Buscar

RESOLUÇAO FISICA 3

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes
Você viu 3, do total de 72 páginas

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes
Você viu 6, do total de 72 páginas

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes
Você viu 9, do total de 72 páginas

Faça como milhares de estudantes: teste grátis o Passei Direto

Esse e outros conteúdos desbloqueados

16 milhões de materiais de várias disciplinas

Impressão de materiais

Agora você pode testar o

Passei Direto grátis

Você também pode ser Premium ajudando estudantes

Prévia do material em texto

���(T�������JLYHV�&RXORPE¶V�/DZ�� ) N T T
U
� �
� ��ZKLFK�ZH�VROYH�IRU�WKH�GLVWDQFH��
� � � � �
� �
���� �� 1 P & ���� �� & ���� �� &_ __ _ ����P�
����1
N T TU
)
��� �D��:LWK�D� XQGHUVWRRG� WR�PHDQ� WKH�PDJQLWXGH�RI� DFFHOHUDWLRQ��1HZWRQ¶V� VHFRQG�DQG�
WKLUG�ODZV�OHDG�WR�
P D P D P� � � � �
�
�� � �� � �
� �
� � ��
� �
�
�
NJ P V
P V
NJ�
�
�
2 72 7
�E��7KH�PDJQLWXGH�RI�WKH��RQO\��IRUFH�RQ�SDUWLFOH���LV�
�
� � � � �
� � � ����� �� 1 P & ���������P�
T T T
) P D N
U
,QVHUWLQJ�WKH�YDOXHV�IRU�P��DQG�D���VHH�SDUW��D���ZH�REWDLQ�_T_� ����� ���±���&��
���7KH�PDJQLWXGH�RI�WKH�PXWXDO�IRUFH�RI�DWWUDFWLRQ�DW�U� �������P�LV�
� �
� � � � �
� �
���� �� & ���� �� &
���� �� 1 P & ����1�
�������P�
T T
) N
U
���7KH�IDFW�WKDW�WKH�VSKHUHV�DUH�LGHQWLFDO�DOORZV�XV�WR�FRQFOXGH�WKDW�ZKHQ�WZR�VSKHUHV�DUH�
LQ� FRQWDFW�� WKH\� VKDUH� HTXDO� FKDUJH�� 7KHUHIRUH�� ZKHQ� D� FKDUJHG� VSKHUH� �T�� WRXFKHV� DQ�
XQFKDUJHG�RQH��WKH\�ZLOO��IDLUO\�TXLFNO\��HDFK�DWWDLQ�KDOI�WKDW�FKDUJH��T�����:H�VWDUW�ZLWK�
VSKHUHV� �� DQG� �� HDFK� KDYLQJ� FKDUJH� T� DQG� H[SHULHQFLQJ� D� PXWXDO� UHSXOVLYH� IRUFH�
� ��) NT U ��:KHQ�WKH�QHXWUDO�VSKHUH���WRXFKHV�VSKHUH����VSKHUH��¶V�FKDUJH�GHFUHDVHV�WR�
T����7KHQ�VSKHUH����QRZ�FDUU\LQJ�FKDUJH�T����LV�EURXJKW�LQWR�FRQWDFW�ZLWK�VSKHUH����D�WRWDO�
DPRXQW�RI�T�����T�EHFRPHV�VKDUHG�HTXDOO\�EHWZHHQ�WKHP��7KHUHIRUH��WKH�FKDUJH�RI�VSKHUH�
��LV��T���LQ�WKH�ILQDO�VLWXDWLRQ��7KH�UHSXOVLYH�IRUFH�EHWZHHQ�VSKHUHV���DQG���LV�ILQDOO\�
�
� �
� � ���� � �� � � 
 �� �� ������
� � �
T T T )) N N )
U U )
)
T 4 T
U
�
� �
�
1 6
ZKHUH�U�LV�WKH�GLVWDQFH�EHWZHHQ�WKH�FKDUJHV��:H�ZDQW�WKH�YDOXH�RI�T�WKDW�PD[LPL]HV�WKH�
IXQFWLRQ�I�T�� �T�4�±�T���6HWWLQJ�WKH�GHULYDWLYH� �G) GT HTXDO�WR�]HUR�OHDGV�WR�4�±��T� ����
RU�T� �4����7KXV��T�4� ��������
���7KH�PDJQLWXGH�RI�WKH�IRUFH�RI�HLWKHU�RI�WKH�FKDUJHV�RQ�WKH�RWKHU�LV�JLYHQ�E\�
���7KH�XQLW�$PSHUH�LV�GLVFXVVHG�LQ�†������8VLQJ�L�IRU�FXUUHQW��WKH�FKDUJH�WUDQVIHUUHG�LV�
� ���� �� $ �� �� V ���� &�T LW
���:H�DVVXPH�WKH�VSKHUHV�DUH�IDU�DSDUW��7KHQ�WKH�FKDUJH�GLVWULEXWLRQ�RQ�HDFK�RI�WKHP�LV�
VSKHULFDOO\� V\PPHWULF� DQG� &RXORPE¶V� ODZ� FDQ� EH� XVHG�� /HW� T�� DQG� T�� EH� WKH� RULJLQDO�
FKDUJHV��:H�FKRRVH�WKH�FRRUGLQDWH�V\VWHP�VR�WKH�IRUFH�RQ�T��LV�SRVLWLYH�LI�LW�LV�UHSHOOHG�E\�
T���7KHQ��WKH�IRUFH�RQ�T��LV�
) T T
U
N T T
UD
�
� �
� �
�
� �
�
ZKHUH�U� �������P��7KH�QHJDWLYH�VLJQ�LQGLFDWHV�WKDW�WKH�VSKHUHV�DWWUDFW�HDFK�RWKHU��$IWHU�
WKH�ZLUH�LV�FRQQHFWHG��WKH�VSKHUHV��EHLQJ�LGHQWLFDO��DFTXLUH�WKH�VDPH�FKDUJH��6LQFH�FKDUJH�
LV�FRQVHUYHG��WKH�WRWDO�FKDUJH�LV�WKH�VDPH�DV�LW�ZDV�RULJLQDOO\��7KLV�PHDQV�WKH�FKDUJH�RQ�
HDFK�VSKHUH�LV��T����T������7KH�IRUFH�LV�QRZ�RQH�RI�UHSXOVLRQ�DQG�LV�JLYHQ�E\�
)
U
N
T T
UE
T T T T
�
� ��
� �
�
� �
�
�
� � � �3 83 8 1 6 � �
:H�VROYH�WKH�WZR�IRUFH�HTXDWLRQV�VLPXOWDQHRXVO\�IRU�T��DQG�T���7KH�ILUVW�JLYHV�WKH�SURGXFW�
T T U )
N
D
� �
� �
�
������ ����
� �� ��
��� ��
� �
�
� �
P 1
1 P &
&� �
�1 6 1 6
DQG�WKH�VHFRQG�JLYHV�WKH�VXP�
T T U )
N
E
� �
�� � ���� � ���� � �� ��� � �P 1
���� �� 1 P &
&� � �1 6
ZKHUH�ZH�KDYH�WDNHQ�WKH�SRVLWLYH�URRW��ZKLFK�DPRXQWV�WR�DVVXPLQJ�T����T� �����7KXV��WKH�
SURGXFW�UHVXOW�SURYLGHV�WKH�UHODWLRQ�
�� �
�
�
���� �� &
T
T
ZKLFK�ZH�VXEVWLWXWH�LQWR�WKH�VXP�UHVXOW��SURGXFLQJ�
T
T�
��
�
���� �� � �� ��� �& &�
�
0XOWLSO\LQJ�E\�T��DQG�UHDUUDQJLQJ��ZH�REWDLQ�D�TXDGUDWLF�HTXDWLRQ�
T T�
� �
�
��� �� �� ��� �� �� � �& &�2 7
7KH�VROXWLRQV�DUH�
T�
� � � ��� �� �� � �� �� � ��� ��
�
� � �
�
& & &�2 7 2 7
,I� WKH� SRVLWLYH� VLJQ� LV� XVHG�� T�� � ����� � ��±�� &�� DQG� LI� WKH� QHJDWLYH� VLJQ� LV� XVHG��
�
� ���� �� �&T ���
�D��8VLQJ�T�� ��±����� ���±����T��ZLWK�T�� ������ ���±��&��ZH�JHW� �� ���� �� �&T ���
�E��,I�ZH�LQVWHDG�ZRUN�ZLWK�WKH�T�� �±����� ���±��&�URRW��WKHQ�ZH�ILQG� �� ���� �� �&T ���
1RWH�WKDW�VLQFH�WKH�VSKHUHV�DUH�LGHQWLFDO��WKH�VROXWLRQV�DUH�HVVHQWLDOO\�WKH�VDPH��RQH�VSKHUH�
RULJLQDOO\�KDG�FKDUJH�±����� ���±��&�DQG�WKH�RWKHU�KDG�FKDUJH������� ���±��&���
:KDW� LI�ZH� KDG� QRW�PDGH� WKH� DVVXPSWLRQ�� DERYH�� WKDW�T�� ��T� � �"� ,I� WKH� VLJQV� RI� WKH�
FKDUJHV�ZHUH�UHYHUVHG��VR�T����T�������� WKHQ�WKH�IRUFHV�UHPDLQ�WKH�VDPH��VR�D�FKDUJH�RI�
������ ���±��&�RQ�RQH�VSKHUH�DQG�D�FKDUJH�RI�±����� ���±��&�RQ�WKH�RWKHU�DOVR�VDWLVILHV�
WKH�FRQGLWLRQV�RI�WKH�SUREOHP��
� �
� � � �
� �
_ _� _ _ _ _ �VLQ ��
� � _ _� ��
\
T 4T T 4)
D D TD
ZKLFK� �LI�ZH�GHPDQG�)�\� ���� OHDGV� WR� � �� � �4 T ���7KH� UHVXOW� LV� LQFRQVLVWHQW�ZLWK�
WKDW�REWDLQHG� LQ�SDUW� �D���7KXV��ZH�DUH�XQDEOH� WR�FRQVWUXFW�DQ�HTXLOLEULXP�FRQILJXUDWLRQ�
ZLWK�WKLV�JHRPHWU\��ZKHUH�WKH�RQO\�IRUFHV�SUHVHQW�DUH�JLYHQ�E\�(T��������
��� )RU� HDVH� RI� SUHVHQWDWLRQ� �RI� WKH� FRPSXWDWLRQV� EHORZ��ZH� DVVXPH�4� !� �� DQG� T� �� ��
�DOWKRXJK�WKH�ILQDO�UHVXOW�GRHV�QRW�GHSHQG�RQ�WKLV�SDUWLFXODU�FKRLFH����
�D��7KH�[�FRPSRQHQW�RI�WKH�IRUFH�H[SHULHQFHG�E\�T�� �4�LV�
� � � �
� �
_ _� _ _ � _ _FRV �� �
� � � ��
[
4 4 T 4 4 T 4 T)
D DD
ZKLFK��XSRQ�UHTXLULQJ�)�[� ����OHDGV�WR� � _ _ � �4 T ��RU� � � � �����4 T
�E��7KH�\�FRPSRQHQW�RI�WKH�QHW�IRUFH�RQ�T�� �T�LV�
���7KH�IRUFH�H[SHULHQFHG�E\�T��LV�
� � � � � �
� �� �� �� � ��
�
_ __ _ _ __ _ _ __ _� Ö Ö Ö ÖM �FRV�� L VLQ �� M� L
� � � �
T T T T T T) ) ) )
D DD
�D��7KHUHIRUH��WKH�[�FRPSRQHQW�RI�WKH�UHVXOWDQW�IRUFH�RQ�T��LV�
��
� � �� �
� �� �
�
� ��� �� &_ _ _ _ �_ _ ���� �� 1 P & � ����1�
� �������P�� � � �[
T T) T
D
