Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

==New Insight==

Following the development of our [[#Better_Organized|above, ''Better Organized'']] discussion, we reverted to several hours of pen &amp; paper derivations, primarily investigating whether it will help us to rewrite various expressions using the [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>] [[User:Tohline/Appendix/Mathematics/ScaleFactors#DirectionCosineRelations|Direction-Cosine Relations]].  We discovered that if we set,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[(xq^2y)\ell_q \ell_{3D}]^2 \, ,</math>
  </td>
</tr>
</table>
then,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2 z}{q^2y} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2z}{x} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr] \, .</math>
  </td>
</tr>
</table>

This seems to be a promising method of attack because &#8212; in all three cases, i = 1,3 &#8212; the derivative of <math>~\lambda_2</math> with respect to <math>~x_i</math> does not depend on <math>~x_i</math>.  Perhaps this simplification will help us identify the function that defines <math>~\lambda_2</math>.  This proposed prescription for <math>~h_2(x, y, z)</math> and some of its implications are reflected in the following "New Insight" table.  (Keep in mind that, although the expressions for <math>~\gamma_{21}, \gamma_{22}, ~\mathrm{and}~ \gamma_{23}</math> remain correct, the tabulated expression is a ''guess'' for <math>~h_2</math> and, hence, the tabulated expressions for all three <math>~\partial \lambda_2/\partial x_i</math> are pure speculation.)
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''New Insight'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\lambda_n</math></td>
  <td align="center"><math>~h_n</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
  <td align="center"><math>~\gamma_{n1}</math></td>
  <td align="center"><math>~\gamma_{n2}</math></td>
  <td align="center"><math>~\gamma_{n3}</math></td>
</tr>

<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
  <td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
  <td align="center"><math>~(x) \ell_{3D}</math></td>
  <td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
  <td align="center"><math>~(p^2z) \ell_{3D}</math></td>
</tr>

<tr>
  <td align="center"><math>~2</math></td>
  <td align="center">---</td>
  <td align="center"><math>~\ell_q \ell_{3D} (xq^2y)</math></td>
  <td align="center"><math>~\frac{p^2z}{q^2y}</math></td>
  <td align="center"><math>~\frac{p^2z}{x}</math></td>
  <td align="center"><math>~-\biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr]</math></td>
  <td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{p^2z}{q^2y} \biggr]</math></td>
  <td align="center"><math>~\ell_q \ell_{3D} (xq^2y) \biggl[ \frac{ p^2z}{x} \biggr] </math></td>
  <td align="center"><math>~- \ell_q \ell_{3D}  (xq^2y) \biggl[ \frac{x}{q^2y} + \frac{q^2y}{x} \biggr]</math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\frac{y^{1/q^2}}{x} </math></td>
  <td align="center"><math>~\frac{xq^2 y \ell_q}{\lambda_3}</math></td>
  <td align="center"><math>~-\frac{\lambda_3}{x}</math></td>
  <td align="center"><math>~+\frac{\lambda_3}{q^2y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~-q^2 y \ell_q</math></td>
  <td align="center"><math>~x\ell_q</math></td>
  <td align="center"><math>~0</math></td>
</tr>
</table>