\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\artanh}{artanh}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad \gamma
 & = & \frac{1}{\sqrt{1 - \beta^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad \beta
 & = & \frac{v}{c} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad v
 & = & c \tanh \frac{a_{B} t_{B}}{c} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad v
 & = & c \tanh \frac{a_{B} t_{B}}{c} \\
\frac{v}{c}
 & = & \tanh \frac{a_{B} t_{B}}{c} \\
\frac{a_{B} t_{B}}{c}
 & = & \artanh \frac{v}{c} \\
t_{B}
 & = & \frac{c}{a_{B}} \artanh \beta \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad t_{B}
 & = & \frac{c}{a_{B}} \artanh \beta \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad \lim_{a_{B} \to \infty} t_{B}
 & = & \lim_{a_{B} \to \infty} \frac{c}{a_{B}} \artanh \beta \\
& = & c \artanh \beta \lim_{a_{B} \to \infty} \frac{1}{a_{B}} \\
& \approx & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad v
 & = & \frac{a_{B} t}{\sqrt{1 + \left ( \frac{a_{B} t}{c} \right )^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad \qquad \qquad v
 & = & \frac{a_{B} t}{\sqrt{1 + \left ( \frac{a_{B} t}{c} \right )^{2}}} \\
v^{2}
 & = & \frac{a_{B}^{2} t^{2}}{1 + \left ( \frac{a_{B} t}{c} \right )^{2}} \\
a_{B}^{2} t^{2}
 & = & v^{2} \left ( 1 + \left ( \frac{a_{B} t}{c} \right )^{2} \right ) \\
& = & v^{2} \left ( 1 + \frac{a_{B}^{2} t^{2}}{c^{2}} \right ) \\
& = & v^{2} + \frac{v^{2} a_{B}^{2} t^{2}}{c^{2}} \\
t^{2} \left ( a_{B}^{2} - \frac{v^{2} a_{B}^{2}}{c^{2}} \right )
 & = & v^{2} \\
t^{2}
 & = & \frac{v^{2}}{a_{B}^{2} - \frac{v^{2} a_{B}^{2}}{c^{2}}} \\
& = & \frac{v^{2}}{a_{B}^{2} \left ( 1 - \frac{v^{2}}{c^{2}} \right )} \\
t
 & = & \frac{v}{a_{B} \sqrt{1 - \beta^{2}}} \\
& = & \frac{\gamma v}{a_{B}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad \lim_{a_{B} \to \infty} t
 & = & \lim_{a_{B} \to \infty} \frac{\gamma v}{a_{B}} \\
& = & \gamma v \lim_{a_{B} \to \infty} \frac{1}{a_{B}} \\
& \approx & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9)} \qquad \mbox{Volgens A, heenweg: } \qquad \mbox{Afstand die B overbrugt}
 & = & x_{A} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad \mbox{Volgens A, heenweg: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad \mbox{Volgens A, terugweg: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad \mbox{Volgens A, totale reis: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{v} + \frac{x_{A}}{v} \\
& = & \frac{2 x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad \mbox{Volgens B, heenweg: } \qquad \mbox{Afstand die B overbrugt}
 & = & \frac{x_{A}}{\gamma} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad \mbox{Volgens B, heenweg: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{\gamma v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad \mbox{Volgens B, terugweg: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{\gamma v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad \mbox{Volgens B, totale reis: } \qquad \mbox{Tijd die B nodig heeft}
 & = & \frac{x_{A}}{\gamma v} + \frac{x_{A}}{\gamma v} \\
& = & \frac{2 x_{A}}{\gamma v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad \Delta t
 & = & \Delta t_{A} - \Delta t_{B} \\
& = & \frac{2 x_{A}}{v} - \frac{2 x_{A}}{\gamma v} \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \frac{1}{\gamma} \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad f^{\prime}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, f \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad f
