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

====Fourth Approximation====

Let's assume that we know the four quantities, <math>x_{J-1}, x_J,(x_J)^' \equiv (dx/d\tilde{r})_J</math>, and <math>(x_{J-1})^' \equiv (dx/d\tilde{r})_{J-1}</math> and want to project forward to determine, <math>x_{J+1}</math>.  We should assume that, locally, the displacement function <math>x</math> is cubic in <math>\tilde{r}</math>, that is,

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

<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b\tilde{r} + c\tilde{r}^2 + e\tilde{r}^3
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{dx}{d\tilde{r}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b + 2c\tilde{r} + 3e\tilde{r}^2 \, ,
</math>
  </td>
</tr>
</table>
where we have four unknowns, <math>a, b, c, e</math>.  These can be determined by appropriately combining the four relations,

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

<tr>
  <td align="right"><math>(x_J)^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b + 2c\tilde{r}_J + 3e\tilde{r}_J^2\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>(x_{J-1})^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b + 2c(\tilde{r}_J - \Delta\tilde{r}) + 3e(\tilde{r}_J - \Delta\tilde{r})^2\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>x_J</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b\tilde{r}_J + c\tilde{r}_J^2 + e\tilde{r}_J^3\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>x_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 + e(\tilde{r}_J - \Delta\tilde{r})^3 \, ,
</math>
  </td>
</tr>
</table>
The difference between the first 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>
[2c\tilde{r}_J + 3e\tilde{r}_J^2]
-
[2c(\tilde{r}_J - \Delta\tilde{r}) + 3e(\tilde{r}_J - \Delta\tilde{r})^2]
</math>
  </td>
</tr>

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

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

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

And 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>
\biggl[ b\tilde{r}_J + c\tilde{r}_J^2 + e\tilde{r}_J^3\biggr]
-
\biggl[ b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 + e(\tilde{r}_J - \Delta\tilde{r})^3\biggr]
</math>
  </td>
</tr>

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

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

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

+ 3e\Delta\tilde{r}
\biggl[  \frac{\Delta\tilde{r}^2}{2} - \tilde{r}_J \Delta\tilde{r}\biggr]

- 3e\Delta\tilde{r}
\biggl[  \tilde{r}_J\Delta\tilde{r} - 2\tilde{r}_J^2\biggr]

- 3e\Delta\tilde{r} \biggl[
\tilde{r}_J^2 - \tilde{r}_J\Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{3} 
\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}\biggr]
+ 
\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \frac{\Delta\tilde{r}}{2} - \tilde{r}_J\biggr] 

- 3e\Delta\tilde{r}\biggl\{
\biggl[  \tilde{r}_J \Delta\tilde{r} - \frac{\Delta\tilde{r}^2}{2} \biggr]

+
\biggl[  \tilde{r}_J\Delta\tilde{r} - 2\tilde{r}_J^2\biggr]

+ \biggl[
\tilde{r}_J^2 - \tilde{r}_J\Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{3} 
\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}\biggr]
+ 
\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \frac{\Delta\tilde{r}}{2} - \tilde{r}_J\biggr] 

+ e\Delta\tilde{r}\biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
\, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left">
<div align="center"><b>Summary #1:</b></div>
In terms of the coefficient, <math>e</math> &hellip;

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

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

+ e\Delta\tilde{r}\biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
\, ,
</math>
  </td>
</tr>

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

</td></tr></table>
Hence, from the first of the four relations, we find that,

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

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

+ e\Delta\tilde{r}\biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
+
\biggl\{ 
\biggl[ (x_J)^' - (x_{J-1})^'\biggr] + e\Delta\tilde{r}\biggl[ 3\Delta\tilde{r} - 6\tilde{r}_J \biggr]
\biggr\} \tilde{r}_J 
</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}\biggr]
+ 
\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \frac{\Delta\tilde{r}}{2} \biggr] 

+ e\Delta\tilde{r}\biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
+
e\Delta\tilde{r}\biggl[ 3\tilde{r}_J\Delta\tilde{r} - 6\tilde{r}_J^2 \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}\biggr]
+ 
\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \frac{\Delta\tilde{r}}{2} \biggr] 

+ e\Delta\tilde{r}\biggl[
-3\tilde{r}_J^2
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
</math>
  </td>
</tr>

