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Classical Mechanics
Prof. J. J. Binney
Michaelmas Term 2006
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AS ISW 7+ +/S; 0 [XHA191+ [ ] \.^T +/S IS ' + +/S - 0 [1# 7[F2 L; I : 95::%+>/.(C 7 957 C 7 `: [ W 7S; X'0 ,/.FC 7[F2 L;=A7 /? ,/:E](:=*./.6/?5:E(A/? 7S; 2I :DCDA;>E : /?5:BT f :%$ 9 7[F2 JK;T(5,/. 7/+ :%+A8:%CDA"F :=/?5: E]#DE/7 C/7 .F9E 1S ! ' ! + +/S !S - [ IS ' + +/S - 0 ["# 7[F2 "; X 7 9= :3+ . :E 95."0+ :'(:91+ .F9' )@/?5: .K(/7 CDA1B (A/?(E/A/7>E-:E K ++/S + +/S S 7+ +/S; 0 ["! ! A?57>=@? 7 C (5B 7 :E 7S; ^] \T # S[ 4 # S +/S ! A?5:,/: # 7>E A = .F9E@A19.107 9/:'!F,@A/7 .F92a 9/:,/CDE.10 A1,/7>A"D5B : E 7 9 0$ [ # S8/?5: A19E]A8:, 7>E ' [#&% $ 4 .0 E 7 9 I $(' # 7[F2 \; a 0cA8:.A,/7 /:*)[ I $ /?57>E CDA;D:pA,/7 ]/:9=S_7[ 4 = . E )i;+1I # @ 7,) 4 E 7 9-)i;+1I # @1A?57>=@? 7>E AD= ; = B .F7/+A7 /?B/?5: .F,/7 !F7 9 A87 @E8=E](2 # CDA; D:=+ :/:,/C 7 95:%+ED ;8-,@E E .FB 7 9#!D T.S TV 7,)T 4 E 7 9-)TP;+7[ 4 = . E )T;0 .F,-)TA191+B/?5:96E 7 9#!8/?57>E A1B 5: 7 9 # .0 7[ 4 = . E )TP;+S TF2 J K0/21 L * ; L #43 * | EA AFE%E@A/:%+ A"D.F:K@i/?5: (A/? >c7];0@A"F :9>/?5,/.K#!F? = .F9-1!K5,@A/7 .F9 E](AF= :8D ; A9+ ; 9A1C 7>=A1B-E ; E/:C =A19 D: 0 .K591+~D ; 7/+ :9/7 0 ; 7 9#!B/?5:p(A/?~/?A :/,/:C 7 `:E AE$A19/7 3;GFW >c7]; XD:A8::9*E](:= 7 -:%+(B . =A/7 .F9E >c7%.'; A191+>c7E0P; .10/?5: E ; E/:C A8!F7 F:9 /7 C :E%.!E0F2?G 7>E =A1B B :%+ /?5: * A191+ 7>E:E]A1B B ; 7eD# 95." 7 9 A1,/7>A"D5B ;#; C 7 957 C 7 `:%+wD ;Y/?5:3+ ; 9A1C 7>=A1B5(A/?2 < :9= :6/?5:*7/+ :A /?AE/?5:>+ ; 9A1C 7>=A1B(A/?=A19wD:3+ :/:,/C 7 95:%+wD ; :/,/:C 7 `7 9#!HG 7>E =A1B B :%+B/?5: * ;5 # * 2G @A"F :E/?5:%0 .F,/C.10A19 7 9/:'!F,@A1B .F:, 8.10A 0e59=*/7 .F965 .108> A191+ A>? 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(5,/7 C :%+ A191+>59#(5,/7 C :%+ = . .F, + 7 9A/:E%E ?A1,/:8/?5: E/A1C : .F,/7 !F7 9 @)D# /?5:p(5,/7 C :%+ = . .F, + 7 9A/:E,/."