�E��6LPLODUO\��WKH�\�FRPSRQHQW�RI�WKH�QHW�IRUFH�RQ�T��LV�
��
� � �� �
� �� �
�
� ��� �� &_ _ _ _ �_ _ ���� �� 1 P & � �����1�
� �������P�� � � �\
T T) T
D
�����D��7KH�LQGLYLGXDO�IRUFH�PDJQLWXGHV��DFWLQJ�RQ�4��DUH��E\�(T��������
� �
� �
� �
� �
� �� � � �
T 4 T 4
D D D D
ZKLFK�OHDGV�WR�_T�_� �����_T�_��6LQFH�4�LV�ORFDWHG�EHWZHHQ�T��DQG�T���ZH�FRQFOXGH�T��DQG�T�
DUH�OLNH�VLJQ��&RQVHTXHQWO\��T��T�� ������
�E��1RZ�ZH�KDYH�
� �
� �
� �
� �
� �� � � � � �
T 4 T 4
D D D D
ZKLFK�\LHOGV� _T�_� ���� _T�_��1RZ��4� LV�QRW� ORFDWHG�EHWZHHQ�T��DQG�T���RQH�RI� WKHP�PXVW�
SXVK�DQG�WKH�RWKHU�PXVW�SXOO��7KXV��WKH\�DUH�XQOLNH�VLJQ��VR�T��T�� �±����
� � � �
� �� �� � �
���� ��
�T T T T) ) ) N N
// /
:H� QRWH� WKDW� HDFK� WHUP� H[KLELWV� WKH� SURSHU� VLJQ� �SRVLWLYH� IRU� ULJKWZDUG�� QHJDWLYH� IRU�
OHIWZDUG�� IRU� DOO� SRVVLEOH� VLJQV� RI� WKH� FKDUJHV�� )RU� H[DPSOH�� WKH� ILUVW� WHUP� �WKH� IRUFH�
H[HUWHG� RQ� T�� E\� T��� LV� QHJDWLYH� LI� WKH\� DUH� XQOLNH� FKDUJHV�� LQGLFDWLQJ� WKDW� T�� LV� EHLQJ�
SXOOHG�WRZDUG�T���DQG�LW�LV�SRVLWLYH�LI�WKH\�DUH�OLNH�FKDUJHV��VR�T��ZRXOG�EH�UHSHOOHG�IURP�
T����6HWWLQJ�WKH�QHW�IRUFH�HTXDO�WR�]HUR�/�� �/���DQG�FDQFHOLQJ�N��T��DQG�/���OHDGV�WR�
� �
�
�
���� �� �����
����
T TT
T
����:LWK�ULJKWZDUGV�SRVLWLYH��WKH�QHW�IRUFH�RQ�T��LV�
����$V�D�UHVXOW�RI�WKH�ILUVW�DFWLRQ��ERWK�VSKHUH�:�DQG�VSKHUH�$�SRVVHVV�FKDUJH��� T$���ZKHUH�
T$�LV�WKH�LQLWLDO�FKDUJH�RI�VSKHUH�$���$V�D�UHVXOW�RI�WKH�VHFRQG�DFWLRQ��VSKHUH�:�KDV�FKDUJH�
� ��
� �
$T H ���
$V�D�UHVXOW�RI�WKH�ILQDO�DFWLRQ��VSKHUH�:�QRZ�KDV�FKDUJH�HTXDO�WR�
� � �� ��
� � �
$T H H ���
6HWWLQJ� WKLV� ILQDO� H[SUHVVLRQ� HTXDO� WR� ���H� DV� UHTXLUHG� E\� WKH� SUREOHP� OHDGV� �DIWHU� D�
FRXSOH�RI�DOJHEUD�VWHSV��WR�WKH�DQVZHU��T$� ����H��
�����D��(T�������JLYHV�
��
� � �� �
�� ��
���� �� &
���� �� 1 P & ����1�
����P
T T) N
G
�E��2Q�WKH�ULJKW��D�IRUFH�GLDJUDP�LV�VKRZQ�DV�ZHOO�DV�RXU�FKRLFH�RI�\
D[LV��WKH�GDVKHG�OLQH���
7KH�\�D[LV�LV�PHDQW�WR�ELVHFW�WKH�OLQH�EHWZHHQ�T��DQG�T��LQ�RUGHU�WR�
PDNH� XVH� RI� WKH� V\PPHWU\� LQ� WKH� SUREOHP� �HTXLODWHUDO� WULDQJOH� RI�
VLGH� OHQJWK�G�� HTXDO�PDJQLWXGH�FKDUJHV�T�� �T�� �T�� �T���:H�VHH�
WKDW�WKH�UHVXOWDQW�IRUFH�LV�DORQJ�WKLV�V\PPHWU\�D[LV��DQG�ZH�REWDLQ����
� � �
��
���� �� &
� FRV�� � ���� �� 1 P & FRV�� ����1
����P\
T) N
G
��
���� �D�� $FFRUGLQJ� WR� WKH� JUDSK�� ZKHQ� T�� LV� YHU\� FORVH� WR� T�� �DW� ZKLFK� SRLQW� ZH� FDQ�
FRQVLGHU� WKH�IRUFH�H[HUWHG�E\�SDUWLFOH���RQ���WR�GRPLQDWH�� WKHUH�LV�D��ODUJH��IRUFH�LQ�WKH�
SRVLWLYH�[�GLUHFWLRQ���7KLV�LV�D�UHSXOVLYH�IRUFH��WKHQ��VR�ZH�FRQFOXGH�T��KDV�WKH�VDPH�VLJQ�
DV�T����7KXV��T��LV�D�SRVLWLYH�YDOXHG�FKDUJH��
�E��6LQFH� WKH�JUDSK�FURVVHV�]HUR�DQG�SDUWLFOH��� LV�EHWZHHQ� WKH�RWKHUV��T��PXVW�KDYH� WKH�
VDPH�VLJQ�DV�T���ZKLFK�PHDQV�LW�LV�DOVR�SRVLWLYH�YDOXHG���:H�QRWH�WKDW�LW�FURVVHV�]HUR�DW�U
 �������P��ZKLFK�LV�D�GLVWDQFH�G� �������P�IURP�T�����8VLQJ�&RXORPE¶V�ODZ�DW�WKDW�SRLQW��
ZH�KDYH�
� �
� � � �
� � � �� �
� �
� � ������P ���
� � ������P
T T T T GT T T T
U G U
��
RU�T��T�� ������
�����D��7KHUH�LV�QR�HTXLOLEULXP�SRVLWLRQ�IRU�T� EHWZHHQ�WKH�WZR�IL[HG�FKDUJHV��EHFDXVH�LW�LV�
EHLQJ�SXOOHG�E\�RQH�DQG�SXVKHG�E\�WKH�RWKHU��VLQFH�T��DQG�T��KDYH�GLIIHUHQW�VLJQV���LQ�WKLV�
UHJLRQ�WKLV�PHDQV�WKH�WZR�IRUFH�DUURZV�RQ�T��DUH�LQ�WKH�VDPH�GLUHFWLRQ�DQG�FDQQRW�FDQFHO���
,W�VKRXOG�DOVR�EH�FOHDU�WKDW�RII�D[LV��ZLWK�WKH�D[LV�GHILQHG�DV�WKDW�ZKLFK�SDVVHV�WKURXJK�WKH�
WZR�IL[HG�FKDUJHV��WKHUH�DUH�QR�HTXLOLEULXP�SRVLWLRQV��2Q�WKH�VHPL�LQILQLWH�UHJLRQ�RI�WKH�
D[LV�ZKLFK� LV� QHDUHVW�T�� DQG� IXUWKHVW� IURP�T�� DQ� HTXLOLEULXP� SRVLWLRQ� IRU�T�� FDQQRW� EH�
IRXQG�EHFDXVH�_T�_���_T�_�DQG�WKH�PDJQLWXGH�RI�IRUFH�H[HUWHG�E\�T��LV�HYHU\ZKHUH��LQ�WKDW�
UHJLRQ�� VWURQJHU� WKDQ� WKDW� H[HUWHG�E\�T�� RQ�T���7KXV��ZH�PXVW� ORRN� LQ� WKH� VHPL�LQILQLWH�
UHJLRQ�RI�WKH�D[LV�ZKLFK�LV�QHDUHVW�T��DQG�IXUWKHVW�IURP�T���ZKHUH�WKH�QHW�IRUFH�RQ�T��KDV�
PDJQLWXGH�
� � � �
��
� �
T T T T
N N
/ / /
ZLWK�/� ����FP�DQG� �/ �LV DVVXPHG�WR�EH�SRVLWLYH��:H�VHW�WKLV�HTXDO�WR�]HUR��DV�UHTXLUHG�E\�
WKH�SUREOHP��DQG�FDQFHO�N�DQG�T���7KXV��ZH�REWDLQ�
�
� � � �
��
� � ��
��� &� ���
��� &
T T / / T
/ / T/ /
ZKLFK�\LHOGV��DIWHU�WDNLQJ�WKH�VTXDUH�URRW��
�
�
�
���FP� ��FP
� � � �
/ / //
/
IRU�WKH�GLVWDQFH�EHWZHHQ�T��DQG�T�� 7KDW�LV�� �T �VKRXOG�EH�SODFHG�DW� ���FP[ �DORQJ�WKH�
[�D[LV��
�E��$V�VWDWHG�DERYH��\� �����
����6LQFH� WKH� IRUFHV� LQYROYHG�DUH�SURSRUWLRQDO� WR�T��ZH�VHH� WKDW� WKH�HVVHQWLDO�GLIIHUHQFH�
EHWZHHQ�WKH�WZR�VLWXDWLRQV�LV�)D T%���T&���ZKHQ�WKRVH�WZR�FKDUJHV�DUH�RQ�WKH�VDPH�VLGH��
YHUVXV�)E T%���T&���ZKHQ�WKH\�DUH�RQ�RSSRVLWH�VLGHV����6HWWLQJ�XS�UDWLRV��ZH�KDYH�
)D
�)E�� ��
T%���T&
��T%���T&
��
��
� ������ �� 1
����� �� 1 � �
& %
& %
T T
T T
���
$IWHU�QRWLQJ�WKDW�WKH�UDWLR�RQ�WKH�OHIW�KDQG�VLGH�LV�YHU\�FORVH�WR�±����WKHQ��DIWHU�D�FRXSOH�RI�
DOJHEUD�VWHSV��ZH�DUH�OHG�WR�
� � � ������
� � �
&
%
T
T
�����D��7KH�GLVWDQFH�EHWZHHQ�T��DQG�T��LV�
� � � �
�� � � � � ������P ������P ������P ������P ������P�U [ [ \ \
7KH�PDJQLWXGH�RI�WKH�IRUFH�H[HUWHG�E\�T��RQ�T��LV�
� � � � �
� �
�� � �
��
���� �� 1 P & ��� �� & ��� �� &_ _ �� �1�
�������P�
T T) N
U
�E��7KH�YHFWRU� )�� ��LV�GLUHFWHG�WRZDUGV�T��DQG�PDNHV�DQ�DQJOH� �ZLWK�WKH��[�D[LV��ZKHUH�
� �� �
� �
����FP ����FPWDQ WDQ ���� �� �
����FP ����FP
\ \
[ [
�F��/HW�WKH�WKLUG�FKDUJH�EH�ORFDWHG�DW��[���\����D�GLVWDQFH�U�IURP�T���:H�QRWH�WKDW�T���T��DQG�
T��PXVW� EH� FROOLQHDU�� RWKHUZLVH�� DQ� HTXLOLEULXP�SRVLWLRQ� IRU� DQ\�RQH�RI� WKHP�ZRXOG�EH�
LPSRVVLEOH�WR�ILQG��)XUWKHUPRUH��ZH�FDQQRW�SODFH�T��RQ�WKH�VDPH�VLGH�RI�T��ZKHUH�ZH�DOVR�
ILQG�T���VLQFH�LQ�WKDW�UHJLRQ�ERWK�IRUFHV��H[HUWHG�RQ�T��E\�T��DQG�T���ZRXOG�EH�LQ�WKH�VDPH�
GLUHFWLRQ��VLQFH�T��LV�DWWUDFWHG�WR�ERWK�RI�WKHP���7KXV��LQ�WHUPV�RI�WKH�DQJOH�IRXQG�LQ�SDUW�
�D���ZH�KDYH�[�� �[��±�U�FRV �DQG�\�� �\��±�U�VLQ ��ZKLFK�PHDQV�\��!�\��VLQFH� �LV�QHJDWLYH���
7KH�PDJQLWXGH�RI�IRUFH�H[HUWHG�RQ�T��E\�T��LV� ��� � �_ _) N T T U ��ZKLFK�PXVW�HTXDO�WKDW�RI�
