_ =-L>+9)6=&> ="Tt9 a m a l p > =& "@12 :)^q_+^ > a=&@s:)^ 2"@4819)@47+2cL> =&84@19 h:.=f 1:)L+2">+9O_+7h7 = R0b7 =&@"= π m π l m l p L>+9$:$9)?7o = 9Q@4> =&7+:)819)>]w=wK = 602.R@12 9Q 1:)L+2">+9O_+7 =p7 =p819)6@12"> = < =p:ok 9O_+8 =& -=&@"=n>+:) 076=p@12"8 =p:.= p > =&8477@ "@12T<7 =&8 =&@`:)^ 2"@4819)@47+2L>+9)];L> =&84@19 SÏ _+84 0L+24:$2"?7RÏH819)6@12"> d π l π m m l p 4Ð t

31 R:)7+2n944?+9Q> =&]$L+2&K 2.à :.="Tt2 ')(+* '),# 7 = p (QC04A&", e"cq')!$*+%0 â< &C:#&,# ' e"cq')!.')%0(+*+ã+ˆ(+ * zed;(+.w+#"á *+%0(+# e"cq')!$*+%0ˆ* z t l l π l a l t m Òp349O_ 9)@19\:$9\:.="T<<?+9f]w= _ =c7 =&>+:) 08 ="Tt9f7R602.RZ2"@S> =&8477@19 K+ 9ˆ_ 24:)@"=&@4^?7+2S_ = π l π L+24:)@4>+2"+Tœ 1:)L+2">+9O_+7 =&@"=ˆ 7 = L>+249)6oR E K+ M7 =9O_+7+2"+Tt9)77 =&@"=f L> =&8 = m p m l l n9)6= kœp>+9$:$tt9)@47+9)@19 0> =&847+9)7R@"=h7 =[@4> =&7+:)819)>]w=wK =&@"=H 1:)L+2">+9O_+7 =7 = m 7 = p = (1, 2, 3) p L> =&84@19H:J 0> =&847+9)7R Ó l : l = ( 1, 0, 2)+t(1, 1, 2) m : m = (1, 1, 1)+τ(2, 1, 1) l : l = (0, 0, 1)+ t(2, 1, 2) m : m = (1, 0, 1)+τ(1, 2, 2) l : l = (3, 4, 2)+t( 3, 1, 2) m : m = (1, 0, 1)+τ(2, 1, 0) Åq qs Œn2"7R602" )=pl>+9o_ :)@"=&8"K+R8 =&@7@19)>+9$: EG7 =&L>+Tt9)>-8n:)@4>+2"@19OK+7 =&@"=p7 =$ 06=1M<T<7+2.à 9$:)@4812 =&7+:)819)>]w=wK+F7 =mn60:)> =&7\6>^:)@124:.=&7FL> =&840602"@12`+TU=&@`7R6=&6812:)812"}+:)@4812t =&@12L>+Tt9)> 9\> =&]$ DK 9O_ ="Tt9c81:)?6š@4> =&7+:)819)>]w=wK+y602"@12h:.= 1:)L+2">+9O_+7œ7 =_ =w_ 9)7 =S> =&8477 =p:)7+294u?+9 =&602\7602.R2"@\L> =&84@19 7+9`9 1:)L+2">+9O_+7 =\7 =ˆ> =&8477 +/1C04A&(+*+(+ 602.R@12F9 l m α α 1:)L+2">+9O_+7 =F7 = L>+9$:)? =F81:DR6= 2"@FL> =&84@19 8ƒEGL+2 9O_+7 =1M;@12"?6= > =&].3"> = :$94 u α m l M u =&7+:)819)>]w=wK =&@"= 9p 1:)L+2">+9O_+7 =-7 = EG84aÖŸ Ä0 vufeg80mômo u M u L u α u Š=\> =&]$ DK 9O_ ="Tt9f6=&@12 L>+Tt9)>h 1:)L+2">+9O_+7@19ˆ> =&8477 L> =&84@19 α u : x + y + z u = 0 l : l = (0, 0, 0) + τ(1, 0, 0) m : m = (1, 1, 0) + τ(0, 0, 1) =ˆL>+24342$_ =f7 = : L+2&K+ 0? =&8 ="Tt9 :)^ 2"@4819)@47+2 l α u t = u L(u, 0, 0) τ = u 2 M(1, 1, u 2) EGL>+24:)@12-]w="Tt9$:)@19)@ ">1_+7 =&@4@19`7 = l 8-0> =&847+9)7+9)@12ˆ7 = MO m α =;_ =;7 =&LÕ9$T 0> =&847+9)7+9)@12n7 =;@4> =":)819)>]w=wK =&@"= L>+9$:$T<R@"="Tt9J60242">1_+7 =&@4@19;7 = u Ó M u L u M u L u M u L u = OL u 7 =$?