@A/:A7 /?*= .F9E@A19 A19#!K5B>A1,F:B . = 7 3; A7 /?*,/:E](:=* /.8/?5:.59#(5,/7 C :%+ = . .F, + 7 9A/:E'@A?57>=@?(A1,/:%7 95:,]/7>A1B 2G ?5:9 A @ @ # X . A,/7 ]/:9 7 9 /:,/CDE .10/?5:.(5,/7 C :%+ = . .F, + 7 9A/:E/?5:.F 2 :F2 7>E = .0 ! 0 .0 #"" A @ $ @ "" 0 .0 c A @ c 0 A @ d7% @ ;B .0 c @ c 0 7[F2 [IK; G ?5:.(2 :F2 7>Ee E >B7 @b!]; E . 5 .0 c A @ c 0 A @ d7% @ ;$ .0 c @ c 0 4 >I# 7[F2 [%d;a 96A,/7 /7 9#!E+ .A9/?5:pT f :%$ 9EA8: ,/:=A1B B /?A A @ d7% @n;I @ dK7 A @ !.; 2cI :/?5:9 -91+ K ++ ( 5( A @ 4 ( 5( @ ++ 7% A @ ! @ ; 4 8 A @ !6 (( @ ' .0 c @ c 0 4 >5-:# 7[F2 [*<; b'.FB B :=*/7 9#! :'F:,/;/?57 9#!:/.K!F:/?5:,A8:%?APF:=-9A1B B ; @ I& A @ ! 4 ( > ('( @ A?5:,/: > ('[ > 4 .0 c @ c 0 # 7[F2 [JK; a 9A ,/."@A/7 9#! 0 ,@A1C :c/?5:,/:87>EA = .F9/,/7 D#/7 .F9/.0/?5:*)/AF== :B :,@A/7 .F9,+ @ 0 ,/.FC /?5:.- F; *K I&!/ A @ @1A191+ /?5:0(."/:9/7>A1B 95::%+5Ec/.:D:%A"#!FC :9/:%+ D ;DA./:,/C /?A!F7 F:E-,/7>E :5/.p/?5: *K ); & # I; *K 10 0 4 7% d @n; 25.F,@= :EE]=@? AFE/?5:E :K@A?57>=@?(A"(#(:A1,D:=A"E :%.F95:K4 E80 ,@A1C :%7>E 95.F9 7 95:,]/7>A1Bl@5A1,/: =A1B B :%+ /K ; *K 2|JE := .F91+ :5A1C (5B : 7 B B E/,@A/:E./?AE HA"!F,@A19#!F7>A19EA8.F,CF :'F:9 0 .F, = . .F, + 7 9A/:E./?A + :'(:91+ :(5B 7>= 7 /B ; .F9>/7 C :F2Ea 9 = . E C .FB .K!F; 7 7>E%?A191+ ;~/.~E :UN = .FC . 7 9#!14 = . .F, + 7 9A/:E%E]=@?>/?A./?5: E](A/7>A1B= . .F, + 7 9A/:E .10 (A1,]/7>= B :ET/?AC .F:%A"(A1,]AFE/?5: { 957 F:,@E : :(A191+5E A1,/:%= .F9E@A192c :h/?5:.(5,/7 C :%+ E ; E/:C D:%7 95:,]/7>A1BA191+ /?5:.59#(5,/7 C :%+ E ; E/:C = .FC . 7 9#!2G ?5:9 @Y32i7]; @#A?5:,/:42i7];87>Eh/?5: = . E C 7>=%E/=A1B : 0AF=*/.F,2X . = .0 A @ 0 .0 728A A2 ; 0 # 7[F2 [L; e QjQ W Q ; ; I ,/7 /7 9#! /?5:.(."