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$@C+@n@6N4(B $@Lu(B
\end{flushright}

\noindent
{\it Acta Mathematica} {\bf 47} (1925), 297-311.

\begin{center}
{\large 
{\bf $@%]%"%s%+%l$N:G8e$N4v2?3XDjM}$N3HD%(B} \\
{\bf An extension of Poincar\'{e}'s last geometric theorem}
}

\vspace{0.5cm}
George D. Birkhoff
\end{center}

\vspace{0.5cm}
\noindent
{\bf 1. $@=x(B}

Acta Mathematica$@$N(B13$@4,$K7G:\$5$l$?%]%"%s%+%l$N<u>^O@J8!V;0BNLdBj$HNO3X$NJ}Dx<0!W(B(Le probl$\grave{\rm e}$me de trois corps et les \'{e}quations de la dynamique)$@$ONO3X$NHs2D@QJ,LdBj$X$N:G=i$N0NBg$JD)@o$G$"$C$?!%(BMittag-Leffler$@65<x$N;XF3$N2<$K!"(BActa Mathematica$@;o$OB?$/$N$9$0$l$?O@J8$r7G:\$7$?!#$7$+$7!"%]%"%s%+%l$N$3$NO@J8$[$I2J3XE*$K=EMW$JO@J8$O$J$$$@$m$&!%$=$NB?$/$NH/A[!"$J$+$s$:$/$=$NCf?4ItJ,$r$J$9<~4|1?F0$K4X$9$kH/A[$O$=$N8eH`$r=EMW$JNO3XE*8&5f$X$H<+A3$KF3$$$?!%(B

Rendiconti del Circolo Matematico di Palermo$@;o$N(B33$@4,$K;`$ND>A0$KH/I=$7$?!"Hs>o$KLLGr$$O@J8!V4v2?3X$N$"$kDjM}$K$D$$$F!W(B(Sur un th\'{e}or$\grave{\rm e}$me de g\'{e}om\'{e}trie)$@$K$*$$$F!"%]%"%s%+%l$O!"$"$k<o$N4v2?3XDjM}(B($@FCJL$J>l9g$KH`$,>ZL@$7$F$$$k(B)$@$,<~4|50F;$K4X$9$k$$$/$D$+$N=EMW$JLdBj$NEz$($rF3$-F@$k$3$H$r<($7$?!%$=$ND>8e$K(B\footnote[1]{Proof of Poincar\'{e}'s Geometric Theorem, {\it Trans. Amer. Math. Soc.} {\bf 14} $@$^$?$O(B {\it Bull. Soc. Math. France} {\bf 42}$@$r;2>H$;$h!%(B}$@$o$?$7$,$=$NDjM}$N@5$7$5$r>ZL@$7$?J}K!$NFC<l@-$^$?DjM}<+?H$NNO3XE*AG@-$+$i!"DjM}$N0lHL2=$,<(:6$5$l$?$,!"$=$l$r$3$3$G<($9!%(B

$@$3$3$K(BN\"{o}rlund$@65<x$N>7$-$K1~$8$k$3$H$K$h$j!"(BActa Mathematica$@;o$N;I7cE*$JEAE}$rBG$AN)$F$?(BMittag-Leffler$@65<x$K7I0U$rI=$7$?$$!%(B

\vspace{0.5cm}
\noindent
{\bf 2. $@DjM}$NDD=R(B}

$r, \theta$$@$rJ?LL$N6K:BI8$H$9$k!%$9$k$H(B$r = a > 0$$@$OH>7B(B$a$$@$N1_(B$C$$@$NJ}Dx<0$G$"$k!%1_(B$C$$@$*$h$S$=$l$r<h$j0O$`JD6J@~(B
\footnote[2]{$@JD6J@~$H$OM-8B$NC1O"7k$J3+O"B3BN$H$=$N30B&$NJd3+O"B3BN$N6&DL$N6-3&$H$7$FDj5A$5$l$k!%1_4D$OF~$l;R$K$J$C$?(B2$@$D$NJD6J@~$K8B$i$l$kNN0h$G$"$k!%$3$N(B2$@$D$N6J@~$,@\$7$J$1$l$P1_4D$OFs=EO"7k$N3+O"B3BN$G$"$k!%:G8e$N(B8$@@a0J30$G$O$[$+$N%?%$%W$N1_4D$O=P$F$3$J$$!%(B}
$\Gamma$$@$K$O$5$^$l$?Fs=EO"7k$J1_4DNN0h$r(B$R$$@!"F1$81_(B$C$$@$*$h$S$=$l$r<h$j0O$`F1MM$JJD6J@~(B$\Gamma_1$$@$K$O$5$^$l$?1_4DNN0h$r(B$R_1$$@$H$9$k!%$3$N(B2$@$D$,$o$l$o$l$N6=L#$NBP>]$G$"$k!%0lBP0l!"J}8~J]B8(B(direct)$@!"O"B3$JE@JQ49(B$T$$@$K$h$C$F(B$R$$@$,(B$R_1$$@$K0\$k$H$$$&$3$H$G(B2$@$D$N1_4D(B$R$$@$H(B$R_1$$@$r4X78$5$;$k!%$9$J$o$A!"(B
$$ \begin{array}{lll}
  C_1 = T(C), & \Gamma_1 = T (\Gamma), & R_1 = T(R), \\
  C = T^{-1}(C_1), & \Gamma = T^{-1} (\Gamma_1), & R = T^{-1}(R_1).
   \end{array}
$$
$@5-9f$N0UL#$OL@$i$+$G$"$m$&!%(B

