\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Trudy Inst. Theor. Astron.  SSSR} {\bf 14} (1969), 48-84

\vspace{0.5cm}
\begin{center}
{\Large
{\bf $BCO5e50F;MWAG$N1JG/@]F0$H5$8uJQF0$NE7J83XM}O@(B \\
Secular perturbations in the elements of \\
the Earth's orbit and the astronomical theory \\

\vspace{0.3cm}
of climate variations}
}

\vspace{0.3cm}
S. G. Sharaf and N. A. Budnikova
\end{center}

\vspace{0.5cm}
\noindent
{\bf $BMWLs(B}

$B$3$NO@J8$G$O5$8uJQF0$NE7J83XM}O@$NBhFsCJ3,$N7k2L$r07$&(B
\footnote{$BBh0lCJ3,$N7k2L$O(BITA Bulletin Vol.11, No.4(1967), 127$B$KH/I=$7$?(B.}. 
$BCO5e50F;MWAG$N1JG/JQ2=$K0MB8$7$FCO5e$N<u$1$kB@M[J|<M$NJQ2=$K4X$9$kLdBj$r(B
$B9M;!$9$k(B. $B5$8uJQF0$NE7J83XM}O@$NM}O@E*4pAC$K$D$$$F=R$Y$k(B. 
$BCO5e50F;MWAG$N1JG/JQ2=$r(B1950.0$B$N(B3$BI4K|G/A0$+$iI4K|G/8e$^$GI=$K$7$?(B. 
$BFnKLN>H>5e$N0^EY(B$65^{\circ}$$B$K$*$1$kB@M[J|<M$N1JG/E*$U$k$^$$$rF1$8(B
$B4|4V$KBP$7$F(B($BI8=`E*C10L$G(B)$BI=$*$h$S%0%i%U$N7A$G@bL@$7(B, $B$^$?(B3$B@iK|G/A0$+$iI4K|G/8e(B
$B$^$G$O%0%i%U$G@bL@$9$k(B($BEy2A0^EY$G(B). 
$BBg5$>eC<$X$NF|<MNL$NJQF0$HN&CO$NJ?6QE*9b$5$G$N5$29JQF0$N4X78$rD4$Y$k(B.

\vspace{0.3cm}
\begin{center}
{\bf $B=x(B} 
\end{center}

$BE7J83XE*$JMW0x$NJQ2=$K$h$C$FCO5e$N5$8u$,JQF0$9$kLdBj$OE7J83X<T$d?t3X<T$N(B
$B6=L#$r<f$$$F$-$?(B. Meach, Croll, Boll, Hargreaves$B$d$=$NB>$N?M!9$,(B
$B5$8uJQF0$HB@M[$KAjBPE*$JCO5e50F;$N7A$d0LCV$NJQ2=$H$N4X78$r<yN)$7$h$&$H(B
$BEXNO$7$F$-$?(B. 
$B$?$@%f!<%4%9%i%S%"$NE7J83X<T(BM.$B%_%i%s%3%S%C%A$@$1$,$=$N$[$H$s$I$N@836$r(B
$B$3$NLdBj$KJ{$2(B($B:G=i$N;E;v$,(B1913$BG/$G:G8e$N;E;v$,(B1957$BG/(B), 
$BB@M[J|<M$H(B3$B$D$NE7BNNO3XE*MW0x(B$-$ 
$B2+F;79<P3Q(B, $BCO5e50F;$NN%?4N($H6aF|E@7PEY(B$-$$B$K$h$C$F<f$-5/$3$5$l$k(B
$BCO5e$N5$8uJQF0$NAH?%$@$C$?E7J83XM}O@$rBG$AN)$F$k$3$H$K@.8y$7$?(B.  

$BB@M[Dj?t$*$h$SCO5e$N8xE><~4|$O1JG/JQF0$N>.$5$$NL$J$N$G(B, $B$3$NM}O@$G$O(B
$B0lDj$H$_$J$5$l$k(B. $B$"$k4|4V$N5$8uJQF0$NFCD'$H$7$F(BM.$B%_%i%s%3%S%C%A$,9M$($?$N$O(B, 
$B$=$N4|4V$K$*$$$FG.E*H>G/$N4V$K$"$k;XDj$7$?0^EY$NCOI=$NC10LLL@Q$"$?$j$K(B
$B>H<M$9$kB@M[J|<MNL$G$"$k(B. $BG.E*H>G/$H$O2F$NG$0U$NF|$K0^EY(B$\varphi$$B$N(B
$BC10LLL@Q$GF@$i$l$kG.NL$,(B, $BE_$NG$0U$NF|$KF1$8>l=j$GF@$i$l$kG.NL$r1[$($k$H$$$&(B
$B>r7o$GDj5A$5$l$k(B. 

$BCO<A;~Be$N$"$kG/$NG.E*H>G/$N4V$K0^EY(B$\varphi$$B$NC10LLL@Q$KF~<M$9$k(B
$BA4J|<M$H(B, $B8=Be$NG.E*H>G/$N4V$KF1$8>l=j$G<u$1<h$kA4J|<M$rHf3S$9$l$P(B, 
$BCO<A;~Be$N$=$NG/$KF@$?G.NL$,8=Be$h$jB?$+$C$?$N$+>/$J$+$C$?$N$+(B, $B$D$^$j$3$N4V$K(B
$B5$8uJQF0$,$"$C$?$N$+$I$&$+$rH=CG$9$k$3$H$,$G$-$k(B. 

M. $B%_%i%s%3%S%C%A(B(Milankovitch, 1939, 1941)$B$O(B, V. $B%_%7%3%S%C%A(B
(Mishkovitch, 1931)$B$N7W;;$7$?CO5e50F;$N1JG/@]F0MWAG$+$i=PH/$7$F(B, 
1800$BG/0JA0$N(B60$BK|G/4V$KCO5e$,B@M[$+$i<u$1$kJ|<M$NJQ2=$NI=$H?^$r:n$C$?(B. 
M. $B%_%i%s%3%S%C%A$N7k2L$r:G6a(BWoerkom(1958)$B$,CO5e50F;MWAG$N1JG/@]F0$N?7$7$$(B
$B%G!<%?(B(Brouwer \& Woerkom, 1950)$B$r;H$C$F7W;;$7$J$*$7$?(B. 
$BH`$O(B1950$BG/0JA0$NI4K|G/4V$NFnKLH>5e$N0^EY(B$65^{\circ}$$B$K$*$1$k2F$NF|<MNL6J@~(B
$B$r:n$C$?(B. 

