Radiaci n del cuerpo negro. Hip tesis de Planck.

 1 47.,*3 )* 1& C8.(& :A39.(& 9:;4 1:,&7 (43 *1 *89:).4 )* 1& 7&).&(.E3 9B72.(& !4)4
(:*754 547 *1 -*(-4 )* *89&7 & :3& (.*79& 9*25*7&9:7& *2.9* *3*7,C& *1*(9742&,3B9.(&
)*342.3&)& 7&).&(.E3 9B72.(& = 6:* 8* ,*3*7& 547 *1 24;.2.*394 &(*1*7&)4 6:* 8:+7*3 1&8
5&79C(:1&8 (&7,&)&8 6:* (4389.9:=*3 *1 (:*754 :&3)4 *1 (:*754 8* *3(:*397& *3 *6:.1.'7.4
9B72.(4 1& 7&).&(.E3 6:* *2.9* 547 :3.)&) )* 9.*254 *8 .,:&1 & 1& 6:* &'847'* . 1&
9*25*7&9:7& )*1 (:*754 *8 )* :348 9*25*7&9:7& &2'.*39* 34 ;*248 1& 7&).&(.E3
6:* *2.9* =& 6:* B89& 8* *3(:*397& *3 *1 .3+7&774/4 = *1 4/4 -:2&34 34 1& )*9*(9& .3
*2'&7,4 8. &:2*39&248 1& 9*25*7&9:7& 11*,& :3 242*394 *3 6:* *25*>&248 & ;*7 *1
(:*754 )* (4147 74/4 8* ).(* 6:* *89A &1 74/4 ;.;4 = 8. 8* 8.,:* &:2*39&3)4 8* ;*7A '1&3(4
*8 )*(.7 8* *2.9* 7&).&(.E3 *3 94)4 *1 *85*(974 ;.8.'1* 1 *85*(974 )* 1& 7&).&(.E3 9B72.(&
)*5*3)* *3 ,*3*7&1 )* 1& (42548.(.E3 )*1 (:*754 8.3 *2'&7,4 *<.89*3 :3& 8*7.* )* (:*7548
6:* *2.9*3 :3 *85*(974 )* 7&).&(.E3 :3.;*78&1 (:&3)4 *89A3 *3 *6:.1.'7.4 9B72.(4 = 843 148
)*342.3&)48 (:*7548 3*,748 "3 (:*754 3*,74 *8 &6:*1 6:* 34 7*@*/& 7&).&(.E3 *8 )*(.7
6:* &'847'* 94)& 1& 7&).&(.E3 6:* 7*(.'* <.89* :3& +472& 2:= 8.251* )* (43897:.7 :3
(:*754 3*,74 = *8 7*&1.>&3)4 :3& (&;.)&) (424 1& 6:* 8* 2:*897& *3 1& ?,:7& )* 5&7*)*8
&'847'*39*8 )* 24)4 6:* 1& 7&).&(.E3 6:* *397& ).+C(.12*39* 5:*)* 8&1.7 )* 1& (&;.)&)
3 1& 8.,:.*39* ?,:7& 8* 2:*897& *1 *85*(974 (&7&(9*7C89.(4 )* :3 (:*754 3*,74 & ).89.39&8
9*25*7&9:7&8 & +:3(.E3 6:* 8* 7*57*8*39& *8 9&1 6:* *8 1& *3*7,C& )* 1&
7&).&(.E3 *1*(9742&,3B9.(& 547 :3.)&) )* ;41:2*3 (:=& +7*(:*3(.& *89A (4257*3).)& *397*
= U(ν) (1019J/m3Hz)
10
500K
8
6
400K
4
T=300K
2
0
0
2
4
ν
6
(1013Hz)
8
10
?3&1*8 )*1 8.,14 %% 8* (434(C& *<5*7.2*39&12*39* *89* *85*(974 )* 7&).&(.E3 = )48
1*=*8 7*1&(.43&)&8 (43 *89* *85*(974 * 8&'C& 6:* 1& +7*(:*3(.& 5&7& 1& (:&1 1& +:3(.E3 *8 2A<.2& *8 574547(.43&1 & 1& 9*25*7&9:7& (434(.)& (424 1& 1*= )*1 )*851&>&2.*394 )*
$.*3
>
3 1& ?,:7& 54)*248 4'8*7;&7 6:* (:&3)4 &:2*39& 1& 9*25*7&9:7& *1 2A<.24 8*
