\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arcosh}{arcosh}
\DeclareMathOperator{\arcsec}{arcsec}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \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{(2)} \qquad \frac{\mbox{d}r}{\mbox{d}t}
 & = & - 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{(3)} \qquad R_{s}
 & = & \frac{2 G M}{c^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad P
 & = & \frac{J}{c K} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5a)} \qquad J
 & = & r^{2} \, \frac{\mbox{d}\varphi}{\mbox{d}s} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5b)} \qquad K
 & = & \left ( 1 - \frac{2 G M}{r c^{2}} \right ) \frac{\mbox{d}t}{\mbox{d}s} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad \frac{\mbox{d}r}{\mbox{d}t} \, \biggr |_{A}
 & = & - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
0
 & = & - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
& = & \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{(7)} \qquad \qquad \qquad 0
 & = & \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
& = & \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
& = & 1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right ) \\
\frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 & = & 1 \\
\frac{P^{2}}{r_{A}^{2}} \left ( 1 - \frac{R_{s}}{r_{A}} \right )
 & = & 1 \\
P^{2}
 & = & \frac{r_{A}^{2}}{1 - \frac{R_{s}}{r_{A}}} \\
P
 & = & r_{A} \frac{1}{\sqrt{1 - \frac{R_{s}}{r_{A}}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9)} \qquad \frac{\mbox{d}r}{\mbox{d}t}
 & = & - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{P^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )} \\
& = & - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad \frac{\mbox{d}r}{\mbox{d}t}
 & = & - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \\
\mbox{d}t
 & = & \left ( - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad \int\limits_{t = t_{0}}^{t = t_{A}} \mbox{d}t
 & = & \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( - c \left ( 1 - \frac{R_{s}}{r} \right ) \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
\left [ t \right ]_{t = t_{0}}^{t = t_{A}}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 - \frac{R_{s}}{r} \right )^{-1} \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{1 - \frac{R_{s}}{r}} \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad \frac{1}{1 - x}
 & = & \sum_{n = 0}^{n = \infty} x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{1 - \frac{R_{s}}{r}} \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( \sum_{n = 0}^{n = \infty} \left ( \frac{R_{s}}{r} \right )^{n} \right )
 \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( \sum_{n = 0}^{n = \infty} \left ( \frac{R_{s}}{r} \right )^{n} \right )
 \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \left ( \frac{R_{s}}{r} \right )^{2} + \left ( \frac{R_{s}}{r} \right )^{3}
 + \left ( \frac{R_{s}}{r} \right )^{4} + \ldots \right ) \\
 & & \qquad \qquad \qquad \qquad \cdot \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
& \approx & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \left ( \frac{R_{s}}{r} \right )^{2} \right )
 \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \frac{R_{s}^{2}}{r^{2}} \right )
 \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad \frac{1}{\sqrt{1 - (ax)^{2}}}
 & = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (ax)^{2n} \\
& = & 1 + \frac{1}{2} (ax)^{2} + \frac{3}{8} (ax)^{4} + \frac{5}{16} (ax)^{6} + \frac{35}{128} (ax)^{8} + \frac{63}{256} (ax)^{10} + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad \frac{1}{\sqrt{1 - (ax)^{2}}}
 & = & \frac{1}{\sqrt{1 - x^{2}}} \times \mbox{een of andere reeks die een functie is van } a \mbox{ en } x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad \frac{1}{\sqrt{1 - (ax)^{2}}}
 & = & \frac{1}{\sqrt{1 - x^{2}}} \sum_{n = 0}^{n = \infty} c_{n} x^{2n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad a
 & = & \sqrt{\frac{1 - \frac{R_{s}}{r}}{1 - \frac{R_{s}}{r_{A}}}} \\
a^{2}