$B%=O"CO<A3X8&5f=j(B(VSEGEI)$B$N%$%K%7%c%A%V$N$b$H(B, $B%=O"2J3X%"%+%G%_!<M}O@E7J83X(B
$B8&5f=j(B(ITA)$B$G<B9T$5$l$F$$$k$5$i$K?J$s$@5$8u$NE7J83XM}O@8&5f$O$b$C$HD94|$K$o$?$k(B
$BF|<MNL$N7W;;$rL\;X$7$F$$$k(B. 

$B$3$N8&5f$NBh0lCJ3,$OM}O@$NE7J83XE*4pAC$N=`Hw$G$"$C$?(B. $B$3$NItJ,$r>\$7$/(B
$B=R$Y$k$3$H$O$7$J$$(B. $B4pK\E*7k2L$OCx<T$i$NO@J8(B(Sharaf \& Budnikova, 1967)
$B$G=PHG$5$l$F$$$k(B. $B$?$@<!$N$3$H$r;XE&$7$F$*$3$&(B. 
$B2f!9$O(B, $BCO5e50F;$NN%?4N($H50F;79<P3Q$N(B2$B<!$^$G$N@:EY$G:P:9$N;03Q4X?t8x<0(B
$B$rF3$-(B, $B@QJ,Dj?t$H78?t$r7hDj$7$?(B. $B$=$N:](B, $BE7J8Dj?t$O9q:]E7J8O"9g$N(B
$BCM$r;H$$(B, Brouwer \& Woerkom(1950)$B$NOG@150F;$N1JG/@]F0O@$r;H$C$?(B.  
$B$^$?(B, 1950$BG/0JA0$N(B3$B@iK|G/4V$K$o$?$C$F(B5$B@iG/$4$H$K(B, $B2+F;79<P3Q(B, 
$BCO5e50F;$NN%?4N($H6aE@7PEY$N@]F0CM$rF@$?(B. 

$B:#2s$N8&5f$G$O5$8uJQF0$NE7J83XM}O@$NBhFsCJ3,$N4pK\7k2L$r=R$Y$k(B. $B$9$J$o$A(B, 
1950$BG/0JA0$N(B3$B@iK|G/4V$NFnKLN>H>5e$N0^EY(B$65^{\circ}$$B$NBg5$>eC<$K$*$1$k(B
$BF|<MNL$N8+@Q$b$j$G$"$k(B. 
$B$=$N:](B, $BF|<MNL$N4pK\8x<0$rF3$/$3$H(B, $B:G$b1F6ANO$NBg$-$J<~4|$rL@$i$+$K(B
$B$9$k$3$H(B, $BJd=uE*Dj?t$N7hDj(B, $B$*$h$S2f!9$N7k2L$H@h9T7k2L$NHf3S$KB?Bg$N(B
$BCm0U$r$O$i$C$?(B. 
$B$^$?6/D4$7$?$$$N$O(B, $B7W;;$7$?J|<M$N$U$k$^$$$H4QB,$5$l$kJ?6Q5$29$H$N4X78$rBG$A(B
$BN)$F$kEXNO$r$7$?$3$H$G$"$k(B. 

M. $B%_%i%s%3%S%C%A$OIT2DHr$N4JC12=$N$"$H$GM}O@E*$K(B, $BBg5$>eC<$NF|<MNLJQF0$H(B
$BN&CO$NJ?6QE*$J9bEY$K$*$1$k5$29JQF0$N4V$K4JC1$JAj4X$rF@$?(B. 
M. $B%_%i%s%3%S%C%A$K$h$l$P(B, $B5$29JQF0$OF|<MNLJQF0$K@5HfNc$9$k(B. 
$B9qFb30$N8&5f<T$N2??M$+$O(B, M. $B%_%i%s%3%S%C%A$N5a$a$?HfNc(B
$B78?t$OBg$-$9$.$k$H9M$($F$*$j(B, $B$7$?$,$C$FE7J83XE*$JMW0x$K$h$kF|<MNLJQF0$O(B
$B5$8uJQF0$KBg$-$J0UL#$r;}$?$J$$$H9M$($F$$$k(B. $B$3$l$i$NCx<T$O$U$D$&(B
Simpson(1940)$B$r0zMQ$9$k(B. $B2f!9$N<($7$?$H$3$m$K$h$k$H(B, Simpson$B$N8&5f$O(B 
$BAjL7=b$9$k2>Dj$r4^$s$G$*$j(B, $B$7$?$,$C$F5$29JQF0$,(B$1.5 - 2^{\circ}.0$$B$r(B
$B1[$($J$$$H$9$kH`$N5"7k$O$^$C$?$/@5$7$$$H$O1>$($J$$(B. 

$B$o$l$o$l$N7W;;$K$h$l$P(B, $B2F$N5$29$NJQF0$KBP$7$FF@$i$l$?78?t$O(B
M. $B%_%i%s%3%S%C%A$N%G!<%?$H$h$/0lCW$9$k(B. 
$BE_H>G/$N>l9g(B, $B78?t$O2F$KHf$Y$FH>J,$G$"$k(B(M. $B%_%i%s%3%S%C%A$G$O2F$bE_$bF10l$N(B
$BHfNc78?t$G$"$C$?(B). 
$B$3$l$i$N%G!<%?$r4p$K(B, $B:G6a$N(B3$BI4K|G/$N4V$K2F$N5$29$O(B$5-6^{\circ}$,
$BE_$N5$29$O(B$2-3^{\circ}$$B>e$,$C$?$j2<$,$C$?$j$7$?$H<gD%$G$-$k(B. 

$B$3$NO@J8$O(B3$B$D$N>O$HJd0d$+$i@.$C$F$$$k(B. 

I$B>O$G$O5$8uJQF0$NE7J83XM}O@$NLdBjE@$r@bL@$9$k(B. 

