\vspace{0.5cm}
\noindent
{\bf 10. $BFCJL$J>l9g(B}

$B$3$N@a$G$O!"30B&(B2$B$D$N<ANL$,Ey$7$/FbB&$N<ANL$,>.$5$$>l9g$r9M$(<!$N$h$&$K$*$/!%(B
$$ \begin{array}{ll}
 m_1 = m_3 = m, & m_2 = \varepsilon m.
   \end{array}
$$

\noindent
${\bf a}$$B$H(B${\bf b}$$B$NDj5A$O(B4$B@a$N$O$8$a$K!"$^$?(B$\lambda$$B$NDj5A$O(B(4.7)$B<0$K$"$k$3$H$r;W$$$@$=$&!%$3$N>l9g(B
$$ \begin{array}{l}
 {\bf a} = 
( m(1+\varepsilon)(2+\varepsilon))^{-1/2} (-1, -1, 1+\varepsilon ), \\
 {\bf b} = 
( m(1+\varepsilon)(2+\varepsilon))^{-1/2} (-1-\varepsilon, 1, 1 ),
   \end{array}
$$
$B$H$"$+$i$5$^$K=q$1$k!%(B${\bf a}^T M {\bf b} = (1+\varepsilon)^{-1}$$B$H7W;;$G$-$k$+$i!"(B$\lambda$$B$O(B
$$
  \cos 2 \lambda = \displaystyle \frac{1}{1+\varepsilon}, 
$$
$B$rK~$?$9:G>.$N@5?t$G$"$k!%$@$+$i(B$\varepsilon \rightarrow 0$$B$N$H$-(B$\lambda \rightarrow 0$$B$G$"$k!%(B

$B$3$N$h$&$K<ANL$rA*$V$H!"(B(4.14)$B<0$r<!$N$h$&$K=q$-49$($k$3$H$,$G$-$k!%(B
$$
 V(s) = \displaystyle m^{3/2} 
\left( \frac{1+\varepsilon}{2+\varepsilon} \right)^{1/2} 
\sin 2 \lambda \left\{
\frac{\varepsilon}{\sin \lambda (1+s)} +  
\frac{\varepsilon}{\sin \lambda (1-s)} +  
\frac{1}{\sin \lambda (1+s) + \sin \lambda (1-s)} \right\}. 
\eqno(10.1)
$$
$B>e$NI=<0$*$h$S(B(5.1)$B<0$N(B$W(s)$$B$NDj5A$h$j!"(B$V$$B$b(B$W$$B$b6v4X?t$G$"$k$3$H$,$o$+$k!%(B

\vspace{1.0cm}
\hspace*{5cm} $B?^(B7

\vspace{9.0cm}
\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B10.1.} $BG$0U$K8GDj$7$?(B$m$$B$KBP$7$F(B$C$$B>e$NN.$l(B$\varphi$$B$,A4=LB`$G$J$$$h$&$J(B$\varepsilon$$B$NCM$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{$B>ZL@(B.} $c$$B$NIT0BDjB?MMBN$NJ,;^$N(B1$B$D$OC10l$N50F;(B$\varphi(x_0,(-\infty,\infty))$$B$G$"$C$F!"(B$\tau \rightarrow - \infty$$B$N$H$-(B$\varphi(x_0,\tau) \rightarrow c$$B$G$"$k!%(B$\tau_1 \in (-\infty, \infty]$$B$O(B$\varphi(x_0, \tau_1)$$B$K$*$1$k(B$v$$B$NCM$,(B$v_c$$B$G$"$k$H$-$N(B$\tau$$B$NCM$H$9$k!%(B$\tau \rightarrow +\infty$$B$N$H$-(B$\varphi(x_0, \tau) \rightarrow d$$B$N$3$H$r9M$($F!"(B$\tau_1=+\infty$$B$r4^$a$?!%(B$\varphi$$B$N8{G[E*@-<A$h$j!"(B$\tau_1$$B$O0l0U$G$"$k!%(B(6.2c)$B<0$h$j!"D>@~(B$\{ s=+1 \}$$B$O(B$C$$B>e$NN.$l$NCGLL$G$"$k!%(B$Z(\varepsilon)$$B$O(B$\varphi(x_0,(-\infty,\tau_1))$$B$,D>@~(B$\{ s=+1 \}$$B$r2#@Z$k2s?t$H$9$k!%$U$?$?$S(B$\varphi$$B$N8{G[E*@-<A$h$j!"(B$Z(\varepsilon)$$B$OM-8B$J@5@0?t$G$"$k!%(B

$B$5$F!"$9$Y$F$N(B$\varepsilon$$B$KBP$7$F(B$\varphi$$B$OA4=LB`$G$"$k$H2>Dj$9$k!%$3$N$H$-(B$Z(\varepsilon)$$B$O0lDj$G$"$k!%(B$\varepsilon \rightarrow 0$$B$N$H$-(B$Z(\varepsilon) \rightarrow \infty$$B$G$"$k$3$H$r<($=$&!%$@$+$i2>Dj$KL7=b$7!"L?Bj$,>ZL@$5$l$?$3$H$K$J$k!%(B

$BL?Bj(B6.2$B$N>ZL@$K$*$$$F(B$V$$B$,(B$[-1,1]$$B>e$GM#0l$N:G>.$r;}$D$3$H$r<($7$?!%:#$N>l9g!"(B$V$$B$O6v4X?t$G$"$k$+$i!":G>.$O(B$s=0$$B$G@8$8$k$O$:$G$"$k!%(B(10.1)$B<0$h$j(B
$$ \begin{array}{lll}
 V(0) \rightarrow 2^{-1/2} m^{5/2} & {\rm as} & 
\varepsilon \rightarrow 0 , 
   \end{array}
$$
$B$G$"$k$3$H$,$o$+$k!%$=$3$G<!$N$h$&$K$*$/!%(B
$$
  \mu = v_c^{-2} = (2V(0))^{-1} = W(0)^{-1}. 
$$
$B$3$N$H$-(B$\mu$$B$O<ANL$K0MB8$9$kDj?t$G!"(B$\varepsilon \rightarrow 0$$B$N$H$-@5$N6K8B$K6a$E$/!%(B