 & = & \frac{1 - \frac{R_{s}}{r}}{1 - \frac{R_{s}}{r_{A}}} \\
& = & \frac{1 - \frac{R_{s}}{r_{A}} + \frac{R_{s}}{r_{A}} - \frac{R_{s}}{r}}{1 - \frac{R_{s}}{r_{A}}} \\
& = & \frac{1 - \frac{R_{s}}{r_{A}}}{1 - \frac{R_{s}}{r_{A}}} + \frac{\frac{R_{s}}{r_{A}} - \frac{R_{s}}{r}}{1 - \frac{R_{s}}{r_{A}}} \\
& = & 1 + R_{s} \frac{\frac{1}{r_{A}} - \frac{1}{r}}{1 - \frac{R_{s}}{r_{A}}} \\
& = & 1 + R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad B
 & = & \frac{r - r_{A}}{r (r_{A} - R_{s})} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad a^{2}
 & = & 1 + R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \\
& = & 1 + R_{s} B \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21)} \qquad x
 & = & \frac{r_{A}}{r} \\
x^{2}
 & = & \frac{r_{A}^{2}}{r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22a)} \qquad c_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22b)} \qquad c_{1}
 & = & - \frac{1}{2} (1 - a^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22c)} \qquad c_{2}
 & = & - \frac{1}{8} (1 + 2 a^{2} - 3 a^{4}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22d)} \qquad c_{3}
 & = & - \frac{1}{16} (1 + a^{2} + 3 a^{4} - 5 a^{6}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22e)} \qquad c_{4}
 & = & - \frac{1}{128} (5 + 4 a^{2} + 6 a^{4} + 20 a^{6} - 35 a^{8}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22f)} \qquad c_{5}
 & = & - \frac{1}{256} (7 + 5 a^{2} + 6 a^{4} + 10 a^{6} + 35 a^{8} - 63 a^{10}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22g)} \qquad c_{n}
 & = & \frac{1}{b_{0}} \left ( a_{n} - \sum_{k = 1}^{k = n} b_{k} c_{n - k} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23a)} \qquad c_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23b)} \qquad c_{1}
 & = & - \frac{1}{2} (1 - a^{2}) \\
& = & - \frac{1}{2} (1 - (1 + R_{s} B)) \\
& = & - \frac{1}{2} (1 - 1 - R_{s} B) \\
& = & - \frac{1}{2} (- R_{s} B) \\
& = & \frac{1}{2} R_{s} B \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23c)} \qquad c_{2}
 & = & - \frac{1}{8} (1 + 2 a^{2} - 3 a^{4}) \\
& = & - \frac{1}{8} (1 + 2 (1 + R_{s} B) - 3 (1 + R_{s} B)^{2}) \\
& = & - \frac{1}{8} (1 + 2 + 2 R_{s} B - 3 (1 + 2 R_{s} B + R_{s}^{2} B^{2})) \\
& = & - \frac{1}{8} (1 + 2 + 2 R_{s} B - 3 - 6 R_{s} B - 3 R_{s}^{2} B^{2}) \\
& = & - \frac{1}{8} (- 4 R_{s} B - 3 R_{s}^{2} B^{2}) \\
& = & \frac{1}{2} R_{s} B + \frac{3}{8} R_{s}^{2} B^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23d)} \qquad c_{3}
 & = & - \frac{1}{16} (1 + a^{2} + 3 a^{4} - 5 a^{6}) \\
& = & - \frac{1}{16} (1 + (1 + R_{s} B) + 3 (1 + R_{s} B)^{2} - 5 (1 + R_{s} B)^{3}) \\
& = & - \frac{1}{16} (1 + 1 + R_{s} B + 3 (1 + 2 R_{s} B + R_{s}^{2} B^{2}) - 5 (1 + 3 R_{s} B + 3 R_{s}^{2} B^{2} + R_{s}^{3} B^{3})) \\
& = & - \frac{1}{16} (1 + 1 + R_{s} B + 3 + 6 R_{s} B + 3 R_{s}^{2} B^{2} - 5 - 15 R_{s} B - 15 R_{s}^{2} B^{2} - 5 R_{s}^{3} B^{3}) \\
& = & - \frac{1}{16} (- 8 R_{s} B - 12 R_{s}^{2} B^{2} - 5 R_{s}^{3} B^{3}) \\
& \approx & - \frac{1}{16} (- 8 R_{s} B - 12 R_{s}^{2} B^{2}) \\
& = & \frac{1}{2} R_{s} B + \frac{6}{8} R_{s}^{2} B^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23e)} \qquad c_{4}
 & = & - \frac{1}{128} (5 + 4 a^{2} + 6 a^{4} + 20 a^{6} - 35 a^{8}) \\
& = & - \frac{1}{128} (5 + 4 (1 + R_{s} B) + 6 (1 + R_{s} B)^{2} + 20 (1 + R_{s} B)^{3} - 35 (1 + R_{s} B)^{4}) \\
& = & - \frac{1}{128} (5 + 4 + 4 R_{s} B + 6 (1 + 2 R_{s} B + R_{s}^{2} B^{2}) + 20 (1 + 3 R_{s} B + 3 R_{s}^{2} B^{2} + R_{s}^{3} B^{3}) \\
 & & \qquad \qquad - \, 35 (1 + 4 R_{s} B + 6 R_{s}^{2} B^{2} + 4 R_{s}^{3} B^{3} + R_{s}^{4} B^{4})) \\
& = & - \frac{1}{128} (5 + 4 + 4 R_{s} B + 6 + 12 R_{s} B + 6 R_{s}^{2} B^{2} + 20 + 60 R_{s} B + 60 R_{s}^{2} B^{2} + 20 R_{s}^{3} B^{3} \\
 & & \qquad \qquad - \, 35 - 140 R_{s} B - 210 R_{s}^{2} B^{2} - 140 R_{s}^{3} B^{3} - 35 R_{s}^{4} B^{4}) \\
& = & - \frac{1}{128} (- 64 R_{s} B - 144 R_{s}^{2} B^{2} - 120 R_{s}^{3} B^{3} - 35 R_{s}^{4} B^{4}) \\
& \approx & - \frac{1}{128} (- 64 R_{s} B - 144 R_{s}^{2} B^{2}) \\
