Editing Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2

__FORCETOC__
=Better Interface for 51BiPolytrope Stability Study=

This is Part 2 of an extended chapter discussion.  For Part 1, go [[Appendix/Ramblings/51BiPolytropeStability/BetterInterface|here]].
==Discretize for Numerical Integration (continued)==

===General Discretization===
====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>

====Fifth Approximation====

Let's assume that we know the three quantities, <math>x_{J-1}, x_J,(x_J)^' \equiv (dx/d\tilde{r})_J</math>, and  want to project forward to determine, <math>x_{J+1}</math>.  Here we will assume that, locally, the displacement function <math>x</math> has only an even-power dependence on  <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}^2 + c\tilde{r}^4 
</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>
2b\tilde{r} + 4c\tilde{r}^3 \, ,
</math>
  </td>
</tr>
</table>
where we have three unknowns, <math>a, b, c</math>.  These can be determined by appropriately combining the three 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>
2b\tilde{r}_J + 4c\tilde{r}_J^3 \, ,
</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^2 + c\tilde{r}_J^4 \, ,
</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})^2 + c(\tilde{r}_{J}-\Delta\tilde{r})^4 \, ,
</math>
  </td>
</tr>
</table>

=====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>

=====Project Forward=====

Let's now determine the expression for <math>x_{J+1}</math>.  We begin by writing &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>
a + 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>
a + b(\tilde{r}_{J}^2 +2\tilde{r}_J\Delta\tilde{r} + \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>
a + b\tilde{r}_{J}^2 \biggl[1 + \frac{2\Delta\tilde{r}}{\tilde{r}_J} + \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggr] 
+ c \biggl[(\tilde{r}_{J}^4 +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)
\biggr] 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right"><math>\mathcal{A} \cdot x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a \cdot \mathcal{A} 
</math>
  </td>
</tr>

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

Keep going &hellip;

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

<tr>
  <td align="right"><math>\mathcal{A} \cdot x_{J+1}</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]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
(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]
\biggl[\frac{1}{2} \biggr] 
</math>
  </td>
</tr>

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

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

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

<tr><td align="center" colspan="3">---- ''next in line'' ----</td></tr>

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

<tr>
  <td align="right"><math>\Rightarrow~~~ \mathcal{A}\cdot x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J - x_{J-1})\biggl[ 
- 2\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3\biggr]
+ 
\mathcal{A}\cdot x_{J-1} 
+ 
2(x_J)^' \Delta\tilde{r} 
\biggl[ 
\biggl(\frac{\Delta\tilde{r}}{\tilde{r}}\biggr)^2
- \frac{1}{4}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggr] \, .
</math>
  </td>
</tr>
</table>
<span id="FirstGrouping">Grouping terms</span> with like powers of <math>(\Delta\tilde{r}/\tilde{r}_J)</math> we find,

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

<tr>
  <td align="right"><math>0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 \biggl[
\biggl(x_{J-1} - x_{J+1}\biggr)+ 2(x_J)^' \Delta\tilde{r} 
\biggr]
+
\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 \biggl[
x_{J-1} - 2x_J + x_{J+1}  
\biggr]
+
\frac{1}{4}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 \biggl[
\biggl(x_{J-1} - x_{J+1}\biggr) - 2(x_J)^' \Delta\tilde{r} 
\biggr]
\, .
</math>
  </td>
</tr>
</table>


----

<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)^' \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>
\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">
<math>
4c\tilde{r}_{J}^4 \cdot \mathcal{A}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
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]
\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>

Try again &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>
a + 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>
a 
+ 
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>
a 
+ 
b(\tilde{r}_{J}^2 +2 \tilde{r}_J \Delta\tilde{r} + \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>
a 
+ 
b(\tilde{r}_{J}^2 +2 \tilde{r}_J \Delta\tilde{r} + \Delta\tilde{r}^2) 
+ 
c \biggl[(\tilde{r}_{J}^4 +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) 
\biggr]
</math>
  </td>
</tr>

