From: Francis Russell <francis@unchartedbackwaters.co.uk>
Date: Fri, 2 Dec 2011 17:31:09 +0000 (+0000)
Subject: Replace instances of eqnarray with align.
X-Git-Url: https://git.unchartedbackwaters.co.uk/w/?a=commitdiff_plain;h=HEAD;p=francis%2Frelativity.git

Replace instances of eqnarray with align.
---

diff --git a/relat10.tex b/relat10.tex
index 63c5f1d..dc7f123 100644
--- a/relat10.tex
+++ b/relat10.tex
@@ -1093,12 +1093,12 @@ the relative orientation in space of the co-ordinate systems indicated
 in the diagram (Fig \ref{fig:2}), this problem is solved by means of the
 equations:
 
-\begin{eqnarray*} 
-x' &=& \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\
-y' &=& y \\
-z' &=& z \\
-t' &=& \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \\
-\end{eqnarray*}
+\begin{align*} 
+x' &= \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\
+y' &= y \\
+z' &= z \\
+t' &= \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \\
+\end{align*}
 
 \noindent This system of equations is known as the ``Lorentz transformation."\footnotemark
 
@@ -1107,12 +1107,12 @@ basis the tacit assumptions of the older mechanics as to the absolute
 character of times and lengths, then instead of the above we should
 have obtained the following equations:
 
-\begin{eqnarray*} 
-x' &=& x - vt \\
-y' &=& y \\
-z' &=& z \\
-t' &=& t \\
-\end{eqnarray*}
+\begin{align*} 
+x' &= x - vt \\
+y' &= y \\
+z' &= z \\
+t' &= t \\
+\end{align*}
 
 \noindent This system of equations is often termed the ``Galilei
 transformation." The Galilei transformation can be obtained from the
@@ -1136,10 +1136,10 @@ relation between $x'$ and $t'$. In point of fact, if we substitute for $x$
 the value $ct$ in the first and fourth equations of the Lorentz
 transformation, we obtain:
 
-\begin{eqnarray*} 
-x' &=& \frac{(c-v)t}{\sqrt{I-\frac{v^2}{c^2}}} \\
-t' &=& \frac{(I-\frac{v}{c})t}{\sqrt{I-\frac{v^2}{c^2}}}
-\end{eqnarray*}
+\begin{align*} 
+x' &= \frac{(c-v)t}{\sqrt{I-\frac{v^2}{c^2}}} \\
+t' &= \frac{(I-\frac{v}{c})t}{\sqrt{I-\frac{v^2}{c^2}}}
+\end{align*}
 
 \noindent from which, by division, the expression
 
@@ -1175,10 +1175,10 @@ lie with respect to $K$ at a particular time $t$ of the system $K$. By means
 of the first equation of the Lorentz transformation the values of
 these two points at the time $t = 0$ can be shown to be
 
-\begin{eqnarray*} 
-x_{\mbox{(begining of rod)}} &=& 0 \sqrt{I-\frac{v^2}{c^2}} \\
-x_{\mbox{(end of rod)}} &=& 1 \sqrt{I-\frac{v^2}{c^2}}
-\end{eqnarray*}
+\begin{align*} 
+x_{\mbox{(begining of rod)}} &= 0 \sqrt{I-\frac{v^2}{c^2}} \\
+x_{\mbox{(end of rod)}} &= 1 \sqrt{I-\frac{v^2}{c^2}}
+\end{align*}
 ~
 
 \noindent the distance between the points being $\sqrt{I-v^2/c^2}$.
@@ -2649,10 +2649,12 @@ the table. We call these two numbers the co-ordinates of the surface
 of the table (Gaussian co-ordinates). For example, the point $P$ in the
 diagram has the Gaussian co-ordinates $u=3$, $v=1$. Two neighbouring
 points $P$ and $P_1$ on the surface then correspond to the co-ordinates
-\begin{eqnarray*} 
-P: & u ~~,~~v \\
-P': & u + du , v + dv
-\end{eqnarray*} 
+
+\begin{align*} 
+P:&~u,~v \\
+P':&~u + du,~v + dv
+\end{align*} 
+
 where $du$ and $dv$ signify very small numbers. In a similar manner we may
 indicate the distance (line-interval) between $P$ and $P_1$, as measured
 with a little rod, by means of the very small number $ds$. Then
@@ -3650,12 +3652,14 @@ right-hand side.
 We can characterise the Lorentz transformation still more simply if we
 introduce the imaginary $\sqrt{-I} \cdot ct$ in place of $t$, as time-variable. If, in
 accordance with this, we insert
-\begin{eqnarray*}
-                              x_1 & = & x \\
-                              x_2 & = & y \\
-                              x_3 & = & z \\
-                              x_4 & = & \sqrt{-I} \cdot ct
-\end{eqnarray*}
+
+\begin{align*}
+                              x_1 & =  x \\
+                              x_2 & =  y \\
+                              x_3 & =  z \\
+                              x_4 & =  \sqrt{-I} \cdot ct
+\end{align*}
+
 and similarly for the accented system $K^1$, then the condition which is
 identically satisfied by the transformation can be expressed thus: