Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

====Third====
Noticing that <math>~h_1^2</math> is proportional to <math>~\lambda_1^2</math> and that <math>~h_3^2</math> is inversely proportional to <math>~\lambda_3^2</math>, let's consider both as possible behaviors for the 2<sup>nd</sup> scale factor.  Let's try the first of these behaviors.  Specifically, what if we assume &hellip;

<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{xp^2 z}{\lambda_2} \, ,</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{q^2y p^2z}{\lambda_2} \, ,</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>~-\frac{(x^2 + q^4y^2)}{\lambda_2} \, .</math>
  </td>
</tr>
</table>
Then,
<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>~
\biggl( \frac{\partial \lambda_2}{\partial x}\biggr)^2
+ \biggl( \frac{\partial \lambda_2}{\partial y}\biggr)^2
+\biggl( \frac{\partial \lambda_2}{\partial z}\biggr)^2
=
[\lambda_2 \ell_q \ell_{3D} ]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda_2 \ell_q \ell_{3D} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Primary implication:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21} = h_2 \biggl(\frac{\partial \lambda_2}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(xp^2 z) \ell_q \ell_{3D} \ ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22} = h_2 \biggl(\frac{\partial \lambda_2}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(q^2 y p^2 z) \ell_q \ell_{3D} \ ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23} = h_2 \biggl(\frac{\partial \lambda_2}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(x^2 + q^4 y^2) \ell_q \ell_{3D} \ .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
These perfectly match the direction-cosine expressions (<math>~\gamma_{2i}</math> for i = 1, 3)<br />that have been summarized in our above [[#Table1DaringAttack|Daring Attack Table]].
  </td>
</tr>
</table>

Secondary implication:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{21}
=
\lambda_2 \ell_q^2 \ell_{3D}^2 (xp^2z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{22}
=
\lambda_2 \ell_q^2 \ell_{3D}^2 (q^2 y p^2 z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{23}
=
- \lambda_2 \ell_q^2 \ell_{3D}^2(x^2 + q^4y^2) \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>


Now, what specifically is the function, <math>~\lambda_2(x, y, z)</math> ?  Start by rewriting the three partial derivatives as,

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

<tr>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2^2)}{\partial x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~xp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2)^2}{\partial y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2y p^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2)^2}{\partial z} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(x^2 + q^4y^2) \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\lambda_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2y^2)p^2z \, .</math>
  </td>
</tr>
</table>

Then we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2xp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2q^2 yp^2z \, .</math>&nbsp; &nbsp; &nbsp; <font color="red">Great!</font>
  </td>
</tr>
</table>

But this cannot be the correct expression for <math>~\lambda_2^2</math> because,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2y^2)p^2 \, ,</math>
  </td>
</tr>
</table>
which does not match the desired partial derivative with respect to <math>~z</math>.