/:9/7>A1B :95:,C!F; AFE > w/?5:pT :%$ 9EA1,/: K ++ 728A A2 ; 2 4 728A A2 ; A256( ( # b'B :A1957 9#! #(6A8:!F: I A22 A 22 4 [ 2 0 ( ( # 7[F2 [ "; | -9A1B :5A1C (5B : 7 B B E/,@A/:E?5.A /.!F:= 7 9 A=A8:7 , + =5,C 7 B 7 95:A1,= . .F, + 7 9A/: E ; E/:C 2 f D5B>A/:E](5?5:,/.F7/+5A1B = . .F, + 7 9A/:E7 :! ! )i; A1,/:%,/:B>A/:%+B/. ,/:'!K5B>A1,= ; B 7 91+ ,/7>=A1B (.FB>A1,@E.7!S ! )i;TD ; = . E ? = . E & S E 7 95? E 7 9 # 7[F2 [%\; X B 7 !F?/B ; =@?A19#!F7 9#! @ A191+ ) 7 9 C5,/9 A?57 B : B :AP 7 9#! /?5: ."/?5:, = . .F, + 7 9A/:E A1B .F95:K@!F:95:,@A/:E E CDA1B B + 7>E](5B>AF= :C :9@E 7H 7E 7 95? = . E + = . E ? E 7 9 ( ; 7H i7 4 = . E ? E 7 9 + E 7 95? = . E ( ; GH.) # a 7>E :AFE ;6/. =@?5:=,F6/?A=/?5:E :8/?5,/:: + 7>E](5B>AF= :C :9F:=*/.F,@E A1,/: C:CA1B B ;~(:,C(:91+ 7>=5B>A1,2 X ./?5:E+ 7>E@A19= : .F95:!F. :E .F9 =@?A19#!F7 9#! A1B B.10h7 :! ! )i;E 7 C:5B @A195:.KE B ;D7>E +! 0 c c 0 H 0 ]H 0 ]H 0 0 7H ; 0 7E 7 95? 0 = . E 0 5 = . E ? 0 E 7 9 0 ; 7H ; 0 7= . E ? 0 E 7 9 0 5 E 7 95? 0 = . E 0 ;$ 7H.)i; 0 = . E ? 0 = . E 0 0#" 7= . E ? 0 4 = . E 0 ;W 7H ; 0 7H ; 0 X = . E ? 0 = . E 0 i7H.)i; 0$ # 7[F2 [ P; ^ 7 7/+ 7 9#! /?5,/.K#!F?9D ;+ 0 A8:!F:/?5:.F 7 95:/7>= :95:,C!F; 7 9 /:,/CDE .107 A ! A Y! A)i;*? = .0 ' +! + - 0 .0 0" 7= . E ? 0 4 = . E 0 ;W A 0 A 0 X = . E ? 0 = . E 0 A) 0#$ # 7[F2 I1K; G ?5: :%$ 9E .10C ."/7 .F9(A1,/:0/?5:,/: 0 .F,/: 0% ++ 7]7= . E ? 0 4 = . E 0 ; A 4 .0 E 7 95? I % A 0 A 0 = . E 0 A) 0'& )( >( K 0 % ++ 7]7= . E ? 0 4 = . E 0 ; A 4 .0 E 7 9 I& % A 0 A 0 4 = . E ? 0 A) 0' & ( >( K 0 0 ++ % = . E ? 0 = . E 0 A) ' 3 )( >( ) K # J K ' LT & 5 & 5 ; & ' )( HA"!F,@A19#!F7>A19 + ; 9A1C 7>=E,/:A1B B ; = .FC :E7 9/. 7 @E-.A9 0 .F,/?5:0+ ; 9A1C 7>=E-.10A ,/7 !F7/+ D.+ ;p:/?A87>E'A19D.KD 3:=*8E]=@? AFE8A E](A19595:,T/?A= .F9@A17 9E8Ap AFE 95C:D:, $ .10)(A1,]/7>= B :E/?AA1,/: E . E/,/.F9#!FB ; = .K#(5B :%+E/. :AF=@? ."/?5:,T/?A A8: CDA; = .F9E 7/+ :,/?5:+ 7>E@A19= :ED:A8::9p/?5:C /.D:- :%+ 2a 9:/?57>E-A"(#(5,/./ 7 CDA/7 .F9 @%/?5:= . .F, + 7 9A/:E.10:'F:,/; (A1,]/7>= B :A1,/:.F 95.A9*AFE E . .F9(AFE5A8: ?APF:p+ :/:,/C 7 95:%+6/?5: E 7 6!F:95:,@A1B 7 `:%+ = . .F, + 7 9A/:Eh/?A%A1,/: ,/:%$57 ,/:%+/. ( * l `Fc,+ `.