$@%]%"%s%+%l$N:G8e$NDjM}$O<!$N$h$&$K3HD%$G$-$k!%(B

\vspace{0.5cm}
\noindent
{\bf $@DjM}(B.} $\Gamma$$@$*$h$S(B$\Gamma_1$$@$OG$0U$NF07B@~(B$\theta=$$@0lDj$H#12s$7$+$b#12s$@$18r$o$j!"(B$T$$@$K$h$C$F(B$C$$@$*$h$S(B$\Gamma$$@>e$NE@$O$=$l$>$l(B$C$$@>e$*$h$S(B$\Gamma_1$$@>e$NE@$K0\$k$H$7!"$7$+$b(B$\theta$$@$K4X$7$F5UJ}8~$KF0$/$H$9$k!%$3$N$H$-<!$N(B(a)$@!"(B(b)$@$N$I$A$i$+$,<B8=$5$l$k(B
\footnote[1]{$@6J@~(B$\Gamma$$@$*$h$S(B$\Gamma_1$$@$K4X$9$k@)8B$O!"$3$l$i$N6J@~$,!V1&B&$+$i$N$_E~C#2DG=!W$^$?!V:8B&$+$i$N$_E~C#2DG=!W$H$9$k$3$H$K$h$C$F<e$a$k$3$H$,$G$-$k!%$3$NMQ8l$O(BActa Mathematica, Vol.43$@$N$o$?$7$NO@J8(B"Surface Transformation and their Dynamical Applications"$@$GDj5A$5$l$F$$$k!%$7$+$7!"$=$l$[$I0lHLE*$G$J$/C1=c$JDjM}$G$bF1$8%?%$%W$N3HD%$N@bL@$K$O==J,$G$"$j!"NO3X$X$N1~MQ$K$U$5$o$7$$!%(B}$@!%(B

(a) $R$$@$*$h$S(B$R_1$$@Fb$K(B$T$$@$N$b$H$G0[$J$kITF0E@$,(B2$@$DB8:_$9$k!%(B

(b) $R$($@$^$?$O(B$R_1$)$@Fb$K(B$C$$@$K@\$9$k1_4D$,$"$C$F(B$T$($@$^$?$O(B$T^{-1}$)$@$K$h$C$F$=$l<+?H$N0lIt$K(B \\
\hspace*{1.2cm}$@0\$5$l$k!%(B

\vspace{0.3cm}
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$$ \begin{array}{cc}
\displaystyle 
    \int \int Prdr d \theta,  & (P>0)
   \end{array}
$$
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\footnote[2]{$@CmL\$9$Y$-(BH.Bohr$@$N8&5f$G$3$N$h$&$J1?F0$N2r@OE*$JI=8=$,<h$j07$o$l$F$$$k!%$?$H$($P$+$l$N0J2<$N:G6a$NO@J8$r;2>H$5$l$?$$!%(BZur Theorie der fast periodischer Functionen, {\it Acta Math.} Vol.45; Einige Fourierreihe fastperiodischer Functionen, {\it Math. Zeit.} Vol.23.}$@!%(B
$@$3$NO@J8$G$O$3$N$h$&$JNO3XE*1~MQ$K$ON)$AF~$i$J$$!%(B

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\footnote[3]{$@NO3X7O$K4X$9$k$o$?$7$N(BChicago Colloquium$@$N9V5A$O6a!9K\$H$7$F=PHG$5$l$k!%$3$NK\$G$3$l$i$N<gD%$r3NN)$9$k!%(B}$@!%(B
$n$$@<!85$X$&$^$/3HD%$9$k$?$a$K$O(B$n$$@<!85$N2r@OE*<h$j07$$$NDj@-E*$KK\<AE*$JMWAG$r7hDj$9$kI,MW$,$"$k!%$?$V$s!"$3$l$OC1=c$JJ}K!$G$G$-$k$@$m$&!%(B

\vspace{0.5cm}
\noindent
{\bf 3. $\delta-$$@:?(B. $@JdBj(B 1.}

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$$ 
P_0, P_1, P_2, \ldots,
$$
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\vspace{0.3cm}
\noindent
{\bf $@JdBj(B 1.} $@M-8B(B$\delta-$$@:?$,B8:_$7$J$$$?$a$NI,MW==J,>r7o$O(B$R$$@Fb$K(B$C$$@$K@\$9$k3+1_4D(B$\Sigma$$@$,$"$C$F!"$=$N(B$\Sigma$$@$r(B$T$$@$GJQ49$7$?(B$T(\Sigma)$$@$,(B$\Sigma$$@$K4^$^$l!"$7$+$b(B$\Sigma$$@$N30B&$N6-3&$+$i$N5wN%$,>/$J$/$H$b(B$\delta$$@$"$k$3$H$G$"$k!%(B

