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====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"> </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"> </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"> </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"> </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"> </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 … <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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </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 … <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"> </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"> </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"> </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"> </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"> </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 … <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"> </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"> </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"> </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"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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 … <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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </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 … <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"> </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"> </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"> </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"> </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"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </td> <td align="center"> </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"> </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"> </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]]!
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