$B$5$F<!$r9M$($h$&!%(B
$$
  g(v,s) = \displaystyle \frac{1 - (1-s^2) v^2 W(s)^{-1}}{1 - \mu v^2},    \eqno(10.2)
$$
$B$3$N$H$-(B$g$$B$O(B$(-v_c, v_c) \times [-1,1]$$B>e$N@5$NO"B34X?t$G$"$k!%?7$7$$JQ?t(B
$$
  \eta = g(v,s)^{-1/2} w,
$$
$B$rDj5A$9$k!%$9$J$o$A!"JQ49(B
$$ \begin{array}{rl}
  (-v_c,v_c) \times [-1,1] \times R^1 & 
       \rightarrow (-v_c,v_c) \times [-1,1] \times R^1 : \\
 (v,s,w)  & \mapsto (v,s,g(v,s)^{-1/2}w),
   \end{array}
$$
$B$rDj5A$9$k!%$=$3$G<!$N$h$&$K$*$/!%(B
$$
  C_0 = \{ (v,s,w) \in C: -v_c < v < v_c \}.
$$
(6.1)$B<0$h$j!">e$NJQ49$O(B$C_0$$B$+$i(B
$$
  C' = \{ (v,s,\eta) \in (-\mu,\mu) \times [-1,1] \times R: 
\eta^2 - (1-s^2)(1 - \mu v^2) = 0 \},
$$
$B$X$NHyJ,F1Aj$G$"$k$3$H$,$o$+$k!%%Y%/%H%k>l(B(6.2)$B$O(B$C'$$B$XJQ49$5$l$k$H<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
\displaystyle \frac{dv}{d \tau} = \displaystyle 
\frac{\lambda}{2} W(s)^{1/2} g(v,s) (1- \mu v^2), \\ 
\displaystyle \frac{ds}{d \tau} = g(v,s)^{1/2} \eta, \\
\displaystyle 
\frac{d \eta}{d \tau} = \displaystyle 
-g(v,s)^{1/2} s(1- \mu v^2) - \frac{\lambda}{2} \mu 
W(s)^{1/2} g(v,s) v \eta. 
   \end{array}            \eqno(10.3)
$$
$B;~4VJQ49(B
$$
  d \tau = g(v,s)^{1/2} d \tau',
$$
$B$r9T$J$&$H!"(B$C'$$B>e$N%Y%/%H%k>l$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
\displaystyle 
\frac{dv}{d \tau'} = \displaystyle 
\frac{\lambda}{2} W(s)^{1/2} g(v,s)^{1/2} (1- \mu v^2), \\ 
\displaystyle 
\frac{ds}{d \tau'} = \eta, \\
\displaystyle 
\frac{d \eta}{d \tau'} = \displaystyle 
- s(1- \mu v^2) - \frac{\lambda}{2} \mu W(s)^{1/2} g(v,s)^{1/2} v \eta. 
   \end{array}            \eqno(10.4)
$$
$B$5$F(B(5.2)$B<0$h$j!"(B$[-1,1]$$B>e$G0lMM$K(B
$$ \begin{array}{lll}
 W(s) \rightarrow \sqrt{2} m^{5/2} (1-s^2) &  {\rm as}  & 
\lambda \rightarrow 0 ,
   \end{array}
$$
$B$G$"$k$3$H$,$o$+$k!%$@$+$i(B(10.2)$B<0$+$i!"(B$(-v_c,v_c) \times [-1,1]$$B$N%3%s%Q%/%H$JItJ,=89g>e$G0lMM$K(B
$$ \begin{array}{lll}
 W(s) g(v,s) \rightarrow \sqrt{2} m^{5/2} (1-s^2)  &  {\rm as}  & 
\lambda \rightarrow 0 ,
   \end{array}
$$
$B$H$J$k!%$7$?$,$C$F%3%s%Q%/%H$JItJ,=89g>e$G0lMM$K(B
$$ \begin{array}{lll}
 \lambda W(s)^{1/2} g(v,s)^{1/2} \rightarrow 0  &  {\rm as}  & 
\lambda \rightarrow 0. 
   \end{array}
$$
$B$h$C$F%Y%/%H%k>l(B(10.4)$B$N50F;$O(B$\lambda \rightarrow 0$$B$N$H$-%Y%/%H%k>l(B
$$ \begin{array}{l}
\displaystyle \frac{dv}{d \tau'} = 0, \\
\displaystyle \frac{ds}{d \tau'} = \eta, \\
\displaystyle \frac{d \eta}{d \tau'} = - s(1- \mu v^2), 
   \end{array}            \eqno(10.5)
$$
$B$N50F;$K6a$E$/!%$H$3$m$,>e$N%Y%/%H%k>l$N50F;$O!"(B$C'$$B>e$N(B$v$$B0lDj$N1_$=$N$b$N$G$"$k!%$@$+$i%Y%/%H%k>l(B(10.4)$B$N50F;$O(B$\lambda \rightarrow 0$$B$N$H$-%7%j%s%@!<(B$C'$$B$N$^$o$j$rG$0U$KBg$-$J2s?t4,$/$O$:$G$"$k!%$7$?$,$C$FF1$8$3$H$,%Y%/%H%k>l(B(10.3)$B$G$b@.$jN)$D!%(B(10.3)$B$GM?$($i$l$k(B$C'$$B>e$NN.$l$O(B(6.2)$B$GM?$($i$l$k(B$C_0$$B>e$NN.$l$KF1Aj$G$"$k$+$i!"(B$\varepsilon \rightarrow 0$$B$N$H$-$K(B$Z(\varepsilon) \rightarrow \infty$$B$H$$$&<($9$Y$-DD=R$rF@$?$3$H$K$J$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B%Y%/%H%k>l(B(10.4)$B$O<!$N$h$&$K2r<a$G$-$k!%N3;R$N<ANL$O(B$m_1=m_3=m$$B$*$h$S(B$m_2=\varepsilon m$$B$G$"$k!%(B$\lambda \rightarrow 0$$B$N$H$-(B$\varepsilon \rightarrow 0$$B$+$D(B$m_2 \rightarrow 0$$B$G$"$k!%$@$+$i(B(10.5)$B<0$OCf?4N3;R$N<ANL$,%<%m$N$H$-$N;0BN>WFM$rI=$o$9!%3F<o$NJQ49$,FC0[$K$J$k$+$i(B$\varepsilon=0$$B$rJ*M}E*$K2r<a$9$k$N$OFq$7$$!%$7$+$7$J$,$i(B$\varepsilon$$B$,>.$5$$$H$-!"(B$C'$$B$N$^$o$j$K$-$D$/4,$-$D$$$F$$$k$3$H$O!"Cf?4N3;R$,(B2$B$D$N30B&$NN3;R$N4V$r9T$C$?$jMh$?$j$7$J$,$iB??t2sD7$MJV$j!"$7$+$b;0BN>WFM$N6a$/$rDL$k$3$H$rI=$o$7$F$$$k!%D7$MJV$j$N2s?t$O(B$\varepsilon \rightarrow 0$$B$N$H$-L58BBg$K$J$k!%(B