\vspace{0.3cm}
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$$
  a \leq r < a + \delta.
$$
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$S$$@$,(B$R$$@Fb$K$"$k$+$iL@$i$+$K(B$S + \overline{S}$$@$O(B$R$$@Fb$K$"$k!%$@$+$i(B$S + \overline{S}$$@$K$O(B$T$$@$r$[$I$3$9$3$H$,$G$-$k!%$^$?(B$S + \overline{S}$$@$K(B$T$$@$r$[$I$3$9$H!"$=$l<+?H$^$?$O$=$l<+?H$N0lIt$K0\$5$l$k!%$H$$$&$N$O!"E@$,(B$S$$@$KB0$;$P!"(B$T$$@$K$h$C$F(B$S$$@$NE@$K0\$k$3$H$OH=$C$F$$$k!%0lJ}!"E@$,(B$\overline{S}$$@$KB0$;$P!"(B$S$$@$KJD$8$3$a$i$l$F$$$k$+$i(B$T$$@$K$h$C$F(B$T(S)$$@$KJD$8$3$a$i$l$?E@$K0\$j!"$7$?$,$C$F$b$A$m$s(B$S$$@$KJD$8$3$a$i$l$F$$$k!%$@$+$i$=$NE@$O(B$S + \overline{S}$$@$KB0$9$k!%$5$i$KF1MM$J5DO@$+$i!"(B$T(S + \overline{S})$$@$N$I$NE@$b(B$S + \overline{S}$$@$N6-3&$+$iH>7BJ}8~$K>/$J$/$H$b(B$\delta$$@$@$1N%$l$F$$$k!%$J$<$J$i!"$b$7!"$=$NE@$,(B$T(S)$$@$KB0$7$F$$$l$P(B$S$$@$N6-3&$KBP$7$F$=$N@-<A$r;}$C$F$$$k$N$G$b$A$m$s(B$S + \overline{S}$$@$N6-3&$KBP$7$F$=$N@-<A$r;}$C$F$$$k!%0lJ}!"$=$NE@$,(B$T(\overline{S})$$@$KB0$7$F$$$l$P!"$3$NE@$O(B$S$$@$KJD$8$3$a$i$l$?E@$+$iF@$i$l$?$N$G$"$C$F!"(B$T(S)$$@$KJD$8$3$a$i$l$F$$$k!%$@$+$i(B$\delta$$@L$K~$NH>7BJ}8~308~$-$N0\F0$K$h$C$F(B$S$$@$KJD$8$3$a$i$l$?E@$K0\$k$+$i(B$S + \overline{S}$$@$K4^$^$l$k!%$3$N:G8e$N%9%F%C%W$O$9$G$KF3$$$?(B$S$$@$H(B$T(S)$$@$N4X78$H$+$i$s$G$$$k!%(B

$@$f$($K$I$N>l9g$b(B$S + \overline{S}$$@$O1_4D(B$\Sigma$$@$r9=@.$7!"JdBj(B1$@$N@-<A$rK~$?$9!%>ZL@$O407k$7$?!%(B

\vspace{0.5cm}
\noindent
{\bf 4. $@6K>.(B$\delta-$$@:?(B.}

$@>/$J$/$H$b(B1$@8D$NM-8B(B$\delta-$$@:?$,$"$k$H$7$h$&!%$3$N$H$-:G>.$N@5?t(B$n$$@$,$"$C$F(B$\delta-$$@:?(B$P_0, P_1, \cdots,$ $P_n$$@$,B8:_$7!"(B$P_n$$@$O(B$R$$@$N30$K=P$k!%(B

$@$3$N$h$&$J(B\underline{$@6K>.(B}$\delta-$$@:?$K$O$$$/$D$+$*$b$7$m$$@-<A$,$"$k!%$?$H$($P!"L@$i$+$K$=$N(B$\delta-$$@:?$NE@(B$P_i$$@$O(B$M_i$$@$KB0$9$,!"(B$M_j (j<i)$$@$KB0$5$J$$!%<B:]!"$=$&$G$J$$$H$9$k$H!"$b$C$H>/$J$$MWAG$+$i@.$kM-8B(B$\delta-$$@:?$r9=@.$G$-$k!%$@$+$i(B$P_0$$@$O$3$N(B$\delta-$$@:?$NE@$N$&$A(B$C$$@>e$K$"$kM#0l$NE@$G$"$j!"(B$P_1$$@$O$3$N(B$\delta-$$@:?$NE@$N$&$A3+1_4D(B$a < r < a + \delta$$@Fb$K$"$kM#0l$NE@$G$"$j!"0J2<F1MM$G$"$k!%(B

$@$b$&$R$H$DI,MW$J@-<A$b$9$0<($;$k!%(B$P_i$$@$H(B$P_j (i \geq 1, j \geq 1)$$@$,F1$8F07B>e$K$"$l$P!"(B$T(P_{i-1})$$@$H(B$T(P_{j-1})$$@$bF1$8F07B>e$K$"$k$,!"$3$N$H$-(B$T(P_{i-1}), T(P_{j-1})$$@$O(B$P_i, P_j$$@$HF1$8=g=x$GJB$s$G$$$k!%(B