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

Multiplying through by <math>4\mathcal{A}</math> gives,

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

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

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

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4 \mathcal{A}x_J + 4\biggl(x_J - x_{J-1} \biggr)
\biggl\{
- 
\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] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ (x_J)^' \Delta\tilde{r}
\biggl\{
\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">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+
\biggl[-2 
+ 3\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">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+
\biggl[-4\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr) 
+ 6\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
- 4\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ \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>
+
\biggl[-2\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
+ 3\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
- 2\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 
+ \frac{1}{2}\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^5 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
\biggl[1 +4 \biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr) + 6\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
+ 
4\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
+ 
\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 
\biggr]
+
\biggl[- \frac{1}{2}\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr) 
-2 \biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 -3\biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^3 
- 2 \biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 
- \frac{1}{2} \biggl(\frac{ \Delta\tilde{r}}{\tilde{r}_J}\biggr)^5 
\biggr]

\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4 \mathcal{A}x_J + 4\biggl(x_J - x_{J-1} \biggr)
\biggl\{
- 
\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] 
\biggr\}
+ (x_J)^' \Delta\tilde{r}
\biggl\{
\biggl[ 
8\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
-2\biggl( \frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4 
\biggr]
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Grouping terms with like powers of <math>(\Delta\tilde{r}/\tilde{r}_J)</math> we find,

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

<tr>
  <td align="right"><math>0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(x_J - x_{J+1} \biggr)
\cdot \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]

+
\biggl(x_{J-1} - x_J \biggr)
\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[ 
2\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2 
-\frac{1}{2}\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>
\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^2\biggl[
x_{J-1} + 2(x_J)^'\Delta\tilde{r} - x_{J+1}
\biggr]
+ \biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^3\biggl[
x_{J-1} - 2x_J + x_{J+1}
\biggr]
+ \frac{1}{4}\biggl(\frac{\Delta\tilde{r}}{\tilde{r}_J}\biggr)^4\biggl[
x_{J-1} - 2(x_J)^'\Delta\tilde{r} - x_{J+1}

\biggr]
</math>
  </td>
</tr>
</table>
This <font color="red"><b>EXACTLY MATCHES</b></font> our [[#FirstGrouping|above first derivation and grouping]]!

====Improve First Approximation====

After slogging through the preceding "Second thru Fifth" approximations, I have come to appreciate that the approach used way back in the "[[Appendix/Ramblings/51BiPolytropeStability/BetterInterface#First_Approximation|First Approximation]]" was a good one, but that the attempt to introduce an implicit dependence was misguided.  I now think I have discovered the preferable implicit treatment.  Let's repeat, while improving this approach.

=====2<sup>nd</sup>-Order Explicit Approach=====
As was done in our earlier [[Appendix/Ramblings/51BiPolytropeStability/BetterInterface#First_Approximation|First Approximation]], let's set up a grid associated with a uniformly spaced spherical radius, where the subscript <math>J</math> denotes the grid zone at which all terms in the finite-difference representation of the governing relations will be evaluated.  More specifically,

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

<tr>
  <td align="right"><math>\tilde{r}_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\tilde{r}_J - \Delta\tilde{r}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\tilde{r}_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\tilde{r}_J + \Delta\tilde{r} \, ;
</math>
  </td>
</tr>
</table>
also,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d\tilde{r}}\biggr)_{J}</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{(x_{J+1} - x_{J-1})}{2\Delta\tilde{r}}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d\tilde{r}}\biggr)_{J}</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\frac{(p_{J+1} - p_{J-1})}{2\Delta\tilde{r}} \, .
</math>
  </td>
</tr>
</table>
And at each grid location, the governing relations establish the local evaluation of the derivatives, that is,