II$B>O$G$O2a5n(B3$B@iK|G/4V(B, $BL$MhI4K|G/4V$NN>H>5e$N0^EY(B$65^{\circ}$$B$K$*$1$k2F$HE_$N(B
$BF|<MNL$N8+@Q$b$j$r9T$J$&(B. 
$B$^$?$$$/$D$+$N?^$d5i?t$r=P$9(B. 

III$B>O$G$O(BSimpson$B$N8&5f$rHcH=E*$K8!F$$7(B, $BBg5$>eC<$K$*$1$kF|<MNLJQF0$H(B
$BN&CO$NJ?6QE*9b$5$K$*$1$k5$29JQF0$N4X78$r3NN)$9$k(B. 

$B@bL@$N$?$a$K(B, $BF|<MNL$N1JG/E*$U$k$^$$$N%0%i%U$rM?$($k(B. 

$BCx<T$O?4$+$i$N46<U$N5$;}$A$r(B, $B$3$N;E;v$KCmL\$7$F$/$l$?(BVSEGEI, A.V. Khabakov, 
G.C. Ganeshin, I.I. Krasnov$B$*$h$S(BV.A. Zubakov$B$K(B, $B$^$?7W;;$NJd=u$r$7$F$/$l(B
$B%0%i%U$r(B
$BIA$$$F$/$l$?2J3X%"%+%G%_!<M}O@E7J83X8&5f=j(B(ITA)$B$NF1N=(BP.K. Sadriev$B$KJ{$2$k(B.  

\vspace{0.3cm}
\begin{center}
{\bf I. $B5$8uJQF0$NE7J83XM}O@$N4pK\E*H/A[(B} 
\end{center}

\vspace{0.2cm}
\begin{center}
{\bf 1. $BE7J83XE*(B1$BG/$K$*$1$kB@M[J|<MNL(B} 
\end{center}

$W$$B$r;~9o(B$t_1$$B$+$i(B$t_2$$B$^$G$KCOI=$N0^EY(B$\varphi$$B$NC10LLL@Q$,<u$1$kJ?6QJ|<MNL(B
$B$H$7(B, $w$$B$rF1$80^EY$GC10LLL@Q$"$?$j<u$1$kJ|<M$NG/4V$NJ?6QE*$U$k$^$$$H$9$k$H(B, 
$$
   W = \displaystyle \int_{t_1}^{t_2} w dt .    \eqno(1)
$$

$w$$B$r7h$a$k<0$H$7$F(B, M. $B%_%i%s%3%S%C%A(B(1939)$B$O<!$N<0$rM?$($?(B.
$$
  w = \displaystyle \frac{1}{\pi} \frac{J_0}{\rho^2} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) .
$$
$B$?$@$7(B
$$ \begin{array}{rcl}
 J_0 & - & $BB@M[Dj?t(B, \\
 \varphi & - & $B0^EY(B, \\
 \rho & - & $BB@M[$NF07B%Y%/%H%k(B, \\
 \delta & - & $BB@M[$N@V0^(B, 
   \end{array}
$$
$$
    \cos \psi_0 = - \tan \varphi \tan \delta.   \eqno(2)
$$
$-\psi_0$$B$H(B$+\psi_0$$B$O0^EY(B$\varphi$$B$K$*$$$FB@M[$K>H$i$5$l$kItJ,$H>H$i$5$l$J$$(B
$BItJ,$rJ,$1$k6-3&$r<($7(B, $B;R8a@~$N0LCV$r7h$a$k(B. $BB@M[$O;R8a@~(B$\psi=0$$B$K$"$k(B. 
$B0^EY(B$\varphi$$B$K$*$$$F>H$i$5$l$kItJ,$O(B$2 \psi_0$$B$NI}$G$"$k(B. 

$B<0(B(1)$B$O<!$N7A$K=q$1$k(B. 
$$
  W = \displaystyle \frac{1}{\pi} \int_{t_1}^{t_2} \frac{J_0}{\rho_2} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) dt . 
\eqno(3)
$$

$B<!$N4X78<0$r;H$&(B. 
$$
\displaystyle \rho^2 \frac{d \lambda}{dt} = 
\displaystyle \frac{2 \pi a^2}{T} \sqrt{1 - e^2} . \eqno(3')
$$
$B$?$@$7(B
$$ \begin{array}{rcl}
  a=1 & - & $BCO5e$N50F;H>D97B(B, \\
  \lambda & - & $BB@M[$N7PEY(B, \\
  T & - & $BCO5e$N8xE><~4|(B, \\
  e & - & $BCO5e$N50F;N%?4N((B.
   \end{array}
$$
$B$3$N<0$r;H$&$H(B, (3)$B<0$NFHN)JQ?t$O(B$t$$B$+$i(B$\lambda$$B$K$J$k(B. $B@QJ,HO0O$O(B
$t_1, t_2$$B$+$i(B$\lambda', \lambda''$$B$KJQ$o$k(B. $B$3$&$7$F(B
$$
  W = \displaystyle \frac{1}{2 \pi^2} \int_{\lambda'}^{\lambda''} 
\frac{T J_0}{\sqrt{1 - e^2}} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda .        \eqno(4)
$$

$\psi_0$$B$r(B$\delta$$B$H(B$\lambda$$B$GI=$9$?$a$K<!$N8x<0$rMxMQ$9$k(B. 
$$ \begin{array}{ll}
  \sin \delta = \sin \varepsilon \sin \lambda, & 
  \cos \psi_0 = - \tan \varphi \tan \delta .
   \end{array}
$$
$B$3$3$G(B$\varepsilon$$B$O2+F;79<P3Q$G$"$k(B. 

