Editing Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2 (section)

=====Determine Coefficients=====
The difference between the last two expressions gives,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_J - x_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
[b\tilde{r}_J^2 + c\tilde{r}_J^4] 
-
[b(\tilde{r}_{J}-\Delta\tilde{r})^2 + c(\tilde{r}_{J}-\Delta\tilde{r})^4]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b\tilde{r}_J^2 + c\tilde{r}_J^4 
-
b(\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2) 
- 
c(\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2)(\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
c\tilde{r}_J^4 
+
2b\tilde{r}_J\Delta\tilde{r} - b\Delta\tilde{r}^2 
- 
c\biggl[
\tilde{r}_{J}^2 (\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2)
- 2\tilde{r}_J\Delta\tilde{r}(\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2)
+ \Delta\tilde{r}^2(\tilde{r}_{J}^2 - 2\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
c\tilde{r}_J^4 
+
2b\tilde{r}_J\Delta\tilde{r} - b\Delta\tilde{r}^2 
- 
c\biggl[
\tilde{r}_{J}^4 - 2\tilde{r}_J^3\Delta\tilde{r} + \tilde{r}^2\Delta\tilde{r}^2
-2\tilde{r}_J^3\Delta\tilde{r} + 4\tilde{r}_J^2\Delta\tilde{r}^2 - 2\tilde{r}_J\Delta\tilde{r}^3
+ \tilde{r}_{J}^2\Delta\tilde{r}^2 - 2\tilde{r}_J\Delta\tilde{r}^3 + \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b \biggl[ 2\tilde{r}_J\Delta\tilde{r} - \Delta\tilde{r}^2 \biggr]
+c\biggl[
4\tilde{r}_J^3\Delta\tilde{r} 
- 6\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2b\tilde{r}_J\Delta\tilde{r} \biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]
+c\biggl[
4\tilde{r}_J^3\Delta\tilde{r} 
- 6\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
\, .
</math>
  </td>
</tr>
</table>

<!-- ************************** -->

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
Repeat, to check &hellip;

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
x_J - x_{J-1} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
[b\tilde{r}_J^2 + c\tilde{r}_J^4]
-[b(\tilde{r}_{J}-\Delta\tilde{r})^2 + c(\tilde{r}_{J}-\Delta\tilde{r})^4]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
[b\tilde{r}_J^2 + c\tilde{r}_J^4]
-[b(\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2) 
+ 
c(\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2)(\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
[c\tilde{r}_J^4]
+ b[2\tilde{r}_{J}\Delta\tilde{r} - \Delta\tilde{r}^2] 
- c[ 
\tilde{r}_{J}^2 (\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2)
- 2\tilde{r}_{J}\Delta\tilde{r} (\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2)
+ 
\Delta\tilde{r}^2(\tilde{r}_{J}^2 - 2\tilde{r}_{J}\Delta\tilde{r} + \Delta\tilde{r}^2)
]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b[2\tilde{r}_{J}\Delta\tilde{r} - \Delta\tilde{r}^2] 
- c[ 
(- 2\tilde{r}_{J}^3\Delta\tilde{r} + \tilde{r}_{J}^2 \Delta\tilde{r}^2)
+ (-2\tilde{r}_{J}^3\Delta\tilde{r}  + 4\tilde{r}_{J}^2\Delta\tilde{r}^2  - 2\tilde{r}_{J}\Delta\tilde{r}^3 )
+ 
(\tilde{r}_{J}^2\Delta\tilde{r}^2 - 2\tilde{r}_{J}\Delta\tilde{r}^3 + \Delta\tilde{r}^4)
]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2b\tilde{r}_J \Delta\tilde{r}\biggl[1 - \frac{\Delta\tilde{r}}{2\tilde{r}_{J}} \biggr] 
+ c\biggl[ 
4\tilde{r}_{J}^3\Delta\tilde{r} - 6\tilde{r}_{J}^2 \Delta\tilde{r}^2
+ 4\tilde{r}_{J}\Delta\tilde{r}^3 - \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
x_J - x_{J-1} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}- 4c\tilde{r}_J^3 \Delta\tilde{r}\biggr]
\biggl[1 - \frac{\Delta\tilde{r}}{2\tilde{r}_{J}} \biggr] 
+ c\biggl[ 
4\tilde{r}_{J}^3\Delta\tilde{r} - 6\tilde{r}_{J}^2 \Delta\tilde{r}^2
+ 4\tilde{r}_{J}\Delta\tilde{r}^3 - \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[1 - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) \biggr] 
+ 
c\biggl[4\tilde{r}_J^3 \Delta\tilde{r}\biggr]\biggl[\frac{\Delta\tilde{r}}{2\tilde{r}_{J}} - 1\biggr] 
+ 
c\biggl[ 
4\tilde{r}_{J}^3\Delta\tilde{r} - 6\tilde{r}_{J}^2 \Delta\tilde{r}^2
+ 4\tilde{r}_{J}\Delta\tilde{r}^3 - \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ x_J - x_{J-1} - (x_J)^' \Delta\tilde{r}
\biggl[1 - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) \biggr] 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
c \biggl[2\tilde{r}_J^2\Delta\tilde{r}^2 - 4\tilde{r}_J^3 \Delta\tilde{r}\biggr] 
+ 
c\biggl[ 
4\tilde{r}_{J}^3\Delta\tilde{r} - 6\tilde{r}_{J}^2 \Delta\tilde{r}^2
+ 4\tilde{r}_{J}\Delta\tilde{r}^3 - \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
c\biggl[ 
- 4\tilde{r}_{J}^2 \Delta\tilde{r}^2
+ 4\tilde{r}_{J}\Delta\tilde{r}^3 - \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
4c\tilde{r}_{J}^4 \cdot \mathcal{A}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1}-x_J + (x_J)^' \Delta\tilde{r}
\biggl[1 - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1}-x_J + (x_J)^' \tilde{r}_J
\biggl[\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr] 
</math>
  </td>
</tr>
</table>
where,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\mathcal{A}
</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[ 
\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4
\biggr] \, .
</math>
  </td>
</tr>
</table>

Also,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
2b\tilde{r}_J^2  
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \tilde{r}_J - 4c\tilde{r}_J^4 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 2b\tilde{r}_J^2 \cdot \mathcal{A} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \tilde{r}_J\cdot \mathcal{A} 
- \biggl\{ 
x_{J-1}-x_J + (x_J)^' \tilde{r}_J
\biggl[\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J-x_{J-1} 
+ (x_J)^' \tilde{r}_J
\biggl[\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
\, . 
</math>
  </td>
</tr>
</table>

</td></tr></table>

<!-- ************************** -->

From the first expression, we also see that,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
2b\tilde{r}_J \Delta\tilde{r} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \Delta\tilde{r}- 4c\tilde{r}_J^3 \Delta\tilde{r}
\, .
</math>
  </td>
</tr>
</table>
Therefore we have,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_J - x_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[
(x_J)^' \Delta\tilde{r}- 4c\tilde{r}_J^3 \Delta\tilde{r}
\biggr]
\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]
+c\biggl[
4\tilde{r}_J^3\Delta\tilde{r} 
- 6\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]
\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]
+
\biggl[2c\tilde{r}_J^2 \Delta\tilde{r}\biggr]
\biggl[ \Delta\tilde{r} - 2\tilde{r}_J\biggr]
+
c\biggl[
4\tilde{r}_J^3\Delta\tilde{r} 
- 6\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]
\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]
+
c\biggl[2\tilde{r}_J^2 \Delta\tilde{r}^2
-
4\tilde{r}_J^3 \Delta\tilde{r}\biggr]
+
c\biggl[
4\tilde{r}_J^3\Delta\tilde{r} 
- 6\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]
\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]
+
c\biggl[
- 4\tilde{r}_J^2\Delta\tilde{r}^2 
+ 4\tilde{r}_J\Delta\tilde{r}^3 
- \Delta\tilde{r}^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
c\Delta\tilde{r}^2
\biggl[ 4\tilde{r}_J^2 - 4\tilde{r}_J\Delta\tilde{r} + \Delta\tilde{r}^2 \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
4c \tilde{r}_J^2 \Delta\tilde{r}^2
\biggl[ 1 - \frac{\Delta\tilde{r}}{\tilde{r}_J} + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
4c \tilde{r}_J^4 
\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1} 
\, .
</math>
  </td>
</tr>
</table>
Hence also,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
2b\tilde{r}_J \Delta\tilde{r} \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \Delta\tilde{r}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)
- \biggl\{ 
4c\tilde{r}_J^2 \Delta\tilde{r}^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
2b\tilde{r}_J \Delta\tilde{r} \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)
\biggl[ 1 - \frac{\Delta\tilde{r}}{\tilde{r}_J} + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \Delta\tilde{r}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)
\biggl[ 1 - \frac{\Delta\tilde{r}}{\tilde{r}_J} + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr]
- \biggl\{ 
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} 
+
(x_J)^' \biggl(\frac{\Delta\tilde{r}^2}{\tilde{r}_J}\biggr)
\biggl[ 1 - \frac{\Delta\tilde{r}}{\tilde{r}_J} + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr]
-\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} 
+
(x_J)^' \biggl(\frac{\Delta\tilde{r}^2}{\tilde{r}_J}\biggr)
\biggl[ 1 - \frac{\Delta\tilde{r}}{\tilde{r}_J} + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr]
- (x_J)^' \Delta\tilde{r}
+\frac{1}{2} (x_J)^' \biggl( \frac{\Delta\tilde{r}^2}{\tilde{r}_J} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
2b\tilde{r}_J^2 
\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} 
+
(x_J)^' \Delta\tilde{r} 
\biggl[-1 + \frac{3}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
</math>
  </td>
</tr>
</table>

