\vspace{0.5cm}
\noindent
{\bf 3. $B>WFM50F;(B}

$B$3$N@a$G$OA0@a$N7k2L$r;H$C$F;0BN>WFM$K;O$^$C$?$j=*$o$C$?$j$9$k50F;$N=89g$r5-=R$9$k!%>WFM$K$O$8$^$k50F;$O!VJ|=P!W(B(ejection)$B50F;$H8F$P$l$k!%;0BN>WFM$K=*$o$k50F;$O!V>WFM!W50F;$H8F$P$l$k!%>WFM$K$O$8$^$j>WFM$K=*$o$k50F;$O!VJ|=P(B$-$$B>WFM!W(B(ejection-collision)$B50F;$H8F$P$l$k!%A0@a$G<($7$?$h$&$K!"$3$N$h$&$J50F;$O$I$l$bCf?4?^7A$KF1H<$9$kJ?9UE@$N$I$l$+$KA26a$9$k$O$:$G$"$k!%$D$^$j!"$3$l$i$OJ?9UE@$N0BDj$^$?$OIT0BDjB?MMBN>e$K$"$k!%(B$\theta_0$$B$K$*$1$kJ|=P50F;$*$h$S>WFM50F;$N=89g$r$=$l$>$l(B$E(\theta_0)$$B$*$h$S(B$C(\theta_0)$$B$GI=$o$9!%F1MM$K!"(B$E(\theta_{\pm})$$B$*$h$S(B$C(\theta_{\pm})$$B$O@5;03Q7ACf?4?^7A$K$*$1$kJ|=P50F;$*$h$S>WFM50F;$N=89g$rI=$o$9!%(B

$B@~7A2=(B(2.1)$B$r;H$($P!">WFMB?MMBN>e$NJ?9UE@$O$9$Y$FAP6JE*$G$"$C$F!"J|=P50F;$H>WFM50F;$OAj6u4V$K$O$a9~$^$l$?ItJ,B?MMBN$G$"$k$3$H$,<($;$k!%$3$l$r8+$k$?$a$K!"(B$v \cos \theta/ W(\theta)^{1/2}$$B$,(B(2.1)$B$N8GM-%Y%/%H%k(B$(v, e, 0, 0)$$B$KF1H<$9$k8GM-CM$G$"$k$3$H$r3NG'$7$FM_$7$$!%$3$N8GM-%Y%/%H%k$O(B$e$$B$KBP1~$9$k%(%M%k%.!<%l%Y%k$K@\$9$k!%$3$l$O(B(1.13)$B$rHyJ,$9$l$PJ,$+$k!%$f$($K!";0BN>WFMB?MMBNFb$N<!85$K2C$($F!"(B$v_0>0$$B$J$i$$$D$bIT0BDj8GM-CM!"$^$?(B$v_0<0$$B$J$i$$$D$b0BDj8GM-CM$r$R$H$D;}$D!%(B

$B$f$($K<!$N7k2L$rF@$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B 5.} $BG$0U$N%(%M%k%.!<%l%Y%k(B$H = e$$B$K$*$$$F!"(B$E(\theta_{\pm})$$B$*$h$S(B$C(\theta_{\pm})$$B$O$H$b$K#2<!85B?MMBN$G$"$k$,!"(B$E(\theta_0)$$B$*$h$S(B$C(\theta_0)$$B$O#1<!85$G$"$k!%J|=P50F;$O$9$Y$F(B$v_0>0$$B$NJ?9UE@$+$i=PH/$7!">WFM50F;$O$9$Y$F(B$v_0<0$$B$NJ?9UE@$KA26a$9$k!%(B

\vspace{0.3cm}
$B$3$l$O!">WFM$*$h$SJ|=P50F;$N=89g$O3F%(%M%k%.!<%l%Y%k$NDc<!85ItJ,B?MMBN$NOB$r9=@.$9$k$H$$$&(BSiegel$B$N8EE5E*7k2L$r:F>ZL@$7$?$3$H$K$J$k!%$3$l$O$^$?!"FsEyJULdBj$N>l9g!";0BN>WFM$O%j%e%Y!<%0B,EY$N0UL#$GM-$jF@$J$$$H$$$&(BSaari[6]$B$N7k2L$r$b:F>ZL@$7$F$$$k!%(B($BFsBN>WFM$O=i4|>r7o$NBg$-$J3+=89g$KBP$7$F5/$3$jF@$k$3$H$rCm0U$7$F$*$/(B.)