$B$3$l$+$i=R$Y$kM}O@$G$O(B, $T$$B$H(B$J_0$$B$O0lDj$H$_$J$9(B. $BN%?4N((B$e$$B$H2+F;79<P3Q(B
$\varepsilon$$B$O69$$HO0O$GJQ2=$9$k(B($e$$B$O(B0$B$+$i(B0.067$B$^$G(B, $\varepsilon$$B$O(B
$22^{\circ}.068$$B$+$i(B$24^{\circ}.568$$B$^$G(B)(Sharaf \& Budnikova, 1967).
$B$7$?$,$C$F(B(4)$B<0$r@QJ,$9$k$H$-$K(B$e$$B$N(B2$B<!$OL5;k$G$-$k(B. $B$^$?(B
$\varepsilon = \varepsilon_0 + \Delta \varepsilon$$B$HCV$-(B, $W$$B$r(B
$\Delta \varepsilon$$B$N6R$GE83+$7(B, $\Delta \varepsilon$$B$K4X$7(B1$B<!$N9`$@$1$r(B
$B9MN8$9$l$P$h$$(B. $B$3$&$7$F(B
$$
  W = \displaystyle W_0 +  
\frac{\partial W}{\partial \varepsilon} \Delta \varepsilon 
 \eqno(5)
$$
$B$H$J$k(B. 

$W_0$$B$O(B(4)$B<0$G7h$^$k$,(B, $B@QJ,$r<B9T$9$k$H$-$K$O(B$T, J_0, e, \varepsilon$$B$O(B
$\lambda$$B$K0MB8$7$J$$(B, $B$D$^$j;~4V$K0MB8$7$J$$NL$H9M$($k(B. 

$B<!$N>r7o$,@.$jN)$D(B. 
$$ \begin{array}{ll}
\displaystyle \frac{\partial w}{\partial \psi_0} = 0, &
\displaystyle \frac{\partial \delta}{\partial \varepsilon} = 
\cot \varepsilon \tan \delta .  
   \end{array}
$$
$B$3$l$r9MN8$7$F<!$rF@$k(B. 
$$ \begin{array}{rl}
\displaystyle \frac{\partial W}{\partial \varepsilon} = 
& \displaystyle \frac{T J_0}{\pi^2 \sqrt{1 - e^2}} 
\left[ \int_{\lambda'}^{\lambda''} 
(\psi_0 \sin \varphi \cos \delta - \cos \varphi \sin \delta \sin \psi_0)
\cot \varepsilon_0 \tan \delta d \lambda \right. \\
& \displaystyle \left. 
+ \left( \frac{\partial \lambda''}{\partial \varepsilon} - 
\frac{\partial \lambda'}{\partial \varepsilon} \right)
 \int_{\lambda'}^{\lambda''} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0)
d \lambda \right] .
   \end{array}             \eqno(6)
$$

(4)$B$*$h$S(B(6)$B<0$K$h$l$P(B, $\varphi$$B$r(B$-\varphi$$B$KCV$-49$((B, $\lambda$$B$r(B
$\lambda + \pi$$B$KCV$-49$($F$b(B($BBP1~$7$F(B$\lambda', \lambda''$$B$r(B
$\lambda'+\pi, \lambda''+\pi$$B$KCV$-49$($k(B), $W$$B$NCM$OJQ$o$i$J$$(B. 
$B$3$l$+$iH=$k$H$*$j(B, $B0^EY(B$\varphi$$B$K$*$$$FB@M[$,2+F;>e(B$\lambda'$$B$+$i(B
$\lambda''$$B$^$G$N8L$rF0$/$H$-$K<u$1<h$kJ|<MNL$O(B, $B0^EY(B$-\varphi$$B$K$*$$$FB@M[$,(B
$\lambda'+\pi$$B$+$i(B
$\lambda''+\pi$$B$^$G$N8L$rF0$/$H$-$K<u$1<h$kJ|<MNL$KEy$7$$(B. 

$B$$$m$$$m$JE7J83XE*$J5(@a$K;XDj$7$?0^EY(B$\varphi$$B$K$*$$$FC10LLL@Q$"$?$j$K(B
$B<u$1<h$kG.NL(B$W$$B$r7hDj$9$kLdBj$K0\$m$&(B. $W_I, W_{II}, W_{III}, W_{IV}$$B$r(B
$B$=$l$>$l=U(B, $B2F(B, $B=)(B, $BE_$KKLH>5e$N0^EY(B$\varphi$$B$NC10LLL@Q$,<u$1<h$kG.NL(B$W$$B$N(B
$BCM$H$9$k(B. 

$W$$B$NCM$r7hDj$9$k$?$a$K$O0^EY(B$\varphi$$B$,6K0h30$K$"$k$+6K0h$K$"$k$+$K(B
$B1~$8$F(B2$B$D$N>l9g$r9M$($J$1$l$P$J$i$J$$(B. 

\vspace{0.3cm}
\noindent
\underline{$B6K0h30(B}. $B6K0h30$G$OB@M[$O0lG/Cf(B, $BCOJ?@~$+$i$N=PKW$r7+$jJV$9(B. 
$BE7J83XE*=U$N4V$K(B, $BB@M[$N7PEY$O(B$0^{\circ}$$B$+$i(B$90^{\circ}$$B$K(B, $B2F$N4V$K(B
$90^{\circ}$$B$+$i(B$180^{\circ}$$B$K0\$k(B. $B$7$?$,$C$F(B, (4)$B$H(B(6)$B<0$G(B$W_I$$B$r7h$a$k(B
$B$K$O(B, $\lambda'$$B$r(B$0$$B$K(B, $\lambda''$$B$r(B$\pi/2$$B$KCV$-49$((B, $W_{II}$$B$r7h$a$k(B
$B$K$O(B$\lambda'$$B$r(B$\pi/2$$B$K(B, $\lambda''$$B$r(B$\pi$$B$KCV$-49$($J$1$l$P$J$i$J$$(B. 
$B$9$k$H(B$W_I = W_{II}$$B$G$"$k$3$H$,H=$k(B. $W_s = W_I + W_{II}$$B$rE7J82FH>G/$N(B
$BJ|<M$H$9$l$P<!$rF@$k(B. 
$$ \begin{array}{rl}
 W_s = 
& \displaystyle \frac{T J_0}{\pi^2 \sqrt{1 - e^2}}\int_0^{\pi/2} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda\\
& \displaystyle + \frac{\pi}{180^{\circ}}
\frac{T J_0}{\pi^2 \sqrt{1 - e^2}} \left[ \int_0^{\pi/2}
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0)
d \lambda \right. \\
& \hspace{5cm} \displaystyle \left. 
- \cos \varphi \int_0^{\pi/2} \frac{\sin \psi_0}{\cos \delta} 
d \lambda \right] \cot \varepsilon_0 (\Delta)^{\circ} .  
   \end{array}     \eqno(7)
$$ 