& = & \frac{1}{2} R_{s} B + \frac{9}{8} R_{s}^{2} B^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23f)} \qquad c_{5}
 & = & - \frac{1}{256} (7 + 5 a^{2} + 6 a^{4} + 10 a^{6} + 35 a^{8} - 63 a^{10}) \\
& = & - \frac{1}{256} (7 + 5 (1 + R_{s} B) + 6 (1 + R_{s} B)^{2} + 10 (1 + R_{s} B)^{3} + 35 (1 + R_{s} B)^{4} - 63 (1 + R_{s} B)^{5}) \\
& = & - \frac{1}{256} (7 + 5 + 5 R_{s} B + 6 (1 + 2 R_{s} B + R_{s}^{2} B^{2}) + 10 (1 + 3 R_{s} B + 3 R_{s}^{2} B^{2} + R_{s}^{3} B^{3}) \\
 & & \qquad \qquad + \, 35 (1 + 4 R_{s} B + 6 R_{s}^{2} B^{2} + 4 R_{s}^{3} B^{3} + R_{s}^{4} B^{4}) \\
 & & \qquad \qquad \qquad - \, 63 (1 + 5 R_{s} B + 10 R_{s}^{2} B^{2} + 10 R_{s}^{3} B^{3} + 5 R_{s}^{4} B^{4} + R_{s}^{5} B^{5})) \\
& = & - \frac{1}{256} (7 + 5 + 5 R_{s} B + 6 + 12 R_{s} B + 6 R_{s}^{2} B^{2} + 10 + 30 R_{s} B + 30 R_{s}^{2} B^{2} + 10 R_{s}^{3} B^{3} \\
 & & \qquad \qquad + \, 35 + 140 R_{s} B + 210 R_{s}^{2} B^{2} + 140 R_{s}^{3} B^{3} + 35 R_{s}^{4} B^{4} \\
 & & \qquad \qquad \qquad - \, 63 - 315 R_{s} B - 630 R_{s}^{2} B^{2} - 630 R_{s}^{3} B^{3} - 315 R_{s}^{4} B^{4} - 63 R_{s}^{5} B^{5}) \\
& = & - \frac{1}{256} (- 128 R_{s} B - 384 R_{s}^{2} B^{2} - 480 R_{s}^{3} B^{3} - 280 R_{s}^{4} B^{4} - 63 R_{s}^{5} B^{5}) \\
& \approx & - \frac{1}{256} (- 128 R_{s} B - 384 R_{s}^{2} B^{2}) \\
& = & \frac{1}{2} R_{s} B + \frac{12}{8} R_{s}^{2} B^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24a)} \qquad c_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24b)} \qquad c_{n}
 & = & \frac{1}{2} R_{s} B + \frac{3 (n - 1)}{8} R_{s}^{2} B^{2} \qquad n \, \, \, > \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25)} \qquad c_{n} (n \, \, > \, \, 0)
 & = & \frac{1}{2} R_{s} B + \frac{3 (n - 1)}{8} R_{s}^{2} B^{2} \\
& = & \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 + \frac{3 (n - 1)}{8} R_{s}^{2} \left ( \frac{r - r_{A}}{r (r_{A} - R_{s})} \right )^{2} \\
& = & \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 + \frac{3 (n - 1)}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad W
 & = & \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27)} \qquad W
 & = & \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \\
& = & \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}}} \, \right )^{-1} \sum_{n = 0}^{n = \infty} c_{n} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \\
& = & \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}}} \, \right )^{-1} \left (
 1 + \sum_{n = 1}^{n = \infty} \left ( \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 + \frac{3 (n - 1)}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \right ) \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \sum_{n = 1}^{n = \infty} \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 + \sum_{n = 1}^{n = \infty} \frac{3 (n - 1)}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 + R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \sum_{n = 1}^{n = \infty} \frac{3 (n - 1)}{8} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right . \\
 & & \qquad \left . + \, R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} \frac{3 n}{8} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \sum_{n = 1}^{n = \infty} \frac{3}{8} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad W
 & = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}} \sum_{n = 1}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 \left ( - 1 + \sum_{n = 0}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right ) \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \left ( - 1 + \sum_{n = 0}^{n = \infty} \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n} \right ) \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 \left ( - 1 + \frac{1}{1 - \frac{r_{A}^{2}}{r^{2}}} \right ) \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \left ( - 1 + \frac{1}{1 - \frac{r_{A}^{2}}{r^{2}}} \right ) \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 \left ( - \frac{r^{2} - r_{A}^{2}}{r^{2} - r_{A}^{2}} + \frac{r^{2}}{r^{2} - r_{A}^{2}} \right ) \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \left ( - \frac{r^{2} - r_{A}^{2}}{r^{2} - r_{A}^{2}} + \frac{r^{2}}{r^{2} - r_{A}^{2}} \right ) \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 \frac{- r^{2} + r_{A}^{2} + r^{2}}{r^{2} - r_{A}^{2}} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \frac{- r^{2} + r_{A}^{2} + r^{2}}{r^{2} - r_{A}^{2}} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{1}{2} R_{s} \frac{r - r_{A}}{r (r_{A} - R_{s})}