$@$3$l$r<($9$?$a$K!"$^$:(B$T(P_{i-1})$$@$H(B$T(P_{j-1})$$@$O0lCW$7$J$$$3$H$KCm0U$7$h$&!%<B:]!"0lCW$7$?$H$9$l$P(B$P_{i-1}$$@$H(B$P_{j-1}$$@$O0lCW$7!":?$NE@$N$&$A(B$P_{i-1}$$@$H(B$P_{j-1}$$@$N4V$NE@$O$9$Y$F$O$V$1$k$7!"(B$P_{i-1}$$@$H(B$P_{j-1}$$@$N$&$A$I$A$i$+$b$O$V$1$k!%$3$l$O$*$+$7$$!%F1MM$NM}M3$G(B$P_i$$@$H(B$P_j$$@$b0lCW$7$J$$!%(B

$@$5$F(B$T(P_{i-1})$$@$N(B$r$$@:BI8$O(B$T(P_{j-1})$$@$N(B$r$$@:BI8$h$j>.$5$$$H$9$k!%(B$i<j$$@$J$i$3$N>r7o$OK~$?$5$l$k!%(B4$@E@(B$T(P_{i-1}), P_i, T(P_{j-1}), P_j$$@$NF07BJ}8~$NJB$SJ}$N=g=x$G!">ZL@$9$Y$-DD=R$K9g$o$J$$=g=x$O(B
$$
T(P_{i-1}), T(P_{j-1}), P_j, P_i, 
$$
$@$G$"$k!%$?$@$7!"(B$r$$@:BI8$O:8$+$i1&$X$U$($k!%;v<B!"(B$P_i$$@$O(B$P_j$$@$h$j30$K$J$1$l$P$J$i$:!"(B$P_j$$@$O$b$A$m$s(B$T(P_{j-1})$$@0J1s$K$"$k!%(B($@$3$N=g=x$G$O(B$T(P_{j-1})$$@$H(B$P_j$$@$,0lCW$9$k$3$H$b9M$($i$l$k(B.) $@$H$3$m$,$3$N$H$-L@$i$+$K(B$P_j$$@$O(B$T(P_{i-1})$$@$rH>7BJ}8~308~$-$K(B$\delta$$@L$K~0\F0$7$FF@$i$l$F$7$^$&!%F1$8$/(B$P_i$$@$O(B$T(P_{j-1})$$@$+$iF@$i$l$F$7$^$&!%$3$l$O(B$T(P_{i-1})$$@$,(B$P_i$$@$H(B$\delta$$@L$K~$7$+N%$l$F$$$J$$$3$H$+$i2DG=$G$"$k!%$7$?$,$C$F(B$P_j$$@$O(B$M_i$$@$NE@$G$"$j!"(B$P_i$$@$O(B$M_j$$@$NE@$G$"$k$3$H$K$J$k!%$H$3$m$,$9$G$K<($7$?6K>.(B$\delta-$$@:?$N@-<A$K$h$j!"$3$N$&$A$I$A$i$+$O@.$jN)$?$J$$!%$h$C$F(B4$@E@$NJB$SJ}$O#2CJMnA0$K=R$Y$?$b$N$K$J$k!%(B

%\vspace{0.5cm}
\newpage
\noindent
{\bf 5. $@Jd=uJQ49(B$E$.}

$@$5$F(B$P_0, P_1, \cdots, P_n$$@$rG$0U$N6K>.(B$\delta-$$@:?$NE@$H$9$k!%>e$G3NN)$7$?@-<A$+$i!"(B$P_i, P_j, P_k, \ldots$ $(i \geq 1, j \geq 1, k \geq 1, \ldots)$$@$,$3$N:?$NE@$GF10lF07B>e$K$"$l$P!"(B$T(P_{i-1}), T(P_{j-1}), T(P_{k-1}), \ldots$$@$b$^$C$?$/F1$8=g=x$GF1$8F07B>e$KJB$s$G$$$k!%(B

\vspace{1cm}
\hspace*{6cm} $@?^(B1

\vspace{5cm}
$@$3$NF07B>e$r$"$kE@(B$Q$$@$,(B$r=a$$@$+$i308~$-$KF0$$$?$H$9$k!%BhFs$NE@(B$\overline{Q}$$@$rF1$8F07B>e$K$H$C$FF0$+$=$&!%$?$@$7(B$r=a$$@$+$i$N5wN%$O(B$Q$$@$^$G$N5wN%0J>e$G$"$C$F!"(B$Q$$@$+$i$N5wN%$O(B$\delta$$@L$K~$G$"$j!"(B$Q$$@$,(B$T(P_{i-1}), T(P_{j-1}), T(P_{k-1}), \ldots$$@$K0lCW$9$k$H$-$O(B$P_i, P_j, P_k, \ldots$$@$K0lCW$9$k$h$&$K$9$k!%(B