$B3FCf?4?^7A$KH<$&;0BNLdBj$N8EE5E*2r$,$$$/$D$+$"$k!%(B$V'(\theta)=0$$B$H2>Dj$9$k!%$9$k$H(B$\theta$$B$OCf?4?^7A$N$I$l$+$G$"$k!%$5$i$K(B$w=0$$B$H2>Dj$9$k!%$3$N$h$&$JE@$G!"7O(B(1.12)$B$O<!$N$h$&$K4JC1$K$J$k!%(B

%\displaystyle
$$ \begin{array}{ll}
\displaystyle \frac{dr}{dt} = \displaystyle
   \frac{\cos \theta}{\sqrt{W(\theta)}} rv, \\
\displaystyle \frac{dv}{dt} = \displaystyle
   \frac{re \cos \theta}{\sqrt{W(\theta)}}, \\
\displaystyle \frac{d \theta}{dt} = 0 = \displaystyle \frac{dw}{dt}. 
  \end{array}             \eqno(3.1)
$$
$B$@$+$i(B2$B<!85(B$r,v-$$BJ?LL$O!"(B$V'(\theta)=0$$B$+$D(B$w=0$$B$N$H$-$O$$$D$bITJQ$G$"$k!%(B(3.1)$B$N2r$N35N,$r(BFig.7$B$K<($7$?!%(B

\vspace{5cm}
$B$3$l$i$N6J@~$O%(%M%k%.!<4X78$r(B$r, v-$$BLL$K@)8B$7$?%l%Y%k=89g$H$7$FF@$i$l$k$3$H$KCm0U$7$h$&!%$3$l$i$O<!<0$GM?$($i$l$k!%(B
$$
  \frac{\cos \theta}{2 W(\theta)} v^2 - 1 = 
\frac{\cos \theta}{W(\theta)} re.
$$

Fig.7$B$N#2$D$N?eJ?$J2r$O(B$e=0$$B$N$H$-$K@8$8$k!%(B$e<0$$B$N>l9g!"(B$v>0$$B$NJ?9UE@$+$i$O$8$^$C$F(B$v<0$$B$NJ?9UE@$G=*$o$kM#0l$N2r$,B8:_$9$k!%$3$N2r$O7O$N(B\underline{$BJ?9TAj;w(B}(homothetic)$B2r$H8F$P$l$F$$$k!%$3$l$O!"$9$Y$F$N;~4V$KBP$7$F<AE@$,(B($B<\EY$NITDj@-$r=|$$$F(B)$BF1$8G[0L$r<h$k2r$G$"$k!%>\$7$/$O(BWintner[14]$B$r;2>H$7$FM_$7$$!%(B

$B$H$/$K!">e$N9M;!$+$i!"#1<!85=89g(B$C(\theta_0)$$B$*$h$S(B$E(\theta_0)$$B$,$^$5$K;0BNLdBj$N$3$NFC<l$J2r$+$i@.$C$F$$$k$3$H$,=P$k!%$7$?$,$C$F!"Ii%(%M%k%.!<$N>l9g!"(B$C(\theta_0) = E(\theta_0)$$B$G$"$j!"(B$\theta_0$$B$KH<$&(B2$B$D$NJ?9UE@$r7k$V%X%F%m%/%j%K%C%/$J2r$,F@$i$l$k!%$3$N50F;$rG[0L6u4V$KIA$/$N$O4JC1$G$"$k!%Bh;0BN$O86E@$K8GDj$7!";D$j$NFsBN$O$=$NN>B&$+$iJ|=P$5$l!"M-8B5wN%(B$x$$B<4$K1h$C$FF0$$$?8e$KB.EY%<%m$KC#$7!":G=*E*$K86E@$KLa$C$F$-$F;0BN>WFM$9$k!%(B