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

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_J </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{\tilde{r}_J}\biggl[ 
3x + \frac{p}{\gamma_g}\biggr]_J \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_J </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\tilde{\rho}_J}{\tilde{P}_J}\biggl[ 
(4x + p)\frac{\tilde{M}_r}{\tilde{r}^2}
+
\tau_c^2 \omega^2 \tilde{r} x \biggr]_J \, .
</math>
  </td>
</tr>
</table>
<span id="1stapprox">So, integrating</span> step-by-step from the center of the configuration, outward, once all the variable values are known at grid locations <math>J</math> and <math>(J-1)</math>, the values of <math>x</math> and <math>p</math> at <math>(J+1)</math> are given by the expressions,
<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>
x_{J-1} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{\gamma_g}\biggr]
\biggr\}_J \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + 2\Delta\tilde{r}
\biggl\{
\frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_J\, .
</math>
  </td>
</tr>
</table>

Then we will obtain the "<math>x_J</math>" and "<math>p_J</math>" values via the ''average'' expressions,
<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>
\frac{1}{2}(x_{J-1} + x_{J+1})
\, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{J}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(p_{J-1} + p_{J+1})
\, .
</math>
  </td>
</tr>
</table>

=====Convert to Implicit Approach=====

Consider implementing an ''implicit'' finite-difference analysis that improves on our earlier [[Appendix/Ramblings/51BiPolytropeStability/BetterInterface#First_Approximation|First Approximation]].  The general form of the source term expressions is,
<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>
x_{J-1} + 2\Delta\tilde{r} \biggl\{
\mathfrak{A}x_J + \mathfrak{B}p_J
\biggr\} 
</math>
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\mathfrak{A}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
- \biggl\{ \frac{3}{\tilde{r}} \biggr\}_J \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\mathfrak{B}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
- \biggl\{ \frac{1}{\gamma_g \tilde{r}} \biggr\}_J \, ;
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right"><math>p_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + 2\Delta\tilde{r} \biggl\{
\mathfrak{C}x_J + \mathfrak{D}p_J
\biggr\} 
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right"><math>\mathfrak{C}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl\{
\mathfrak{D}
\biggl[ 4
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr)  \biggr] 
\biggr\}_J\, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\mathfrak{D}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl\{
\frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggr\}_J\, .
</math>
  </td>
</tr>
</table>

Now, wherever a "<math>J+1</math>" index appears in the source term, replace it with the ''average expressions''; specifically, <math>x_{J+1} \rightarrow (2 x_J - x_{J-1})</math> and <math>p_{J+1} \rightarrow (2 p_J - p_{J-1})</math>.  For the fractional radial displacement, we have,
<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>
x_{J-1} + \Delta\tilde{r} \biggl\{
\mathfrak{A}x_J + \mathfrak{B}p_J
\biggr\} \, ;
</math>
  </td>
</tr>
</table>
and for the fractional pressure displacement,

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

<tr>
  <td align="right"><math>p_{J} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + \Delta\tilde{r} \biggl\{
\mathfrak{C}x_J + \mathfrak{D}p_J
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Solving for <math>p_J</math> in this second expression, we obtain,