-2hji/?d f`c-10 0 m2+ac AFE . A191+ 0F2TX . B :%4 E =@?5. . E : .5/.ED:/?5: A 7>EA"D.KA?57>=@?6A8:%,/."@A/:%+B/?5,/.K#!F?65B :, A19#!FB : $ 2-G ?5:9 A$ = .F9/,/7 D#/:E A$ .0/. 25| 9(7 9= ,/:C :9 7 9 )*,/."@A/:Eh/?5: E ; E/:C A"D.K H 2 G ?57>E B 7 :E 7 96/?5: (5B>A195: .10 0A191+ 2DA191+*7>E 7 9= B 7 95:%+ A A19#!FB : $ /. 212E< :9= : A) = .F9/,/7 D#/:E A)7= . E $ 2 4 E 7 9 $ 0%;0/. 2 |0+#+ 7 9#!*A1B B/?5,/:: = .F9/,/7 D#/7 .F9ET/.K!F:/?5:,/.0 .F,/C A191+ E]#DE/7 C/7 9#!8/?5:%,/:E]5B 7 9/.67[F2 I P;A8:0-91+B/?Ah/?5:.F 7 95:/7>= :95:,C!F; .10A19 A 7>E ; C C :/,/7>= D.+ ; 7>E = .0 ."7 A) 0 E 7 9 0 $ A$ 0 ;B .0 27 A) = . 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A1B 5:E .10 A )@ AR A191+ ASDAh/7 C := A191+ A/7 C :.`:,/.2X /$Y 5 + :'(:91+5E.F95B ; .F9:/?5:5+ 7 } :,/:9= :D:A8::9 = . .F, + 7 9A/:E'@1E . 7 -7>E7 9 A1,/7>A19591+ :,7 !.RY!S; 7 !.R5 !S *; 2-G ?5: AFE/E . = 7>A/:%+ 7 9 A1,/7>A197>E( 5( A )( 5( AR )( 5( AS @.7 A AR#;$'0 AS? # 7]; 7[F2 LQJK; E . # 7]; 4 # 7K;I 4 @.P7 A 4 A T AR 4 AR TP; 4 '0K7 AS 4 AS TP; # 7[F2 LFL;< :,/:K4 E-A19DA"(#(5B 7>=A/7 .F98/. C ."/7 .F9 7 9 A5957 0 .F,/C CDA"!F95:/7>=-:B/+ G H 2c :%4 E-=@?5. . E : 3 7 4 RY!/K !/K; 2G ?5:9 D ;>7[F2 d#[P;:5 .0 A 0 4 R A 7>E 7 9 A1,/7>A19h591+ :,A8.T.A E'?h7 7n; R R H A191+ 7 7 7n; R8 R H H 2< :9= :.A8:%?APF:0A8. 7 9 A1,/7>A19@E [ ( 5( A ! 4 R & \ [ ( 5( AS ! \ # 7[F2 L A; b'?5. . E 7 9#! 3 7K !D !/K;A8:=-91+ A8/?57 , + 7 9 A1,/7>A19 0 .F,h/?5: E/A1C :(5? ; E 7>=A1B (5,/.KD5B :C6? [ ( 5( AR ! D # 7[F2 L PD G ?5:.(5? ; E 7>=A1B C :A1957 9#! .10 \ 7>E .KD 7 .KE'@D#5A?A5+ . A191+ C :A19 (5? ; E 7>=A1B B ;#6|0+#+/?5:C#( ? [ 7 7% 7 ;B 7 7 4 R#; A 7 A?5:,/: [ 7 7 RY# 7[F2 LK\; G ?57>E7>EA(B 7 95:A1,p-,@E .F, + :,:+ 2 :F2H0 .F, &'7 ?AFE 7 9/:'!F,@A/7 9#! 