\vspace{0.5cm}
\noindent
{\bf 4. $B6a@\>WFM50F;(B}

$B$3$N@a$G$O!";0BN>WFMB?MMBN>e$NN.$l$r$I$NMM$K;H$($P;0BN>WFM$N6a$/$rDL2a$9$k50F;$NDj@-E*$U$k$^$$$r$[$H$s$I40A4$K5-=R$G$-$k$+$r<($9!%4JC1$N$?$a!"Ii$N%(%M%k%.!<6JLL#1$D$KOC$r8B$m$&!%0J2<$N7k2L$N$$$/$D$+$O%(%M%k%.!<%l%Y%k$,%<%m$^$?$O@5$N$H$3$m$XE,Ev$J=$@5$N$b$H$G3HD%$G$-$k!%(B

$B$O$8$a$K!"A0@a$N:G8e$K5-=R$7$?D>@~J?9TAj;w50F;$N6aK5$r9M$($h$&!%(B$v_0<0$$B$NN>J}$N%i%0%i%s%8%eJ?9UE@$N0BDjB?MMBN$NJ,;^$N#1$D$O(B$\theta_0$$B$KBP1~$9$k8;E@$+$i=P$k!%(B$m_3$$B$,==J,>.$5$$>l9g!"$3$l$i$NJ,;^$N3F!9$O!"L?Bj(B4$B$G8+$?$h$&$K!"8;E@$N$^$o$j$rL58B2s$^$o$k!%$7$?$,$C$F!"(B$C(\theta_0)$$B$O(B$C(\theta_+)$$B$*$h$S(B$C(\theta_-)$$B$NJDJq$NCf$K$"$j!"(B$C(\theta_0)$$B$r6I=jE*2#CGLL(B$S$$B$G@Z$l$P(B$C(\theta_{\pm}) \cap S$$B$O!"(B$m_3$$B$,==J,>.$5$1$l$P!"$d$O$j(B$C(\theta_0) \cap S$$B$XL58B2s$^$o$j$J$,$iMn$A$F$$$/!%(B

$BF1MM$J$3$H$,(B$E(\theta_0)$$B$*$h$S(B$E(\theta_{\pm})$$B$KBP$7$F$b@.$jN)$D!%(B

$B$5$F2#CGLL(B$S$$BFb$K=i4|>r7o(B$p$$B$r9M$($k!%(B$S$$B$,==J,>.$5$1$l$P!"(B$p$$B$rDL$k50F;$O(B$C(\theta_{\pm}) \cap S$$B$H8r$o$k$+!"(B$C(\theta_0) \cap S$$B$H8r$o$k$+!"$"$k$$$O;0BN>WFM$K6a$E$/$,>WFM$O$7$J$$!%:G8e$N>l9g!"50F;$O8;E@$NIT0BDjB?MMBNFb$N50F;$rDI?o$9$k$h$&$K8~$+$&$,!"(B$v_0<0$$B$N%5%I%k$N0BDjB?MMBN$K$O$J$$!%(BFig.5$B$,<($9$H$3$m$K$h$l$P!"5v$5$l$k>l9g!"$3$N$h$&$J50F;$O$9$Y$F;0BN>WFMB?MMBN$N>eB&$NOS$rEP$k!%$f$($K(B$p$$B$rDL$k50F;$O(B$S$$B$r==J,>.$5$/<h$l$PF1MM$K$U$k$^$&!%(B

$B$3$l$O<!$N$h$&$K2r<a$G$-$k!%D>@~J?9TAj;w2r$K==J,6a$$50F;$O;0BN>WFM$G=*$o$k$+!";0BN>WFM$N6a$/$rDL2a$7$F!"<!$N#2$D$N$I$A$i$+$N;EJ}$GN%$l$F$$$/!%(B$m_3$$B$,(B$y-$$B<4$N>e$K8~$$O"@1$,(B$y-$$B<4$r2<$K8~$+$&$+$=$N5U!%50F;$,;0BN>WFM$K6a$E$1$P6a$E$/$[$I!";0BN>WFM$N6aK5$rN%$l$k$H$-Bg$-$J(B$v$$B$NCM$rC#@.$9$k!%$3$l$O3F<AE@$,Hs>o$KBg$-$JB.EY$GN%$l$F$$$/$3$H$r0UL#$7$F$$$k!%;0BN>WFMB?MMBNFb$N50F;$,OS$rL58B2s4,$/$h$&$K8~$+$&$H$$$&;v<B$OA4BN$NN.$l$K4X$9$k2r<a$bM?$($k!%$9$J$o$A!";0BN>WFM$N==J,6a$/$KMh$k50F;$O!";0BN>WFM$N6aK5$+$iN%$l$F$$$/4V$K!"(B$m_1$$B$H(B$m_2$$B$N4V$NG$0U$KBg$-$J2s?t$NFsBN>WFM$r7+$jJV$9!%(B