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

+ e\biggl[\frac{\Delta\tilde{r}^3}{2} \biggr]
</math>
  </td>
</tr>

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

+ (x_J)^'\Delta\tilde{r}
</math>
  </td>
</tr>

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

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

Finally, from the third of the four relations, we can evaluate the coefficient, <math>a</math>; specifically,

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

<tr>
  <td align="right"><math>x_J - a - e\tilde{r}_J^3</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b\tilde{r}_J + c\tilde{r}_J^2 
</math>
  </td>
</tr>

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

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

+ e\Delta\tilde{r}\biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
\biggr\} 
+ 
\frac{\tilde{r}_J^2}{2\Delta\tilde{r}} \biggl\{
\biggl[ (x_J)^' - (x_{J-1})^'\biggr] + e\Delta\tilde{r}\biggl[ 3\Delta\tilde{r} - 6\tilde{r}_J \biggr]
\biggr\}
</math>
  </td>
</tr>

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

+ e\tilde{r}_J \biggl[
3\tilde{r}_J^2
- 3\tilde{r}_J \Delta\tilde{r} 
+ \frac{\Delta\tilde{r}^2}{2} 
\biggr]
\biggr\} 
+ 
\biggl\{
\frac{\tilde{r}_J^2}{2\Delta\tilde{r}} \biggl[ (x_J)^' - (x_{J-1})^'\biggr] 
+ \frac{e\tilde{r}_J^2}{2} \biggl[ 3\Delta\tilde{r} - 6\tilde{r}_J \biggr]
\biggr\}
</math>
  </td>
</tr>

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

+ e \biggl[
3\tilde{r}_J^3
- 3\tilde{r}_J^2 \Delta\tilde{r} 
+ \frac{\tilde{r}_J\Delta\tilde{r}^2}{2} 
\biggr]
+ 
e\biggl[ \frac{3\tilde{r}_J^2 \Delta\tilde{r}}{2} - 3\tilde{r}_J^3 \biggr]
</math>
  </td>
</tr>

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

+ \frac{e\Delta\tilde{r}}{2} \biggl[
\tilde{r}_J\Delta\tilde{r} 
-3\tilde{r}_J^2 \biggr] \, .
</math>
  </td>
</tr>
</table>
That is,

<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  -\frac{\tilde{r}_J}{\Delta\tilde{r}} \biggl[x_J - x_{J-1}\biggr]
- 
\frac{1}{2\Delta\tilde{r}}\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \tilde{r}_J \Delta\tilde{r} - \tilde{r}_J^2\biggr] 

- e\biggl\{
\frac{\Delta\tilde{r}}{2} \biggl[
\tilde{r}_J\Delta\tilde{r} 
-3\tilde{r}_J^2 \biggr]
+ \tilde{r}_J^3
\biggr\}
</math>
  </td>
</tr>

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

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

<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left">
<div align="center"><b>Summary #2:</b></div>
In terms of the coefficient, <math>e</math> &hellip;

<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  -\frac{\tilde{r}_J}{\Delta\tilde{r}} \biggl[x_J - x_{J-1}\biggr]
- 
\frac{1}{2\Delta\tilde{r}}\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[  \tilde{r}_J \Delta\tilde{r} - \tilde{r}_J^2\biggr] 

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

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

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

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

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

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

<table border=1 align="center" cellpadding="10" width="80%"><tr><td bgcolor="pink" align="left">
This is test ...
<table border="1" align="center" cellpadding="5">

  <tr>
<td align="center" bgcolor="white"><math>\tilde{r}_J = \tilde{r}_i + \Delta\tilde{r}</math></td>
<td align="center" bgcolor="white"><math>\Delta\tilde{r}</math></td>
<td align="center" bgcolor="white"><math>x_J</math></td>
<td align="center" bgcolor="white"><math>x_{J-1}</math></td>
<td align="center" bgcolor="white"><math>(x_J)^'</math></td>
<td align="center" bgcolor="white"><math>(x_{J-1})^'</math></td>
  </tr>

  <tr>
<td align="center" bgcolor="white">0.01740039</td>
<td align="center" bgcolor="white">0.001936393</td>
<td align="center" bgcolor="white">-4.695376</td>
<td align="center" bgcolor="white">-4.547832</td>
<td align="center" bgcolor="white">-116.0119</td>
<td align="center" bgcolor="white">-76.19513</td>
  </tr>
</table>
</td></tr>
<tr><td bgcolor="white" align="left">