$W_{III}$$B$H(B$W_{IV}$$B$r7h$a$k$K$O(B, (4)$B$*$h$S(B(6)$B<0$G@QJ,HO0O$r$=$l$>$l=)$J$i(B
$\pi$$B$H(B$\frac{3}{2} \pi$$B$K(B, $BE_$J$i(B$\frac{3}{2} \pi$$B$H(B$2 \pi$$B$KJQ$($l$P$h$$(B. 
$W_{III} = W_{IV}$$B$KCm0U$7$J$,$i(B, $B$$$/$i$+7W;;$9$l$P(B, 
$BE7J8E_H>G/$K$*$1$kJ|<M(B
$$
  W_w = W_{III} + W_{IV}
$$
$B$H$7$F<!$rF@$k(B. 
$$
  W_w = \displaystyle W_s - \frac{T J_0}{\pi \sqrt{1 - e^2}}
\left[ \sin \varphi \sin \varepsilon_0 + \frac{\pi}{180^{\circ}} 
\sin \varphi \cos \varepsilon_0 (\Delta \varepsilon)^{\circ} \right] . 
 \eqno(8)
$$

\vspace{0.3cm}
\noindent
\underline{$B6K0h(B}. $BKL6K0h$G$O4X78<0(B$\varphi > \frac{\pi}{2} - \varepsilon$
$B$,K~$?$5$l(B, $BFn6K0h$G$O(B$\varphi < - \frac{\pi}{2} + \varepsilon$$B$,K~$?$5$l$k(B. 
$B$3$3$G$O(B, $BB@M[$,=PKW$9$k>l9g(B, $BB@M[$,D@$^$J$$>l9g(B, $BB@M[$,=P8=$7$J$$>l9g$,$"$k(B. 
$BB@M[$NFnCf$HKLCf$N9bEY$O<!<0$G7h$^$k(B. 
$$ 
\left.
   \begin{array}{l}
 h_u = \displaystyle \frac{\pi}{2} - \varphi + \delta , \\
 h_l = \displaystyle \varphi + \delta - \frac{\pi}{2} .
   \end{array} 
\right\}  \eqno(9)
$$
$B$7$?$,$C$FB@M[$,=PKW$9$kF|$K$O<!<0$,@.$jN)$D$O$:$G$"$k(B. 
$$ \begin{array}{ll}
  h_u > 0, & h_l < 0,
   \end{array} 
$$
$B$9$J$o$A(B
$$
\displaystyle - \left( \frac{\pi}{2} - \varphi \right) < \delta < 
\frac{\pi}{2} + \varphi .
$$
$BB@M[$,D@$^$J$$F|$K$O(B
$$ \begin{array}{ll}
  h_l > 0, & \delta > \frac{\pi}{2} - \varphi .
   \end{array}
$$
$BB@M[$,>:$i$J$$F|$K$O(B
$$ \begin{array}{ll}
  h_u < 0, & \delta < - \frac{\pi}{2} + \varphi .
   \end{array}
$$

$B=U$N4V$K@V0^(B$\delta$$B$O<!$N$h$&$KJQ2=$9$k(B. $BB@M[$,=PKW$9$kF|$K$O(B
$\delta_1=0$$B$+$i(B$\delta_2 = \frac{\pi}{2} - \varphi$$B$^$GF0$/(B. $BBP1~$7$F(B
$BF1$84|4V$KB@M[$O7PEY$G(B$\lambda'=0$$B$+$i(B$\lambda''= \lambda_1$$B$^$GF0$/(B. 
$B$?$@$7(B
$$ \begin{array}{c}
 \sin \lambda'' = \displaystyle \frac{\sin \delta_2}{\sin \varepsilon} 
= \displaystyle \frac{\cos \varphi}{\sin \varepsilon} , \\
 \lambda'' = \lambda_1 < \displaystyle \frac{\pi}{2} .
   \end{array}
$$

$BB@M[$,D@$^$J$$F|$K$O(B, $B@V0^(B$\delta$$B$O(B$\delta'_1 = \frac{\pi}{2} - \varphi$$B$+$i(B
$\delta'_2 = \varepsilon$$B$^$GF0$-(B, $BBP1~$7$F(B$\lambda'=\lambda_1$$B$+$i(B
$\lambda'' = \frac{\pi}{2}$$B$^$GF0$/(B.

$BE7J82F$N>l9g(B, $BB@M[$,D@$^$J$$F|$K$O(B
$$ \begin{array}{llll}
  \delta'_1 = \varepsilon, & \delta'_2 = \frac{\pi}{2} - \varphi; & 
  \lambda' = \frac{\pi}{2}, & \lambda'' = \pi - \lambda_1 , 
   \end{array}
$$
$BB@M[$,=PKW$9$kF|$K$O(B
$$ \begin{array}{llll}
  \delta_1 = \frac{\pi}{2} - \varphi, & \delta_2 = ; & 
  \lambda' = \pi - \lambda_1, & \lambda''_2 = \pi ,
   \end{array}
$$
$BE7J8=)$N>l9g(B, $BB@M[$,=PKW$9$kF|$K$O(B
$$ \begin{array}{llll}
  \delta_1 = 0, & \delta_2 = - \frac{\pi}{2} + \varphi; & 
  \lambda' = \pi, & \lambda'' = \pi + \lambda_1 ,
   \end{array}
$$
$BB@M[$,>:$i$J$$F|$K$O(B
$$ \begin{array}{llll}
  \delta''_1 = - \frac{\pi}{2} + \varphi, & \delta''_2 = - \varepsilon; & 
  \lambda' = \pi + \lambda_1, & \lambda''_2 = \frac{3}{2} \pi ,
   \end{array}
$$
$BE7J8E_$N>l9g(B, $BB@M[$,>:$i$J$$F|$K$O(B
$$ \begin{array}{llll}
  \delta''_1 = - \varepsilon, & \delta''_2 = - \frac{\pi}{2} + \varphi; & 
  \lambda' = \frac{3}{2} \pi, & \lambda'' = 2 \pi - \lambda_1 , 
   \end{array}
$$
$BB@M[$,=PKW$9$kF|$K$O(B
$$ \begin{array}{llll}
  \delta_1 = - \frac{\pi}{2} + \varphi, & \delta_2 = 0; & 
  \lambda' = 2 \pi - \lambda_1, & \lambda'' = 2 \pi .
   \end{array}
$$