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

<tr>
  <td align="right"><math>p_{J}\biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr] </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) x_J 
</math>
  </td>
</tr>
</table>
in which case the first expression gives,
<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>
x_{J-1} + (\mathfrak{A} \Delta\tilde{r})x_J 
+ (\mathfrak{B} \Delta\tilde{r}) \biggl[ p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) x_J\biggr]\biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ x_J \biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
x_{J-1} + (\mathfrak{A} \Delta\tilde{r})x_J 
\biggr\}\biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr]
+ (\mathfrak{B} \Delta\tilde{r}) \biggl[ p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) x_J\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} \biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr]
+ (\mathfrak{B} \Delta\tilde{r}) p_{J-1} 
+
(\mathfrak{A} \Delta\tilde{r})x_J \biggl[ 1 - \mathfrak{D}\Delta\tilde{r}\biggr]
+ (\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r}) x_J
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} ( 1 - \mathfrak{D}\Delta\tilde{r})
+ (\mathfrak{B} \Delta\tilde{r}) p_{J-1} 
+\biggl[
(\mathfrak{A} \Delta\tilde{r}) ( 1 - \mathfrak{D}\Delta\tilde{r})
+ (\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r}) 
\biggr] x_J
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ 
x_J \biggl\{ ( 1 - \mathfrak{D}\Delta\tilde{r} )- \biggl[
(\mathfrak{A} \Delta\tilde{r}) ( 1 - \mathfrak{D}\Delta\tilde{r})
+ (\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r}) 
\biggr] \biggr\}
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} ( 1 - \mathfrak{D}\Delta\tilde{r})
+ (\mathfrak{B} \Delta\tilde{r}) p_{J-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ 
x_J \biggl\{ 1 - \biggl[
(\mathfrak{A} \Delta\tilde{r}) 
+ \frac{(\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r})}{ ( 1 - \mathfrak{D}\Delta\tilde{r} ) } 
\biggr] \biggr\}
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} 
+ \biggl[\frac{(\mathfrak{B} \Delta\tilde{r})}{( 1 - \mathfrak{D}\Delta\tilde{r} )}\biggr] p_{J-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ 
x_J 
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ x_{J-1} 
+ \biggl[\frac{(\mathfrak{B} \Delta\tilde{r})}{( 1 - \mathfrak{D}\Delta\tilde{r} )}\biggr] p_{J-1} \biggr\}
\biggl\{ 1 - \biggl[
(\mathfrak{A} \Delta\tilde{r}) 
+ \frac{(\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r})}{ ( 1 - \mathfrak{D}\Delta\tilde{r} ) } 
\biggr] \biggr\}^{-1}
\, .
</math>
  </td>
</tr>
</table>
Then,

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

<tr>
  <td align="right"><math>p_{J}( 1 - \mathfrak{D}\Delta\tilde{r} ) </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) x_J 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) 
\biggl\{ x_{J-1} 
+ \biggl[\frac{(\mathfrak{B} \Delta\tilde{r})}{( 1 - \mathfrak{D}\Delta\tilde{r} )}\biggr] p_{J-1} \biggr\}
\biggl\{ 1 - \biggl[
(\mathfrak{A} \Delta\tilde{r}) 
+ \frac{(\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r})}{ ( 1 - \mathfrak{D}\Delta\tilde{r} ) } 
\biggr] \biggr\}^{-1}

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

<table border=1 align="center" cellpadding="10" width="80%"><tr><td bgcolor="lightblue" align="left">
This is test of  our "implicit" scheme for the <math>(n_c, n_e) = (5, 1)</math> bipolytrope with <math>\mu_e/\mu_c = 0.31</math> and (Model A) <math>\xi_i = 9.12744</math>; here, we also assume <math>\sigma_c^2 = 0.000109</math> and <math>J = i+2</math>.  Here are the quantities that we assume are <b>known</b> &hellip;
<table border="1" align="center" cellpadding="5">

  <tr>
<td align="center" bgcolor="white"><math>\tilde{r}</math></td>
<td align="center" bgcolor="white"><math>\tilde\rho</math></td>
<td align="center" bgcolor="white"><math>\tilde{P}</math></td>
<td align="center" bgcolor="white"><math>\tilde{M}_r</math></td>
<td align="center" bgcolor="white"><math>\tilde{\rho}_c</math></td>

<td align="center" bgcolor="white"><math>\mathfrak{A}</math></td>
<td align="center" bgcolor="white"><math>\mathfrak{B}</math></td>
<td align="center" bgcolor="white"><math>\frac{\mathfrak{C}}{\mathfrak{D}}</math></td>
<td align="center" bgcolor="white"><math>\mathfrak{D}</math></td>
  </tr>