 & = & \frac{1}{T} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad f^{\prime}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, f \\
\frac{1}{T^{\prime}}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, \frac{1}{T} \\
T
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, T^{\prime} \\
T^{\prime}
 & = & \sqrt{\frac{1 + \beta}{1 - \beta}} \, T \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21a)} \qquad t_{A}
 & = & \gamma \left ( t_{B} + \frac{\beta}{c} x_{B} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21b)} \qquad x_{A}
 & = & \gamma (x_{B} + v t_{B}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22a)} \qquad t_{B}
 & = & \gamma \left ( t_{A} - \frac{\beta}{c} x_{A} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22b)} \qquad x_{B}
 & = & \gamma (x_{A} - v t_{A}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23a)} \qquad x_{A}
 & = & \gamma (x_{B} + v t_{B}) \\
0
 & = & \gamma (x_{B} + v t_{B}) \\
& = & x_{B} + v t_{B} \\
x_{B}
 & = & - v t_{B} \\
t_{B}
 & = & - \frac{x_{B}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23b)} \qquad x_{B}
 & = & \gamma (x_{A} - v t_{A}) \\
0
 & = & \gamma (x_{A} - v t_{A}) \\
& = & x_{A} - v t_{A} \\
x_{A}
 & = & + v t_{A} \\
t_{A}
 & = & + \frac{x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24a)} \qquad t_{A}
 & = & + \frac{x_{A}}{c} + n_{A} \qquad (n = 1, 2, 3, 4, \ldots) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24b)} \qquad t_{B}
 & = & - \frac{x_{B}}{c} + n_{B} \qquad (n = 1, 2, 3, 4, \ldots) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25a)} \qquad + \frac{x_{A}}{v}
 & = & + \frac{x_{A}}{c} + n_{A} \\
x_{A} \left ( \frac{1}{v} - \frac{1}{c} \right )
 & = & n_{A} \\
x_{A} \left ( 1 - \frac{v}{c} \right )
 & = & v n_{A} \\
x_{A} (1 - \beta)
 & = & v n_{A} \\
x_{A}
 & = & \frac{v n_{A}}{1 - \beta} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25b)} \qquad - \frac{x_{B}}{v}
 & = & - \frac{x_{B}}{c} + n_{B} \\
x_{B} \left ( \frac{1}{v} - \frac{1}{c} \right )
 & = & n_{B} \\
x_{B} \left ( 1 - \frac{v}{c} \right )
 & = & v n_{B} \\
x_{B} (1 - \beta)
 & = & v n_{B} \\
x_{B}
 & = & \frac{v n_{B}}{1 - \beta} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26a)} \qquad t_{A}
 & = & + \frac{x_{A}}{v} \\
& = & + \frac{1}{v} \frac{v n_{A}}{1 - \beta} \\
& = & + \frac{n_{A}}{1 - \beta} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26b)} \qquad t_{B}
 & = & - \frac{x_{B}}{v} \\
& = & - \frac{1}{v} \frac{v n_{B}}{1 - \beta} \\
& = & - \frac{n_{B}}{1 - \beta} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27)} \qquad t_{B}
 & = & \gamma \left ( t_{A} - \frac{\beta}{c} x_{A} \right ) \\
& = & \gamma \left ( t_{A} - \frac{\beta}{c} v t_{A} \right ) \\
& = & \gamma \left ( 1 - \frac{\beta v}{c} \right ) t_{A} \\
& = & \frac{1}{\sqrt{1 - \beta^{2}}} (1 - \beta^{2}) \frac{n_{A}}{1 - \beta} \\
& = & \sqrt{1 - \beta^{2}} \frac{n_{A}}{1 - \beta} \\
& = & \sqrt{\frac{1 - \beta^{2}}{(1 - \beta)^{2}}} \, n_{A} \\
& = & \sqrt{\frac{(1 - \beta) (1 + \beta)}{(1 - \beta) (1 - \beta)}} \, n_{A} \\
& = & \sqrt{\frac{1 + \beta}{1 - \beta}} \, n_{A} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad t_{B}
 & = & \sqrt{\frac{1 + \beta}{1 - \beta}} \, n_{A} \\
n_{A}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad n_{A}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B} \\
t_{A}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30)} \qquad t_{A, heen}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B, heen} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad t_{A, terug}
