\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad \mbox{d}s^{2}
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \sin^{2} \theta \, \mbox{d}\varphi^{2}
 - r^{2} \, \mbox{d}\theta^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad \mbox{d}s
 & = & \left ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \sin^{2} \theta \, \mbox{d}\varphi^{2}
 - r^{2} \, \mbox{d}\theta^{2} \right )^{\frac{1}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad \mbox{d}s
 & = & \sqrt{g_{\mu\nu} \, \mbox{d}x^{\mu} \, \mbox{d}x^{\nu}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad s
 & = & \int\limits_{A}^{B} \, \mbox{d}s \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad \delta s
 & = & 0 \\
\delta \int\limits_{A}^{B} \, \mbox{d}s
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad \qquad \qquad \delta \int\limits_{A}^{B} \, \mbox{d}s
 & = & 0 \\
\delta \int\limits_{A}^{B} \sqrt{g_{\mu\nu} \, \mbox{d}x^{\mu} \, \mbox{d}x^{\nu}}
 & = & 0 \\
\delta \int\limits_{A}^{B} \sqrt{g_{\mu\nu} \, \frac{\mbox{d}x^{\mu}}{\mbox{d}s} \, \frac{\mbox{d}x^{\nu}}{\mbox{d}s}} \, \mbox{d}s
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
 \delta \int\limits_{A}^{B} \sqrt{g_{\mu\nu} \, \frac{\mbox{d}x^{\mu}}{\mbox{d}s} \, \frac{\mbox{d}x^{\nu}}{\mbox{d}s}} \, \mbox{d}s
 & = & 0 \\
\delta \int\limits_{A}^{B} \left ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \left ( \frac{\mbox{d}t}{\mbox{d}s} \right )^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}s} \right )^{2}
 - r^{2} \sin^{2} \theta \left ( \frac{\mbox{d}\varphi}{\mbox{d}s} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\theta}{\mbox{d}s} \right )^{2} \right )^{\frac{1}{2}} \, \mbox{d}s
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{\partial f}{\partial y_{n}^{\prime}} \right )
 & = & \frac{\partial f}{\partial y_{n}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9a)} \qquad \frac{\mbox{d}}{\mbox{d}s} \Biggl ( \frac{\partial}{\partial (\mbox{d}t/\mbox{d}s)}
 \Biggl ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \left ( \frac{\mbox{d}t}{\mbox{d}s} \right )^{2} \qquad \qquad \quad \\
 & & \\
- \, \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}s} \right )^{2}
 - r^{2} \sin^{2} \theta \left ( \frac{\mbox{d}\varphi}{\mbox{d}s} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\theta}{\mbox{d}s} \right )^{2} \Biggr ) \Biggr )
 & = & \frac{\partial }{\partial t}
 \Biggl ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \left ( \frac{\mbox{d}t}{\mbox{d}s} \right )^{2} \\
 & & \qquad - \, \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}s} \right )^{2} \\
 & & \qquad \qquad - \, r^{2} \sin^{2} \theta \left ( \frac{\mbox{d}\varphi}{\mbox{d}s} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\theta}{\mbox{d}s} \right )^{2} \Biggr ) \\
\frac{\mbox{d}}{\mbox{d}s} \left ( 2 \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \frac{\mbox{d}t}{\mbox{d}s} + 0 + 0 + 0 \right )
 & = & 0 + 0 + 0 + 0 \\
\frac{\mbox{d}}{\mbox{d}s} \left ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \frac{\mbox{d}t}{\mbox{d}s} \right )
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9b)} \qquad \frac{\mbox{d}}{\mbox{d}s} \Biggl ( \frac{\partial}{\partial (\mbox{d}\varphi/\mbox{d}s)}
 \Biggl ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \left ( \frac{\mbox{d}t}{\mbox{d}s} \right )^{2} \qquad \qquad \quad \\
 & & \\
- \, \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}s} \right )^{2}
 - r^{2} \sin^{2} \theta \left ( \frac{\mbox{d}\varphi}{\mbox{d}s} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\theta}{\mbox{d}s} \right )^{2} \Biggr ) \Biggr )
 & = & \frac{\partial }{\partial \varphi}
 \Biggl ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \left ( \frac{\mbox{d}t}{\mbox{d}s} \right )^{2} \\
 & & \qquad - \, \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}s} \right )^{2} \\
 & & \qquad \qquad - \, r^{2} \sin^{2} \theta \left ( \frac{\mbox{d}\varphi}{\mbox{d}s} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\theta}{\mbox{d}s} \right )^{2} \Biggr ) \\
\frac{\mbox{d}}{\mbox{d}s} \left ( 0 + 0 - 2 r^{2} \sin^{2} \theta \, \frac{\mbox{d}\varphi}{\mbox{d}s} + 0 \right )
 & = & 0 + 0 + 0 + 0 \\
\frac{\mbox{d}}{\mbox{d}s} \left ( r^{2} \sin^{2} \theta \, \frac{\mbox{d}\varphi}{\mbox{d}s} \right )
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad \theta
 & = & \frac{\pi}{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11a)} \qquad \sin^{2} \theta