0AF=*/.F,: _ @A?5:,/: 0 [ 7>E./?5:9 HA1,/C .F,0 ,/:%$5:9= ;F2<~5B /7 (5B ; 7 9#!8/?5,/.K#!F?D ;D: _ @ 7 9/:'!F,@A/7 9#!:A 2 ,2 2'A191+B/?5:9 C:5B /7 (5B ; 7 9#!8/?5,/.K#!F?D ;D: _ A8:%.KD@A17 9 7]; 7 0 *: _ ! 7[F2 L P; A?5:,/: 7>E%A = .F9E@A19%.10'7 9/:'!F,@A/7 .F92I : 95.A E :: /?A/?5: ,/:A1BA191+*7 CDA"!F7 9A1,/;9(A1,]@E .10 :9= .+ : /?5:R A191+' = . .F, + 7 9A/:E .10/?5:p!K57/+ 7 9#! = :9/,/: A1,/.K591+9A?57>=@?9/?5:p(A1,]/7>= B :p!F; ,@A/:E'@A?57 B : c c 7>E/?5: ,@AK+ 7 E .10 !F; ,@A/7 .F9 @A?57>=@?~+ :'(:91+5E.F9/?5:.(A1,]/7>= B :K4 E :95:,C!F;F2 J K - 5 % X .FC :/7 C :E7 7>E = .F9F:957 :9h/.EA8.F,CFBA7 /? C .F,/: = . .F, + 7 9A/:ET/?A19*ADE ; E/:C?AFE5+ :'!F,/::E .100 ,/::%+ .FC 2X#( (. E :K@ 0 .F,%:5A1C (5B :K@ /?A=/?5:DE ; E/:C = .F9E 7>E@E .108A9+5C:D:B B-.10'B :9#!"/? :/?A 7>E 0 ,/:: /.(E B 7/+ : .F9 A E C . ."/? @A"D5B :F2 G ?57>EE ; E/:C ?AFE./?5,/::+ :'!F,/::E .100 ,/::%+ .FC6@9A1C :B ;>/?5:(. E 7 /7 .F9 .10h/?5: = :9/,/: .10 CDAFE/EA191+ /?5: .F,/7 :9@A/7 .F9 .10/?5:p+5C:D:B B 2860A8: C 7 !F?5(5,/: 0 :,h/.B+ :E/= ,/7 D:/?5:E ; E/:C7 96/:,/CDE.10/?5:= A191+UR = . .F, +5E .10/?5:p+5C:D:B Bl4 E(A1,]/7>= B :E2G ?5:E : A1,/:%95."7 91+ :'(:91+ :9%@1D# E/A/7>E30 ;E/?5: = .F9E/,@A17 9 7 f. 4 0P; 0 7R-. 4 R"0%; 0 0 # 7[F2 K; G ?5:=+ ; 9A1C 7>=E'.10 /?5:%E ; E/:C A1,/: .KD@A17 95:%+ED ;D:/,/:C 7 `7 9#!./?5:%AF=*/7 .F9 E]#D 3:=*/.p/?57>E = .F9E/,@A17 9':%$A/7 .F92 HA"!F,@A19#!F: C:5B /7 (5B 7 :,@E=768./ IK;:9A"D5B :5E/.8+ . /?57>E E 7 C (5B ;F2I :0A,/7 /:5/?5: = .F9E/,@A17 9-:%$A/7 .F9 AFE 7)> ; K A191+ :' A1B A/: KIH1G 4 ] + H! ] _` _ba + H 8 ( 5( 4 + + ' ( 5( A - "( ( :# 7[F2 [P; b *j C G R / LT & 5 & X#(#(. E : A8: A1,/:>!F7 F:9 /?5: (5,/."-#E7 !.RY!S;BA?5:9 A8: E :B B E .FC :*0 . .+ A7 /? A1C .K59@EZ )@?RA191+ S .10%AK+#+ 7 /7 F:E8/?A A1,/: = .F9E/,@A17 95:%+YD ; /?5:(?5:A1B /?"A191+ E/A 0 :3; ,/:'!K5B>A/7 .F9E E]=@? /?AEA8: C:ED?APF:UE7 !.RY!S;IK#@KA?5:,/: U 7>E'A E](:= 7 -:%+ 0e59=*/7 .F92G ?5: ,/:'!K5B>A/7 .F9E.KD5B 7 !F:5Ec/. CDA19 0AF=*C5,/: A(5,/.