Finally,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>a</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - b\tilde{r}_J^2 - c\tilde{r}_J^4 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ 
a\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
  </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
- b\tilde{r}_J^2 \biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]

- c\tilde{r}_J^4 \biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \frac{1}{2}\biggl\{
x_J - x_{J-1} 
+
(x_J)^' \Delta\tilde{r} 
\biggl[-1 + \frac{3}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \frac{1}{4}\biggl\{
\biggl[(x_J)^' \Delta\tilde{r}\biggr]\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
- \frac{1}{2}\biggl\{
x_J - x_{J-1} 
\biggr\}
- \frac{1}{4}\biggl\{
-x_J + x_{J-1}
\biggr\}

</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \frac{1}{2}\biggl\{
(x_J)^' \Delta\tilde{r} 
\biggl[-1 + \frac{3}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\biggr\}
- \frac{1}{4}\biggl\{
(x_J)^' \Delta\tilde{r}\biggl[ 1 - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]

+\frac{x_{J-1} - x_J}{4}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
(x_J)^' \Delta\tilde{r} 
\biggl[\frac{1}{2}- \frac{1}{4} + \frac{1}{8}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) - \frac{3}{4}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 - \frac{1}{8}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{x_{J-1} - x_J}{4}
+
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
+ 
(x_J)^' \Delta\tilde{r} 
\biggl[\frac{1}{4} - \frac{5}{8}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)  
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 - \frac{1}{8}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
<div align="center"><b>OLD Summary:</b></div>

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>a \cdot \mathcal{A}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{x_{J-1} - x_J}{4}
+
x_J\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
+ 
(x_J)^' \Delta\tilde{r} 
\biggl[\frac{1}{4} - \frac{5}{8}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)  
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 - \frac{1}{8}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math> 
2b\tilde{r}_J^2 \cdot \mathcal{A}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} 
+
(x_J)^' \Delta\tilde{r} 
\biggl[-1 + \frac{3}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 + \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
4c \tilde{r}_J^4 \cdot \mathcal{A}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' \Delta\tilde{r}\biggl[ 1 - \frac{\Delta\tilde{r}}{2\tilde{r}_J} \biggr]-x_J + x_{J-1} 
\, ,
</math>
  </td>
</tr>
</table>
where,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\mathcal{A}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

<!-- 333333333333333333 -->

Repeat, to check &hellip;

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>4\biggl[ x_J - a \biggr]\cdot \mathcal{A}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4b\tilde{r}_J^2\cdot \mathcal{A} + 4c\tilde{r}_J^4 \cdot \mathcal{A}
=
2\biggl\{ 2b\tilde{r}_J^2 \cdot \mathcal{A} \biggr\} 
+ 
\biggl\{ 4c\tilde{r}_J^4 \cdot \mathcal{A} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl\{ x_J-x_{J-1} 
+ (x_J)^' \tilde{r}_J
\biggl[\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
\biggr\} 
+ 
\biggl\{ 
x_{J-1}-x_J + (x_J)^' \tilde{r}_J
\biggl[\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ x_J-x_{J-1} 
+ (x_J)^' \tilde{r}_J
\biggl[2\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~  4 a \cdot \mathcal{A}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4 x_J \cdot \mathcal{A} - \biggl\{ x_J-x_{J-1} 
+ (x_J)^' \tilde{r}_J
\biggl[2\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(4 \mathcal{A} - 1) x_J+x_{J-1} - (x_J)^' \tilde{r}_J
\biggl[2\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
</math>
  </td>
</tr>
</table>

<!-- 444444444444444 -->

<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
<div align="center"><b>New Summary:</b></div>

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>4a\cdot \mathcal{A}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(4 \mathcal{A} - 1) x_J+x_{J-1} - (x_J)^' \tilde{r}_J
\biggl[2\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(4 \mathcal{A} - 1) x_J+x_{J-1} + (x_J)^' \Delta\tilde{r}
\biggl\{ 1 
- \frac{5}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
+ 2\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \frac{1}{2}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
\biggr\}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
2b\tilde{r}_J^2 \cdot \mathcal{A} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J-x_{J-1} 
+ (x_J)^' \tilde{r}_J
\biggl[\mathcal{A} -\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) 
+ \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr)^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J-x_{J-1} 
+ (x_J)^' \Delta\tilde{r}
\biggl\{-1 
+ \frac{3}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr) 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
\biggr\}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
4c\tilde{r}_{J}^4 \cdot \mathcal{A}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1}-x_J + (x_J)^' \Delta\tilde{r}
\biggl[ 1 - \frac{1}{2}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_{J}}\biggr) \biggr]
\, , 
</math>
  </td>
</tr>
</table>
where,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\mathcal{A}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \frac{1}{4}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr]
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>