Editing Appendix/Ramblings/ConcentricEllipsoidalT8Coordinates (section)

===Associated h<sub>3</sub> Scale Factor===
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[[File:EUREKA 21Jan2021 sm.png|350px|Whiteboard EUREKA moment]]</td></tr></table>
After working through various scenarios on my whiteboard today (21 January 2021), I propose that,
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<tr>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{xp^2z}{(x^2 + q^4y^2)} \, ;</math>
  </td>
<td align="center>&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2 y p^2z}{(x^2 + q^4y^2)} \, ;</math>
  </td>
<td align="center>&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-1 \, .</math>
  </td>
</tr>
</table>

This means that,
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  <td align="right">
<math>~h_3^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_{i=1}^3 \biggl( \frac{\partial \lambda_3}{\partial x_i}\biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xp^2z}{(x^2 + q^4y^2)} \biggr]^2
+
\biggl[ \frac{q^2 y p^2z}{(x^2 + q^4y^2)} \biggr]^2
+
\biggl[ -1 \biggr]^2
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(x^2 + q^4y^2 +p^4z^2)}{(x^2 + q^4y^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\ell_{3D}^2 (x^2 + q^4y^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ h_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ell_{3D} (x^2 + q^4y^2)^{1 / 2} \, . 
</math>
  </td>
</tr>
</table>
This seems to work well because, when combined with the three separate expressions for <math>~\partial \lambda_3/\partial x_i</math>,  this single expression for <math>~h_3</math> generates all three components of the third unit vector, that is, all three direction cosines, <math>~\gamma_{3i}</math>.  All of the elements of this new "EUREKA moment" result have been entered into the following table.


<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T8 Coordinates'''</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"><math>~\frac{x}{  y^{1/q^2}}</math></td>
  <td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y }{(x^2 + q^4y^2)^{1 / 2}}\biggr] </math></td>
  <td align="center"><math>~\frac{\lambda_2}{x}</math></td>
  <td align="center"><math>~-\frac{\lambda_2}{q^2 y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{q^2 y }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~- \frac{x }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">---</td>
  <td align="center"><math>~\ell_{3D}(x^2 + q^4 y^2)^{1 / 2}</math></td>
  <td align="center"><math>~\frac{xp^2z}{(x^2 + q^4y^2)} </math></td>
  <td align="center"><math>~\frac{q^2 y p^2z}{(x^2 + q^4y^2)}</math></td>
  <td align="center"><math>~-1</math></td>
  <td align="center"><math>~\frac{x p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~\frac{q^2 y p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~-\frac{(x^2 + q^4 y^2)\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
</tr>

<tr>
  <td align="left" colspan="9">
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<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} </math>
  </td>
</tr>

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