 \frac{r_{A}^{2}}{(r + r_{A}) (r - r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \frac{r_{A}^{2}}{(r + r_{A}) (r - r_{A})} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad \frac{x}{(1 - x)^{2}}
 & = & \sum_{n = 1}^{n = \infty} n x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30)} \qquad W
 & = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \sum_{n = 1}^{n = \infty} n \left ( \frac{r_{A}^{2}}{r^{2}} \right )^{n}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \frac{\frac{r_{A}^{2}}{r^{2}}}{\left ( 1 - \frac{r_{A}^{2}}{r^{2}} \right )^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{r^{2} (r_{A} - R_{s})^{2}}
 \frac{r_{A}^{2} r^{2}}{(r^{2} - r_{A}^{2})^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3}{8} R_{s}^{2} \frac{(r - r_{A})^{2}}{(r_{A} - R_{s})^{2}}
 \frac{r_{A}^{2}}{((r + r_{A}) (r - r_{A}))^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
& = & \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \left . + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \frac{R_{s}^{2}}{r^{2}} \right )
 \left ( \sqrt{1 - \frac{r_{A}^{2}}{r^{2}} \left ( 1 - \frac{R_{s}}{r} \right )
 \left ( 1 - \frac{R_{s}}{r_{A}} \right )^{-1}} \, \right )^{-1} \, \mbox{d}r \\
& \approx & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \frac{R_{s}^{2}}{r^{2}} \right )
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \qquad \qquad \left . + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \left ( 1 + \frac{R_{s}}{r} + \frac{R_{s}^{2}}{r^{2}} \right )
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \right . \\
 & & \qquad \qquad \qquad \left . + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \right ) \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})}
 + \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}} \right . \\
 & & \qquad \qquad \qquad
 - \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})}
 + \frac{R_{s}}{r}
 + \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r^{2} (r + r_{A})} \\
 & & \qquad \qquad \qquad \qquad
 + \, \frac{3 R_{s}^{3} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{r (r + r_{A})^{2}}
 - \frac{3 R_{s}^{3} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{3} (r + r_{A})}
 + \frac{R_{s}^{2}}{r^{2}} \\
 & & \qquad \qquad \qquad \qquad \qquad
 + \, \frac{R_{s}^{3} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r^{3} (r + r_{A})}
 + \frac{3 R_{s}^{4} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{r^{2} (r + r_{A})^{2}} \\
 & & \qquad \qquad \qquad \qquad \qquad \qquad \left .
 - \, \frac{3 R_{s}^{4} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{4} (r + r_{A})} \right ) \, \mbox{d}r \\
& \approx & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \left (
 1 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})}
 + \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}} \right . \\
 & & \qquad \qquad \qquad \left . - \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})}
 + \frac{R_{s}}{r} + \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r^{2} (r + r_{A})}
 + \frac{R_{s}^{2}}{r^{2}} \right ) \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}} \left (
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A})} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \right . \\
 & & \qquad \qquad \qquad
 + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{1}{(r + r_{A})^{2}} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r^{2} (r + r_{A})} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad
 + \, \frac{R_{s}}{r} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r^{2} (r + r_{A})} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad \qquad \left .