$B:#$N>l9g!"$D$^$j(B$m_1=m_3$$B$N$H$-!"(B$C$$B>e$NN.$l$KBP>N@-$,$"$k!%(B$W$$B$,6v4X?t$G$"$k$+$i!"(B(6.2)$B<0$OJQ49(B
$$
  (v,s,w) \rightarrow (v,-s,-w),
$$
$B$N$b$H$GITJQ$G$"$k!%$7$?$,$C$F(B$c$$B$NIT0BDjB?MMBN$N(B2$B$D$NJ,;^$O8_$$$NH?E>$G$"$k!%(B9$B@a$N:G8e$N(B$C_+(\nu)$$B$H(B$C_-(\nu)$$B$NDj5A$r;W$$$@$=$&!%BP>N@-$K$h$j!"(B$c$$B$NIT0BDjB?MMBN$NJ,;^$N(B1$B$D$,(B$C_+(\nu)$$B$H8r$o$l$P!"$b$&(B1$B$D$NJ,;^$O(B$C_-(\nu)$$B$H8r$o$k$O$:$G$"$k!%$@$+$iL?Bj(B10.1$B$K$h$j(BFig.7$B$K<($5$l$kN.$l$NB8:_$,3NN)$5$l$?!%(B

\vspace{0.5cm}
\noindent
{\bf 11. $B=j8+$*$h$S?dB,(B}

$B$?$/$5$s$NLdBj$,L$2r7h$N$^$^;D$5$l$?!%$?$H$($P!";0BN>WFM$,@5B'2=IT2DG=$G$"$k$h$&$J<ANL$NAH9g$;$N=89g$r5-=R$G$-$k$+(B?$B$3$NLdBj$O!"$b$H$N1?F0J}Dx<0(B(2.1)$B$,(B$\mu>0$$B$J$kJQ49(B
$$ \begin{array}{l}
 M = \mu M', \\
{\bf q} = \mu^{1/2} {\bf q}',
   \end{array}
$$
$B$N$b$H$GITJQ$G$"$k$3$H$KCm0U$9$l$P$$$/$i$+4JC1$K$J$k!%$3$&$7$F!";0BN>WFM$,<ANL(B$(m_1,m_2,$ $m_3)$$B$KBP$7$F@5B'2=IT2DG=$G$"$k$?$a$NI,MW==J,>r7o$O!"G$0U$N(B$\mu>0$$B$KBP$7$F;0BN>WFM$,<ANL(B$(\mu m_1,\mu m_2,\mu m_3)$$B$KBP$7$F@5B'2=IT2DG=$G$"$k$3$H$G$"$k!%$7$?$,$C$F!"C1BN(B
$$
{\cal M} = \{ (m_1,m_2,m_3): m_1 + m_2 + m_3 = 1, $B$9$Y$F$N(B k $B$KBP$7$F(B m_k>0 \},
$$
$B>e$N<ANL$NCM$r9M$($k$@$1$G$h$$!%(B