$@$3$N;v<B$O$b$C$H$o$+$j0W$/$D$.$N$h$&$K%0%i%U2=$G$-$k!%(B$r_1, r_2, \ldots$$@$r(B$T(P_{i-1}), T(P_{j-1}), \ldots$$@$NF07B5wN%$H$7!"(B$s_1, s_2, \ldots$$@$r(B$P_i, P_j, \ldots$$@$NF07B5wN%$H$9$k!%$9$k$H$D$.$NITEy<0$,F@$i$l$k!%(B
$$ \begin{array}{l}
 r_1 < r_2 < r_3 < \ldots, \\
 s_1 < s_2 < s_3 < \ldots, \\
0 \leq s_1 - r_1 < \delta, \ 0 \leq s_2 - r_2 < \delta, \ldots .
   \end{array}
$$
$@?t$NAH(B$(r_1,s_1), (r_2,s_2), \ldots$$@$rJ?LL>e$NE@$ND>3Q:BI8$K$H$C$F!"$3$l$i$NE@$r<!!9$H@~J,$G7k$S(B(Fig.1)$@!"$3$&$7$FF@$i$l$?@^$l@~$NN>C<E@$K(B$r$$@<4$H(B$45$$@EY$r$J$9D>@~$r$D$12C$($F:81&$K1dD9$9$k!%$3$&$7$F4X?t(B$s = f(r)$$@$N%0%i%U$,@^$l@~$GM?$($i$l$k!%(B$r$$@$r(B$Q$$@$NF07B:BI8!"(B$s$$@$r(B$\overline{Q}$$@$NF07B:BI8$H$9$l$P!"(B$Q$$@$H(B$\overline{Q}$$@$N4V$NBP1~$OMW@A$5$l$?@-<A$r;}$D!%(B

$r=a$$@$G(B$Q$$@$H(B$\overline{Q}$$@$,0lCW$9$k$3$H$b9M$($i$l$k!%$?$@$7$=$N>l9g$K$O(B$Q$$@$O$b$A$m$sE@(B$T(P_{i-1})$$@$G$O$J$$!%$J$<$J$i!"$b$70lCW$7$?$H$9$k$H(B$Q$$@$O(B$T(P_0)$$@$G$J$1$l$P$J$i$:(B$\overline{Q}$$@$O(B$Q$$@$H0[$J$k(B$P_1$$@$G$J$1$l$P$J$i$L!%$$$:$l$K$7$m!"(B$r \leq r_1$$@$N%0%i%U$ND>@~ItJ,$r>/$7$J$@$i$+$J@~J,$GCV$-49$($l$P(B$r=a$$@$N=PH/E@$G(B$\overline{Q}$$@$,(B$Q$$@$h$j>e$K$/$kJL$NBP1~$,F@$i$l$k!%0J2<$G$O$3$l$,9T$J$o$l$?$H$9$k$N$,ET9g$h$$!%(B

$@$3$N$h$&$K$7$F2f!9$N9M$($F$$$k6K>.(B$\delta-$$@:?$NE@(B$P_i, P_j, \ldots$$@$,>h$C$F$$$k$I$NF07B>e$K$b0lBP0lO"B3$N!"0\F05wN%$,(B$\delta$$@L$K~$NH>7B308~$-$N1?F0$,Dj5A$5$l!"$3$N1?F0$K$h$C$F$I$NE@(B$T(P_{i-1}), T(P_{j-1}), \ldots$$@$bBP1~$9$k(B$P_i, P_j, \ldots$$@$K0\$k!%(B

$@$3$l$i$9$Y$F$NH>7B308~$-$ND>@~1?F0$O!"(B$r \geq a$$@$GDj5A$5$l!"0\F05wN%$,(B$\delta$$@L$K~$N!"J?LL>e$N0lBP0lO"B3$JC10l$N1?F0$K$h$C$F9T$J$($k!%$H$$$&$N$O!">e$N?^(B(Figure 1)$@$G!"(B$r,s$$@<4$K?bD>$JBh;0$N(B$\theta$$@<4$r9M$(!"%0%i%U$N$9$Y$F$OE,Ev$J(B$\theta=$$@0lDjLL>e$KIA$+$l$?$H9M$($F$_$h$&!%$3$l$i$N@^$l@~$O$9$Y$F(B$r$$@J}8~$KA}Bg$7!"$+$DLL(B$s=r$$@$+$i(B$\delta$$@L$K~$N5wN%$@$1>eB&$K$"$k!%6aNY$N@^$l@~$NF1$8(B$r$$@:BI8$NE@$NAH$r@~J,$G7k$V!%$3$l$K$h$C$F4X?t(B$s=f(r, \theta)$$@$,Dj5A$5$l!"MW@A$5$l$?@-<A$rM-$9$k(B$r \geq a$$@$G$N308~$-$N1?F0(B$E$$@$,F@$i$l$k!%(B

$@A0(B2$@@a$N7k2L$O<!$N$h$&$K$^$H$a$k$3$H$,$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@JdBj(B 2.} $@M-8B(B$\delta-$$@:?$,B8:_$9$k$H$9$k!%$7$?$,$C$F6K>.(B$\delta-$$@:?(B$P_0, P_1, \ldots, P_n$($n$$@$O6K>.(B)$@$,B8:_$9$k!%$3$N$H$-0\F05wN%(B$\delta$$@L$K~$NH>7B308~$-$N0lBP0lO"B3$J1?F0(B$E$$@$,(B$r \geq a$$@$GDj5A$G$-!"$3$l$K$h$C$F(B$C$$@$O30B&$K0\F0$7!"$H$/$K(B
$$
   T(P_0), T(P_1), \ldots, T(P_{n-1}),
$$
$@$r$=$l$>$l(B
$$
  P_1, P_2, \ldots, P_n,
$$
$@$K0\$9!%(B