 + \, \frac{R_{s}^{2}}{r^{2}} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}} \left (
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \right . \\
 & & \qquad \qquad \qquad
 + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad
 + \, R_{s} \frac{1}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad \qquad \left .
 + \, R_{s}^{2} \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}} \left (
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \right . \\
 & & \qquad \qquad \qquad
 + \, \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \frac{r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad
 + \, R_{s} \frac{1}{\sqrt{r^{2} - r_{A}^{2}}}
 + \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \frac{1}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \\
 & & \qquad \qquad \qquad \qquad \qquad \left .
 + \, R_{s}^{2} \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 - \frac{1}{c} \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad
 - \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad
 + \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad \qquad
 - \, \frac{1}{c} R_{s} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{\sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{c} \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad \qquad \qquad
 - \, \frac{1}{c} R_{s}^{2} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33a)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \left [ \sqrt{r^{2} - r_{A}^{2}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \sqrt{r_{A}^{2} - r_{A}^{2}} - \sqrt{r_{0}^{2} - r_{A}^{2}} \\
& = & 0 - \sqrt{r_{0}^{2} - r_{A}^{2}} \\
& = & - \sqrt{r_{0}^{2} - r_{A}^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33b)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \left [ \frac{1}{r_{A}} \frac{r - r_{A}}{\sqrt{r^{2} - r_{A}^{2}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \frac{r - r_{A}}{\sqrt{(r + r_{A}) (r - r_{A})}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \sqrt{\frac{(r - r_{A}) (r - r_{A})}{(r + r_{A}) (r - r_{A})}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \sqrt{\frac{r - r_{A}}{r + r_{A}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \frac{1}{r_{A}} \sqrt{\frac{r_{A} - r_{A}}{r_{A} + r_{A}}} - \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & \frac{1}{r_{A}} \cdot 0 - \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & - \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33c)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{r}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \int\limits_{r = r_{0}}^{r = r_{A}} \frac{r + r_{A} - r_{A}}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{r + r_{A}}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{r_{A}}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}}
 - r_{A} \frac{1}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & \left [ \frac{1}{r_{A}} \frac{r - r_{A}}{\sqrt{r^{2} - r_{A}^{2}}}
 - r_{A} \frac{1}{3 r_{A}^{2}} \frac{r^{2} + r_{A} r - 2 r_{A}^{2}}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \frac{r - r_{A}}{\sqrt{(r + r_{A}) (r - r_{A})}}
 - \frac{1}{3 r_{A}} \frac{(r + 2 r_{A}) (r - r_{A})}{(r + r_{A}) \sqrt{(r + r_{A}) (r - r_{A})}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \sqrt{\frac{(r - r_{A}) (r - r_{A})}{(r + r_{A}) (r - r_{A})}}
 - \frac{1}{3 r_{A}} \frac{(r + 2 r_{A}) \sqrt{r - r_{A}}}{(r + r_{A}) \sqrt{r + r_{A}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}} \sqrt{\frac{r - r_{A}}{r + r_{A}}}
 - \frac{1}{3 r_{A}} \frac{r + 2 r_{A}}{r + r_{A}} \sqrt{\frac{r - r_{A}}{r + r_{A}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left ( \frac{1}{r_{A}} \sqrt{\frac{r_{A} - r_{A}}{r_{A} + r_{A}}}
 - \frac{1}{3 r_{A}} \frac{r_{A} + 2 r_{A}}{r_{A} + r_{A}} \sqrt{\frac{r_{A} - r_{A}}{r_{A} + r_{A}}} \, \right ) \\
 & & \qquad - \, \left ( \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 - \frac{1}{3 r_{A}} \frac{r_{0} + 2 r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \\
& = & \frac{1}{r_{A}} \cdot 0 - \frac{1}{3 r_{A}} \cdot 0