 & = & \sqrt{\frac{1 + \beta}{1 - \beta}} \, t_{B, terug} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad \Delta t_{A}
 & = & t_{A, heen} + t_{A, terug} \\
& = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B, heen} + \sqrt{\frac{1 + \beta}{1 - \beta}} \, t_{B, terug} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33)} \qquad \Delta t_{A}
 & = & \sqrt{\frac{1 - \beta}{1 + \beta}} \, t_{B, heen} + \sqrt{\frac{1 + \beta}{1 - \beta}} \, t_{B, terug} \\
& = & \sqrt{\frac{1 - \beta}{1 + \beta}} \frac{x_{A}}{\gamma v} + \sqrt{\frac{1 + \beta}{1 - \beta}} \frac{x_{A}}{\gamma v} \\
& = & \frac{x_{A}}{\gamma v} \left ( \sqrt{\frac{(1 - \beta) (1 - \beta)}{(1 + \beta) (1 - \beta)}}
 + \sqrt{\frac{(1 + \beta) (1 + \beta)}{(1 - \beta) (1 + \beta)}} \, \right ) \\
& = & \frac{x_{A}}{\gamma v} \left ( \sqrt{\frac{(1 - \beta)^{2}}{1 - \beta^{2}}} + \sqrt{\frac{(1 + \beta)^{2}}{1 - \beta^{2}}} \, \right ) \\
& = & \frac{x_{A}}{\gamma v} \left ( \frac{1 - \beta}{\sqrt{1 - \beta^{2}}} + \frac{1 + \beta}{\sqrt{1 - \beta^{2}}} \right ) \\
& = & \frac{x_{A}}{\gamma v} \frac{1 - \beta + 1 + \beta}{\sqrt{1 - \beta^{2}}} \\
& = & \frac{x_{A}}{\gamma v} \frac{2}{\sqrt{1 - \beta^{2}}} \\
& = & \frac{2 x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad \Delta t
 & = & \Delta t_{A} - \Delta t_{B} \\
& = & \Delta t_{A} - (t_{B, heen} + t_{B, terug}) \\
& = & \frac{2 x_{A}}{v} - \left ( \frac{x_{A}}{\gamma v} + \frac{x_{A}}{\gamma v} \right ) \\
& = & \frac{2 x_{A}}{v} - \frac{2 x_{A}}{\gamma v} \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \frac{1}{\gamma} \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35a)} \qquad t_{Ah}
 & = & \gamma \left ( t_{Bh} + \frac{\beta_{h}}{c} x_{Bh} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35b)} \qquad x_{Ah}
 & = & \gamma (x_{Bh} + v_{h} t_{Bh}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36a)} \qquad t_{Bh}
 & = & \gamma \left ( t_{Ah} - \frac{\beta_{h}}{c} x_{Ah} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36b)} \qquad x_{Bh}
 & = & \gamma (x_{Ah} - v_{h} t_{Ah}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37a)} \qquad t_{At}
 & = & \gamma \left ( t_{Bt} + \frac{\beta_{t}}{c} x_{Bt} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37b)} \qquad x_{At}
 & = & \gamma (x_{Bt} + v_{t} t_{Bt}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(38a)} \qquad t_{Bt}
 & = & \gamma \left ( t_{At} - \frac{\beta_{t}}{c} x_{At} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(38b)} \qquad x_{Bt}
 & = & \gamma (x_{At} - v_{t} t_{At}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(39a)} \qquad t_{Ah}
 & = & \gamma \left ( t_{Bh} + \frac{\beta_{h}}{c} x_{Bh} \right ) \\
& = & \gamma \left ( t_{Bh} + \frac{\beta_{h}}{c} \cdot 0 \right ) \\
& = & \gamma (t_{Bh} + 0) \\
& = & \gamma t_{Bh} \\
t_{Bh}
 & = & \frac{t_{Ah}}{\gamma} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(39b)} \qquad x_{Ah}
 & = & \gamma (x_{Bh} + v_{h} t_{Bh}) \\
& = & \gamma (0 + v_{h} t_{Bh}) \\
& = & \gamma v_{h} t_{Bh} \\
t_{Bh}
 & = & \frac{x_{Ah}}{\gamma v_{h}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(40a)} \qquad t_{Bh}
 & = & \gamma \left ( t_{Ah} - \frac{\beta_{h}}{c} x_{Ah} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(40b)} \qquad x_{Bh}
 & = & \gamma (x_{Ah} - v_{h} t_{Ah}) \\
0
 & = & \gamma (x_{Ah} - v_{h} t_{Ah}) \\
& = & x_{Ah} - v_{h} t_{Ah} \\
t_{Ah}
 & = & \frac{x_{Ah}}{v_{h}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(41a)} \qquad t_{At}