+=* A?5. E : ,/:'(5,/:E :9@A/7 F: (.F7 9 7 9 7 !.RY!S; E](AF= : B 7 :E .F9 /?5:BA8.1 + 7 C :9E 7 .F9A1B'E]5, 0AF= :HU K#@A191+ A8: CDA 7 C 7 `: .K5,h(5,/."-#5D ;E-91+ 7 9#! /?5:.(.F7 9.F9 /?57>E E]5, 0AF= :%A5A?57>=@? 7>ED57 !K!F:E2 a 0A8: CDA"F :%E CDA1B BH=@?A19#!F:E87 9 /?5: 7 9#(#@E'@5.K5,(5,/."-# =@?A19#!F:ED ; +Q ( ( + )( ( R +R )( ( S +/S # 76I 2 [P; { 9 0 .F,]C59A/:B ;K@#A8: A1,/:%.KD5B 7 !F:%+B/. ,/:CDA17 9 .F9 /?5: E]5, 0AF= :U K#@5E . .K5, =@?A19#!F:E8?APF:=/.DE/A/7>E30 ; K + U ( U( + )( U( R +R5)( U( S +/S # 76I 2 IK; I : C:5B /7 (5B ;:/?57>E-:%$A/7 .F98D ; A19DA1,CD57 /,@A1,/;%0e59=*/7 .F9 )7 !.RY!S;A191+ E]#D/,@AF=*c/?5: ,/:E]5B 0 ,/.FC :%$A/7 .F9 76I 2 [P; 2cI :/?5:9 ?APF: K ' ( ( 4 "( U( -i+ ' ( ( R 4 "( U( R - +R5 ' ( ( S 4 "( U( S -i+/S # 76I 2 d; I : 95.A =@?5. . E :Y/?5:"0e59=*/7 .F9 /. CDA"F : /?5:= . :, = 7 :9 .106+/S A1957>E ? /?A 7>E A8: E : 7 ( ( S;+7 ( U- ( S; 2cX . A8:%95.A ?APF: KD ' ( ( 4 "( U( -i+ ' ( ( R 4 "( U( R - +RY# 76I 2 <; G ?5:=@?A19#!F:E5+ A191+~+R =A19~D:=@?5. E :9 7 91+ :'(:91+ :9/B ;6D:=A"E :.A?A/:'F:,5 A1B 5:EA8:AK+ .K( 0 .F,/?5:E : A1,/7>A"D5B :E'@K/?5: = .F9E/,@A17 9076I 2 IK;cA7 B B D: E/A/7>E-:%+ 0 .F,8A19 A"(#(5,/.K(5,/7>A/: A1B 5: .10 +/S2 f 95: A1B B .A8:%+D=@?5.F7>= : 7>E + K>A7 /? +R K#@ A191+ 0 .F,8/?57>E =@?5.F7>= : :%$A/7 .F9 76I 2 <; ?5.FB/+5E .F95B ; 7 0=/?5:(= . :, = 7 :9D.10+ A1957>E ?5:E2~X 7 C 7 B>A1,/B ;K@=@?5. . E 7 9#!9/. E ::+ K~A7 /?Y+R K6A8:D7 9 0 :,p/?Ap/?5: = . :, = 7 :9 .105+R A1B>E . A1957>E ?5:E2cI : 95.A ?APF: 0 .K5,:%$A/7 .F9E/?A C:E ?5.FB/+ A/?5:.(.F7 9h/?A CDA 7 C 7 `:E @59A1C :B ; U KD ( ( 4 "( U( & KD ( ( R 4 "( U( R &K ( ( S 4 "( U( S # 76I 2 JK; a 93(5,/7 9= 7 (5B : A8:=A19*E .FB F:./?5:E : 0 .K5, :%$A/7 .F9E0 .F,5/?5: 0 .K5,059#F 95.A9E'?T/?5:8 A1B 5:E .108 )@CR(A191+,S A /?5: E@A/7 .F9A1,/; (.F7 9%@A191+B/?5:%95C :,/7>=A1Bi A1B 5:%.10)/?5:%0e59=*/7 .F9 (Ah/?A(.F7 92-G ?57>Eh(5,/. = :%+5,/:=A AFE 7 9F:9/:%+D ;B/?5:p HA"!F,@A19#!F:K@ E . 