 - \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{3 r_{A}} \frac{r_{0} + 2 r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & - \frac{1}{3 r_{A}} \frac{3 (r_{0} + r_{A})}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{3 r_{A}} \frac{r_{0} + 2 r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & - \frac{1}{3 r_{A}} \frac{3 r_{0} + 3 r_{A} - r_{0} - 2 r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & - \frac{1}{3 r_{A}} \frac{2 r_{0} + r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33d)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \int\limits_{r = r_{0}}^{r = r_{A}} \frac{r + r - r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \frac{2 r - (r + r_{A})}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{2 r}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{r + r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \right ) \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{2}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \, \right ) \, \mbox{d}r \\
& = & \left [ \frac{2}{r_{A}} \frac{r - r_{A}}{\sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{2}{r_{A}} \frac{r - r_{A}}{\sqrt{(r + r_{A}) (r - r_{A})}}
 - \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{2}{r_{A}} \sqrt{\frac{(r - r_{A}) (r - r_{A})}{(r + r_{A}) (r - r_{A})}}
 - \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{2}{r_{A}} \sqrt{\frac{r - r_{A}}{r + r_{A}}}
 - \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left ( \frac{2}{r_{A}} \sqrt{\frac{r_{A} - r_{A}}{r_{A} + r_{A}}}
 - \frac{1}{r_{A}} \arcsec \frac{r_{A}}{r_{A}} \right ) \\
 & & \qquad - \, \left ( \frac{2}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \right ) \\
& = & \frac{2}{r_{A}} \cdot 0 - \frac{1}{r_{A}} \arcsec 1
 - \frac{2}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & 0 - \frac{2}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} + \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & - \frac{2}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} + \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33e)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{\sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \left [ \arcosh \frac{r}{r_{A}} \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \arcosh \frac{r_{A}}{r_{A}} - \arcosh \frac{r_{0}}{r_{A}} \\
& = & \arcosh 1 - \arcosh \frac{r_{0}}{r_{A}} \\
& = & 0 - \arcosh \frac{r_{0}}{r_{A}} \\
& = & - \arcosh \frac{r_{0}}{r_{A}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33f)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{r_{A}} \frac{r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{r_{A}} \frac{r + r_{A} - r}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{1}{r_{A}} \frac{r + r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{r_{A}} \frac{r}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & \int\limits_{r = r_{0}}^{r = r_{A}} \left ( \frac{1}{r_{A}} \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{r_{A}} \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \right ) \, \mbox{d}r \\
& = & \left [ \frac{1}{r_{A}} \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}}
 - \frac{1}{r_{A}} \frac{1}{r_{A}} \frac{r - r_{A}}{\sqrt{r^{2} - r_{A}^{2}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}^{2}} \arcsec \frac{r}{r_{A}}
 - \frac{1}{r_{A}^{2}} \frac{r - r_{A}}{\sqrt{(r + r_{A}) (r - r_{A})}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}^{2}} \arcsec \frac{r}{r_{A}}
 - \frac{1}{r_{A}^{2}} \sqrt{\frac{(r - r_{A}) (r - r_{A})}{(r + r_{A}) (r - r_{A})}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left [ \frac{1}{r_{A}^{2}} \arcsec \frac{r}{r_{A}}
 - \frac{1}{r_{A}^{2}} \sqrt{\frac{r - r_{A}}{r + r_{A}}} \, \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \left ( \frac{1}{r_{A}^{2}} \arcsec \frac{r_{A}}{r_{A}}
 - \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{A} - r_{A}}{r_{A} + r_{A}}} \, \right ) \\
 & & \qquad - \, \left ( \frac{1}{r_{A}^{2}} \arcsec \frac{r_{0}}{r_{A}}
 - \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \\
& = & \frac{1}{r_{A}^{2}} \arcsec 1 - \frac{1}{r_{A}^{2}} \cdot 0
 - \frac{1}{r_{A}^{2}} \arcsec \frac{r_{0}}{r_{A}}
 + \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & \frac{1}{r_{A}^{2}} \cdot 0 - \frac{1}{r_{A}^{2}} \arcsec \frac{r_{0}}{r_{A}}
 + \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
& = & - \frac{1}{r_{A}^{2}} \arcsec \frac{r_{0}}{r_{A}} + \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33g)} \qquad \int\limits_{r = r_{0}}^{r = r_{A}} \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 & = & \left [ \frac{1}{r_{A}} \arcsec \frac{r}{r_{A}} \right ]_{r = r_{0}}^{r = r_{A}} \\