+TU="Tt9 OM u = (1 u, 1, u 2) (H) x = u + λ(1 u) y = λ z = λ(u 2) =&]$c:)+:)@19$tu=` 0> =&847+9)7R 2"L+:)8 =T<7+2.à 9$:)@4812"@12 =&7+:)@4812"@12+602"@12 (H) K 9)a=&@ 7 =;@4> =":)819)>]w=wK+@194 1:)L+2">+9O_+7ˆ7 = <Œ =&> ="Tt9)@4>@19 Tt2" )=&@n_ =;L>+9$TU=&@nL>+2"]$812&K+7 α u λ u :)@12"}7+24:)@4nL+2J@"=&6^ 8H7 =$?7`:$9<L+2&K+ 0? =&8 =&@;@12"?6@19U2"@ =y_ =y7 ="Tt9)>+Tc6=&68 =H9<]w=&84+:)+Tt24:)@4@"= (H) Tt9)a_+ 9_ 24:)@"=&@4^?7+2_ =ú* &/&WHGJ%0*0 L =&> ="Tt9)@4>@19 2"@ f 2"8 =c2"]$7 =$? =&8 = _ = x y z u λ (H) >+9)Õ`+Ţ _+819c2"@ 0> =&847+9)7R@"= Z_ =S]w="Tt9$:)@4+T >+9)]$ wk+@"=&@"=8@4>+9)@12"@12ó (H) λ = y u = ]w="tt9$:)@48 ="Tt9f8cL^ >812"@12c 0> =&847+9)7+94Ó z y + 2 x = z y y(1 2 z ) fœn24:ok 9O_+7+2"@12 y xy = y + z yz > =&819)7+:)@4812f:$9p7 =&>? =ž )C04A&( ')(+* ' 7 = ~ 3"67+2"819)7+2ˆ]w=&L+:)8 =@"=&6=Ó (H) u

32 H : xy + yz y z = 0. r<=&].3"> =ƒ:$94.2">7+2"@12h 0> =&847+9)7+92"L+:)8 =[.9$24Tt9)@4>?7+2"@12 T<R:)@12h7 =[@19)]$œ@12"?6y602"@12K 9)a=&@ 7 =":)819)>]w=wK+ 1:)L+2">+9O_+7 7 = áp±7 =&Õ`Rº:OK+ 0? =&}º@12"8 =Ž9 L+2"84^ >1b77 = E«bL+9)>+342&K+?+9)7 α L =&> ="342&K 2"_qMO u Åq qs s =&LÕ9)@19p 0> =&847+9)7+9)@12-7 =`L+2"84^ >1b77@194+7 =&>+:) 08 =&7 7 =:OK 9O_+8 =& -=&@"=>+:) 076= ="TB31=&]w=&@"=c:$9-:)^:)@12"ƒ2"@c819)6@12">@19 a = OA h = 9> =& O_+>+9)7 OAB OA = OB = a OH = h 9tL+9)>L+9)7_+6 wk+r>+9)7`7 =Q> =&8477 =&@"=H7 =J24:)7+2"8 =&@"= 9t 1:)L+2">+9O_+7 = OH l 7 =-Tt9O_+ =&7 =&@"=EGc^ DK 2"L+2&K 2"84R -=&@"=1MOq= :.=-:)@4> =&7@19n7 =@4>^ $^qk+76= m m OAB Åq qs «Å =&LÕ9)@19c 0> =&847+9)7R@"=h7 =[L+2"84^ >1b77@194J].243"> =&].9)7Ž7 =:OK 9O_+7 =&@"=h>+:) 076= EG =&7+94ÓU].349)>+9)@19 L+2$_b 2$_+R -=-60242">1_+7 =&@47 =f:)+:)@19$tu=1m 4Ÿ