\vspace{0.5cm}
\noindent
{\bf 6. $@Jd=u6J@~(B. $@JdBj(B III.}

$@<!$K(B$T$$@$N$"$H$K(B$E$$@$r$[$I$3$99g@.JQ49(B$TE$$@$r9M$($h$&!%L@$i$+$K(B$TE$$@$O(B$R$$@$r1_4D(B$E(R_1)$$@$K0\$90lBP0l!"J}8~J]B8(B(direct)$@JQ49$G$"$j!"1_(B$C$$@$O$3$l$K$h$C$F(B$C$$@$r0O$`JL$NO"B3JD6J@~(B$C_1$$@$K0\$k!%$5$i$K(B$E$$@$KBP1~$9$k6K>.(B$\delta-$$@:?$N3FE@(B$P_0, P_1, \ldots, P_{n-1}$$@$O(B$TE$$@$K$h$C$F$=$l$>$l(B$P_1, P_2, \ldots, P_n$$@$K0\$k!%<B:]!"G$0U$NE@(B$P_{i-1}$$@$O(B$T$$@$K$h$C$F(B$T(P_{i-1})$$@$K0\$j!"<!$K(B$E$$@$K$h$C$F(B$P_i$$@$K0\$k!%(B$P_0$$@$O(B$C_0=C$$@>e$K$"$k$+$i!"(B$P_1$$@$O(B$C_1$$@>e$K$"$k!%(B

$TE$$@$r$[$I$3$9$3$H$K$h$C$F!"(B$C_0$$@$H(B$C_1$$@$K$O$5$^$l$?Fs=EO"7k$N1_4D$O(B$C_1$$@$H(B$C_2$$@$K$O$5$^$l$?F1MM$N1_4D$K0\$5$l$k!%BhFs$N1_4D(B$C_1C_2$$@$OBh0l$N1_4D(B$C_0C_1$$@$K(B$C_1$$@$G@\$7$F$*$j!"(B$P_2$$@$O(B$C_2$$@>e$K$"$k!%$@$+$i(B$TE$$@$r<!!9$H$[$I$3$9$3$H$K$h$C$F3HBg$9$k1_4D$NNs(B$C_0C_1, C_1C_2, \ldots, C_{n-1}C_n$$@$,F@$i$l$k!%3F1_4D$OA0$N$b$N$K@\$7!"E@(B$P_0, P_1, \ldots, P_n$$@$O$=$l$>$l(B$C_0, C_1, \ldots, C_n$$@>e$K$"$k!%(B

$@$b$A$m$s$N$3$H$J$,$i!"$3$N2aDx$O$I$l$+$N1_4D(B$C_{r-1}C_r$ $(r < n)$$@$,(B$R$$@$N30$K$O$_=P$F$7$^$($P=*$o$k!%$H$3$m$,(B$C_0C_1$$@$NE@$O$9$Y$F(B$M_1$$@$KB0$7!"(B$C_1C_2$$@$NE@$O$9$Y$F(B$M_2$$@$KB0$7!"0J2<F1MM$K$7$F!"(B$C_{r-1}C_r$$@$NE@$O$9$Y$F(B$M_r$$@$KB0$9$k$+$i6K>.(B$\delta-$$@:?$NDj5A$=$N$b$N$K$h$C$F(B$R$$@$N30$K=P$i$l$J$$!%0lJ}!"(B$C_n$$@>e$N(B$P_n$$@$O(B$R$$@$N30$K$"$k!%$@$+$i1_4D(B$C_{n-1}C_n$$@$N0lIt$,(B$R$$@$N30$K=P$k!%(B

$@$3$N;~E@$G(B$r, \theta$$@LL$NE@$ND>8r:BI8$H$7$F(B$r$$@$H(B$\theta$$@$r$H$k$N$,ET9g$h$$!%$3$NLL>e$GJQ49(B$T$$@$r$I$N$h$&$KI=8=$7$F$b!"$=$NB>$NI=8=$O(B$\theta$$@J}8~$X$N(B$2k \pi (k = 1,2, \ldots)$$@$NJ?9T0\F0$GF@$i$l$k!%1_(B$C$$@$O(B$\theta$$@<4$KJ?9T$JD>@~(B$r=a$$@$H$J$j!"(B$\Gamma$$@$H(B$\Gamma_1$$@$O(B$r=a$$@$N>e$K0LCV$7!":81&$KL58B$KB3$/6J@~$H$J$k!%0lJ}(B$C_1, C_2, \ldots$$@$bF1MM$N6J@~$G$"$C$F(B$C_1$$@$,(B$C$$@$N>e$K!"(B$C_2$$@$,(B$C_1$$@$N>e$K$"$j!"0J2<F1MM$G$"$k!%$3$l$i$N6J@~$O3F6h4V(B
$$
   2k \pi \leq \theta \leq 2(k+1) \pi,
$$
$@$G9gF1$G$"$k!%1_4D(B$CC_1, C_1C_2, \ldots$$@$ONY$j9g$&BS$K$J$k!%9g@.JQ49(B$TE$$@$O3FBS$r$9$0>e$NBS$K0\$9!%(B