(4)$B$*$h$S(B(6)$B<0$K>e$GF@$i$l$?@QJ,8B3&(B$\lambda'$$B$H(B$\lambda''$$B$rBeF~$7(B, 
$BB@M[$,D@$^$J$$F|$K$O(B$\psi_0=\pi$$B$G$"$j(B, $BB@M[$,>:$i$J$$F|$K$O(B$\psi_0=0$$B$G(B
$B$"$k$3$H$r9MN8$9$l$P(B, $BKL6K0h$N0^EY(B$\varphi$$B$G$O0J2<$NG.NL$r<u$1<h$k(B. 
$$ \begin{array}{rl}
 W_s = 
& \displaystyle \frac{T J_0}{\pi^2 \sqrt{1 - e^2}}
\left[ \int_0^{\lambda_1} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda + \pi \sin \varphi \sin \varepsilon \cos \lambda_1 \right] \\ 
& \displaystyle + \frac{T J_0}{\pi \sqrt{1 - e^2} 180^{\circ}}
\cot \varepsilon (\Delta \varepsilon)^{\circ} 
\left[ \int_0^{\lambda_1} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda \right. \\
& \hspace*{5cm} \displaystyle \left.  
+ \pi \sin \varphi \sin \varepsilon \cos \lambda_1 - 
\cos \varphi \int_0^{\lambda_1} \frac{\cos \psi_0}{\cos \delta} d \lambda 
\right] ,
   \end{array} \eqno(10)
$$
$$
 W_w = \displaystyle W_s - \frac{T J_0}{\pi \sqrt{1 - e^2}} \sin \varphi 
\sin \varepsilon - \frac{T J_0}{\pi \sqrt{1 - e^2} 180^{\circ}}
\sin \varphi \cos \varepsilon (\Delta \varepsilon)^{\circ} . \eqno(11)
$$
$B$9$G$K;XE&$7$?$H$*$j(B, $\sin \lambda_1 = \frac{\cos \varphi}{\sin \varepsilon}$
$B$G$"$j(B, $\Delta \varepsilon$$B$OEY$GI=$5$l$F$$$k(B. 
$W_s$$B$*$h$S(B$W_w$$B$N=i4|CM$r(B$W_s^0$$B$*$h$S(B$W_w^0$$B$GI=$7(B, $\Delta \varepsilon$$B$N(B
$B78?t$r(B$\Delta W_s$$B$*$h$S(B$\Delta W_w$$B$GI=$9(B.
$B$9$k$H(B
$$
\left.
   \begin{array}{l}
 W_s = W_s^0 + \Delta W_s \Delta \varepsilon, \\
 W_w = W_w^0 + \Delta W_w \Delta \varepsilon. 
   \end{array}
\right\} \eqno(12)
$$

$B6K0h30$KBP$7$F$O(B, $B<0(B(7)$B$H(B(8)$B$h$j<!$rF@$k(B. 
$$
\left.
   \begin{array}{c}
 W_s^0 = \displaystyle \frac{T J_0}{\pi^2 \sqrt{1 - e^2}} \int_0^{\pi/2} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda , \\
 W_w^0 = \displaystyle W_s^0 - \frac{T J_0}{\pi \sqrt{1 - e^2}} 
\sin \varphi \sin \varepsilon 
   \end{array}
\right\}    \eqno(13)
$$
$B$*$h$S(B
$$
\left.
   \begin{array}{c}
 \Delta W_s = \displaystyle \frac{\pi}{180^{\circ}} \cot \varepsilon 
\left[ W_s^0 - \frac{T J_0 \cos \varphi}{\pi^2 \sqrt{1 - e^2}} 
\int_0^{\pi/2} \frac{\sin \psi_0}{\cos \delta} d \lambda \right] , \\
 \Delta W_w = \displaystyle \Delta W_s - \frac{\pi}{180^{\circ}} 
\cot \varepsilon (W_w^0 - W_s^0) .
   \end{array}
\right\} \eqno(14)
$$  

$B6K0h$KBP$7$F$O(B(10)$B$H(B(11)$B<0$h$j<!$rF@$k(B. 
$$
\left.
   \begin{array}{c}
 W_s^0 = \displaystyle \frac{T J_0}{\pi^2 \sqrt{1 - e^2}} 
\left[ \int_0^{\lambda_1} 
(\psi_0 \sin \varphi \sin \delta + \cos \varphi \cos \delta \sin \psi_0) 
d \lambda + \pi \sin \varphi \sin \varepsilon \cos \lambda_1 \right], \\
 W_w^0 = \displaystyle W_s^0 - \frac{T J_0}{\pi \sqrt{1 - e^2}} 
\sin \varphi \sin \varepsilon ,
   \end{array}
\right\}    \eqno(15)
$$
$B$*$h$S(B
$$
\left.
   \begin{array}{c}
 \Delta W_s = \displaystyle \frac{\pi}{180^{\circ}} \cot \varepsilon 
\left[ W_s^0 - \frac{T J_0 \cos \varphi}{\pi^2 \sqrt{1 - e^2}} 
\int_0^{\lambda_1} \frac{\sin \psi_0}{\cos \delta} d \lambda \right] , \\

 \Delta W_w = \displaystyle \Delta W_s - \frac{T J_0}{180^{\circ} 
\sqrt{1 - e^2}} \sin \varphi \cos \varepsilon ,
   \end{array}
\right\} \eqno(16)
$$  
$B$^$?$O(B
$$
  \Delta W_w = \displaystyle \Delta W_s + \frac{\pi}{180^{\circ}}
\cot \varepsilon (W_w^0 - W_s^0) . 
$$

$J_0$$B$H(B$T$$B$K4X$9$k2f!9$N2>Dj$N$b$H$G$O(B, $W_s^0, W_w^0, \Delta W_s^0$$B$*$h$S(B
$\Delta W_w^0$$B$O0^EY$,M?$($i$l$l$P0lDj$G$"$k(B. (13)-(16)$B<0$+$i(B
$B$3$l$i$r7W;;$9$k$?$a$K$O(B, M. $B%_%i%s%3%S%C%A$,$7$?$h$&$K(B, $BHs@QJ,4X?t$r(B
$B>.$5$JNL$N6R5i?t$KE83+$9$l$P$h$$(B. $B$?$@$3$N6R5i?t$O(B$\varphi$$B$,(B
$BBg$-$/$J$k$H<}B+$,CY$/$J$k$N$G(B, $B2?$i$+$N5a@QK!$r;H$&J}$,L\E*$K3p$C$F$$$k(B. 