$p$$B$rDL$k8e8~$-$N50F;$KBP$7$F$bF1MM$N$3$H$,8@$($k!%$3$N$h$&$J50F;$O8;E@$+$iJ|=P$5$l$?$+!"$"$k$$$O>e$G=R$Y$?$N$HF1MM$J;EJ}$G;0BN>WFM$N6aK5$K6a$E$$$F$O$J$l$F$$$/$+$G$"$k!%$3$l$+$i!";0BN>WFM$K;O$^$j$+$D=*$o$kD>@~J?9TAj;w50F;$K6a$E$/!"$D$^$j?7$?$JJ|=P(B$-$$B>WFM50F;$N!"2DG=@-$,=P$k!%$3$l$i$N50F;$O$b$A$m$sD>@~G[0L$G$J$/!"@5;03Q7ACf?4?^7A$K$O$8$^$j$+$D=*$o$k$O$:$G$"$k!%(B

\vspace{8.0cm}
$m_3$$B$,==J,$K>.$5$/$F8;E@$,J#AG8GM-CM$r;}$D$H2>Dj$7$h$&!%6I=j2#CGLL(B$S_0$$B$rA*$s$G!"%(%M%k%.!<LL$N%<%mB.EY6J@~$N3j$i$+$J>.JR$r4^$_!"D>@~J?9TAj;w50F;$H%<%mB.EY6J@~$N8rE@$K$"$kE@(B$q$$B$r4^$`$h$&$K$9$k!%>e$N7k2L$K$h$l$P!"(B$C(\theta_{\pm}) \cap S_0$$B$O$H$b$K(B$S_0$$B$K$*$$$F(B$q$$B$K<}B+$9$k3j$i$+$JMf@{$G$"$k!%$7$?$,$C$F$3$l$i$NMf@{$O$I$l$b%<%mB.EY6J@~$HL58B2s8r$o$k$O$:$G$"$k!%BP>N@-$K$h$j!"$3$N$h$&$J8rE@$O$I$l$b(B$E(\theta_{\pm})$$BFb$K$bB0$9$O$:$G$"$k!%$f$($K$3$N$h$&$J=i4|>r7o$OJ|=P(B$-$$B>WFM50F;$KF3$/!%(BFig.8$B$r;2>H$7$FM_$7$$!%(B

$B$3$l$i$N50F;$N#1$D$,;0BN>WFM$K6a$E$/$K$D$l$F!"B??t2sD66JLL(B$\theta=0$$B$rDL2a$9$k$3$H$r3NG'$7$FM_$7$$!%$3$l$O!"3FDL2a$K$*$$$F(B$x_2=0$$B$r0UL#$9$k!%$f$($K$3$l$i$NJ|=P(B$-$$B>WFM50F;$OO"@1$N=E?4$rDL$C$F(B$m_3$$B$,?6F0$9$k$3$H$rFCD'$E$1$F$$$k!%(B

$B$^$?!"$3$N$h$&$J50F;$O!";0BN>WFM$ND>A0$K<AE@$,;03Q7AG[0L$K%8%c%s%W$9$k$^$G$O$[$H$s$ID>@~>WFM50F;$K;w$F$$$k!%(B

$B:G8e$K!";0BN>WFM$N6a$/$rDL2a$9$k50F;$,!"$O$8$a(B$m_3$$B$,(B$y$$B<4$r>e$K8~$+$C$FF0$$$F$$$k$H$7$F!"$3$N50F;$N$U$k$^$$$N2DG=$J$9$Y$F$r5-=R$7$h$&!%(B$\gamma$$B$O;0BN>WFMB?MMBN>e$N50F;$G(B$\theta = - \pi/2$$B$NDc$$OS$rEP$k50F;$H$9$k!%(B$\gamma$$B$K6a$$Hs>WFM50F;$r5-=R$7$h$&!%(B