& = & \frac{1}{r_{A}} \arcsec \frac{r_{A}}{r_{A}} - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & \frac{1}{r_{A}} \arcsec 1 - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & 0 - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad t_{A} - t_{0}
 & = & - \frac{1}{c} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{\sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r
 - \frac{1}{c} \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{(r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad
 - \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r}{(r + r_{A})^{2} \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad
 + \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{r - r_{A}}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad \qquad
 - \, \frac{1}{c} R_{s} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{\sqrt{r^{2} - r_{A}^{2}}}
 - \frac{1}{c} \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{r (r + r_{A}) \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
 & & \qquad \qquad \qquad \qquad
 - \, \frac{1}{c} R_{s}^{2} \int\limits_{r = r_{0}}^{r = r_{A}}
 \frac{1}{r \sqrt{r^{2} - r_{A}^{2}}} \, \mbox{d}r \\
& = & - \frac{1}{c} \left ( - \sqrt{r_{0}^{2} - r_{A}^{2}} \, \right )
 - \frac{1}{c} \frac{R_{s} r_{A}^{2}}{2 (r_{A} - R_{s})} \left ( - \frac{1}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \\
 & & \qquad
 - \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}}
 \left ( - \frac{1}{3 r_{A}} \frac{2 r_{0} + r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \\
 & & \qquad \qquad
 + \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}^{2}}{8 (r_{A} - R_{s})^{2}}
 \left ( - \frac{2}{r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} + \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \right )
 - \frac{1}{c} R_{s} \left ( - \arcosh \frac{r_{0}}{r_{A}} \right ) \\
 & & \qquad \qquad \qquad
 - \, \frac{1}{c} \frac{R_{s}^{2} r_{A}^{2}}{2 (r_{A} - R_{s})}
 \left ( - \frac{1}{r_{A}^{2}} \arcsec \frac{r_{0}}{r_{A}} + \frac{1}{r_{A}^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \\
 & & \qquad \qquad \qquad \qquad
 - \, \frac{1}{c} R_{s}^{2} \left ( - \frac{1}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \right ) \\
& = & \frac{1}{c} \sqrt{r_{0}^{2} - r_{A}^{2}}
 + \frac{1}{c} \frac{R_{s} r_{A}}{2 (r_{A} - R_{s})} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
 & & \qquad
 + \, \frac{1}{c} \frac{R_{s}^{2} r_{A}}{8 (r_{A} - R_{s})^{2}}
 \frac{2 r_{0} + r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
 & & \qquad \qquad
 - \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}}{4 (r_{A} - R_{s})^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{c} \frac{3 R_{s}^{2} r_{A}}{8 (r_{A} - R_{s})^{2}} \arcsec \frac{r_{0}}{r_{A}}
 + \frac{1}{c} R_{s} \arcosh \frac{r_{0}}{r_{A}} \\
 & & \qquad \qquad \qquad
 + \, \frac{1}{c} \frac{R_{s}^{2}}{2 (r_{A} - R_{s})} \arcsec \frac{r_{0}}{r_{A}}
 - \frac{1}{c} \frac{R_{s}^{2}}{2 (r_{A} - R_{s})} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{c} \frac{R_{s}^{2}}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad t_{A} - t_{0}
 & = & \frac{1}{c} \sqrt{r_{0}^{2} - r_{A}^{2}}
 + \frac{1}{c} \frac{R_{s} r_{A}}{2 (r_{A} - R_{s})} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
 & & \qquad
 + \, \frac{1}{c} \frac{R_{s}^{2} r_{A}}{8 (r_{A} - R_{s})^{2}}
 \frac{2 r_{0} + r_{A}}{r_{0} + r_{A}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \\
 & & \qquad \qquad
 - \, \frac{1}{c} \frac{3 R_{s}^{2} r_{A}}{4 (r_{A} - R_{s})^{2}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{c} \frac{3 R_{s}^{2} r_{A}}{8 (r_{A} - R_{s})^{2}} \arcsec \frac{r_{0}}{r_{A}}
 + \frac{1}{c} R_{s} \arcosh \frac{r_{0}}{r_{A}} \\
 & & \qquad \qquad \qquad
 + \, \frac{1}{c} \frac{R_{s}^{2}}{2 (r_{A} - R_{s})} \arcsec \frac{r_{0}}{r_{A}}
 - \frac{1}{c} \frac{R_{s}^{2}}{2 (r_{A} - R_{s})} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 + \frac{1}{c} \frac{R_{s}^{2}}{r_{A}} \arcsec \frac{r_{0}}{r_{A}} \\
& = & \frac{1}{c} \sqrt{r_{0}^{2} - r_{A}^{2}}
 + \frac{R_{s}}{c} \left ( \left (
 \arcosh \frac{r_{0}}{r_{A}}
 + \frac{1}{2} \frac{r_{A}}{r_{A} - R_{s}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 \, \right ) \right . \\
 & & \qquad
 + \, \frac{R_{s}}{(r_{A} - R_{s})^{2}} \left (
 \left ( \frac{(r_{A} - R_{s})^{2}}{r_{A}} + \frac{1}{2} (r_{A} - R_{s}) + \frac{3}{8} r_{A} \right ) \arcsec \frac{r_{0}}{r_{A}}
 \right . \\
 & & \qquad \qquad \left . \left .