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11b)} \qquad \mbox{d}\theta
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad \mbox{d}s^{2}
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \sin^{2} \theta \, \mbox{d}\varphi^{2}
 - r^{2} \, \mbox{d}\theta^{2} \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \cdot 1 \cdot \, \mbox{d}\varphi^{2}
 - r^{2} \cdot 0 \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \, \mbox{d}\varphi^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad \frac{\mbox{d}}{\mbox{d}s} \left ( r^{2} \sin^{2} \theta \, \frac{\mbox{d}\varphi}{\mbox{d}s} \right )
 & = & 0 \\
\frac{\mbox{d}}{\mbox{d}s} \left ( r^{2} \cdot 1 \cdot \frac{\mbox{d}\varphi}{\mbox{d}s} \right )
 & = & 0 \\
\frac{\mbox{d}}{\mbox{d}s} \left ( r^{2} \, \frac{\mbox{d}\varphi}{\mbox{d}s} \right )
 & = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad \int \frac{\mbox{d}}{\mbox{d}s} \left ( r^{2} \, \frac{\mbox{d}\varphi}{\mbox{d}s} \right ) \, \mbox{d}s
 & = & \int 0 \cdot \, \mbox{d}s \\
r^{2} \, \frac{\mbox{d}\varphi}{\mbox{d}s}
 & = & J \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad r^{2} \, \frac{\mbox{d}\varphi}{\mbox{d}s}
 & = & J \\
\frac{\mbox{d}\varphi}{\mbox{d}s}
 & = & \frac{J}{r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad \int \frac{\mbox{d}}{\mbox{d}s} \left ( \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \frac{\mbox{d}t}{\mbox{d}s} \right ) \, \mbox{d}s
 & = & \int 0 \cdot \, \mbox{d}s \\
\left ( 1 - \frac{2 G M}{r c^{2}} \right ) \frac{\mbox{d}t}{\mbox{d}s}
 & = & K \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \frac{\mbox{d}t}{\mbox{d}s}
 & = & K \\
\frac{\mbox{d}t}{\mbox{d}s}
 & = & \frac{K}{1 - \frac{2 G M}{r c^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad \frac{\frac{\mbox{d}\varphi}{\mbox{d}s}}{\frac{\mbox{d}t}{\mbox{d}s}}
 & = & \frac{\frac{J}{r^{2}}}{\frac{K}{1 - \frac{2 G M}{r c^{2}}}} \\
\frac{\mbox{d}\varphi}{\mbox{d}t}
 & = & \frac{J}{K r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad \mbox{d}s^{2}
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \, \mbox{d}\varphi^{2} \\
0
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \, \mbox{d}\varphi^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad 0
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2} \, \mbox{d}t^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \, \mbox{d}r^{2}
 - r^{2} \, \mbox{d}\varphi^{2} \\
0
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\varphi}{\mbox{d}t} \right )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21)} \qquad 0
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 - r^{2} \left ( \frac{\mbox{d}\varphi}{\mbox{d}t} \right )^{2} \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 - r^{2} \left ( \frac{J}{K r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \right )^{2} \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) c^{2}
 - \frac{1}{1 - \frac{2 G M}{r c^{2}}} \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 - \frac{J^{2}}{K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} c^{2}
 - \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 - \frac{J^{2}}{K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{3} \\
\left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} c^{2}
 - \frac{J^{2}}{K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{3} \\
& = & \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} \left ( c^{2} - \frac{J^{2}}{K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \right ) \\
& = & c^{2} \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} \left ( 1 - \frac{J^{2}}{c^{2} K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22)} \qquad R_{s}
 & = & \frac{2 G M}{c^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23)} \qquad \frac{\mbox{d}\varphi}{\mbox{d}t}
 & = & \frac{J}{K r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \\
& = & \frac{J}{K r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24)} \qquad \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 & = & c^{2} \left ( 1 - \frac{2 G M}{r c^{2}} \right )^{2} \left ( 1 - \frac{J^{2}}{c^{2} K^{2} r^{2}} \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \right ) \\
& = & c^{2} \left ( 1 - \frac{R_{s}}{r} \right )^{2} \left ( 1 - \frac{J^{2}}{c^{2} K^{2} r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25a)} \qquad E
 & = & c K \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25b)} \qquad P