)*851&>& -&(.& ;&147*8 )* +7*(:*3(.& 2&=47*8 97& 1*= 6:* 8* (434(C& (438.89* *3 6:* 1&
*3*7,C& 949&1 547 :3.)&) )* ;41:2*3 6:* 8* 4'9.*3* .39*,7&)4 *89& +:3(.E3 5&7& 94)&8 1&
+7*(:*3(.&8 *8 574547(.43&1 & 1& (:&79& 549*3(.& )* 1& 9*25*7&9:7& (434(.)& (424 1& 1*=
)* 9*+&3419>2&33
! 2 424 )./.248 &39*7.472*39* *1 *85*(974 )* 1& 7&).&(.E3 34 )*5*3)* )* 1& (42548.(.E3
)*1 (:*754 8.34 F3.(&2*39* )* 8: 9*25*7&9:7& 1 *<51.(&7 1& 7&).&(.E3 )*1 (:*754 3*,74
+:* :3 7*94 5&7& 148 +C8.(48 )* ?3&1*8 )*1 8.,14 %% =& 6:* &1 34 )*5*3)*7 )* 1& (42548.(.E3
)*1 (:*754 34 )*5*3)*7A )* (E24 8* *2.9* = &'847'* 1& 7&).&(.E3 547 5&79* )* 148 *1*(9743*8
)* 1&8 5&7*)*8 )* 1& (&;.)&) = 8* )*'.*7& 54)*7 *<51.(&7 *3 '&8* &1 *1*(9742&,3*9.824 =
1& 9*724).3A2.(& .3 *2'&7,4 94)48 148 .39*3948 6:* 8* 7*&1.>&743 5&7& *<51.(&7 *89&
7&).&(.E3 2*).&39* 1&8 9*47C&8 (1A8.(&8 +:*743 ;&348 #&248 & &3&1.>&7 *1 2A8 &(*79&)4 )*
*8948 .39*3948 6:* +:* 14 6:* 8* (434(* (424 *1 24)*14 )* &=1*.,-*&38
43 *89* 24)*14 8* 57*9*3)* &3&1.>&7 1& 7&).&(.E3 *1*(9742&,3B9.(& 6:* 8* 5745&,&
)*3974 )* :3& (&;.)&) (:&3)4 8* *3(:*397& *3 *6:.1.'7.4 9B72.(4 &'*248 6:* *1 *85*(974
)* 1& 7&).&(.E3 34 )*5*3)* )* 1& +472& )* 1& (&;.)&) 3. )* 1& (42548.(.E3 )* 1&8 5&7*)*8
47 9&394 9&254(4 )*5*3)*7A )* 1&8 (43).(.43*8 )* (4394734 *3 1&8 5&7*)*8 =& 6:* B89&8
843 2:= ).89.39&8 547 */*2514 5&7& :3 (43):(947 6:* 5&7& :3 ).*1B(97.(4 47 *89& 7&>E3
84248 1.'7*8 )* *8(4,*7 9&394 1& +472& (424 1&8 (43).(.43*8 )* (4394734 )* 24)4 6:* ;&248
& *8(4,*7 1&8 2A8 8*3(.11&8 6:* (438.89*3 *3 942&7 :3 (:'4 )* 1&)4 (43 (43).(.43*8 )*
(4394734 5*7.E).(&8 *3 1&8 5&7*)*8
a
:*7*248 *89:).&7 1& 7&).&(.E3 *1*(9742&,3B9.(& )*3974 )* 1& (&;.)&) 4)*248 7*8
97.3,.7 3:*8974 *89:).4 &1 (&254 *1B(97.(4 =& 6:* 5&7& :3& 43)& *1*(9742&,3B9.(& & 5&79.7
)*1 (&254 *1B(97.(4 54)*248 4'9*3*7 *1 2&,3B9.(4 1 (&254 *1B(97.(4 )*3974 )* 1& (&;.)&)
8* 54)7A *8(7.'.7 (424 :3& 8:5*7548.(.E3 )* 43)&8 51&3&8 (424 1& 8.,:.*39*
)43)* 148 ;*(947*8 = 843 )48 ;*(947*8 :3.9&7.48 5*75*3).(:1&7*8 *397* 8. = 5*75*3).