<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> 
-3.36955 -2.76645 - e\Delta\tilde{r}^3(608.9698) = -232.7874
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>b \Delta\tilde{r}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0.5067329 + e\Delta\tilde{r}^3 (215.7856) = +80.819698
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>c\Delta\tilde{r}^2</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-0.0385505 + e\Delta\tilde{r}^3 ( -25.45794 ) = -9.51370
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>e \Delta\tilde{r}^3 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0.3721883
\, .
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right"><math>x_J </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a 
+ b\Delta\tilde{r} \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} \biggr)
+ c\Delta\tilde{r}^2 \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} \biggr)^2
+ e\Delta\tilde{r}^3 \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} \biggr)^3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-232.7874 +726.2442 -768.2108 +270.0593 = -4.68369 \, .
</math>
  </td>
</tr>
</table>
Higher precision value (from Excel) is  <math>x_J = -4.695376 \, ,</math> which precisely matches the input value.  Also from Excel, <math>x_{J-1} = -4.547832</math> and <math>x_{J+1} = -3.803455 \, .</math>
</td></tr></table>

As a result,

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

<tr>
  <td align="right"><math>x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
a 
\biggr\}
+ (\tilde{r}_J +\Delta\tilde{r}) \biggl\{
b
\biggr\}
+ (\tilde{r}_J+\Delta\tilde{r})^2 \biggl\{
c
\biggr\}
+ (\tilde{r}_J+\Delta\tilde{r})^3 \biggl\{
e
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
a 
\biggr\}
+ \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr) \biggl\{
b\Delta\tilde{r}
\biggr\}
+ \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr)^2 \biggl\{
c\Delta\tilde{r}^2
\biggr\}
+ \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr)^3 \biggl\{
e\Delta\tilde{r}^3
\biggr\}
</math>
  </td>
</tr>

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

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

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

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

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

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

<!-- CONTINUED equation development -->
<table border="0" align="center" cellpadding="5">

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

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

+ 
e\Delta\tilde{r}^3\biggl[ \frac{3}{2} - \frac{3\tilde{r}_J}{\Delta\tilde{r}} \biggr]
\biggl[ \biggl(\frac{\tilde{r}_J}{\Delta\tilde{r}}\biggr)^2 + \frac{2\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr]

+ e\Delta\tilde{r}^3\biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr) 
\biggl[ \biggl(\frac{\tilde{r}_J}{\Delta\tilde{r}}\biggr)^2 + \frac{2\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr]

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

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

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

+ 
\biggl[ \frac{3}{2} - \frac{3\tilde{r}_J}{\Delta\tilde{r}} \biggr]
\biggl[ \biggl(\frac{\tilde{r}_J}{\Delta\tilde{r}}\biggr)^2 + \frac{2\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr]

+ \biggl( \frac{\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr) 
\biggl[ \biggl(\frac{\tilde{r}_J}{\Delta\tilde{r}}\biggr)^2 + \frac{2\tilde{r}_J}{\Delta\tilde{r}} + 1\biggr]

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

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

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

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

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

Continuing &hellip;

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

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

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

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

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

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

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

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

Finally we may write,

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

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

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

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

<table border=1 align="center" cellpadding="10" width="80%"><tr><td bgcolor="lightblue" align="left">
This is test ...
<table border="1" align="center" cellpadding="5">

  <tr>
<td align="center" bgcolor="white"><math>\Delta\tilde{r}</math></td>
<td align="center" bgcolor="white"><math>x_J</math></td>
<td align="center" bgcolor="white"><math>x_{J-1}</math></td>
<td align="center" bgcolor="white"><math>(x_J)^'</math></td>
<td align="center" bgcolor="white"><math>(x_{J-1})^'</math></td>
  </tr>

  <tr>
<td align="center" bgcolor="white">0.001936393</td>
<td align="center" bgcolor="white">-4.695376</td>
<td align="center" bgcolor="white">-4.547832</td>
<td align="center" bgcolor="white">-116.0119</td>
<td align="center" bgcolor="white">-76.19513</td>
  </tr>
</table>
<tr><td bgcolor="white" align="center">

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

<tr>
  <td align="right"><math>x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[5x_{J-1} - 4x_J \biggr]
+ 
\biggl[ 4(x_J)^' + 2 (x_{J-1})^'\biggr]\Delta\tilde{r}
=
-5.15132
\, .
</math>
  </td>
</tr>
</table>
</td></tr>
</td></tr></table>