 & = & \frac{J}{E} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2}
 & = & c^{2} \left ( 1 - \frac{R_{s}}{r} \right )^{2} \left ( 1 - \frac{J^{2}}{c^{2} K^{2} r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \right ) \\
& = & c^{2} \left ( 1 - \frac{R_{s}}{r} \right )^{2} \left ( 1 - \frac{J^{2}}{E^{2} r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \right ) \\
& = & c^{2} \left ( 1 - \frac{R_{s}}{r} \right )^{2} \left ( 1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \right ) \\
\frac{\mbox{d}r}{\mbox{d}t}
 & = & \pm c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad \frac{\mbox{d}r}{\mbox{d}t}
 & = & \pm c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27a)} \qquad r
 & = & \frac{1}{u} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27b)} \qquad \mbox{d}r
 & = & - \frac{1}{u^{2}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad \frac{\mbox{d}\varphi}{\mbox{d}t}
 & = & \frac{J}{K r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \\
& = & \frac{J u^{2}}{K} (1 - R_{s} u) \\
& = & c P u^{2} (1 - R_{s} u) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad \frac{\mbox{d}r}{\mbox{d}t}
 & = & \pm c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
- \frac{1}{u^{2}} \, \frac{\mbox{d}u}{\mbox{d}t}
 & = & \pm c (1 - R_{s} u) \sqrt{1 - P^{2} u^{2} (1 - R_{s} u)} \\
\frac{\mbox{d}u}{\mbox{d}t}
 & = & \mp c (1 - R_{s} u) u^{2} \sqrt{1 - P^{2} u^{2} (1 - R_{s} u)} \\
\left ( \frac{\mbox{d}u}{\mbox{d}t} \right )^{2}
 & = & c^{2} (1 - R_{s} u)^{2} u^{4} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30)} \qquad \qquad \qquad \qquad \left ( \frac{\mbox{d}u}{\mbox{d}t} \right )^{2}
 & = & c^{2} (1 - R_{s} u)^{2} u^{4} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
\left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \, \frac{\mbox{d}\varphi}{\mbox{d}t} \right )^{2}
 & = & c^{2} (1 - R_{s} u)^{2} u^{4} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
\left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \, c P u^{2} (1 - R_{s} u) \right )^{2}
 & = & c^{2} (1 - R_{s} u)^{2} u^{4} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
\left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \right )^{2} c^{2} P^{2} u^{4} (1 - R_{s} u)^{2}
 & = & c^{2} (1 - R_{s} u)^{2} u^{4} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
\left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \right )^{2}
 & = & \frac{1}{P^{2}} (1 - P^{2} u^{2} (1 - R_{s} u)) \\
& = & \frac{1}{P^{2}} (1 - P^{2} u^{2} + P^{2} R_{s} u^{3}) \\
& = & \frac{1}{P^{2}} - u^{2} + R_{s} u^{3} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad \left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \right )^{2}
 & = & \frac{1}{P^{2}} - u^{2} + R_{s} u^{3} \\
\frac{\mbox{d}u}{\mbox{d}\varphi}
 & = & \pm \sqrt{\frac{1}{P^{2}} - u^{2} + R_{s} u^{3}} \\
\mbox{d}\varphi
 & = & \pm \frac{1}{\sqrt{P^{-2} - u^{2} + R_{s} u^{3}}} \, \mbox{d}u \\
\varphi
 & = & \pm \int \frac{1}{\sqrt{P^{-2} - u^{2} + R_{s} u^{3}}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad \varphi
 & = & \pm \int \frac{1}{\sqrt{P^{-2} - u^{2} + R_{s} u^{3}}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad \frac{\mbox{d}}{\mbox{d}\varphi} \left ( \left ( \frac{\mbox{d}u}{\mbox{d}\varphi} \right )^{2} \right )
 & = & \frac{\mbox{d}}{\mbox{d}\varphi} \left ( \frac{1}{P^{2}} - u^{2} + R_{s} u^{3} \right ) \\
2 \, \frac{\mbox{d}u}{\mbox{d}\varphi} \, \frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}}
 & = & 0 - 2u \, \frac{\mbox{d}u}{\mbox{d}\varphi} + 3 R_{s} u^{2} \, \frac{\mbox{d}u}{\mbox{d}\varphi} \\
\frac{\mbox{d}u}{\mbox{d}\varphi} \, \frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}}
 & = & - u \, \frac{\mbox{d}u}{\mbox{d}\varphi} + \frac{3}{2} R_{s} u^{2} \, \frac{\mbox{d}u}{\mbox{d}\varphi} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33)} \qquad \frac{\mbox{d}u}{\mbox{d}\varphi} \, \frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}}
 & = & - u \, \frac{\mbox{d}u}{\mbox{d}\varphi} + \frac{3}{2} R_{s} u^{2} \, \frac{\mbox{d}u}{\mbox{d}\varphi} \\
\frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}}
 & = & - u + \frac{3}{2} R_{s} u^{2} \\
\frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}} + u
 & = & \frac{3}{2} R_{s} u^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33)} \qquad \frac{\mbox{d}^{2}u}{\mbox{d}\varphi^{2}} + u
 & = & \frac{3}{2} R_{s} u^{2} \\
\end{eqnarray*}
\end{document}