7>E=A1B B :%+ A LT & 5 & 9 2 < :,/: )7)>e!];7>E A19*A1,CD57 /,@A1,/; 0e59=*/7 .F92h| E 7 9~ HA"!F,@A19#!F:K4 E E@A191+5A1, + A1,C!K5C :9%@A8:=@?5. . E : 6/.D:9E]5,/:/?A /?5: = . :, = 7 :9 .10.F95:%.10/?5: H A1957>E ?5:E'@A191+B/?5:9(= .F9= B 1+ :%0 ,/.FC /?5: 7 91+ :'(:91+ :9= : .10/?5: ,/:CDA17 957 9#! /?A/?5:7 , = . :, = 7 :9@EC:E A1957>E ?9/. .2c< :9= :.A8:%?APF: 0 .F,:'F:,/;c/?A + + ' ( 5( A -9( 5( 4 "( ( # 7[F2 1IK; X(:= 7 -=A1B B ; 0 .F,.K5,+5C:D:B BH:5A1C (5B :K@"5 .0 7% 0. 00 ;*@E .8/?5: :%$A/7 .F9E8.10C ."/7 .F9 A1,/: f. 4 I )7 f. 4 0P; 0 I )7 f. 4 0P; R-.F 4 I )7R-. 4 R"0%; R"0V I )7R-. 4 R"0%; # 7[F2 d; |0+#+ 7 9#! /?5:*B .A8:,8/. /?5:~#(#(:, :%$A/7 .F9E A8:(.KD@A17 9 /?5:(:%$A/7 .F9E .10%C ."/7 .F9".10 /?5: = :9/,/: .10%CDAFE/E'? 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E ?5.A /?A :'!F:91+ ,/:./,@A19E30 .F,/CDE A1,/: 7 9F:,]/7 D5B :F2a 9(0AF=*%AB :'!F:91+ ,/:/,@A19E30 .F,/C 7>E 7 @E .A9 7 9F:,@E :K? 7 i; 7 i; 2 a 7>E .10n/:9 ?5:B ( 0e5B/. = .F9E 7/+ :,0/?5: 0e59=*/7 .F9 7 ! ; [Q 4 7 i; .10TA8. 7 91+ :'(:91+ :9. A1,/7>A"D5B :E7 ! ; 2 ,@A"(5?57>=A1B B ;K@ 7 ! ;7>E:/?5:6F:,]/7>=A1Bh+ 7>E](5B>AF= :C :9DA .F, + 7 9A/:Z wD:A8::9Y/?5:(E/,@A17 !F? B 7 95:R *A191+B/?5:.#(A A1, + =5,C 7 9#! !F,@A"(5? .10 7 i;*? G ?5:8 :'!F:91+ ,/:./,@A19E30 .F,/C 7 ;7>E5/?5:8 A1B 5: .10 "A0/?5: (.F7 9 7 ; A=A?57>=@?~/?5: =5,CF: ,C59E5(A1,@A1B B :B /. /?5: B 7 95:F2TX 7 9= : ( ( 4 ( ( ! 76hd52 IK; 7 ; 7>E/?5:. A1B 5: .10: ~A?57>=@? :/,/:C 7 `:E 0 .F,!F7 F:9 @5AFE 7>E A1B ,/:AK+ ;D:' 7/+ :9 0 ,/.FC /?5:-1!K5,/:F2 { E 7 9#!p/?5:=T *:%$ 9E8A191+ E 7 C (5B 7 0 ; 7 9#!p/?5: 95."@A/7 .F9 @K/?5:0-,@EA8..10 /?5:E : :%$A/7 .F9E-B :AK+EE/. 9 -> A>@ ( ( & A 4 ( ( > # 7 I 2 P; | B .F9#!DA:/,@A 3:=*/.F,/; % >c7]; ! 7]; ' @/?5:p< A1C 7 B /.F957>A19 % >c7]; ! 7]; ! 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