(:1&7*8 & 1& ).7*((.E3 )* 5745&,&(.E3 4 6:* 8* 57*9*3)* (43 *89* 24)*14 *8 (439&7 *1
3F2*74 )* 43)&8 51&3&8 6:* 8* 5745&,&3 547 1& (&;.)&) 5&7& :3& +7*(:*3(.& (4257*3).)&
*397* :3& (.*79& +7*(:*3(.& = = &8.,3&71* :3& *3*7,C& 2*).& & *89&8 43)&8 51&3&8
5&7& 4'9*3*7 &8C 1& *3*7,C& )* 1& 7&).&(.E3 *1*(9742&,3B9.(& (43 +7*(:*3(.& (4257*3).)&
*3 ).(-4 .39*7;&14 #&248 & .2543*7 1&8 (43).(.43*8 )* (4394734 5*7.E).(&8 &1 (&254
*1B(97.(4
89&8 (43).(.43*8 8E14 8* 5:*)*3 8&9.8+&(*7 8. 1&8 (42543*39*8 )*1 ;*(947 )* 43)& 8&9.8
+&(*3 1&8 8.,:.*39*8 7*1&(.43*8
=
8.*3)4 = 3F2*748 *39*748 894 *8 8.2.1&7 & 14 6:* 4(:77* (43 1&8 43)&8 *89&
(.43&7.&8 (:&3)4 .2543*248 (43).(.43*8 )* (4394734 7*897.(9.;&8 *3 148 (43947348 7*8:19&
6:* 34 8* 5:*)* 5745&,&7 :3& 43)& (43 (:&16:.*7 ;&147 )* 1& 143,.9:) )* 43)& 4)*248
(&1(:1&7 &-47& *1 3F2*74 )* 43)&8 51&3&8 6:* 8* 5745&,&3 )*3974 )* 1& (&;.)&) 6:* 9.*3*3
8: ;*(947 )* 43)& (4257*3).)4 *397* :3 (.*794 ;*(947 = = 6:* 8*7A *1 8.,:.*39*
)43)* *1 +&(947 &5&7*(* 5&7& 9*3*7 *3 (:*39& 1&8 )48 548.'1*8 541&7.>&(.43*8 )* 1&8 43
)&8 51&3&8 1&8 )48 ).7*((.43*8 = 434(.*3)4 1& 7*1&(.E3 *397* = = 1&8
(42543*39*8 )*1 ;*(947 )* 43)& 54)*248 *<57*8&7 *89* 3F2*74 )* 1& 8.,:.*39* +472&
5&79.7 )* *89* 3F2*74 54)*248 (&1(:1&7 *1 3F2*74 )* 43)&8 51&3&8 6:* 8* 5745&,&3
)*3974 )* 1& (&;.)&) (:=4 2E):14 )*1 ;*(947 )* 43)& 4 3F2*74 )* 43)& *89A (4257*3).)4
*397* :3 (.*794 ;&147 = 9*3.*3)4 *3 (:*39& 1& 8.,:.*39* 7*1&(.E3 )* 24)4 6:* *1 3F2*74 6:* ':8(&248 *8 *1 8.,:.*39*
47 F19.24 (434(.*3)4 1& 7*1&(.E3 6:* *<.89* *397* 1& +7*(:*3(.& = *1 3F2*74 )* 43)&8
54)*248 *3(4397&7 *1 3F2*74 )* 43)&8 51&3&8 6:* 8* 5745&,&3 )*3974 )* 1&
(&;.)&) (43 +7*(:*3(.& (4257*3).)& *397* = 6:* *8
424 8* 5:*)* ;*7 *1 3F2*74 )* 43)&8 51&3&8 (4257*3).)&8 *3 :3 .39*7;&14 *8 574
547(.43&1 &1 (:&)7&)4 )* 1& +7*(:*3(.& = 547 14 9&394 -&= 2A8 43)&8 51&3&8 )* +7*(:*3(.&
&19& 6:* 5*6:*D& 1 3F2*74 )* 24)48 6:* -*248 4'9*3.)4 *8 .3)*5*3).*39* )* 1& +472&
= 1&8 (43).(.43*8 )* (4394734 = 14 -&'7C&248 4'9*3.)4 9&2'.B3 8. -:'.*7&248 *8(4,.)4 :3
(:'4 )* 5&7*)*8 (43):(947&8
&7& (425&7&7 (43 *1 *85*(974 )*1 (:*754 3*,74 6:* -*248 57*8*39&)4 &39*7.472*39*
9*3*248 6:* (&1(:1&7 1& *3*7,C& 547 :3.)&) )* ;41:2*3 )* 1& 7&).&(.E3 6:* 9.*3* 8: +7*(:*3
(.& (4257*3).)& *397* = )* 24)4 6:* ;&248 & &8.,3&7 & (439.3:&(.E3 :3& *3*7,C&
& 1&8 43)&8 51&3&8 6:* 8* 5745&,&3 )*3974 )* 1& (&;.)&) "3& 43)& *1*(9742&,3B9.(& 6:*
8* 5745&,:* )*3974 )* 1& (&;.)&) 5:*)* 9*3*7 (:&16:.*7 ;&147 )* 1& *3*7,C& &-47& '.*3 8.