 + \left ( \frac{1}{8} r_{A} \frac{2 r_{0} + r_{A}}{r_{0} + r_{A}}
 - \frac{3}{4} r_{A}
 - \frac{1}{2} (r_{A} - R_{s}) \right ) \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 \, \right ) \right ) \\
& = & \frac{1}{c} \sqrt{r_{0}^{2} - r_{A}^{2}}
 + \frac{R_{s}}{c} \left ( \left (
 \arcosh \frac{r_{0}}{r_{A}}
 + \frac{1}{2} \frac{r_{A}}{r_{A} - R_{s}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 \, \right ) \right . \\
 & & \qquad
 + \, \frac{R_{s}}{(r_{A} - R_{s})^{2}} \left (
 \left ( \frac{r_{A}^{2} - 2 r_{A} R_{s} + R_{s}^{2}}{r_{A}} + \frac{1}{2} r_{A} - \frac{1}{2} R_{s} + \frac{3}{8} r_{A} \right ) \arcsec \frac{r_{0}}{r_{A}}
 \right . \\
 & & \qquad \qquad \left . \left .
 + \left ( \frac{1}{8} r_{A} \frac{2 r_{0} + 2 r_{A} - r_{A}}{r_{0} + r_{A}}
 - \frac{3}{4} r_{A}
 - \frac{1}{2} r_{A} + \frac{1}{2} R_{s} \right ) \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 \, \right ) \right ) \\
& = & \frac{1}{c} \sqrt{r_{0}^{2} - r_{A}^{2}}
 + \frac{R_{s}}{c} \left ( \left (
 \arcosh \frac{r_{0}}{r_{A}}
 + \frac{1}{2} \frac{r_{A}}{r_{A} - R_{s}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}}
 \, \right ) \right . \\
 & & \qquad
 + \, \frac{R_{s}}{8 (r_{A} - R_{s})^{2}} \left (
 \left ( 15 r_{A} - 20 R_{s} + \frac{8 R_{s}^{2}}{r_{A}} \right ) \arcsec \frac{r_{0}}{r_{A}} \right . \\
 & & \qquad \qquad \left . \left .
 - \, \left ( \frac{r_{A}^{2}}{r_{0} + r_{A}} + 8 r_{A} - 4 R_{s} \right ) \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35)} \qquad t_{v}
 & = & \frac{R_{s}}{c} \left ( \left ( \arcosh \frac{r_{0}}{r_{A}}
 + \frac{1}{2} \frac{r_{A}}{r_{A} - R_{s}} \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \right . \\
 & & \qquad
 + \, \frac{R_{s}}{8 (r_{A} - R_{s})^{2}} \left (
 \left ( 15 r_{A} - 20 R_{s} + \frac{8 R_{s}^{2}}{r_{A}} \right ) \arcsec \frac{r_{0}}{r_{A}} \right . \\
 & & \qquad \qquad \left . \left .
 - \, \left ( \frac{r_{A}^{2}}{r_{0} + r_{A}} + 8 r_{A} - 4 R_{s} \right ) \sqrt{\frac{r_{0} - r_{A}}{r_{0} + r_{A}}} \, \right ) \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36)} \qquad \Delta \varphi
 & = & \frac{2 R_{s}}{b} \left ( 1 + \frac{R_{s}^{2}}{5 b^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37)} \qquad \Delta \varphi_{A}
 & = & \frac{1}{2} \Delta \varphi \\
& = & \frac{1}{2} \frac{2 R_{s}}{b} \left ( 1 + \frac{R_{s}^{2}}{5 b^{2}} \right ) \\
& = & \frac{R_{s}}{b} \left ( 1 + \frac{R_{s}^{2}}{5 b^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(38)} \qquad \Delta b
 & = & r_{0} \, \Delta \varphi_{A} \\
& = & \frac{r_{0} R_{s}}{b} \left ( 1 + \frac{R_{s}^{2}}{5 b^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(39)} \qquad r_{A}
 & = & b - \Delta b \\
& = & b - \frac{r_{0} R_{s}}{b} \left ( 1 + \frac{R_{s}^{2}}{5 b^{2}} \right ) \\
\end{eqnarray*}
\end{document}