$\gamma$$B$KBP$7$F!"5v$5$l$k>l9g$K$*$$$F<g$?$k2DG=@-$,#3$D$"$k!%(B

i. $\gamma$$B$O(B$v>0, \theta = - \pi/2$$B$NOS$rEP$k!%(B

ii. $\gamma$$B$O(B$v>0, \theta = + \pi/2$$B$NOS$rEP$k!%(B

iii. $\gamma$$B$OD@E@$G;`$L!%(B

$B3F!9$N>l9g!"6a$/$N50F;$O7`E*$K0[$J$k$U$k$^$$$r<($9!%>l9g(Bi$B$G$O!"(B$m_3$$B$O;0BN>WFM$N6aK5$+$i!"Mh$?$H$-$HF1$8J}8~!"$D$^$j(B$y-$$B<4$r2<8~$-$KN%$l$F$$$/!%0lJ}(B$m_1$$B$H(B$m_2$$B$O6a@\O"@1$H$J$C$FB??t$NFsBN>WFM$r7+$jJV$7$J$,$i(B$y-$$B<4$r>eJ}$K?J$`!%(B

$B>l9g(Bii$B$G$O!">u67$O$^$C$?$/5U$G$"$k!%(B$m_3$$B$O(B$y$$B<4$r>e$KF0$-!"O"@1$OFsBN>WFM$r7P83$7$J$,$i(B$y$$B<4$r2<$K8~$+$C$FF0$/!%(B

$B>l9g(Biii$B$O$*$=$i$/:G$b$*$b$7$m$$!%$3$l$i$O!V%S%j%"!<%I(Bshot$B!W$H8F$V50F;$G$"$k!%;0BN>WFM$K6a$E$/$K$D$l$F!"!V%-%e!<5e!W(B$m_3$$B$O(B$y-$$B<4$rEP$j!";D$j$N(B$m_1$$B$H(B$m_2$$B$O6a@\O"@1$H$J$C$F(B$y-$$B<4$r2<$KF0$-$J$,$i?6F0$9$k!%;0BN>WFM$N6a$/$rDL2a$7$?8e$G(B$m_3$$B$O(B$m_1$$B$H(B$m_2$$B$N=E?4$N6a$/$G?6F0$7$J$,$i<h$j;D$5$l!"FsBN$O(B($B$[$\(B)$x-$$B<4$K1h$C$FH?BPJ}8~$KHt$S5n$k!%(B

\vspace{0.5cm}
{\bf References}

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\re
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6. Saari, D.: Improbability of collisions in Newtonian gravitational systems, I and II. {\it Tans. Amer. Math. Soc.} {\bf 162}, 267-271 (1971) and {\bf 181}, 351-368 (1973).

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7. Samarov, K.L.: Regularization of the Isosceles three-body problem for a non-zero area vector. {\it Soviet Math. Dokl.} {\bf 18}, 245-248 (1977).

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8. Siegel, C., Moser, J.: Lectures on Celestial Mechanics. Berlin, Springer-Verlag, 1971.

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9. Simo, C.: Regularization of triple collision in the general three body problem. In Instabilities in Dynamical Sysytems, (V.Szebehely, ed.), pp.301-307. Dordrecht, D.Reidel, 1979.

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10. Simo, C.: Masses for which triple collision is regularizable. To appear in the Proceedings of mathematical Methods in Celestial Mechanics, Oberwolfach, 1978.

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12. Sundman, K.: Nouvelles recherches sur le probleme des trois corps. {\it Acta Soc. Sci. Fenn.} {\bf 35}, No.9 (1909).

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13. Waldvogel, J.: Stable and unstable manifolds in planar triple collision. In Instabilities in Dynamical Sysytems, (V.Szebehely, ed.), pp.263-271. Dordrecht, D.Reidel, 1979.

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14. Wintner, A.: {\it The Analytical Foundation of Celestial mechanics}. Princeton University Press, 1941.

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15. Moeckel, R.: Thesis, University of Wisconsin, Madison.

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16. Simo, C.: Analysis of triple collision in the Isosceles problem. In Classical Mechanics and Dynamical Systems. New York, Marcel Dekker, 1980.