 & = & \gamma \left ( t_{Bt} + \frac{\beta_{t}}{c} x_{Bt} \right ) \\
& = & \gamma \left ( t_{Bt} + \frac{\beta_{t}}{c} \cdot 0 \right ) \\
& = & \gamma (t_{Bt} + 0) \\
& = & \gamma t_{Bt} \\
t_{Bt}
 & = & \frac{t_{At}}{\gamma} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(41b)} \qquad x_{At}
 & = & \gamma (x_{Bt} + v_{t} t_{Bt}) \\
& = & \gamma (0 + v_{t} t_{Bt}) \\
& = & \gamma v_{t} t_{Bt}) \\
t_{Bt}
 & = & \frac{x_{At}}{\gamma v_{t}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(42a)} \qquad t_{Bt}
 & = & \gamma \left ( t_{At} - \frac{\beta_{t}}{c} x_{At} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(42b)} \qquad x_{Bt}
 & = & \gamma (x_{At} - v_{t} t_{At}) \\
0
 & = & \gamma (x_{At} - v_{t} t_{At}) \\
& = & x_{At} - v_{t} t_{At} \\
t_{At}
 & = & \frac{x_{At}}{v_{t}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(43)} \qquad \Delta t_{A}
 & = & t_{Ah} + t_{At} \\
& = & \frac{x_{Ah}}{v_{h}} + \frac{x_{At}}{v_{t}} \\
& = & \frac{x_{A}}{v} + \frac{- x_{A}}{- v} \\
& = & \frac{x_{A}}{v} + \frac{x_{A}}{v} \\
& = & \frac{2 x_{A}}{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(44)} \qquad \Delta t_{B}
 & = & t_{Bh} + t_{Bt} \\
& = & \frac{x_{Ah}}{\gamma v_{h}} + \frac{x_{At}}{\gamma v_{t}} \\
& = & \frac{x_{A}}{\gamma v} + \frac{- x_{A}}{\gamma (- v)} \\
& = & \frac{x_{A}}{\gamma v} + \frac{x_{A}}{\gamma v} \\
& = & \frac{2 x_{A}}{\gamma v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(45)} \qquad \Delta t
 & = & \Delta t_{A} - \Delta t_{B} \\
& = & \frac{2 x_{A}}{v} - \frac{2 x_{A}}{\gamma v} \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \frac{1}{\gamma} \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(46)} \qquad \Delta t
 & = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(47)} \qquad \sqrt{1 - x^{2}}
 & = & 1 - \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{16} x^{6} - \frac{5}{128} x^{8} - \frac{7}{256} x^{10} + \ldots \\
& = & - \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} x^{2 n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(48)} \qquad \Delta t
 & = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 + \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \beta^{2 n} \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 - \left ( 1 - \frac{1}{2} \beta^{2} - \frac{1}{8} \beta^{4} - \frac{1}{16} \beta^{6} - \frac{5}{128} \beta^{8}
 - \frac{7}{256} \beta^{10} + \ldots \right ) \right ) \\
& = & \frac{2 x_{A}}{v} \left ( 1 - 1 + \frac{1}{2} \beta^{2} + \frac{1}{8} \beta^{4} + \frac{1}{16} \beta^{6} + \frac{5}{128} \beta^{8}
 + \frac{7}{256} \beta^{10} + \ldots \right ) \\
& = & \frac{2 x_{A}}{v} \left ( \frac{1}{2} \beta^{2} + \frac{1}{8} \beta^{4} + \frac{1}{16} \beta^{6} + \frac{5}{128} \beta^{8}
 + \frac{7}{256} \beta^{10} + \ldots \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(49)} \qquad \Delta t
 & = & \frac{2 x_{A}}{v} \left ( \frac{1}{2} \beta^{2} + \frac{1}{8} \beta^{4} + \frac{1}{16} \beta^{6} + \frac{5}{128} \beta^{8}
 + \frac{7}{256} \beta^{10} + \ldots \right ) \\
& \approx & \frac{2 x_{A}}{v} \left ( \frac{1}{2} \beta^{2} \right ) \\
& = & \frac{x_{A}}{v} \beta^{2} \\
& = & \frac{x_{A} v}{c^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(50)} \qquad \Delta t
 & = & \frac{2 x_{A}}{v} \left ( 1 - \sqrt{1 - \beta^{2}} \, \right ) \\
& = & \frac{2 x_{A}}{c} \left ( 1 - \sqrt{1 - 1^{2}} \, \right ) \\
& = & \frac{2 x_{A}}{c} (1 - 0) \\
& = & \frac{2 x_{A}}{c} \\
\end{eqnarray*}
\end{document}