(13)-(16)$B<0$OKLH>5e$N0^EY(B$\varphi$$B$KBP$9$k$b$N$G$"$k(B. 
$B8x<0(B(4)$B$H(B(6)$B$+$i=PH/$7$F(B(13)-(16)$B$r=P$7$?$1$l$I$b(B, $B>e$G;XE&$7$?$H$*$j(B, 
$B$3$N<0$G(B$\varphi$$B$r(B$-\varphi$$B$K(B, $\lambda$$B$r(B$\lambda + \pi$$B$KCV$-49$($F$b(B
$W$$B$NCM$OJQ$o$i$J$$(B. $B$3$N$3$H$+$i(B, $BFnH>5e$NG$0U$N0^EY$NC10LLL@Q$,FnH>5e$N(B
$BE7J82FH>G/$K<u$1<h$kJ|<M$O(B, $BKLH>5e$NF1$80^EY$NC10LLL@Q$,KLH>5e$NE7J82FH>G/$K(B
$B<u$1<h$kJ|<M$H$^$C$?$/F1$8$G$"$k$H8@$($k(B. 
$B$3$N>u67$OE7J8E_H>G/$K$D$$$F$b@.$jN)$D(B. 

$B$7$?$,$C$F(B, $BFnH>5e$N0^EY(B$\varphi$$B$K$*$$$FC10LLL@Q$,E7J82F$*$h$SE_H>G/$K<u$1<h$k(B
$BJ|<M$r(B$\overline{W}_s$$B$*$h$S(B$\overline{W}_w$$B$HI=$;$P(B, 
$$
\left.
   \begin{array}{l}
  \overline{W}_s = W_s^0 + \Delta W_s \Delta \varepsilon, \\
  \overline{W}_w = W_w^0 + \Delta W_w \Delta \varepsilon .
   \end{array}
\right\}  \eqno(17)
$$

$B0^EY(B$\varphi$$B$NC10LLL@Q$,(B1$BG/4V$K<u$1<h$kJ|<MNL$O(B
$$
 W_T = W_s + W_w = W_s^0 + W_w^0 + (\Delta W_s + \Delta W_w) \Delta 
\varepsilon .
$$

(17)$B$K$h$j(B,
$$
   \overline{W}_T = W_T .
$$

(17)$B<0$KF~$C$F$$$k(B$W_s^0, W_w^0, \Delta W_s, \Delta W_w$$B$O(B(13)-(16)$B<0$+$i(B
$B7h$^$k(B. 

$B$?$@$7Cm0U$9$Y$-$O(B, $W_s = \overline{W}_s$$B$+$D(B$W_w = \overline{W}_w$$B$G$O(B
$B$"$k$1$l$I$b(B, $BKLH>5e$HFnH>5e$G$OF10l$N0^EY$G$b3FE7J8H>G/$K$*$1$k29EY>r7o$,(B
$B0[$J$k$3$H$G$"$k(B. $B%1%W%i!<$NBhFsK!B'$K$h$l$P(B, $BBJ1_$NF1$8D9$5$N8L$rB@M[$,(B
$BF0$/$KMW$9$k;~4V$OBJ1_>e$N>l=j$K$h$C$F0[$J$k(B. 
$B<B:](B, $B8=:_$G$O2F$NF|$KF~<M$9$kG.$OKLH>5e$NJ}$,FnH>5e$h$j>/$J$$(B. 

$BKLH>5e$NE7J82FH>G/$HE_H>G/$N7QB3;~4V$r(B$T_s$$B$*$h$S(B$T_w$$B$HI=$7(B, $BFnH>5e$N(B
$BBP1~$9$kNL$r(B$\overline{T}_s$$B$*$h$S(B$\overline{T}_w$$B$HI=$9$3$H$K$9$k(B. 

$BL@$i$+$K(B
$$ \begin{array}{lll}
  T_s + T_w = T, & \overline{T}_s = T_w, & \overline{T}_w = T_s . 
   \end{array}
$$

$BCO5e$N50F;N%?4N((B$e$$B$N(B1$B<!$^$G$N@:EY$G<!$N$h$&$K=q$1$k(B. 
$$
\left.
   \begin{array}{l}
 T_s = \displaystyle 
\frac{1}{2} T \left( 1 + \frac{4e}{\pi} \sin \Pi \right) , \\
 T_w = \displaystyle 
\frac{1}{2} T \left( 1 - \frac{4e}{\pi} \sin \Pi \right) , \\  
 T_s - T_w = \displaystyle T \frac{4e}{\pi} \sin \Pi .
   \end{array}
\right\}   \eqno(18)
$$
$B$3$3$G(B$\Pi$$B$O=UJ,E@$+$iB,$C$?CO5e50F;$N6aF|E@7PEY$G$"$k(B. 

(12)$B$*$h$S(B(17)$B<0$O(B(13)-(16)$B<0$HJ;$;$F(B, $BCO<A3XE*2a5n$*$h$SL$Mh$NG$0U$NG/$K(B
$BKLH>5e$^$?$OFnH>5e$NM?$($i$l$?0^EY$NC10LLL@Q$,E7J8H>G/$K<u$1<h$kG.NL(B
$B$r7W;;$G$-$k(B. $B$=$N:](B, $W_s, W_w, \overline{W}_s, \overline{W}_w$$B$O(B
$B2+F;79<P3Q(B$\varepsilon$$B$NJQ2=$N4X?t$G$"$j(B, $B$^$?M?$($i$l$?0^EY$*$h$SM?$($i$l$?(B
$BH>G/$K4X$7$F$O0lDj$NNL$G$"$k(B.