1& 7&).&(.E3 *1*(9742&,3B9.(& 8* *3(:*397& *3 *6:.1.'7.4 9B72.(4 (43 1& (&;.)&) 8*7A 2A8
574'&'1* 6:* 9*3,& :3& *3*7,C& 5*6:*D& 6:* :3& 6:* 8*& 2:(-4 2&=47 6:* 1& *3*7,C&
(&7&(9*7C89.(& )*1 (:*754 6:* ;.*3* )&)& 547 )43)* *8 1& (4389&39* )* 419>2&33
= 1& 9*25*7&9:7& )*1 (:*754 & ).897.':(.E3 )* *3*7,C&8 *3 *1 *6:.1.'7.4 9B72.(4 ;*3)7A
)&)& 547 1& ).897.':(.E3 )* 419>2&33
)43)* *8 1& 574'&'.1.)&) )* 6:* :3& 43)& 9*3,& 8: *3*7,C& (4257*3).)& *397*
= 5&79.7 )* *89& ).897.':(.E3 )* *3*7,C&8 54)*248 (&1(:1&7 *1 ;&147 2*).4 )*
1& *3*7,C&
6:* *8 .3)*5*3)*39* )* 1& +7*(:*3(.& 47 F19.24 54)*248 (&1(:1&7 1& *3*7,C& )* 1& 7&
).&(.E3 9B72.(& 547 :3.)&) )* ;41:2*3 (:=& +7*(:*3(.& *89A (4257*3).)& *397* = 6:* *8 14 6:* -&'C&248 349&)4 (424 1& +:3(.E3 4 6:* 9*3*248 6:* -&(*7 *8
2:19.51.(&7 *1 3F2*74 )* 547 1& *3*7,C& 2*).& = ).;.).7 547 *1 ;41:2*3 )*
1& (&;.)&) = 8* 4'9.*3* *1 8.,:.*39* 7*8:19&)4
89& *(:&(.E3 8* (434(* (424 1& +E72:1& )* &=1*.,-*&38 5&7& *1 *85*(974 )* 1&
7&).&(.E3 )*1 (:*754 3*,74 3 1& 8.,:.*39* ?,:7& 8* 2:*897& :3& (425&7&(.E3 *397*
*1 7*8:19&)4 6:* 57*).(* 1& +E72:1& )* &=1*.,-*&38 = *1 *85*(974 7*&1 )* 1& 7&).&(.E3
)*1 (:*754 3*,74 424 8* 5:*)* 4'8*7;&7 1& +E72:1& )* &=1*.,-*&38 8* &/:89& '.*3
&1 *85*(974 )* 7&).&(.E3 7*&1 5&7& *1 (&84 )* +7*(:*3(.&8 5*6:*D&8 .3 *2'&7,4 5&7&
+7*(:*3(.&8 ,7&3)*8 1& +E72:1& )* &=1*.,-*&38 ).;*7,* 2.*397&8 6:* *1 *<5*(974 7*&1 *8
(43;*7,*39* )*2A8 1& +E72:1& )* &=1*.,-*&38 (43):(* & :3 7*8:19&)4 &'8:7)4 = *8
6:* 8. .39*,7&248 1& +:3(.E3 5&7& 94)&8 1&8 +7*(:*3(.&8 5&7& 4'9*3*7 1& *3*7,C& 949&1
)* 1& 7&).&(.E3 8* 4'9.*3* :3 ;&147 .3?3.94 89* 7*8:19&)4 &'8:7)4 8* (434(* (424 1&
(&9A8974+* )*1 :197&;.41*9& .3,:34 )* 148 .39*3948 6:* 8* 7*&1.>&743 & ?3&1*8 )*1 8.,14 %%
(438.,:.E *<51.(&7 1& 7&).&(.E3 )*1 (:*754 3*,74
Rayleigh-Jeans
Espectro real
$.*3 -&'C& *3(4397&)4 2*).&39* &7,:2*3948 9*724).3A2.(48 6:* 1& +:3(.E3 9*3C&
6:* 8*7 )* 1& +472& 8.3 *2'&7,4 1& +:3(.E3 34 8* 54)C& )*9*72.3&7
2*).&39* &7,:2*3948 9*724).3A2.(48 $.*3 5:)4 &/:89&7 )* +472& &574<.2&)& 1& +:3(.E3