$@$3$N?7$7$$LL$GE@(B$P_0$$@$H(B$P_1$$@$rB?=EE@$r;}$?$J$$8L(B$P_0P_1$$@$G7k$V!%$?$@$7$3$N8L$OBS(B$C_0C_1$$@$r2#CG$7!"$7$+$b(B$C_0,C_1$$@$NE@$H$7$F$O$=$l$>$l(B$P_0,P_1$$@$N$_$rM-$9$k$H$9$k!%8L(B$P_0P_1$$@$O(B$TE$$@$K$h$C$FBhFs$NBS(B$C_1C_2$$@$r2#CG$9$k(B$P_1P_2$$@$K0\$k!%$5$i$K(B$P_1P_2$$@$O(B$TE$$@$K$h$C$FBh;0$NBS>e$N(B$P_2P_3$$@$K0\$j!"0J2<F1MM$G$"$k(B($@?^(B2)$@!%L@$i$+$K!"$3$&$7$FB?=EE@$r;}$?$J$$6J@~(B$P_0P_1P_2 \cdots P_n$$@$,F@$i$l$k!%(B

$@6J@~(B$P_0P_1 \cdots P_n$$@$,$O$8$a$F(B$R$$@$N6-3&(B$\Gamma$$@$H8r$o$kE@$r(B$Q_0$$@$H$9$k!%E@(B$Q_0$$@$OL@$i$+$K(B$P_{n-1}P_n$$@$KB0$7!"$7$+$bC<E@(B$P_n$$@$G$J$$!%(B$TE$$@$K$h$k(B$P_0P_1 \cdots P_{n-1}Q_0$$@$NA|$r9M$($h$&!%JQ498e$N6J@~(B$P_1P_2 \cdots P_nQ_1$$@$O8L(B
$$
  P_1P_2, P_2P_3, \cdots, P_{n-1}P_n, P_nQ_1 , 
$$
$@$+$i@.$j!"L@$i$+$KB?=EE@$r;}$?$J$$!%$3$N6J@~$,(B$P_0P_1$$@$H6&DLE@$r;}$?$J$$$3$H$OL@$i$+$G$"$k!%$=$l$f$(Jd=u6J@~(B$P_0Q_1$$@$OB?=EE@$r;}$?$J$$!%(B$R$$@$r2#CG$9$k6J@~(B$P_0Q_0$$@$O!"(B$TE$$@$K$h$C$F0lIt(B$R$$@$r$O$_=P$7!"(B$E(R_1)$$@$r2#CG$9$k6J@~(B$P_1Q_1$$@$K0\$k!%(B

\vspace{1cm}
\hspace*{6cm} $@?^(B2

\vspace{5cm}
$@87L)$K8@$($P!"(B$r, \theta$$@$rD>8r:BI8$H$7$F9M$($?$H$-!"E@(B$P_0$$@$NI=8=$OL58B$K$"$j!"$I$l$+0l$D$rA*$V$H1&$K$b:8$K$b(B$2k \pi$$@$N5wN%$O$J$l$F9gF1$J6J@~(B$P_0Q_1$$@$,L58B$KF@$i$l$k!%$7$+$7(B$r, \theta$$@$r6K:BI8$H9M$(D>$7!"$3$NLL$G8I(B$P_0P_1$$@$,B?=EE@$r;}$?$J$$$h$&$K$9$l$P!"(B$r, \theta$$@$rD>8r:BI8$H$7$??7$7$$LL$G$b(B$P_0Q_1$$@$O8_$$$K0[$J$j!"B?=EE@$r;}$?$J$$!%(B

$@>e$N7k2L$r$^$H$a$k$H(B

\vspace{0.3cm}
\noindent
{\bf $@JdBj(B 3.} $@JdBj(B2$@$N2>Dj$H5-K!$N$b$H$G!"B?=EE@$r;}$?$J$$O"B36J@~(B
$$
   P_0P_1 \cdots P_{n-1}Q_0P_nQ_1,
$$
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\vspace{0.5cm}
\noindent
{\bf 7. $\delta-$$@DjM}(B.}

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\vspace{0.3cm}
\noindent
{\bf $\delta-$$@DjM}(B.} $\Gamma$$@$H(B$\Gamma_1$$@$OG$0U$NF07B@~(B$\theta=$$@0lDj$H0lEY$7$+$b0lEY$@$18r$o$j!"(B$T$$@$K$h$C$F(B$C$$@>e$NE@$H(B$\Gamma$$@>e$NE@$O(B($\theta$$@$K4X$7(B)$@5UJ}8~$K0\$C$F!"$=$l$>$l(B$C$$@>e$*$h$S(B$\Gamma_1$$@>e$NE@$K$J$k$H$9$k!%$3$N$H$-G$0U$N(B$\delta>0$$@$KBP$7$F<!$N(B$(a)$$@!"(B$(b)$$@$N$I$A$i$+$,@.$jN)$D!%(B