$B$7$+$7CO<A3XE*2a5n$+$i$"$k(B1$BG/$rA*$s$G(B(12)$B$*$h$S(B(17)$B<0$+$i(B
$W_s, W_w, \overline{W}_s, \overline{W}_w$$B$r7W;;$7$F8=:_$NCM(B
$W_s^0, W_w^0, \overline{W}_s^0, \overline{W}_w^0$$B$HHf3S$7$F$b(B
$B5$8uJQF0$NFCD'$r$H$i$($F$$$J$$(B. $B$H$$$&$N$O(B, $B$3$l$i$NNL$O(B
$BE7J8H>G/$N4V$K0^EY(B$\varphi$$B$,<u$1<h$kG.NL$rM?$($k$N$G$"$k$,(B, 
(18)$B<0$,<($9$h$&$K(B, $B2FH>G/$*$h$SE_H>G/$N$OD9$5$,0c$&$N$G$"$k(B. 
$B$=$N>e(B, (18)$B<0$K$O(B$e$$B$H(B$\Pi$$B$,F~$C$F$*$j(B, $B$3$l$i$O;~4V$H$H$b$KJQ2=$9$k(B
$B$N$G$"$k(B.$B$7$?$,$C$FE7J8H>G/$ND9$5$b0lDj$K$H$I$^$C$F$O$$$J$$(B.
$B$3$N$h$&$K(B, $W_s, W_w, \overline{W}_s, \overline{W}_w$$B$*$h$S(B
$W_s^0, W_w^0, \overline{W}_s^0, \overline{W}_w^0$$B$O0[$J$kD9$5$N4|4V$K$o$?$k(B
$BJ|<MNL$rM?$((B, $B$=$N;~4VJQ2=$OM?$($i$l$?0^EY$N5$8uJQF0$r==J,$h$/FCD'$E$1$k(B
$B$H$O8@$$$,$?$$(B. 

$BHf3S$N$?$a$KJL$NNL$rF3F~$G$-$k(B. $B$=$l$OB@M[J|<MJ?6Q6/EY$G$"$k(B. 
$BKLH>5e$N$"$kA*$P$l$?0^EY$K$*$1$kE7J82F$*$h$SE_H>G/$NB@M[J|<M$NJ?6Q6/EY(B
$w_s$$B$*$h$S(B$w_w$$B$O<!<0$GDj5A$5$l$k(B. 
$$ \begin{array}{ll}
  w_s = \displaystyle \frac{W_s}{T_s}, & 
  w_w = \displaystyle \frac{W_w}{T_w} .
   \end{array}
$$
$BF1MM$KFnH>5e$KBP$7$F$O(B
$$ \begin{array}{ll}
  \overline{w}_s = \displaystyle \frac{\overline{W}_s}{T_s}, & 
  \overline{w}_w = \displaystyle \frac{\overline{W}_w}{T_w} ,
   \end{array}
$$
$B$"$k$$$O(B
$$ \begin{array}{ll}
  \overline{w}_w = \displaystyle \frac{W_s}{T_w}, & 
  \overline{w}_s = \displaystyle \frac{W_w}{T_s} .
   \end{array}
$$

(12)$B$*$h$S(B(18)$B<0$rMxMQ$7(B, $BN%?4N((B$e$$B$*$h$S2+F;79<P3Q$NJQ2=(B$\Delta \varepsilon$
$B$N(B1$B<!$N9`$K@)8B$7$F<!$rF@$k(B. 
$$
\left.
   \begin{array}{l}
 w_s = \displaystyle \frac{2}{T} W_s^0 
\left( 1 + \frac{\Delta W_s}{W_s^0} \Delta \varepsilon - 
\frac{4}{\pi} e \sin \Pi \right) , \\
 w_w = \displaystyle \frac{2}{T} W_w^0 
\left( 1 + \frac{\Delta W_w}{W_w^0} \Delta \varepsilon + 
\frac{4}{\pi} e \sin \Pi \right) , \\

 \overline{w}_s = \displaystyle \frac{2}{T} W_s^0 
\left( 1 + \frac{\Delta W_s}{W_s^0} \Delta \varepsilon + 
\frac{4}{\pi} e \sin \Pi \right) , \\
 \overline{w}_w = \displaystyle \frac{2}{T} W_w^0 
\left( 1 + \frac{\Delta W_w}{W_w^0} \Delta \varepsilon - 
\frac{4}{\pi} e \sin \Pi \right) . \\
   \end{array}
\right\}   \eqno(19)
$$

(19)$B<0$K$h$l$P(B, $B2+F;79<P3Q$NJQ2=$O(B, $BKL$^$?$OFnH>5e$NA*$P$l$?0^EY$K$*$$$F(B
$B$=$l$>$l$N2F$^$?$OE_E7J8H>G/$NJ?6QJ|<M6/EY$KF1$81F6A$rM?$($k(B.
2$B<oN`$NE7J8H>G/$N7QB3;~4V$N:9$O(B$e \sin \Pi$$B$KHfNc$7(B, $B$=$N7QB3;~4V$NJQ2=$O(B 
$B0[$J$kH>5e$K5U$N1F6A$rM?$($k(B. 

$B$3$&$7$FFnKLN>H>5e$NF10l$N0^EY$O(B, $B$=$l$>$l$NE7J82F$^$?$OE_H>G/$K(B
$B0[$J$kJ?6QJ|<M6/EY$r<u$1<h$k(B. $BG/J?6QJ|<M6/EY$O(B
$$
\left.
   \begin{array}{l}
 w_T = \displaystyle \frac{1}{T} (W_s + W_w) = \displaystyle \frac{1}{T} 
[ W_s^0 + W_w^0 + (\Delta W_s + \Delta W_w) \Delta \varepsilon ] , \\
 \overline{w}_T = \displaystyle \frac{1}{T}
[ W_s^0 + W_w^0 + (\Delta W_s + \Delta W_w) \Delta \varepsilon ] .

   \end{array}
\right\} \eqno(20)
$$