9$B@a$G!";0BN>WFMB?MMBN>e$NN.$l$,A4=LB`$G$J$$$H$-;0BN>WFM$,@5B'2=IT2DG=$G$"$k$3$H$r<($7$?!%$^$?N.$l$,A4=LB`$G$J$$$h$&$J<ANL$N=89g$O3+$G$"$k$3$H$r8+$?!%$3$N=89g$,Bg$-$$$3$H$,4|BT$5$l$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $BM=A[(B.} $B;0BN>WFMB?MMBN>e$NN.$l$,A4=LB`$G$J$$$h$&$J<ANL$N=89g$O(B${\cal M}$$BFb$GcGL)$G$"$k!%(B

\vspace{0.3cm}
$B$3$NM=A[$,@5$7$1$l$P!";0BN>WFMB?MMBN$,@5B'2=IT2DG=$G$"$k$h$&$J<ANL$N=89g$O(B${\cal M}$$BFb$GcGL)$G$"$k!%$=$l$G$b;0BN>WFMB?MMBN>e$NN.$l$,A4=LB`$N$H$-$K;0BN>WFM$,@5B'2=2DG=$+$H$$$&LdBj$,;D$C$F$$$k!%(B2$B$D$NITF0E@$N8GM-CM$,$J$s$i$+$NLr3d$r2L$?$9$@$m$&$,!"Cx<T$O2?$NM=A[$b;}$?$J$$!%(B

$B$3$NO@J8$N7k2L$HJ}K!$,D>@~;0BNLdBj$N;0BN>WFM0J30$N>u67$KE,MQ$G$-$k$+$H$$$&5?Ld$,@8$8F@$k!%D>@~LdBj$NN.$l$O9b<!85$NLdBj$NN.$l$NCf$KITJQ=89g$H$7$F4^$^$l$k!%$7$?$,$C$FD>@~>e$G;0BN>WFM$,@5B'2=IT2DG=$J$i!"<+F0E*$KJ?LL$G$b(B3$B<!85$G$b@5B'2=IT2DG=$G$"$k!%$7$+$7$J$,$i$"$k<o$N<ANL$OD>@~LdBj$G@5B'2=2DG=$G$"$C$F$bJ?LLLdBj$G$O@5B'2=IT2DG=$G$"$k!%(B

3$B@a$GMQ$$$?JQ49$OG$0U$N<!85$N(B$n$$BBNLdBj$N(B$n$$B=E>WFM$KE,MQ$G$-$k!%$9$J$o$A!"86E@$rKD$i$^$7!"$^$?;~4VJQ?t$NJQ49$G(B$n$$B=E>WFM$KM-8B;~4V$GE~C#$9$k$h$&$J50F;$,(B1$B$D$b$J$$$h$&$K$G$-$k$N$G!"(B$n$$B=E>WFM$OITJQB?MMBN$K$G$-$k!%$3$NJ}K!$K$h$C$F2?$+?7$7$$7k2L$,=P$k$+$I$&$+!"Cx<T$OJQ498e$NJ}Dx<0$r==J,>\$7$/D4$Y$F$$$J$$!%(B