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(b) $R$($@$^$?$O(B$R_1$)$@Fb$K3+1_4D(B$\Sigma$$@$G(B$C$$@$K@\$9$k$b$N$,$"$C$F!"(B$T$($@$^$?$O(B$T^{-1}$)$@$K$h$C$F(B$\Sigma$$@$K4^(B \\
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\vspace{0.3cm}
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$@;v$r$O$C$-$j$5$;$k$?$a!"(B$T$$@$r$[$I$3$9$H(B$C$$@>e$NE@$O(B$\theta$$@$,A}$9J}8~$KF0$-!"2>@b$K$h$j(B$\Gamma$$@>e$NE@$O(B$\theta$$@$,8:$kJ}8~$KF0$/$H$9$k!%%Y%/%H%k(B$P_0P_1$$@$,@5$N(B$\theta$$@<4$H0Y$91T3Q$r(B$\alpha$$@$H$7!"(B$Q_0Q_1$$@$,@5$N(B$\theta$$@<4$H0Y$93QEY$r(B$\beta$$@$H$9$k!%$?$@$7!"(B$\beta$$@$O(B$\pi/2$$@$H(B$3 \pi/2$$@$N4V$N3QEY$G$"$k!%$3$N$H$-LdBj$N2sE>3Q$OL@$i$+$K(B$\beta - \alpha$$@$KEy$7$$$+$^$?$O(B$\beta - \alpha$$@$K(B$2 \pi$$@$N@0?tG\$r2C$($?$b$N$K$J$k!%0J2<$G$O$3$N2sE>3Q$,@53N$K(B$\beta - \alpha$$@$G$"$k$3$H$r<($9$3$H$,:G=EMW2]Bj$G$"$k!%(B

$@Jd=u6J@~(B$P_0Q_1$$@$O(B$C$$@$H(B$E(\Gamma_1)$$@$K$O$5$^$l$?BS$r2#CG$9$k!%$3$NBS$rF07BJ}8~$KJQ7A$7$F(B$E(C)$$@$H(B$E(\Gamma_1)$$@$,D>@~(B$r=b$$@$H(B$r=c$$@$H$J$j!"(B$C$$@$O$=$N$^$^ITJQ$G$"$k$h$&$K$7$h$&(B($E(C)$$@$b(B$E(\Gamma_1)$$@$b(B$\Gamma_1$$@$K4X$9$k2>Dj$K$h$C$FG$0U$NF07B@~$H#12s$@$18r$o$kO"B36J@~$G$"$k(B)$@!%$3$N4VJQ7AESCf$N6J@~$K1h$C$F(B$rotAA_1$$@$OO"B3E*$KJQ2=$9$k!%$f$($K!"JQ7A8e$N6J@~$K1h$C$F$NA0$HF1MM$KB,$C$?(B$\beta - \alpha$$@$NCM$O(B$rotAA_1$$@$^$?$O$=$l$KA0$HF1$8(B$2 \pi$$@$N@0?tG\$r2C$($?CM$N$^$^$G$"$k!%$=$N>e!"(B$\alpha$$@$H(B$\beta$$@$OA0$HF1MM(B
$$  \begin{array}{cc}
  0 < \alpha < \frac{\displaystyle \pi}{\displaystyle 2}, & 
\frac{\displaystyle \pi}{\displaystyle 2} < \beta < 
\frac{\displaystyle 3\pi}{\displaystyle 2},
     \end{array}
$$
$@$J$kITEy<0$rK~$?$9!%(B

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$@$@$+$iJQ49(B$T=TE_0$$@$N$b$H$G$NE@(B$A$$@$+$iA|(B$A_1$$@$K8~$+$&%Y%/%H%k$O(B$A$$@$,Jd=u6J@~(B$P_0Q_0$$@$K1h$C$FF0$/$H$-@5J}8~$K2sE>$9$k!%Jd=u6J@~$rO"B3E*$KJQ$($F!"1_4D(B$R$$@$r2#CG$9$kJL$NG$0U$N6J@~$K$7$?$H$-!"2sE>3Q$bO"B3E*$KJQ$o$k$O$:$G$"$k!%$5$b$J$1$l$PITF0E@$,$"$k$3$H$K$J$j!"(B(a)$@$,F@$i$l$k!%$=$l$f$(2sE>3Q$O7h$7$F(B0$@$K$J$i$J$$!%$J$<$J$iDjM}$N2>Dj$K$h$C$F%Y%/%H%k(B$AA_1$$@$O(B$A$$@$,(B$C$$@>e$K$"$k$H$-(B$\theta$$@$N@5$N@.J,$rM-$7!"(B$A$$@$,(B$\Gamma$$@>e$K$"$k$H$-$O(B$\theta$$@$NIi$N@.J,$rM-$9$k$3$H$,J]>Z$5$l$F$$$k$+$i$G$"$k!%$3$&$7$F!"(B$R$$@$r2#CG$9$kG$0U$N6J@~$K1h$C$F%Y%/%H%k(B$AA_1$$@$NA42sE>3Q$O@5$G$"$k!%(B

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\vspace{0.5cm}
\noindent
{\bf 8. $@>ZL@$N407k(B.}

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\footnote[1]{$@ITF0E@$,L5?t$K$"$k>l9g$O=|$$$F9M$($F$$$k!%(B}$@!%(B
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\end{document}