.3974):(.*3)4 *1 (425479&2.*394 *<543*3(.&1 6:* 57*8*39& *1 *85*(974 7*&1 )* 1&
7&).&(.E3 9B72.(& 5&7& &19&8 +7*(:*3(.&8 )* 24)4 6:* )* &(:*7)4 (43 148 &7,:2*3948 )*
$.*3
8.*3)4 :3& (4389&39* 89& *(:&(.E3 8* &/:89& '.*3 5&7& &19&8 +7*(:*3(.&8 5*74 34 5&7&
5*6:*D&8
.3&12*39* *3 1&3(0 57*8*39E 8:8 7*8:19&)48 84'7* :3 &/:89* *25C7.(4 )*1 *85*(
974 7*&1 )* 1& 7&).&(.E3 )*1 (:*754 3*,74 & +E72:1& 6:* 1&3(0 *3(4397E *25C7.(&2*39*
(438*,:C& 7*574):(.7 *1 *85*(974 7*&1 )* 1& 7&).&(.E3 9B72.(& 5&7& 94)&8 1&8 +7*(:*3(.&8 =
9*25*7&9:7&8 1&3(0 &574;*(-E *1 (425479&2.*394 )* 1& +E72:1& )* &=1*.,-*&38 5&7&
+7*(:*3(.&8 5*6:*D&8 = *1 )* 1& +E72:1& )* $.*3 5&7& +7*(:*3(.&8 ,7&3)*8 & +E72:1& )*
1&3(0 *8 1& 8.,:.*39*
8.*3)4 :3& (4389&39* 8
.3 *2'&7,4 1&3(0 9&7)E &1,F3 9.*254 -&89& 6:* (438.,:.E /:89.?(&7 1& +E72:1& 6:*
-&'C& *3(4397&)4 1&3(0 (434(C& 148 *<5*7.2*3948 )* *79> 2*).&39* 148 (:&1*8 :3& (&7,&
48(.1&39* *2.9* 7&).&(.E3 *1*(9742&,3B9.(& )* 1& 2.82& +7*(:*3(.& 6:* 1& )* 48(.1&(.E3 )*
24)4 6:* 5*38E 6:* 1& 7&).&(.E3 )*3974 )* 1& (&;.)&) 8* )*'C& & 148 *1*(9743*8 )* 1&8
5&7*)*8 )* 1& (&;.)&) 6:* )*8(7.'C&3 :3 24;.2.*394 48(.1&947.4 1&3(0 *3(4397E 6:*
8* 54)C& /:89.?(&7 1& +E72:1& 6:* -&'C& *3(4397&)4 8:543.*3)4 6:* *8948 48(.1&)47*8 34
5:*)*3 9*3*7 (:&16:.*7 *3*7,C& * *89& +472& :3 *1*(97E3 6:* 48(.1* (43 +7*(:*3(.& 8E14 5:*)* 9*3*7 :3& *3*7,C& 2F19.514 *39*74 )* & (4389&39* 1& )*/E (424 :3
5&7A2*974 )* &/:89* 424 (438*(:*3(.& )* 1& -.5E9*8.8 )* 1&3(0 1& *3*7,C& )* :3& 43)&
*1*(9742&,3B9.(& )* +7*(:*3(.& 6:* 8* 5745&,:* *3 1& (&;.)&) 34 942& ;&147*8 (439.3:48
8.34 ).8(7*948 2F19.5148 *39*748 )*1 ;&147 3 *89* (&84 *1 ;&147 2*).4 )* 1& *3*7,C& 8*
(&1(:1& 8:2&3)4 84'7* 94)48 148 548.'1*8 ;&147*8 ).8(7*948 )* 1& *3*7,C&
5&79.7 )* 1& -.5E9*8.8 )* 1&3(0 54)*248 (&1(:1&7 1& +:3(.E3 9&1 (424 -.(.248 *3
*1 24)*14 )* &=1*.,-*&38 4'9*3.*3)4 1& +E72:1& 6:* 8* -& 57*8*39&)4 &39*7.472*39* &
+E72:1& )* 1&3(0 34 8E14 8* &/:89& 5*7+*(9&2*39* &1 *85*(974 7*&1 )* 1& 7&).&(.E3 9B72.(&