WKH�IRUFH�H[HUWHG�RQ�LW�E\�T���IRXQG�LQ�SDUW��D����7KHUHIRUH��
� � � � �
��� �
�� �
������P �����FP��
T T T T TN N U U
U U T
&RQVHTXHQWO\��[�� �[��±�U�FRV � �±����FP�±�������FP��FRV�±��ƒ�� �±����FP��
�G��DQG�\�� ��\��±�U�VLQ � �����FP�±�������FP��VLQ�±��ƒ�� �����FP��
���� �D�� )RU� WKH� QHW� IRUFH� WR� EH� LQ� WKH� �[� GLUHFWLRQ�� WKH� \� FRPSRQHQWV� RI� WKH� LQGLYLGXDO�
IRUFHV� PXVW� FDQFHO�� 7KH� DQJOH� RI� WKH� IRUFH� H[HUWHG� E\� WKH� T�� � ��� &� FKDUJH� RQ�
� ��T & �LV���ƒ��DQG�WKH�DQJOH�RI�IRUFH�H[HUWHG�RQ�T��E\�4�LV�DW�± �ZKHUH�
� ����FPWDQ ���� �
����FP
7KHUHIRUH��FDQFHOODWLRQ�RI�\�FRPSRQHQWV�UHTXLUHV�
� � �
� �
� �
_ _VLQ �� VLQ
���� ��P �������P� �������P�
T T 4 TN N
IURP�ZKLFK�ZH�REWDLQ� _4_� ���� &��&KDUJH�4� LV� ³SXOOLQJ´�RQ�T��� VR� �VLQFH�T��!����ZH�
FRQFOXGH�4� �±��� &��
�E��1RZ��ZH�UHTXLUH�WKDW�WKH�[�FRPSRQHQWV�FDQFHO��DQG�ZH�QRWH�WKDW�LQ�WKLV�FDVH��WKH�DQJOH�
RI�IRUFH�RQ�T��H[HUWHG�E\�4�LV�� ��LW�LV�UHSXOVLYH��DQG�4�LV�SRVLWLYH�YDOXHG���7KHUHIRUH��
� � �
� �
� �
FRV �� FRV
���� ��P �������P� �������P�
T T 4TN N
IURP�ZKLFK�ZH�REWDLQ�4� ������ &� ��� & ��
�
�
� �
�
� ���� �
�T
TT T)
[ /
7KH�VLJQV�DUH�FKRVHQ�VR�WKDW�D�QHJDWLYH�IRUFH�YDOXH�ZRXOG�FDXVH�T�WR�PRYH�OHIWZDUG��:H�
UHTXLUH�)T� ���DQG�VROYH�IRU�T���
�
�
� �
� � �� �� �����
� �
TT[T T
/ T
ZKHUH� [� � /��� LV� XVHG�� 1RWH� WKDW� ZH� PD\� HDVLO\� YHULI\� WKDW� WKH� IRUFH� RQ� ����T� DOVR�
YDQLVKHV��
�� � � �
�
� �� � � � �
� � �
� � ��� � � � � � � �
� � � � �T
TTTT T T T)
/ / / / // [
��
���� �D�� ,I� WKH� V\VWHP�RI� WKUHH� FKDUJHV� LV� WR�EH� LQ� HTXLOLEULXP�� WKH� IRUFH�RQ�HDFK�FKDUJH�
PXVW�EH�]HUR��7KH�WKLUG�FKDUJH�T��PXVW�OLH�EHWZHHQ�WKH�RWKHU�WZR�RU�HOVH�WKH�IRUFHV�DFWLQJ�
RQ� LW� GXH� WR� WKH� RWKHU� FKDUJHV� ZRXOG� EH� LQ� WKH� VDPH� GLUHFWLRQ� DQG� T�� FRXOG� QRW� EH� LQ�
HTXLOLEULXP��6XSSRVH�T��LV�DW�D�GLVWDQFH�[�IURP�T��DQG�/�±�[�IURP�����T��7KH�IRUFH�DFWLQJ�
RQ�LW�LV�WKHQ�JLYHQ�E\�
� �
� ��
�
��
�
TT TT)
[ / [
ZKHUH�WKH�SRVLWLYH�GLUHFWLRQ�LV�ULJKWZDUG��:H�UHTXLUH�)�� ���DQG�VROYH�IRU�[��&DQFHOLQJ�
FRPPRQ�IDFWRUV�\LHOGV���[�� ����/�±�[���DQG�WDNLQJ�WKH�VTXDUH�URRW�\LHOGV���[� ����/�±�[���
7KH�VROXWLRQ�LV�[� �/����:LWK�/� ������FP��ZH�KDYH�[� ������FP��
�E��6LPLODUO\��WKH�\�FRRUGLQDWH�RI�T��LV�\� �����
�F��7KH�IRUFH�RQ�T�LV�
���� �D��:H� QRWH� WKDW� FRV���ž�� ��� ���� VR� WKDW� WKH� GDVKHG� OLQH� GLVWDQFH� LQ� WKH� ILJXUH� LV�
� � �U G ���:H�QHW�IRUFH�RQ�T��GXH�WR�WKH�WZR�FKDUJHV�T��DQG�T���ZLWK�_T�_� �_T�_� ������
�� ���&��RQ�WKH�\�D[LV�KDV�PDJQLWXGH�
� � � �
� �
� �
_ _ � � _ _� FRV��� �
� ��
T T T T
U G
��
7KLV�PXVW�EH�VHW�HTXDO�WR�WKH�PDJQLWXGH�RI�WKH�IRUFH�H[HUWHG�RQ�T��E\�T�� ������ ��� ���&�
 ������_T�_�LQ�RUGHU�WKDW�LWV�QHW�IRUFH�EH�]HUR��
� � � �
� �
� �
� � _ _ _ _
�� � � �
T T T T
G ' G
'� �G � �
� �
�� �� ��������G���
*LYHQ�G� ������FP��WKLV�WKHQ�OHDGV�WR�'� ������FP��
�E��$V�WKH�DQJOH�GHFUHDVHV��LWV�FRVLQH�LQFUHDVHV��UHVXOWLQJ�LQ�D�ODUJHU�FRQWULEXWLRQ�IURP�WKH�
FKDUJHV�RQ�WKH�\�D[LV���7R�RIIVHW�WKLV��WKH�IRUFH�H[HUWHG�E\�T��PXVW�EH�PDGH�VWURQJHU��VR�
WKDW� LW� PXVW� EH� EURXJKW� FORVHU� WR� T�� �NHHS� LQ� PLQG� WKDW� &RXORPE¶V� ODZ� LV� LQYHUVHO\
SURSRUWLRQDO�WR�GLVWDQFH�VTXDUHG����7KXV��'�PXVW�EH�GHFUHDVHG��
:H�QRWH� WKDW�� GXH� WR� WKH� V\PPHWU\� LQ� WKH�SUREOHP�� WKHUH� LV� QR�\� FRPSRQHQW� WR� WKH�QHW�
IRUFH�RQ�WKH�WKLUG�SDUWLFOH���7KXV��)�UHSUHVHQWV�WKH�PDJQLWXGH�RI�IRUFH�H[HUWHG�E\�T��RU�T�
RQ�T���/HW�H� ������� ��� ���&��WKHQ�T�� �T�� ���H�DQG�T�� ����H�DQG�ZH�KDYH�
)QHW� �)�FRV ��� ��
���H���H�
� R��[����G��
[
[����G�
�� ��
�H��[
R��[����G������
����
�D��7R�ILQG�ZKHUH�WKH�IRUFH�LV�DW�DQ�H[WUHPXP��ZH�FDQ�VHW�WKH�GHULYDWLYH�RI�WKLV�H[SUHVVLRQ�
HTXDO�WR�]HUR�DQG�VROYH�IRU�[��EXW�LW�LV�JRRG�LQ�DQ\�FDVH�WR�JUDSK�WKH�IXQFWLRQ�IRU�D�IXOOHU�
XQGHUVWDQGLQJ�RI�LWV�EHKDYLRU�±�DQG�DV�D�TXLFN�ZD\�WR�VHH�ZKHWKHU�DQ�H[WUHPXP�SRLQW�LV�D�
PD[LPXP�RU�D�PLPLQXP���,Q�WKLV�ZD\��ZH�ILQG�WKDW�WKH�YDOXH�FRPLQJ�IURP�WKH�GHULYDWLYH�SURFHGXUH� LV� D� PD[LPXP� �DQG� ZLOO� EH� SUHVHQWHG� LQ� SDUW� �E��� DQG� WKDW� WKH�PLQLPXP� LV�
IRXQG�DW�WKH�ORZHU�OLPLW�RI�WKH�LQWHUYDO���7KXV��WKH�QHW�IRUFH�LV�IRXQG�WR�EH�]HUR�DW�[� ����
ZKLFK�LV�WKH�VPDOOHVW�YDOXH�RI�WKH�QHW�IRUFH�LQ�WKH�LQWHUYDO�����P� [ �����
�E��7KH�PD[LPXP�LV�IRXQG�WR�EH�DW�[� �G� ���RU�URXJKO\����FP��
�F��7KH�YDOXH�RI�WKH�QHW�IRUFH�DW�[� ���LV�)QHW� ����
�G��7KH�YDOXH�RI�WKH�QHW�IRUFH�DW�[� �G� ���LV�)QHW� ���� �� ���1��
����,I� �LV�WKH�DQJOH�EHWZHHQ�WKH�IRUFH�DQG�WKH�[�D[LV��WKHQ��
FRV �� ��
[
[����G�
����
�
� �� �
����ó
�� �ó � ������
2I�FRXUVH�� WKLV�FRXOG�DOVR�EH� ILJXUHG�DV� LOOXVWUDWHG� LQ�SDUW� �D��� ORRNLQJ�DW� WKH�PD[LPXP�
IRUFH�UDWLR�E\�LWVHOI�DQG�VROYLQJ��RU� ORRNLQJ�DW� WKH�PLQLPXP�IRUFH�UDWLR��ô��DW� � ����ž�
DQG�VROYLQJ�IRU� ��
����:H�QRWH�WKDW�WKH�SUREOHP�LV�H[DPLQLQJ�WKH�IRUFH�RQ�FKDUJH�$��VR�WKDW�WKH�UHVSHFWLYH�
GLVWDQFHV��LQYROYHG�LQ�WKH�&RXORPE�IRUFH�H[SUHVVLRQV��EHWZHHQ�%�DQG�$��DQG�EHWZHHQ�&
DQG�$�� GR� QRW� FKDQJH� DV� SDUWLFOH�%� LV�PRYHG� DORQJ� LWV� FLUFXODU� SDWK�� �:H� IRFXV� RQ� WKH�
HQGSRLQWV�� � ��ž�DQG����ž��RI�HDFK�JUDSK��VLQFH�WKH\�UHSUHVHQW�FDVHV�ZKHUH�WKH�IRUFHV��RQ�
$��GXH�WR�%�DQG�&�DUH�HLWKHU�SDUDOOHO�RU�DQWLSDUDOOHO��\LHOGLQJ�PD[LPXP�RU�PLQLPXP�IRUFH�
PDJQLWXGHV�� UHVSHFWLYHO\��� � :H� QRWH�� WRR�� WKDW� VLQFH� &RXORPE¶V� ODZ� LV� LQYHUVHO\�
SURSRUWLRQDO�WR�Uð�WKHQ�WKH��LI��VD\��WKH�FKDUJHV�ZHUH�DOO�WKH�VDPH��IRUFH�GXH�WR�&�ZRXOG�EH�
RQH�IRXUWK�DV�ELJ�DV�WKDW�GXH�WR�%��VLQFH�&�LV�WZLFH�DV�IDU�DZD\�IURP�$����7KH�FKDUJHV��LW�
WXUQV�RXW��DUH�QRW�WKH�VDPH��VR�WKHUH�LV�DOVR�D�IDFWRU�RI�WKH�FKDUJH�UDWLR� ��WKH�FKDUJH�RI�&
GLYLGHG�E\�WKH�FKDUJH�RI�%���DV�ZHOO�DV�WKH�DIRUHPHQWLRQHG�ó�IDFWRU����7KDW�LV��WKH�IRUFH�
H[HUWHG�E\�&�LV��E\�&RXORPE¶V�ODZ�HTXDO�WR�“ó �PXOWLSOLHG�E\�WKH�IRUFH�H[HUWHG�E\�%��
�D��7KH�PD[LPXP�IRUFH�LV��)��DQG�RFFXUV�ZKHQ� � ����ž���%�LV�WR�WKH�OHIW�RI�$��ZKLOH�&�LV�
WKH�ULJKW�RI�$����:H�FKRRVH�WKH�PLQXV�VLJQ�DQG�ZULWH��
��)�� ���� �ó ��)� � �±�����
2QH�ZD\�WR�WKLQN�RI�WKH�PLQXV�VLJQ�FKRLFH�LV�FRV����ž�� �±����7KLV�LV�FHUWDLQO\�FRQVLVWHQW�
ZLWK�WKH�PLQLPXP�IRUFH�UDWLR��]HUR��DW� � ��ž�VLQFH�WKDW�ZRXOG�DOVR�LPSO\�
�� �����ó � �±�����
�E�� 7KH� UDWLR� RI� PD[LPXP� WR� PLQLPXP� IRUFHV� LV� ���������� � ���� LQ� WKLV� FDVH�� ZKLFK�
LPSOLHV�
����7KH�FKDUJH�GT�ZLWKLQ�D�WKLQ�VKHOO�RI�WKLFNQHVV�GU�LV� GT G9 $GU �ZKHUH�$� �� U���
7KXV��ZLWK� � �E�U��ZH�KDYH�
T GT E U GU E U U
U
UII � �
�
�
�
�
�
�2 7� �
:LWK�E� ����� &�P���U�� ������P�DQG�U�� ������P��ZH�REWDLQ�T� ������� &� ����� ��� ��&��
����7KH�PDJQLWXGH�RI�WKH�IRUFH�LV�
) N H
U
�
��
�
��
�
�
�
�� �
�� �
�� �� ��
��� ��
� �� ��
� �� ���
�
�
�1 P
&
&
P
1�
�
�
2 7
2 7
�����D��7KH�PDJQLWXGH�RI�WKH�IRUFH�EHWZHHQ�WKH��SRVLWLYH��LRQV�LV�JLYHQ�E\�
)
T T
U
N T
U
1 61 6
� �
�
�
�
ZKHUH�T�LV�WKH�FKDUJH�RQ�HLWKHU�RI�WKHP�DQG�U�LV�WKH�GLVWDQFH�EHWZHHQ�WKHP��:H�VROYH�IRU�
WKH�FKDUJH��
T U )
N
�� �� �� ��
� �� ��
�� ����
�
�
��� �
�
�P 1
1 P &
&�� �2 7
�E��/HW�Q�EH�WKH�QXPEHU�RI�HOHFWURQV�PLVVLQJ�IURP�HDFK�LRQ��7KHQ��QH� �T��RU�
�
��
��� �� & ��
��� �� &
TQ
H
����.HHSLQJ�LQ�PLQG�WKDW�DQ�$PSHUH�LV�D�&RXORPE�SHU�VHFRQG����$� ���&�V���DQG�WKDW�D�
PLQXWH�LV����VHFRQGV��WKH�FKDUJH��LQ�DEVROXWH�YDOXH��WKDW�SDVVHV�WKURXJK�WKH�FKHVW�LV�
_�T�_�� ���������&�V���������V��� ������&���
7KLV�FKDUJH�FRQVLVWV�RI�Q�HOHFWURQV��HDFK�RI�ZKLFK�KDV�DQ�DEVROXWH�YDOXH�RI�FKDUJH�HTXDO�
WR�H����7KXV��
Q� 
_�T�_
H �� ��
�����&
�����[�������&�� ������ ��
����
����(T���������LQ�DEVROXWH�YDOXH��JLYHV�
Q
T
H
�� ��
�� ��
� � ��
�
��
���
�
� �&
&
�����D��(T�������JLYHV�
)
� �� �� ��� ��
��� ��
� �� ��
� �� �
� �
��
� �
�
�
1 P & &
P
1�
� �2 72 7
2 7
�E��,I�Q�LV�WKH�QXPEHU�RI�H[FHVV�HOHFWURQV��RI�FKDUJH�±H�HDFK��RQ�HDFK�GURS�WKHQ�
Q T
H
��� ��
��� ��
���
��
��
�
�
�&
&
����7KH�XQLW�$PSHUH�LV�GLVFXVVHG�LQ�†������7KH�SURWRQ�IOX[�LV�JLYHQ�DV������SURWRQV�SHU�
VTXDUH�PHWHU� SHU� VHFRQG��ZKHUH� HDFK� SURWRQ� SURYLGHV� D� FKDUJH� RI�T� � �H�� 7KH� FXUUHQW�
WKURXJK�WKH�VSKHULFDO�DUHD�� 5�� �� ������� �����P��� ����� ������P��ZRXOG�EH�
L ���
�
���� �� ���� �� �� ����
�� �
�
��� � � �P SURWRQV
V P
& SURWRQ $2 7 2 7
����6LQFH�WKH�JUDSK�FURVVHV�]HUR��T��PXVW�EH�SRVLWLYH�YDOXHG��T�� ������H���:H�QRWH�WKDW�LW�
FURVVHV�]HUR�DW�U � ������P���1RZ�WKH�DV\PSWRWLF�YDOXH�RI�WKH�IRUFH�\LHOGV�WKH�PDJQLWXGH�
DQG�VLJQ�RI�T���
T��T�
� R�U�
�� �) T�� ��
����[������
N�T� U
� ������� �� ���&�� ����H ���
����7KH�YROXPH�RI�����FP��FRUUHVSRQGV�WR�D�PDVV�RI�����J�VLQFH�WKH�GHQVLW\�RI�ZDWHU�LV�
����J�FP���7KLV�PDVV�FRUUHVSRQGV�WR�������� ����PROHV�VLQFH�WKH�PRODU�PDVV�RI�ZDWHU�LV�
����7KHUH�DUH�WHQ�SURWRQV��HDFK�ZLWK�FKDUJH�T� ��H��LQ�HDFK�PROHFXOH�RI�+�2��VR�
�� �� ��� �� ���� �� �� ���� �� & ��� �� &�$4 1 T
ZLWK�WKH�YDOXHV�RI�WKH�FKDUJHV��VWDWHG�LQ�WKH�SUREOHP��SOXJJHG�LQ���)LQGLQJ�WKH�YDOXH�RI�[
ZKLFK�PLQLPL]HV�WKLV�H[SUHVVLRQ�OHDGV�WR�[� �ó�/���7KXV��[� ������FP��
�E��6XEVWLWXWLQJ�[� �ó�/�EDFN�LQWR�WKH�H[SUHVVLRQ�IRU�WKH�QHW�IRUFH�PDJQLWXGH�DQG�XVLQJ�
WKH�VWDQGDUG�YDOXH�IRU�H�OHDGV�WR�)QHW� ����� �� ���1��
�����D��/HW�[�EH�WKH�GLVWDQFH�EHWZHHQ�SDUWLFOH���DQG�SDUWLFOH�����7KXV��WKH�GLVWDQFH�EHWZHHQ�
SDUWLFOH���DQG�SDUWLFOH���LV�/�±�[��%RWK�SDUWLFOHV�H[HUW�OHIWZDUG�IRUFHV�RQ�T���VR�ORQJ�DV�LW�LV�
RQ�WKH�OLQH�EHWZHHQ�WKHP���VR�WKH�PDJQLWXGH�RI�WKH�QHW�IRUFH�RQ�T��LV�
)QHW� ��_)����� _�����_)����� _�� ��
_T��T�_
� R�[�
����
_T��T�_
� R��/ [��
��� ��
H�
R�
�
�[����
��
�/� ��[��
�����D��:H�QRWH�WKDW�WDQ��� �� ��� ���� �,Q�WKH�LQLWLDO��KLJKO\�V\PPHWULFDO��FRQILJXUDWLRQ��
WKH�QHW�IRUFH�RQ�WKH�FHQWUDO�EHDG�LV�LQ�WKH�±\�GLUHFWLRQ�DQG�KDV�PDJQLWXGH��)�ZKHUH�)�LV�
WKH�&RXORPE¶V�ODZ�IRUFH�RI�RQH�EHDG�RQ�DQRWKHU�DW�GLVWDQFH�G� ����FP���7KLV�LV�GXH�WR�WKH�
IDFW�WKDW�WKH�IRUFHV�H[HUWHG�RQ�WKH�FHQWUDO�EHDG��LQ�WKH�LQLWLDO�VLWXDWLRQ��E\�WKH�EHDGV�RQ�WKH�
[� D[LV� FDQFHO� HDFK� RWKHU�� DOVR�� WKH� IRUFH� H[HUWHG� ³GRZQZDUG´� E\� EHDG� �� RQ� WKH� FHQWUDO�
EHDG�LV�IRXU�WLPHV�ODUJHU�WKDQ�WKH�³XSZDUG´�IRUFH�H[HUWHG�E\�EHDG�����7KLV�QHW�IRUFH�DORQJ�
WKH�\� D[LV�GRHV�QRW� FKDQJH�DV�EHDG��� LV�QRZ�PRYHG�� WKRXJK� WKHUH� LV�QRZ�D�QRQ]HUR�[�
FRPSRQHQW�)[����7KH�FRPSRQHQWV�DUH�QRZ�UHODWHG�E\�
����������������������� � ���������WDQ��� ��� ��
)[
�)\
�
�
���� ���
)[
�)
ZKLFK�LPSOLHV�)[� � ��)���1RZ��EHDG���H[HUWV�D�³OHIWZDUG´�IRUFH�RI�PDJQLWXGH�)�RQ�WKH�
FHQWUDO�EHDG��ZKLOH�EHDG���H[HUWV�D�³ULJKWZDUG´�IRUFH�RI�PDJQLWXGH�) ���7KHUHIRUH��
) )� � ��)������� ) �� ��� ��������)���
7KH�IDFW�WKDW�&RXORPE¶V�ODZ�GHSHQGV�LQYHUVHO\�RQ�GLVWDQFH�VTXDUHG�WKHQ�LPSOLHV�
U�� ���
G�
�����
U� ���
G
�����
�� � ���FP ���FP
����� �
�����FP�
ZKHUH� U� LV� WKH� GLVWDQFH� EHWZHHQ� EHDG� �� DQG� WKH� FHQWUDO� EHDG�� � 7KLV� FRUUHVSRQGV� WR�
�����FP �[
�E�� 7R� UHJDLQ� WKH� FRQGLWLRQ� RI� KLJK� V\PPHWU\� �LQ� SDUWLFXODU�� WKH� FDQFHOODWLRQ� RI� [�
FRPSRQHQWV�� EHDG� �� PXVW� EH� PRYHG� FORVHU� WR� WKH� FHQWUDO� EHDG� VR� WKDW� LW�� WRR�� LV� WKH�
GLVWDQFH�U��DV�FDOFXODWHG�LQ�SDUW�D���DZD\�IURP�LW��
)URP� V\PPHWU\��ZH� VHH� WKDW� WKHUH� LV� QR� QHW� IRUFH� LQ� WKH� YHUWLFDO� GLUHFWLRQ� RQ�T�� � ±H
VLWWLQJ�DW�D�GLVWDQFH�5� WR� WKH� OHIW�RI� WKH�FRRUGLQDWH�RULJLQ�� �:H�QRWH� WKDW� WKH�QHW�[� IRUFH�
FDXVHG�E\�T��DQG�T��RQ�WKH�\�D[LV�ZLOO�KDYH�D�PDJQLWXGH�HTXDO�WR�
�
� � �
� � �
� FRV � FRV� FRV
� � � � FRV � �
TH TH TH
U 5 5
���
&RQVHTXHQWO\�� WR� DFKLHYH� D� ]HUR� QHW� IRUFH� DORQJ� WKH�[� D[LV�� WKH� DERYH� H[SUHVVLRQ�PXVW�
HTXDO�WKH�PDJQLWXGH�RI�WKH�UHSXOVLYH�IRUFH�H[HUWHG�RQ�T��E\�T�� �±H��7KXV��
� �
� � �
� �
� FRV
� � �FRV
TH H HT
5 5
��
%HORZ�ZH�SORW�T�H�DV�D�IXQFWLRQ�RI�WKH�DQJOH��LQ�GHJUHHV����
7KH�JUDSK�VXJJHVWV�WKDW�T�H�����IRU������ž��URXJKO\���:H�FDQ�EH�PRUH�SUHFLVH�E\�VROYLQJ�
WKH�DERYH�HTXDWLRQ���7KH�UHTXLUHPHQW��WKDW�T ��H��OHDGV�WR��
� ���
�� FRV
�FRV ����
H H
ZKLFK�\LHOGV� �����ž�� �7KH�SUREOHP�DVNV� IRU�³SK\VLFDOO\�SRVVLEOH�YDOXHV�´�DQG� LW� LV�
UHDVRQDEOH�WR�VXSSRVH�WKDW�RQO\�SRVLWLYH�LQWHJHU�PXOWLSOH�YDOXHV�RI�H�DUH�DOORZHG�IRU�T���,I�
ZH�OHW�T� �QH�� IRU�Q� ���«���� WKHQ� 1�ZLOO�EH�IRXQG�E\�WDNLQJ�WKH�LQYHUVH�FRVLQH�RI� WKH�
FXEH�URRW�RI�����Q�����
����/HW�G�EH�WKH�YHUWLFDO�GLVWDQFH�IURP�WKH�FRRUGLQDWH�RULJLQ�WR�T�� � T�DQG�T�� � T�RQ�WKH�
�\� D[LV�� ZKHUH� WKH� V\PERO� T� LV� DVVXPHG� WR� EH� D� SRVLWLYH� YDOXH�� � 6LPLODUO\�� G� LV� WKH�
�SRVLWLYH��GLVWDQFH�IURP�WKH�RULJLQ�T�� � �RQ�WKH� \�D[LV���,I�ZH�WDNH�HDFK�DQJOH� �LQ�WKH�
ILJXUH�WR�EH�SRVLWLYH��WKHQ�ZH�KDYH�WDQ � �G�5�DQG�FRV � �5�U��ZKHUH�U�LV�WKH�GDVKHG�OLQH�
GLVWDQFH�VKRZQ�LQ�WKH�ILJXUH����7KH�SUREOHP�DVNV�XV�WR�FRQVLGHU� �WR�EH�D�YDULDEOH�LQ�WKH�
VHQVH�WKDW��RQFH�WKH�FKDUJHV�RQ�WKH�[�D[LV�DUH�IL[HG�LQ�SODFH��ZKLFK�GHWHUPLQHV�5���G�FDQ�
WKHQ�EH�DUUDQJHG�WR�VRPH�PXOWLSOH�RI�5��VLQFH�G� �5�WDQ ����7KH�DLP�RI�WKLV�H[SORUDWLRQ�LV�
WR�VKRZ�WKDW�LI�T�LV�ERXQGHG�WKHQ� �DQG�WKXV�G��LV�DOVR�ERXQGHG��
�D��7KH�VPDOOHVW�YDOXH�RI�DQJOH�LV� � �����ž��RU�������UDG���
�E��7KH�VHFRQG�VPDOOHVW�YDOXH�RI�DQJOH�LV� � ������ž��RU�������UDG���
�F��7KH�WKLUG�VPDOOHVW�YDOXH�RI�DQJOH�LV� � �����ž��RU�������UDG���
�����D��(YHU\�FHVLXP�LRQ�DW�D�FRUQHU�RI�WKH�FXEH�H[HUWV�D�IRUFH�RI�WKH�VDPH�PDJQLWXGH�RQ�
WKH� FKORULQH� LRQ� DW� WKH� FXEH� FHQWHU�� (DFK� IRUFH� LV� D� IRUFH� RI� DWWUDFWLRQ� DQG� LV� GLUHFWHG�
WRZDUG� WKH�FHVLXP� LRQ� WKDW� H[HUWV� LW�� DORQJ� WKH�ERG\�GLDJRQDO�RI� WKH�FXEH��:H�FDQ�SDLU�
HYHU\�FHVLXP�LRQ�ZLWK�DQRWKHU��GLDPHWULFDOO\�SRVLWLRQHG�DW�WKH�RSSRVLWH�FRUQHU�RI�WKH�FXEH��
6LQFH� WKH� WZR� LRQV� LQ� VXFK� D� SDLU� H[HUW� IRUFHV� WKDW� KDYH� WKH� VDPH� PDJQLWXGH� EXW� DUH�
RSSRVLWHO\�GLUHFWHG��WKH�WZR�IRUFHV�VXP�WR�]HUR�DQG��VLQFH�HYHU\�FHVLXP�LRQ�FDQ�EH�SDLUHG�
LQ�WKLV�ZD\��WKH�WRWDO�IRUFH�RQ�WKH�FKORULQH�LRQ�LV�]HUR��
�E�� 5DWKHU� WKDQ� UHPRYH� D� FHVLXP� LRQ�� ZH� VXSHUSRVH� FKDUJH� ±H� DW� WKH� SRVLWLRQ� RI� RQH�
FHVLXP�LRQ��7KLV�QHXWUDOL]HV�WKH�LRQ��DQG�DV�IDU�DV�WKH�HOHFWULFDO�IRUFH�RQ�WKH�FKORULQH�LRQ�
LV�FRQFHUQHG��LW�LV�HTXLYDOHQW�WR�UHPRYLQJ�WKH�LRQ��7KH�IRUFHV�RI�WKH�HLJKW�FHVLXP�LRQV�DW�
WKH� FXEH� FRUQHUV� VXP� WR� ]HUR�� VR� WKH� RQO\� IRUFH� RQ� WKH� FKORULQH� LRQ� LV� WKH� IRUFH� RI� WKH�
DGGHG�FKDUJH��
7KH� OHQJWK�RI� D�ERG\�GLDJRQDO�RI� D� FXEH� LV �D ��ZKHUH�D� LV� WKH� OHQJWK�RI�D�FXEH�HGJH��
7KXV��WKH�GLVWDQFH�IURP�WKH�FHQWHU�RI�WKH�FXEH�WR�D�FRUQHU�LVG D� �3 8 ��7KH�IRUFH�KDV�
PDJQLWXGH�
) N H
G
NH
D
�
�
�
�
� � � �� �
� �
�
� �
� �� �� ��� ��
� � � �� ��
�� ��1 6
2 72 7
1 62 7
� �
�
� �
1 P & &
P
1 �
6LQFH� ERWK� WKH� DGGHG� FKDUJH� DQG� WKH� FKORULQH� LRQ� DUH� QHJDWLYH�� WKH� IRUFH� LV� RQH� RI�
UHSXOVLRQ��7KH�FKORULQH�LRQ�LV�SXVKHG�DZD\�IURP�WKH�VLWH�RI�WKH�PLVVLQJ�FHVLXP�LRQ��
���� �D�� 6LQFH� WKH� SURWRQ� LV� SRVLWLYHO\� FKDUJHG�� WKH� HPLWWHG� SDUWLFOH�PXVW� EH� D� SRVLWURQ�������
�DV� RSSRVHG� WR� WKH� QHJDWLYHO\� FKDUJHG� HOHFWURQ�� LQ� DFFRUGDQFH� ZLWK� WKH� ODZ� RI� FKDUJH�
FRQVHUYDWLRQ��
�E�� ,Q� WKLV�FDVH�� WKH� LQLWLDO� VWDWH�KDG�]HUR�FKDUJH� �WKH�QHXWURQ� LV�QHXWUDO��� VR� WKH� VXP�RI�
FKDUJHV� LQ� WKH� ILQDO� VWDWH�PXVW�EH�]HUR�� �6LQFH� WKHUH� LV� D�SURWRQ� LQ� WKH� ILQDO� VWDWH�� WKHUH�
VKRXOG�DOVR�EH�DQ�HOHFWURQ��DV�RSSRVHG�WR�D�SRVLWURQ��VR�WKDW� T� ����
�F����1�KDV���SURWRQV����HOHFWURQV��DQG����±��� ���QHXWURQV���+�KDV���SURWRQ����HOHFWURQ��
DQG���QHXWURQV��DQG��+H�KDV���SURWRQV����HOHFWURQV��DQG���±��� ���QHXWURQV��VR�;�KDV�����
��±��� ���SURWRQV����HOHFWURQV��DQG�������±��� ���QHXWURQV��,W�PXVW�EH�FDUERQ�ZLWK�D�PRODU�
PDVV�RI������� �������&��
���� 1RQH� RI� WKH� UHDFWLRQV� JLYHQ� LQFOXGH� D� EHWD� GHFD\�� VR� WKH� QXPEHU� RI� SURWRQV�� WKH�
QXPEHU�RI� QHXWURQV�� DQG� WKH�QXPEHU�RI� HOHFWURQV� DUH� HDFK� FRQVHUYHG��$WRPLF�QXPEHUV�
�QXPEHUV�RI�SURWRQV�DQG�QXPEHUV�RI�HOHFWURQV��DQG�PRODU�PDVVHV��FRPELQHG�QXPEHUV�RI�
SURWRQV�DQG�QHXWURQV��FDQ�EH�IRXQG�LQ�$SSHQGL[�)�RI�WKH�WH[W��
�D���+�KDV���SURWRQ����HOHFWURQ��DQG���QHXWURQV�DQG��%H�KDV���SURWRQV����HOHFWURQV��DQG���±�
�� ���QHXWURQV��VR�;�KDV������� ���SURWRQV�������� ���HOHFWURQV��DQG�������±��� ���QHXWURQV��
2QH�RI�WKH�QHXWURQV�LV�IUHHG�LQ�WKH�UHDFWLRQ��;�PXVW�EH�ERURQ�ZLWK�D�PRODU�PDVV�RI�������
 ���J�PRO���%��
�E����&�KDV���SURWRQV����HOHFWURQV��DQG����±��� ���QHXWURQV�DQG��+�KDV���SURWRQ����HOHFWURQ��
DQG���QHXWURQV��VR�;�KDV������� ���SURWRQV�������� ���HOHFWURQV��DQG������� ���QHXWURQV��,W�
PXVW�EH�QLWURJHQ�ZLWK�D�PRODU�PDVV�RI������� ����J�PRO����1��
����/HW� ��) �GHQRWHV�WKH�IRUFH�RQ T��H[HUWHG�E\ T��DQG� ��) EH�LWV�PDJQLWXGH��
�D��:H�FRQVLGHU�WKH�QHW�IRUFH�RQ�T��� ��) �SRLQWV�LQ�WKH��[�GLUHFWLRQ�VLQFH�T��LV�DWWUDFWHG�WR�
T��� ��) DQG ��) �ERWK�SRLQW�LQ�WKH�±[�GLUHFWLRQ�VLQFH�T��LV�UHSHOOHG�E\�T��DQG�T���7KXV��XVLQJ�
G� ��������P��WKH�QHW�IRUFH�LV�
�
� �� �� �� � � � �
� � � �
�� � � ��
��
��
� _ _ �� �� � �� ��� � ��
� � �� � � �� � �� �
���� �� 1 P & ���� �� &�� ���� �� �1
�� ���� �� P
H H H H H H H) ) ) )
G G G G
��
RU� ��� ����� �� �1�L�)
�E�� :H� QRZ� FRQVLGHU� WKH� QHW� IRUFH� RQ� T��� � :H� QRWH� WKDW� �� ��) ) �SRLQWV� LQ� WKH� ±[
GLUHFWLRQ��DQG� ��) DQG� ��) �ERWK�SRLQW�LQ�WKH��[�GLUHFWLRQ��7KH�QHW�IRUFH�LV�
�� �� �� � � �
� � �
� _ _ _ _ � _ _ �
� �� � � �
H H H H H H) ) )
G G G
����,I� �LV�WKH�DQJOH�EHWZHHQ�WKH�IRUFH�DQG�WKH�[�D[LV��WKHQ��
FRV �� ��
G�
G�����G��
����
7KXV��XVLQJ�&RXORPE¶V�ODZ�IRU�)��ZH�KDYH�
)[� ��)�FRV ��� ��
T��T�
� R��G�����G���
G�
G�����G��
�� ����� �� ���1���
:LWK�T� ����� �� ��&��P� ����������NJ��Y� ������P�V��WKLV�OHDGV�WR��
� � �
��
� � � �
� �������P������ �� NJ������ P�V� ���� �� &
����� �� 1 P & ������ �� &�
UPY4
T
��
����)RU�WKH�&RXORPE�IRUFH�WR�EH�VXIILFLHQW�IRU�FLUFXODU�PRWLRQ�DW�WKDW�GLVWDQFH��ZKHUH�U� �
������ P� DQG� WKH� DFFHOHUDWLRQ� QHHGHG� IRU� FLUFXODU� PRWLRQ� LV� D� � Y��U�� WKH� IROORZLQJ�
HTXDOLW\�LV�UHTXLUHG��
�
�
��
4T PY
U U
��
���� 7KH� FKDUJH� GT� ZLWKLQ� D� WKLQ� VHFWLRQ� RI� WKH� URG� �RI� WKLFNQHVV� G[�� LV� $� G[� ZKHUH�
� ����� �� P$ �DQG� �LV�WKH�FKDUJH�SHU�XQLW�YROXPH��7KH�QXPEHU�RI��H[FHVV��HOHFWURQV�
LQ�WKH�URG��RI�OHQJWK�/� ������P��LV�Q� �T��±H��ZKHUH�H�LV�JLYHQ�LQ�(T���������
�D��,Q�WKH�FDVH�ZKHUH� � �±������ ���±��&�P���ZH�KDYH�
��
�
_ _ ���� ��
/T $ $/Q G[
H H H
��
�E��:LWK� � �E[���E� �±����� ���±��&�P���ZH�REWDLQ�
�
� ��
�
_ _ ���� �� �
�
/E $ E $/Q [ G[
H H
����/HW�T��EH�WKH�FKDUJH�RI�RQH�SDUW�DQG�T��WKDW�RI�WKH�RWKHU�SDUW��WKXV��T����T�� �4� ����� &��
7KH�UHSXOVLYH�IRUFH�EHWZHHQ�WKHP�LV�JLYHQ�E\�&RXORPE¶V�ODZ��
� � � �
� �
� �
� �
� �
T T T 4 T)
U U
���
,I� ZH�PD[LPL]H� WKLV� H[SUHVVLRQ� E\� WDNLQJ� WKH� GHULYDWLYH�ZLWK� UHVSHFW� WR� T��DQG� VHWWLQJ�
HTXDO�WR�]HUR��ZH�ILQG�T�� �4�����ZKLFK�PLJKW�KDYH�EHHQ�DQWLFLSDWHG��EDVHG�RQ�V\PPHWU\�
DUJXPHQWV����7KLV�LPSOLHV�T�� ��4�� DOVR��:LWK�U� ��������P�DQG�4� ���� �� ��&��ZH�ILQG�
�� � � ��
�
�� � �
� �
���� �� 1 P & ��� �� &� � ��� � �� � � ��� �� 1
� � � � ���� �� P
4 4 4)
U U
��
����7KHUH�DUH�WZR�SURWRQV��HDFK�ZLWK�FKDUJH�T� ��H��LQ�HDFK�PROHFXOH��VR�
4 1 T$ � �� �� � ��� �� �� ��
�� �� �� � �2 71 62 7& & �����0&��
� �[ / �IURP�WKH�EHDULQJ��7KLV�WRUTXH�LV�DOVR�QHJDWLYH��7KH�FKDUJH�4�RQ�WKH�ULJKW�H[HUWV�
DQ�XSZDUG�IRUFH�RI�PDJQLWXGH����� �����T4�K����DW�D�GLVWDQFH�/���IURP�WKH�EHDULQJ��7KLV�
WRUTXH�LV�SRVLWLYH��7KH�HTXDWLRQ�IRU�URWDWLRQDO�HTXLOLEULXP�LV�
� �
� �
� � � ��
� � � � �
T4 / / T4 /: [
K K
7KH�VROXWLRQ�IRU�[�LV�
[ / T4
K :
�
��
�
��� �
�
� �
� �
�E�� ,I�)1� LV� WKH�PDJQLWXGH� RI� WKH� XSZDUG� IRUFH� H[HUWHG� E\� WKH� EHDULQJ�� WKHQ�1HZWRQ¶V�
VHFRQG�ODZ��ZLWK�]HUR�DFFHOHUDWLRQ��JLYHV�� �
� �
� � � ��
� � 1
T4 T4: )
K K
:H�VROYH�IRU�K�VR�WKDW�)1 ����7KH�UHVXOW�LV�
K T4
:
�
�
�
�
�
�����D��6LQFH�WKH�URG�LV�LQ�HTXLOLEULXP��WKH�QHW�IRUFH�DFWLQJ�RQ�LW�LV�]HUR��DQG�WKH�QHW�WRUTXH�
DERXW�DQ\�SRLQW�LV�DOVR�]HUR��:H�ZULWH�DQ�H[SUHVVLRQ�IRU�WKH�QHW�WRUTXH�DERXW�WKH�EHDULQJ��
HTXDWH� LW� WR� ]HUR�� DQG� VROYH� IRU�[�� 7KH� FKDUJH�4� RQ� WKH� OHIW� H[HUWV� DQ� XSZDUG� IRUFH� RI�
PDJQLWXGH����� ����T4�K����DW�D�GLVWDQFH�/���IURP�WKH�EHDULQJ��:H�WDNH�WKH�WRUTXH�WR�EH�
QHJDWLYH�� 7KH� DWWDFKHG�ZHLJKW� H[HUWV� D� GRZQZDUG� IRUFH� RI�PDJQLWXGH�:�� DW� D� GLVWDQFH�
����&RXORPE¶V�ODZ�JLYHV�
� � � �� �� �
� � �� �
�
���� �� 1 P & ����� �� &�_ _ � �� ���1�
� ����� �� P�
T N H)
U U
� � � � �
�
� �
�
_ _ _ ����� �� 1 P & �� ���� �� &�� ���� �� &� __ _ ���� �� 1�
� �������P�
$ &
$&
T T)
G
�F��:H�DOVR�REWDLQ�
� � � � �
�
� �
�
_ _ _ ����� �� 1 P & �� ���� �� &�� ���� �� &� __ _ ���� �� 1�
� �������P�
% &
%&
T T)
G
�����D��6LQFH�T$� �±�����Q&�DQG�T&� �������Q&�(T�������OHDGV�WR�
� � � � �
�
� �
�
_ _ _ ����� �� 1 P & �� ���� �� &������ �� &� __ _ ���� �� 1�
� �������P�
$ &
$&
T T)
G
�E��$IWHU�PDNLQJ�FRQWDFW�ZLWK�HDFK�RWKHU��ERWK�$�DQG�%�KDYH�D�FKDUJH�RI�
���� ����
Q& �����Q&�
� �
$ %T T
:KHQ�%�LV�JURXQGHG�LWV�FKDUJH�LV�]HUR��$IWHU�PDNLQJ�FRQWDFW�ZLWK�&��ZKLFK�KDV�D�FKDUJH�
RI�������Q&��%�DFTXLUHV�D�FKDUJH�RI�>�����±�����Q&�@��� �±�����Q&��ZKLFK�FKDUJH�&�KDV�DV�
ZHOO��)LQDOO\��ZH�KDYH�4$� �±�����Q&�DQG�4%� �4&� �±�����Q&��7KHUHIRUH��
�����D��8VLQJ�&RXORPE¶V�ODZ��ZH�REWDLQ�
�� � ��
�� �
�� �
�
���� �� 1 P & ����&
���� �� 1�
� ����P
T T NT)
U U
�E��,I�U� ������P��WKHQ�
�� � ��
�� �
�� � �
�
���� �� 1 P & ����&
���� �� 1�
� ���� �� P
T T NT)
U U
���� ,Q� H[SHULPHQW� ��� VSKHUH�&� ILUVW� WRXFKHV� VSKHUH�$�� DQG� WKH\� GLYLGHG� � XS� WKHLU� WRWDO�
FKDUJH��4���SOXV�4��HTXDOO\�EHWZHHQ�WKHP��7KXV��VSKHUH�$�DQG�VSKHUH�&�HDFK�DFTXLUHG�
FKDUJH��4����7KHQ��VSKHUH�&�WRXFKHV�%�DQG�WKRVH�VSKHUHV�VSOLW�XS�WKHLU�WRWDO�FKDUJH���4���
SOXV�±4����VR�WKDW�%�HQGV�XS�ZLWK�FKDUJH�HTXDO�WR�4����7KH�IRUFH�RI�UHSXOVLRQ�EHWZHHQ�$
DQG�%�LV�WKHUHIRUH�
� �
�� � ��� � ��4 4) N
G
DW�WKH�HQG�RI�H[SHULPHQW����1RZ��LQ�H[SHULPHQW����VSKHUH�&�ILUVW�WRXFKHV�%�ZKLFK�OHDYHV�
HDFK�RI�WKHP�ZLWK�FKDUJH�4����:KHQ�&�QH[W�WRXFKHV�$��VSKHUH�$�LV�OHIW�ZLWK�FKDUJH��4�����
&RQVHTXHQWO\��WKH�IRUFH�RI�UHSXOVLRQ�EHWZHHQ�$�DQG�%�LV�
� �
�� ����� � ��4 4) N
G
DW�WKH�HQG�RI�H[SHULPHQW����7KH�UDWLR�LV�
�
�
�� ��������� ������
�� � ����� ��
)
)
T T
H
S H � �������� �
WKHQ�WKH�DFWXDO�GLIIHUHQFH�ZRXOG�EH�T TS H �� ��
��� �& �$PSOLILHG�E\�D�IDFWRU�RI����
�� ������DV�LQGLFDWHG�LQ�WKH�SUREOHP��WKLV�DPRXQWV�WR�D�GHYLDWLRQ�IURP�SHUIHFW�QHXWUDOLW\�RI�
T �� � �� �� �� ����� ��2 72 7� �& & �
LQ�D�FRSSHU�SHQQ\��7ZR�VXFK�SHQQLHV�� DW� U� �����P��ZRXOG� WKHUHIRUH�H[SHULHQFH�D�YHU\�
ODUJH�IRUFH��(T�������JLYHV�
) N
T
U
1 6�
�
��� ��� �1
����,I�WKH�UHODWLYH�GLIIHUHQFH�EHWZHHQ�WKH�SURWRQ�DQG�HOHFWURQ�FKDUJHV��LQ�DEVROXWH�YDOXH��
ZHUH�
����/HWWLQJ�NT��U�� �PJ��ZH�JHW�
� � �
��
�� �
���� �� 1 P &���� �� & �����P�
���� �� NJ ��� P V
NU T
PJ
DOJHEUDLFDOO\� VROYH� IRU� WKH� KDOI�KHLJKW� SRLQWV� �WKLV� LQYROYHV� WKH� XVH� RI� WKH� TXDGUDWLF�
IRUPXOD���7KH�UHVXOWV�DUH�
� �
� � � �� �������DQG��� � �����
� �� �
7KXV��WKH�VPDOOHU�YDOXH�RI� �LV� � ���� ��
�F��DQG�WKH�ODUJHU�YDOXH�RI� �LV� � ���� ��
����7KH�WZR�FKDUJHV�DUH�T� � 4��ZKHUH� �LV�D�SXUH�QXPEHU�SUHVXPDEO\�OHVV�WKDQ���DQG�
JUHDWHU�WKDQ�]HUR��DQG�4�±�T� ����±� �4��7KXV��(T�������JLYHV�
)
4 4
G
4
G
�
�
� �
��
�
�
�
�
1 6 1 62 7 1 6 �
7KH�JUDSK�EHORZ��RI�)�YHUVXV� ��KDV�EHHQ�VFDOHG�VR�WKDW�WKH�PD[LPXP�LV����,Q�DFWXDOLW\��
WKH�PD[LPXP�YDOXH�RI�WKH�IRUFH�LV�)PD[� �4���� � G����
�D��,W�LV�FOHDU�WKDW� �
�
� �����JLYHV�WKH�PD[LPXP�YDOXH�RI�)��
�E��6HHNLQJ�WKH�KDOI�KHLJKW�SRLQWV�RQ�WKH�JUDSK�LV�GLIILFXOW�ZLWKRXW�JULG�OLQHV�RU�VRPH�RI�
WKH�VSHFLDO�WUDFLQJ�IHDWXUHV�IRXQG�LQ�D�YDULHW\�RI�PRGHUQ�FDOFXODWRUV��,W�LV�QRW�GLIILFXOW�WR�
�����D��(T���������LQ�DEVROXWH�YDOXH��JLYHV�
Q
T
H
� �� ��
��� ��
��� ��
�
��
���
�
� �&
&
HOHFWURQV �
�E��6LQFH�\RX�KDYH�WKH�H[FHVV�HOHFWURQV��DQG�HOHFWURQV�DUH�OLJKWHU�DQG�PRUH�PRELOH�WKDQ�
SURWRQV��WKHQ�WKH�HOHFWURQV�³OHDS´�IURP�\RX�WR�WKH�IDXFHW�LQVWHDG�RI�SURWRQV�PRYLQJ�IURP�
WKH�IDXFHW�WR�\RX��LQ�WKH�SURFHVV�RI�QHXWUDOL]LQJ�\RXU�ERG\���
�F��8QOLNH�FKDUJHV�DWWUDFW��DQG�WKH�IDXFHW��ZKLFK�LV�JURXQGHG�DQG�LV�DEOH�WR�JDLQ�RU�ORVH�
DQ\�QXPEHU�RI�HOHFWURQV�GXH�WR�LWV�FRQWDFW�ZLWK�(DUWK¶V�ODUJH�UHVHUYRLU�RI�PRELOH�FKDUJHV��
EHFRPHV�SRVLWLYHO\�FKDUJHG��HVSHFLDOO\�LQ�WKH�UHJLRQ�FORVHVW�WR�\RXU��QHJDWLYHO\�FKDUJHG��
KDQG��MXVW�EHIRUH�WKH�VSDUN��
�G��7KH�FDW�LV�SRVLWLYHO\�FKDUJHG��EHIRUH�WKH�VSDUN���DQG�E\�WKH�UHDVRQLQJ�JLYHQ�LQ�SDUW��E��
WKH�IORZ�RI�FKDUJH��HOHFWURQV��LV�IURP�WKH�IDXFHW�WR�WKH�FDW��
�H��,I�ZH�WKLQN�RI�WKH�QRVH�DV�D�FRQGXFWLQJ�VSKHUH��WKHQ�WKH�VLGH�RI�WKH�VSKHUH�FORVHVW�WR�
WKH�IXU�LV�RI�RQH�VLJQ��RI�FKDUJH��DQG�WKH�VLGH�IXUWKHVW�IURP�WKH�IXU�LV�RI�WKH�RSSRVLWH�VLJQ�
�ZKLFK�� DGGLWLRQDOO\�� LV� RSSRVLWHO\� FKDUJHG� IURP�\RXU� EDUH�KDQG�ZKLFK�KDG� VWURNHG� WKH�
FDW¶V� IXU��� 7KH� FKDUJHV� LQ� \RXU� KDQG� DQG� WKRVH� RI� WKH� IXUWKHVW� VLGH� RI� WKH� ³VSKHUH´�
WKHUHIRUH� DWWUDFW� HDFK� RWKHU�� DQG�ZKHQ� FORVH� HQRXJK��PDQDJH� WR� QHXWUDOL]H� �GXH� WR� WKH�
³MXPS´�PDGH�E\�WKH�HOHFWURQV��LQ�D�SDLQIXO�VSDUN��
$FFRUGLQJ� WR�$SSHQGL[�&�RI� WKH� WH[W��0� ������ ������NJ��DQG�P� ������ ������NJ�� VR�
�XVLQJ�� �� ���N��WKH�FKDUJH�LV�
T
� �� �� � �� �� ��� ��
� �� ��
�� ��
�� �� ��
�
��� � �
�
�
1 P NJ NJ NJ
1 P &
&�
� �
� �
2 72 72 7
�E�� 7KH� GLVWDQFH� U� FDQFHOV� EHFDXVH� ERWK� WKH� HOHFWULF� DQG� JUDYLWDWLRQDO� IRUFHV� DUH�
SURSRUWLRQDO�WR���U���
�F��7KH�FKDUJH�RQ�D�K\GURJHQ�LRQ�LV�H� ������ ���±���&��VR�WKHUH�PXVW�EH�
��
��
��
��� �� & ��� �� LRQV�
��� �� &
TQ
H
(DFK�LRQ�KDV�D�PDVV�RI� LP ����� ���
±���NJ��VR�WKH�WRWDO�PDVV�QHHGHG�LV�
�� �� ���� �� ���� �� NJ ��� �� NJ�LP QP
�����D��7KH�PDJQLWXGHV�RI�WKH�JUDYLWDWLRQDO�DQG�HOHFWULFDO�IRUFHV�PXVW�EH�WKH�VDPH��
�
� �
�
� �
T
U
* P0
U
ZKHUH�T� LV� WKH� FKDUJH� RQ� HLWKHU� ERG\�� U� LV� WKH� FHQWHU�WR�FHQWHU� VHSDUDWLRQ� RI� (DUWK� DQG�
0RRQ��*�LV�WKH�XQLYHUVDO�JUDYLWDWLRQDO�FRQVWDQW��0�LV�WKH�PDVV�RI�(DUWK��DQG�P�LV�WKH�PDVV�
RI�WKH�0RRQ��:H�VROYH�IRU�T��
T *P0� � �
E\� (T�� ������:KHQ� WKHVH� WZR� H[SUHVVLRQV� DUH� XVHG� LQ� WKH� HTXDWLRQ�PJ� WDQ � �)H�� ZH�
REWDLQ�
���� �
�
� �
� �
� � �
PJ[ T T /[
/ [ PJ
�E��:H�VROYH�[�� ��NT�/�PJ�IRU�WKH�FKDUJH��XVLQJ�(T���������
���
�
� � �
�����NJ ���P V �����P
��� �� &�
� � ���� �� 1 P & ����P
PJ[T
N/
7KXV��WKH�PDJQLWXGH�LV� �_ _ ��� �� &�T
�����D��$�IRUFH�GLDJUDP�IRU�RQH�RI�WKH�EDOOV�LV�VKRZQ�RQ�WKH�ULJKW��
7KH�IRUFH�RI�JUDYLW\�PJ �DFWV�GRZQZDUG�� WKH�HOHFWULFDO� IRUFH� )H �RI�
WKH�RWKHU�EDOO�DFWV�WR�WKH�OHIW��DQG�WKH�WHQVLRQ�LQ�WKH�WKUHDG�DFWV�DORQJ�
WKH�WKUHDG��DW�WKH�DQJOH� �WR�WKH�YHUWLFDO��7KH�EDOO�LV�LQ�HTXLOLEULXP��
VR� LWV� DFFHOHUDWLRQ� LV� ]HUR�� 7KH� \� FRPSRQHQW� RI�1HZWRQ¶V� VHFRQG�
ODZ�\LHOGV�7�FRV �±�PJ� ���DQG�WKH�[�FRPSRQHQW�\LHOGV�7�VLQ �±�)H
 ����:H�VROYH�WKH�ILUVW�HTXDWLRQ�IRU�7�DQG�REWDLQ�7� �PJ�FRV ��:H�
VXEVWLWXWH�WKH�UHVXOW�LQWR�WKH�VHFRQG�WR�REWDLQ�PJ�WDQ �±�)H� ����
([DPLQDWLRQ�RI�WKH�JHRPHWU\�RI�)LJXUH�������OHDGV�WR�
WDQ �[
/ [
�
�� �1 6
,I�/�LV�PXFK�ODUJHU�WKDQ�[��ZKLFK�LV�WKH�FDVH�LI� �LV�YHU\�VPDOO���ZH�PD\�QHJOHFW�[���LQ�WKH�
GHQRPLQDWRU�DQG�ZULWH�WDQ [��/��7KLV�LV�HTXLYDOHQW�WR�DSSUR[LPDWLQJ�WDQ �E\�VLQ ��7KH�
PDJQLWXGH�RI�WKH�HOHFWULFDO�IRUFH�RI�RQH�EDOO�RQ�WKH�RWKHU�LV�
) T
[H
�
�
��
���� �D�� ,I�RQH�RI� WKHP�LV�GLVFKDUJHG�� WKHUH�ZRXOG�QR�HOHFWURVWDWLF� UHSXOVLRQ�EHWZHHQ� WKH�
WZR�EDOOV�DQG�WKH\�ZRXOG�ERWK�FRPH�WR�WKH�SRVLWLRQ�� ����PDNLQJ�FRQWDFW�ZLWK�HDFK�RWKHU���
�E�� $� UHGLVWULEXWLRQ� RI� WKH� UHPDLQLQJ� FKDUJH�ZRXOG� WKHQ� RFFXU�� ZLWK� HDFK� RI� WKH� EDOOV�
JHWWLQJ� T���� 7KHQ� WKH\� ZRXOG� DJDLQ� EH� VHSDUDWHG� GXH� WR� HOHFWURVWDWLF� UHSXOVLRQ�� ZKLFK�
UHVXOWV�LQ�WKH�QHZ�HTXLOLEULXP�VHSDUDWLRQ�
���� ��� ���
�
� � � ��� FP ��� FP�
� � �
T /
[ [
PJ
����5HJDUGLQJ�WKH�IRUFHV�RQ�T��H[HUWHG�E\�T��DQG�T���RQH�PXVW�³SXVK´�DQG�WKH�RWKHU�PXVW�
³SXOO´� LQ� RUGHU� WKDW� WKH� QHW� IRUFH� LV� ]HUR�� KHQFH�� T�� DQG� T�� KDYH� RSSRVLWH� VLJQV�� )RU�
LQGLYLGXDO�IRUFHV�WR�FDQFHO��WKHLU�PDJQLWXGHV�PXVW�EH�HTXDO��
� � � �
� �
�� �� ��
_ __ _ _ __ _T T T TN N
/ / /
��
:LWK� �� ������ �/ / �WKH� DERYH� H[SUHVVLRQ� VLPSOLILHV� WR�
_ _ _ _ �T T� �
� �
�7KHUHIRUH���
� �� � �T T ��RU� � �� �����T T
���� 7KH�PDVV� RI� DQ� HOHFWURQ� LV�P� � ����� � ��±��� NJ�� VR� WKH� QXPEHU� RI� HOHFWURQV� LQ� D�
FROOHFWLRQ�ZLWK�WRWDO�PDVV�0� ������NJ�LV�
��
��
����NJ ���� �� HOHFWURQV�
���� �� NJ
0Q
P
7KH�WRWDO�FKDUJH�RI�WKH�FROOHFWLRQ�LV�
�� �� ������ �� ���� �� & ���� �� &�T QH
����:H�QRWH�WKDW��DV�UHVXOW�RI�WKH�IDFW�WKDW�WKH�&RXORPE�IRUFH�LV�LQYHUVHO\�SURSRUWLRQDO�WR�
U���D�SDUWLFOH�RI�FKDUJH�4�ZKLFK�LV�GLVWDQFH�G�IURP�WKH�RULJLQ�ZLOO�H[HUW�D�IRUFH�RQ�VRPH�
FKDUJH�TR�DW�WKH�RULJLQ�RI�HTXDO�VWUHQJWK�DV�D�SDUWLFOH�RI�FKDUJH��4�DW�GLVWDQFH��G�ZRXOG�
H[HUW�RQ�TR���7KHUHIRUH��T�� ���H�RQ�WKH�±\�D[LV�FRXOG�EH�UHSODFHG�ZLWK�D���H�FORVHU�WR�WKH�
RULJLQ��DW�KDOI�WKH�GLVWDQFH���WKLV�ZRXOG�DGG�WR�WKH�T�� ���H�DOUHDG\�WKHUH�DQG�SURGXFH���H
EHORZ�WKH�RULJLQ�ZKLFK�H[DFWO\�FDQFHOV�WKH�IRUFH�GXH�WR�T�� ���H�DERYH�WKH�RULJLQ����
6LPLODUO\��T�� ���H�WR�WKH�IDU�ULJKW�FRXOG�EH�UHSODFHG�E\�D��H�DW�KDOI�WKH�GLVWDQFH��ZKLFK�
ZRXOG� DGG� WR�T�� � �H� DOUHDG\� WKHUH� WR� SURGXFH� D� ��H� DW� GLVWDQFH�G� WR� WKH� ULJKW� RI� WKH�
FHQWUDO�FKDUJH�T���7KH�KRUL]RQWDO�IRUFH�GXH�WR�WKLV���H�LV�FDQFHOOHG�H[DFWO\�E\�WKDW�RI�T�� �
��H�RQ�WKH�±[�D[LV��VR�WKDW�WKH�QHW�IRUFH�RQ�T��LV�]HUR��
���� �D�� &KDUJH� 4�� � ���� � ��±�� &� LV� RQ� WKH� \� D[LV� DW� \� � ������ P�� DQG� FKDUJH�
�
� �� �� &4 �LV�RQ�WKH�\�D[LV�DW�\� �±������P��7KH�IRUFH�RQ�SDUWLFOH����ZKLFK�KDV�D�
FKDUJH�RI�T� ����� ���±��&��LV�GXH�WR�WKH�YHFWRU�VXP�RI�WKH�UHSXOVLYH�IRUFHV�IURP�4��DQG�
4���,Q�V\PEROV�� � � � � � �) ) ) ZKHUH�
� � � �
�� � �� �
� � � �
_ __ _ � _ _ �T T T T) N ) N
U U
8VLQJ� WKH� 3\WKDJRUHDQ� WKHRUHP�� ZH� KDYH� U��� � U��� � ������ P�� ,Q� PDJQLWXGH�DQJOH�
QRWDWLRQ��SDUWLFXODUO\�FRQYHQLHQW� LI�RQH�XVHV�D�YHFWRU�FDSDEOH�FDOFXODWRU�LQ�SRODU�PRGH���
WKH�LQGLFDWHG�YHFWRU�DGGLWLRQ�EHFRPHV�
� ����� �� ����� �� ����� � �)
7KHUHIRUH��WKH�QHW�IRUFH�LV� � �������1�L) ��
�E�� 6ZLWFKLQJ� WKH� VLJQ� RI� 4�� DPRXQWV� WR� UHYHUVLQJ� WKH� GLUHFWLRQ� RI� LWV� IRUFH� RQ� T��
&RQVHTXHQWO\��ZH�KDYH�
� ����� �� ����� ��� ����� �� �)
7KHUHIRUH��WKH�QHW�IRUFH�LV� � �������1�M) ��
� �� �� ��� � �� �� �� ��� �� ��� ��� �� ����� �� �� ��� � � �2 7 2 7 2 7 2 7
�D��7KHUHIRUH��WKH�QHW�IRUFH�KDV�PDJQLWXGH������ ���±���1��
�E�� 7KH� GLUHFWLRQ� RI� WKH� QHW� IRUFH� LV� DW� DQ� DQJOH� RI� ±���ƒ� �RU� ���ƒ� PHDVXUHG�
FRXQWHUFORFNZLVH�IURP�WKH��[�D[LV���
���� 7KH� LQGLYLGXDO� IRUFH� PDJQLWXGHV� DUH� IRXQG� XVLQJ� (T�� ������ ZLWK� 6,� XQLWV� �VR�
�����PD �� DQG�N� DV� LQ�(T��������:H�XVH�PDJQLWXGH�DQJOH�QRWDWLRQ� �FRQYHQLHQW� LI�RQH�
XVHV�D�YHFWRU�FDSDEOH�FDOFXODWRU�LQ�SRODU�PRGH���OLVWLQJ�WKH�IRUFHV�GXH�WR������T�������T��
DQG��±����T�FKDUJHV��
����7KH�PDJQLWXGH�RI�WKH�QHW�IRUFH�RQ�WKH�T� ���� ���±��&�FKDUJH�LV�
N T T N T T� �
�
�� �� � ���
_ _
�
ZKHUH� T�� � ��� � ��±�� &� DQG� _T�_� � ��� � ��±�� &�� 7KLV� \LHOGV� �����1�� 8VLQJ�1HZWRQ¶V�
VHFRQG�ODZ��ZH�REWDLQ�
P )
D
� ��
��
� � ���
�� �1
��� P V
NJ��
����)RU� WKH�QHW� IRUFH�RQ�T�� ��4� WR� YDQLVK�� WKH�[� IRUFH� FRPSRQHQW� GXH� WR�T�� �T�PXVW�
H[DFWO\�FDQFHO�WKH�IRUFH�RI�DWWUDFWLRQ�FDXVHG�E\�T�� ��±�4���&RQVHTXHQWO\��
�
� � �
� � �
_ � _ FRV ��
� � � � � � �
4T 4 4 4
D D D
RU�T� �4� ����7KLV�LPSOLHV�WKDW� � �� � ������T 4
ZKHUH�T��LV�QRZ�XQGHUVWRRG�WR�EH�LQ� &��7KXV��ZH�REWDLQ�T�� �±��� &��
����:H�DUH�ORRNLQJ�IRU�D�FKDUJH�T�ZKLFK��ZKHQ�SODFHG�DW�WKH�RULJLQ��H[SHULHQFHV� )QHW �� �
ZKHUH�
) ) ) )QHW � � � � �
7KH�PDJQLWXGH� RI� WKHVH� LQGLYLGXDO� IRUFHV� DUH� JLYHQ� E\� &RXORPE¶V� ODZ�� (T�� ������ DQG�
ZLWKRXW�ORVV�RI�JHQHUDOLW\�ZH�DVVXPH�T�!����7KH�FKDUJHV�T������ &���T���±�� &���DQG�T�
�XQNQRZQ���DUH�ORFDWHG�RQ�WKH��[�D[LV��VR�WKDW�ZH�NQRZ� )� �SRLQWV�WRZDUGV�±[�� )� �SRLQWV�
WRZDUGV��[��DQG� )� �SRLQWV�WRZDUGV�±[�LI�T��!���DQG�SRLQWV�WRZDUGV��[�LI�T�������7KHUHIRUH��
ZLWK�U�� ���P��U�� ����P�DQG�U�� ����P��ZH�KDYH�
� �
�
�
�
�
�
�
�
�N
T T
U
N T T
U
N T T
U
_ _ � �
6LPSOLI\LQJ��WKLV�EHFRPHV�
� �
�
�
�� ��� �
�
�
T
7DNLQJ� WKH� �SRVLWLYH�� VTXDUH� URRW� DQG� VROYLQJ��ZH�REWDLQ�[� �������P�� ,I� RQH� WDNHV� WKH�
QHJDWLYH�URRW�DQG�µVROYHV¶��RQH�ILQGV�WKH�ORFDWLRQ�ZKHUH�WKH�QHW�IRUFH�ZRXOG�EH�]HUR�LI T�
DQG�T��ZHUH�RI�OLNH�VLJQ��ZKLFK�LV�QRW�WKH�FDVH�KHUH���
�G��)URP�WKH�DERYH��ZH�VHH�WKDW�\� ����
����&KDUJH�T�� �±��� ���±��&�LV�DW�WKH�RULJLQ��DQG�FKDUJH�T�� ����� ���±��&�LV�DW�[� ������
P��7KH�IRUFH�RQ�T�� ����� ���±��&�LV�GXH�WR�WKH�DWWUDFWLYH�DQG�UHSXOVLYH�IRUFHV�IURP�T�
DQG�T���UHVSHFWLYHO\��,Q�V\PEROV�� ) ) )� �� � ��QHW ��ZKHUH�
� � � �
�� ��� �
�� � �
_ _ � _ _ �T T T T) N ) N
U U
�D�� ,Q� WKLV� FDVH� U��� � �����P� DQG� U��� � �����P�� ZLWK� ��) �GLUHFWHG� WRZDUGV� ±[� DQG� ��)
GLUHFWHG�LQ�WKH��[�GLUHFWLRQ��8VLQJ�WKH�YDOXH�RI�N�LQ�(T��������ZH�REWDLQ��
� � � � � �
� QHW �� �� �� � � �
�� � � �� � �
� �
� � � �
� �
_ _ _ _Ö Ö Ö ÖL _ _ L L L
�� �� & �� �� & ����� �� 1 P & ���� �� &� L
�����P� �����P�
������1�L �
T T T T T T) ) ) N N NT
U U U U
�E�� ,Q� WKLV� FDVH� U��� � �����P� DQG� U��� � �����P�� ZLWK� ��) �GLUHFWHG� WRZDUGV� ±[� DQG� )� �
WRZDUGV��[��1RZ�ZH�REWDLQ��
� � � � � �
� QHW �� �� �� � � �
�� � � �� � �
� �
� � � �
� �
_ _ _ _Ö Ö Ö ÖL _ _ L L L
�� �� & �� �� & ����� �� 1 P & ���� �� &� L
�����P� �����P�
������1�L �
T T T T T T) ) ) N N NT
U U U U
�F��%HWZHHQ�WKH�ORFDWLRQV�WUHDWHG�LQ�SDUWV��D��DQG��E���WKHUH�PXVW�EH�RQH�ZKHUH� )� �QHW ��
:ULWLQJ� U��� � [� DQG� U��� � [� ±� ����� P�� ZH� HTXDWH� )�� �DQG� )� � �� DQG� DIWHU� FDQFHOLQJ�
FRPPRQ�IDFWRUV��DUULYH�DW�
� �
��
_ _ �
�����P
T T
[ [
7KLV�FDQ�EH�IXUWKHU�VLPSOLILHG�WR�
�
�
�
�
� �����P� � �
_ _ �
T[
[ T
����:H� DUH� FRQFHUQHG�ZLWK� WKH� FKDUJHV� LQ� WKH� QXFOHXV� �QRW� WKH� ³RUELWLQJ´� HOHFWURQV�� LI�
WKHUH�DUH�DQ\���7KH�QXFOHXV�RI�+HOLXP�KDV���SURWRQV�DQG�WKDW�RI�7KRULXP�KDV�����
�D��(T�������JLYHV�
� � � �� ���
�
� �� �
���� �� 1 P & ������� �� &���������� �� &��
��� �� 1�
���� �� P�
T) N
U
�E��(VWLPDWLQJ� WKH�KHOLXP�QXFOHXV�PDVV�DV� WKDW�RI���SURWRQV��DFWXDOO\�� WKDW�RI���SURWRQV�
DQG���QHXWURQV��EXW� WKH�QHXWURQV�KDYH�DSSUR[LPDWHO\� WKH�VDPH�PDVV���1HZWRQ¶V�VHFRQG�
ODZ�OHDGV�WR�
D )
P
�� ��
� ��� ��
� � ��
�
��
���
�
� �1
NJ
P V�2 7
����/HW�WKH�WZR�FKDUJHV�EH�T��DQG�T���7KHQ�T����T�� �4� ����� ���±��&��:H�XVH�(T��������
� � �
� �
�
���� �� 1 P &
���1 �
���P
T T
:H�VXEVWLWXWH�T�� �4� ±�T�� DQG� VROYH� IRU�T��XVLQJ� WKH�TXDGUDWLF� IRUPXOD��7KH� WZR� URRWV�
REWDLQHG� DUH� WKH� YDOXHV� RI�T�� DQG�T��� VLQFH� LW� GRHV� QRW�PDWWHU�ZKLFK� LV�ZKLFK��:H� JHW�
���� �� �& �DQG����� ���±��&��7KXV��WKH�FKDUJH�RQ�WKH�VSKHUH�ZLWK�WKH�VPDOOHU�FKDUJH�LV�
���� �� �& ��
����:KHQ� VSKHUH�&� WRXFKHV� VSKHUH�$�� WKH\� GLYLGH� XS� WKHLU� WRWDO� FKDUJH� �4��� SOXV�4��
HTXDOO\�EHWZHHQ� WKHP��7KXV�� VSKHUH�$� QRZ�KDV�FKDUJH��4���� DQG� WKH�PDJQLWXGH�RI� WKH�
IRUFH�RI�DWWUDFWLRQ�EHWZHHQ�$�DQG�%�EHFRPHV�
��
�
�� � ��� � �� ���� �� 1�4 4) N
G
����:LWK�)� �PHJ��(T�������OHDGV�WR��� � � ���
�
���
���� �� 1 P & ���� �� &
���� �� NJ ���P VH
NH\
P J
ZKLFK� OHDGV� WR� \� � � ����P��:H� FKRRVH ����P\ VLQFH� WKH� VHFRQG� HOHFWURQ�PXVW� EH�
EHORZ� WKH� ILUVW� RQH�� VR� WKDW� WKH� UHSXOVLYH� IRUFH� �DFWLQJ� RQ� WKH� ILUVW�� LV� LQ� WKH� GLUHFWLRQ�
RSSRVLWH�WR�WKH�SXOO�RI�(DUWK¶V�JUDYLW\��
���� �D�� ,I� D� �QHJDWLYH�� FKDUJHG� SDUWLFOH� LV� SODFHG� D� GLVWDQFH� [� WR� WKH� ULJKW� RI� WKH� ��T
SDUWLFOH�� WKHQ� LWV� DWWUDFWLRQ� WR� WKH���T� SDUWLFOH�ZLOO� EH� H[DFWO\�EDODQFHG�E\� LWV� UHSXOVLRQ�
IURP�WKH�±�T�SDUWLFOH�LV�ZH�UHTXLUH�
� �
� �
� �/ [ [
ZKLFK� LV� REWDLQHG� E\� HTXDWLQJ� WKH� &RXORPE� IRUFH� PDJQLWXGHV� DQG� WKHQ� FDQFHOLQJ�
FRPPRQ�IDFWRUV���&URVV�PXOWLSO\LQJ�DQG�WDNLQJ�WKH�VTXDUH�URRW��ZH�REWDLQ�
�
�
[
/ [
ZKLFK�FDQ�EH�UHDUUDQJHG�WR�SURGXFH�
�����
� � � �
/[ /
�E��7KH�\�FRRUGLQDWH�RI�SDUWLFOH���LV�\� ����
����7KH�QHW�FKDUJH�FDUULHG�E\�-RKQ�ZKRVH�PDVV�LV�P�LV�URXJKO\�
�� ��
�
������
���NJ������ �� PROHFXOHV PRO���� HOHFWURQ�SURWRQ�SDLUV PROHFXOH� ���� �� &�������
����� NJ PRO
��� �� &�
$P1 =HT
0
DQG�WKH�QHW�FKDUJH�FDUULHG�E\�0DU\�LV�KDOI�RI�WKDW��6R�WKH�HOHFWURVWDWLF�IRUFH�EHWZHHQ�WKHP�
LV�HVWLPDWHG�WR�EH�
� �
� � � ��
��
� ���� �� &����� �� 1 P & � �� 1�
� ��P
T T
) N
G
7KXV��WKH�RUGHU�RI�PDJQLWXGH�RI�WKH�HOHFWURVWDWLF�IRUFH�LV� ���� �1 ��

Outros materiais