  <tr>
<td align="center" bgcolor="white">0.0193368</td>
<td align="center" bgcolor="white">192.21728</td>
<td align="center" bgcolor="white">1913.1421</td>
<td align="center" bgcolor="white">0.3403116</td>
<td align="center" bgcolor="white">3359266.406</td>

<td align="center" bgcolor="white">-155.14459</td>
<td align="center" bgcolor="white">-25.85743</td>
<td align="center" bgcolor="white">4.0162932</td>
<td align="center" bgcolor="white">91.443479</td>
  </tr>
</table>
<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-1}</math></td>
<td align="center" bgcolor="white"><math>p_{J-1}</math></td>
<td align="center" bgcolor="white"><font size="-1">determined</font><br /><math>x_J</math></td>
<td align="center" bgcolor="white"><font size="-1">determined</font><br /><math>p_J</math></td>
  </tr>

  <tr>
<td align="center" bgcolor="white">0.001936393</td>
<td align="center" bgcolor="white">-4.755073</td>
<td align="center" bgcolor="white">32.25497</td>
<td align="center" bgcolor="white">-4.999355</td>
<td align="center" bgcolor="white">34.874915</td>
  </tr>
</table>
</td></tr>
<tr><td bgcolor="white" align="center">

<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>
\biggl\{ x_{J-1} 
+ \biggl[\frac{(\mathfrak{B} \Delta\tilde{r})}{( 1 - \mathfrak{D}\Delta\tilde{r} )}\biggr] p_{J-1} \biggr\}
\biggl\{ 1 - \biggl[
(\mathfrak{A} \Delta\tilde{r}) 
+ \frac{(\mathfrak{B} \Delta\tilde{r}) (\mathfrak{C}\Delta\tilde{r})}{ ( 1 - \mathfrak{D}\Delta\tilde{r} ) } 
\biggr] \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ x_{J-1} 
+ \biggl[-0.06084379\biggr] p_{J-1} \biggr\}
\biggl\{ 1 - \biggl[-0.3436910 \biggr] \biggr\}^{-1}
=
-4.999354 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>p_{J}( 1 - \mathfrak{D}\Delta\tilde{r} ) </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ p_{J-1} + (\mathfrak{C}\Delta\tilde{r}) x_J \biggr]( 1 - \mathfrak{D}\Delta\tilde{r} )^{-1}
=
34.87491 \, .
</math>
  </td>
</tr>
</table>
<tr><td bgcolor="lightblue" align="left">
<b>Best values:</b><br />Nodes 0: &nbsp;n/a
<br />Nodes 1: &nbsp;<math>\sigma_c^2 = 8.958784\times 10^{-5}</math>
<br />Nodes 2: &nbsp;<math>\sigma_c^2 = 3.021\times 10^{-4}</math>
<br />Nodes 3: &nbsp;<math>\sigma_c^2 = 6.09\times 10^{-4}</math>
</td></tr>
</table>
</td></tr></table>

===Interface===

<font color="darkgreen">CORE:</font> &nbsp; When <math>J = (i - 1)</math> (where <math>i</math> means interface), we can obtain the fractional displacements at the interface, <math>x_i</math> and <math>p_i</math>, via the expressions,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{i-2} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{\gamma_g}\biggr]
\biggr\}_{i-1} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{i-2} + 2\Delta\tilde{r}
\biggl\{
\frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_{i-1}\, .
</math>
  </td>
</tr>
</table>
Then, setting <math>J = i</math>, the pair of radial derivatives '''at the interface''' and '''<font color="darkgreen">as viewed from the perspective of the core</font>''' is given by the expressions,

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

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_i \biggr|_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{\tilde{r}_i}\biggl[ 
3x_i + \frac{p_i}{6/5}\biggr] \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_i\biggr|_\mathrm{core} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(\tilde{\rho}_i)_\mathrm{core}}{\tilde{P}_i} \cdot \frac{\tilde{M}_\mathrm{core}}{\tilde{r}_i^2}
\biggl[ (4x_i + p_i)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}_i^3 }{\tilde{M}_\mathrm{core}}\biggr) x_i \biggr]\, . 
</math>
  </td>
</tr>
</table>

It is important to recognize that, throughout the core, <math>(dx/d\tilde{r})</math> has been evaluated by setting <math>\gamma_g = 6/5</math>. If we continue to use this value of <math>\gamma_g</math> at the interface, we are determining the slope ''as viewed from the perspective of the core''.  