8.34 6:* 9&2'.B3 5*72.9* 4'9*3*7 1& 1*= )* $.*3 = 1& 1*= )* 9*+&3419>2&33 1 ':*3
&(:*7)4 )* 1& +E72:1& )* 1&3(0 348 5:*)* 11*;&7 & 5*38&7 6:* 1& -.5E9*8.8 )* 1&3(0
)*'* 8*7 :3 7*@*/4 )* 14 6:* 4(:77* *3 1& 7*&1.)&) * -*(-4 -4= *3 )C& 8&'*248 6:* 1&
*3*7,C& )* 1& 7&).&(.E3 *1*(9742&,3B9.(& 8E14 5:*)* 942&7 ;&147*8 ).8(7*948 )* 24)4 6:*
1& *3*7,C& )* :3& 43)& *1*(9742&,3B9.(& )* +7*(:*3(.& 8E14 5:*)* 942&7 ;&147*8 6:* 8*&3
2F19.5148 *39*748 )* *8 )*(.7 4)*248 (4257*3)*7 &-47& 547 6:B
1& +E72:1& )* &=1*.,-*&38 +:3(.43& '.*3 5&7& *1 (&84 )* +7*(:*3(.&8 5*6:*D&8 &7& *1
(&84 )* +7*(:*3(.&8 5*6:*D&8 *1 ;&147 )* 9&2'.B3 8*7A 5*6:*D4 )* 24)4 6:* 8* 5:*)*
(438.)*7&7 1& *3*7,C& )* 1& 7&).&(.E3 *1*(9742&,3B9.(& (424 (439.3:& *3 1:,&7 )* ).8(7*9&
&7& ;&147*8 5*6:*D48 )* 1& +7*(:*3(.& 1& *<543*3(.&1 6:* &5&7*(* *3 1& +E72:1& )* 1&3(0
8* 5:*)* &574<.2&7 )* 1& 8.,:.*39* +472&
=
= 8* 7*(:5*7& 547 9&394 1& +E72:1& )* &=1*.,-*&38 * -*(-4 *1 (&7A(9*7 ).8(7*94 )* 1&
*3*7,C& )* 1& 7&).&(.E3 *1*(9742&,3B9.(& 8* )*'* &1 ;&147 ?3.94 )* 1& (4389&39* )* 1&3(0
. 1& (4389&39* )* 1&3(0 +:*7& 3:1& 1& *3*7,C& )* 1& 7&).&(.E3 8*7C& (439.3:& 47 9&394
8*,F3 *89&248 ;.*3)4 = 8*,F3 ;*7*248 & 14 1&7,4 )* 94)4 *1 (:784 1& 9*47C& (:A39.(& )*'*
(439*3*7 & 1&8 9*47C&8 (1A8.(&8 *3 *1 1C2.9* *3 *1 6:* 1& (4389&39* )* 1&3(0 9.*3)* & (*74
& -.5E9*8.8 )* 1&3(0 +:* *1 3&(.2.*394 )* 1& C8.(& (:A39.(& 8.3 *2'&7,E ;&248 & ;*7
6:* &:36:* 1&3(0 /:89.?(E *1 7*8:19&)4 4'9*3.)4 34 ).4 :3& .39*757*9&(.E3 (477*(9& 48
54)*248 57*,:39&7 547 6:B 9&7)E 9&394 *3 )*8(:'7.78* *1 (&7A(9*7 ).8(7*94 )* 1& *3*7,C& )*
1& 7&).&(.E3 *1*(9742&,3B9.(& = 1& 7*85:*89& *89A *3 *1 ;&147 9&3 5*6:*D4 )* 1& (4389&39*
)* 1&3(0 =& 6:* 8: ;&147 *8 8 47 14 9&394 *3 2:(-48 +*3E2*348 6:*
4(:77*3 & 3.;*1 2&(748(E5.(4 *8 &'841:9&2*39* .2548.'1* 4'8*7;&7 1& ).8(7*9.>&(.E3 )* 1&
*3*7,C& . *3 *1 *89:).4 )* :3 +*3E2*34 )&)4 7*8:19& 6:* *8 2:= 5*6:*D4 (425&7&)4
(43 1& *3*7,C& (&7&(9*7C89.(& )*1 8.89*2& 1& *3*7,C& *1*(9742&,3B9.(& 54)7A 8*7 (438.)*7&)&
(424 (439.3:& = *1 +*3E2*34 54)7A 8*7 *<51.(&)4 (43 1&8 9*47C&8 (1A8.(&8