$B>WFM0J30$NFC0[@-$NB8:_$NLdBj$OD9$$4VL$2r7h$G$"$C$?!%(BPainlev\'{e}[8]$B$O!";0BNLdBj$G$OFC0[@-$O$9$Y$F>WFM$K$h$k$3$H$r>ZL@$7$?!%(BPainlev\'{e}[8]$B!"(BVon Zeipel[16]$B!"$*$h$S(BSperling[13]$B$N;E;v$K$h$l$P!"N3;R$N0LCV$,M-3&$G$"$k8B$j!">WFMFC0[@-$N$_$,(B$n$$BBNLdBj$N2r$K@8$8F@$k!%$7$?$,$C$F>WFM0J30$NFC0[@-$r8+$D$1$k$3$H$O!"$9$/$J$/$H$b$I$l$+(B1$B$D$N0LCV$,M-8B;~4V$KHsM-3&$K$J$k50F;$r8+$D$1$k$3$H$HF1CM$G$"$k!%(B

9$B@a$N7k2L$+$i$9$k$H!"$3$N$h$&$J50F;$OB8:_$7F@$k!%D>@~(B5$BBNLdBj$r9M$(!"FsBN>WFM$O$9$Y$F@5B'2=$7$?$b$N$H$9$k!%(B$m_1, \cdots, m_5$$B$r=gHVDL$j$KD>@~>e$KJB$s$@(B5$BBN$H$9$k!%(B$m$$B$*$h$S(B$\varepsilon$$B$rA*$s$G!"<ANL(B$(m, \varepsilon m, m)$$B$N;0BNLdBj$N;0BN>WFMB?MMBN>e$NN.$l$,A4=LB`$G$J$$$h$&$K$9$k!%(B$m_1=m_3=m_5=m$$B$*$h$S(B$m_2=m_4=\varepsilon m$$B$H$9$k!%(B5$BBNLdBj$N;0BN>WFM$,;0BNLdBj$N;0BN>WFM$HF1$8$h$&$K$U$k$^$&$3$H$,<($;$l$P!"(B9$B@a$G<($7$?$h$&$K!"N3;R(B1$B!"(B2$B!"$*$h$S(B3$B$^$?$ON3;R(B3$B!"(B4$B!"$*$h$S(B5$B$,;0BN>WFM$N6a$/$rDL2a$9$k$?$S$K7O$OG$0U$KBg$-$J1?F0%(%M%k%.!<$r<h$j=P$9$3$H$,$G$-$k!%(B

$BC10L;~4VL$K~$KHsM-3&$K$J$k50F;$r5-=R$7$h$&!%N3;R(B1$B$H(B2$B$O$D$M$K8_$$$K6a$/$K$"$k!%$3$l$rO"@17O(B$A$$B$H8F$V!%N3;R(B4$B$H(B5$B$b$D$M$K8_$$$K6a$/$K$"$k!%$3$l$rO"@17O(B$B$$B$H8F$V!%N3;R(B3$B$O(B2$B$D$NO"@17O$N4V$r9T$C$?$jMh$?$j$9$k!%;~9o(B$t=0$$B$KN3;R(B3$B$OO"@17O(B$A$$B$K8~$$!"(B$t_1<1/2$$B$KDI$$$D$/!%N3;R(B1$B!"(B2$B!"$*$h$S(B3$B$,;0BN>WFM$N6a$/$rDL2a$7$?8e$G!"N3;R(B3$B$OO"@17O(B$B$$B$K;~9o(B$t_2<1/4$$B$KDI$$$D$/$h$&$J9bB.EY$rF@$k!%N3;R(B3$B!"(B4$B!"$*$h$S(B5$B$,;0BN>WFM$N6a$/$rDL2a$7$?8e$G!"N3;R(B3$B$OO"@17O(B$A$$B$K(B$t_3<1/8$$B$KDI$$$D$/$h$&$J9bB.EY$rF@$k!%0J2<F1MM!%C10L;~4VL$K~$KO"@17O(B$A$$B$O(B$-\infty$$B$K9T$-!"O"@17O(B$B$$B$O(B$+\infty$$B$K9T$-!"N3;R(B3$B$O$3$l$i$N4V$rL58B2s9T$C$?$jMh$?$j$9$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $BM=A[(B.} $B>e$G5-=R$7$?50F;$OB8:_$9$k!%(B

\vspace{0.3cm}
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