<font color="darkgreen">ENVELOPE:</font> &nbsp; On the other hand, ''as viewed from the perspective of the envelope'', all parameters used to determine <math>(dx/d\tilde{r})_i</math> at the interface (and throughout the entire envelope) are the same ''except'' <math>\gamma_g</math>, which equals 2 instead of 6/5.  Specifically at the interface, we have,

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

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_i \biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{\tilde{r}_i}\biggl[ 
3x_i + \frac{p_i}{2}\biggr] \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_i\biggr|_\mathrm{env} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(\tilde{\rho}_i)_\mathrm{env}}{\tilde{P}_i} \cdot \frac{\tilde{M}_\mathrm{core}}{\tilde{r}_i^2}
\biggl[ (4x_i + p_i)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}_i^3 }{\tilde{M}_\mathrm{core}}\biggr) x_i \biggr]\, . 
</math>
  </td>
</tr>
</table>

(See, for example, our [[SSC/Stability/BiPolytropes#Interface_Conditions|related discussion]].)  Hence, we appreciate that there is a discontinuous change in the value of this slope at the interface.  We note as well &#8212; <font color="red">for the first time (8/17/2023)!</font> &#8212; that there must also be a discontinuous jump in the slope of the "pressure perturbation."  All of the variables used to evaluate <math>(dp/d\tilde{r})_i</math> are the same irrespective of your core/envelope point of view ''except'' the leading density term.  As viewed from the perspective of the core, <math>(\tilde{\rho}_i)|_\mathrm{core} = m_\mathrm{surf}^5 (\mu_e/\mu_c)^{-10} \theta_i^5</math> whereas, from the perspective of the envelope, <math>(\tilde{\rho}_i)|_\mathrm{env} = m_\mathrm{surf}^5 (\mu_e/\mu_c)^{-9} \theta_i^5\phi_i</math>.  Appreciating that <math>\phi_i = 1</math>, this means that the slope of the "pressure perturbation" is a factor of <math>\mu_e/\mu_c</math> smaller as viewed from the perspective of the envelope.

Then the value of the fractional radial displacement and the value of the pressure perturbation at the first zone outside of the interface are obtained by setting <math>J = i</math>.  That is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{i-1} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{2}\biggr]
\biggr\}_i \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{i-1} + 2\Delta\tilde{r}
\biggl\{
\frac{(\tilde{\rho}_i)|_\mathrm{env}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_i\, .
</math>
  </td>
</tr>
</table>
But, as written, these two expressions are unacceptable because the values just inside the interface, <math>x_{i-1}</math> and <math>p_{i-1}</math>, are not known '''as viewed from the perspective of the envelope.''' However, we can fix this by drawing from the "average" expressions as replacements, namely,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(x_{i-1} + x_{i+1}) ~~~ \Rightarrow~~~ x_{i-1} = (2x_i - x_{i+1})
\, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(p_{i-1} + p_{i+1}) ~~~ \Rightarrow~~~ p_{i-1} = (2p_i - p_{i+1})
\, ,
</math>
  </td>
</tr>
</table>
in which case we have,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>2x_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2x_{i} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{2}\biggr]
\biggr\}_i \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>2p_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2p_{i} + 2\Delta\tilde{r}
\biggl\{
\frac{(\tilde{\rho}_i)|_\mathrm{env}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_i\, .
</math>
  </td>
</tr>
</table>

==Compare Core With Analytic Displacement Functions==

=See Also=


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