Editing Appendix/Ramblings/T6CoordinatesPt3

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=Concentric Ellipsoidal (T6) Coordinates (Part 3)=

==Best Thus Far==

===Part A===

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T6 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}}{ z^{2/p^2}}</math></td>
  <td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\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>~-\frac{2\lambda_2}{p^2 z}</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{x}\biggr)</math></td>
  <td align="center"><math>~ \frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{q^2y}\biggr)</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(-\frac{2}{p^2z}\biggr)</math></td>
</tr>

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

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<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>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>

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

<span id="ABderivatives">&nbsp;</span>
<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A \equiv \ell_{3D}^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 ) \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~B \equiv \mathcal{D}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial A}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial A}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2q^4 y \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial A}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2p^4 z\, ;</math>
  </td>
</tr>

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

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

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

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

Try &hellip;

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

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{B}{A}\biggr)^{m/2} = (D \ell_{3D})^m
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{AB}{m\lambda_3} \biggr] \frac{\partial \lambda_3}{\partial x_i}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln x_i}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{x_i}{2}\biggl\{ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

In this case we find,
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
<math>~\frac{z}{2}\biggl\{~~\biggr\}_z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^4 z^2(x^2 - q^4y^2)^2 \, .
</math>
  </td>
</tr>
</table>

The scale factor is, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>~
\sum_{i=1}^3 \biggl\{ 
\biggl[ \frac{m\lambda_3}{2AB} \biggr] \biggl[ 
A \cdot \frac{\partial B}{\partial x_i} - B \cdot \frac{\partial A}{\partial x_i}
\biggr]
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{m\lambda_3}{AB} \biggr]^2 
\biggl\{ 
\biggl[ 
x(2q^4 y^2 + p^4z^2)^2 
\biggr]^2
+
\biggl[ 
q^4y(2x^2 + p^4 z^2)^2
\biggr]^2
+
\biggl[ 
p^4 z(x^2 - q^4y^2)^2
\biggr]^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~h_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{AB}{m\lambda_3} \biggr] 
\biggl\{ 
\biggl[ 
x(2q^4 y^2 + p^4z^2)^2 
\biggr]^2
+
\biggl[ 
q^4y(2x^2 + p^4 z^2)^2
\biggr]^2
+
\biggl[ 
p^4 z(x^2 - q^4y^2)^2
\biggr]^2
\biggr\}^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

===Part B (25 February 2021)===

Now, from [[#Eureka|above]], we know that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} = AB</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]^2
+
\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]^2
+
\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]^2 \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="4">
'''Example:'''  <br /><math>~(q^2, p^2) = (2, 5)</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>~(x, y, z) = (0.7, \sqrt{0.23}, 0.1)~~\Rightarrow~~ \lambda_1 = 1.0</math>
  </td>
</tr>

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

<tr>
  <td align="center">2.14037</td>
  <td align="center">1.39187</td>
  <td align="center">0.04623</td>
  <td align="center">3.57847</td>
</tr>
</table>

As an aside, note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~AB</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln x}
+
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln y}
+
\biggl[ \frac{AB}{m} \biggr]  \frac{\partial \ln \lambda_3}{\partial \ln z}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \ln \lambda_3}{\partial \ln x}
+
\frac{\partial \ln \lambda_3}{\partial \ln y}
+
\frac{\partial \ln \lambda_3}{\partial \ln z} \, .
</math>
  </td>
</tr>
</table>

We realize that this ratio of lengths may also be written in the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
6x^2 q^4y^2 p^4 z^2
+
x^4(4q^4y^2 + p^4z^2)
+
q^8y^4(4x^2 + p^4z^2)
+
p^8z^4(x^2 + q^4y^2) \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="5">
'''Same Example, but Different Expression:'''  <br /><math>~(q^2, p^2) = (2, 5)</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>~(x, y, z) = (0.7, \sqrt{0.23}, 0.1)~~\Rightarrow~~ \lambda_1 = 1.0</math>
  </td>
</tr>

<tr>
  <td align="center"><math>~6x^2 q^4y^2 p^4 z^2</math></td>
  <td align="center"><math>~x^4(4q^4y^2 + p^4z^2)</math></td>
  <td align="center"><math>~q^8y^4(4x^2 + p^4z^2)</math></td>
  <td align="center"><math>~p^8z^4(x^2 + q^4y^2)</math></td>
  <td align="center"><math>~\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2}</math></td>
</tr>

<tr>
  <td align="center">0.67620</td>
  <td align="center">0.94359</td>
  <td align="center">1.87054</td>
  <td align="center">0.08813</td>
  <td align="center">3.57847</td>
</tr>
</table>


Let's try &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x(2 q^4y^2 + p^4z^2 ) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 y(p^4z^2 + 2x^2 ) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2z( x^2 - q^4y^2 ) \, .</math>
  </td>
</tr>
</table>
This means that the relevant scale factor is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_5^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -x(2 q^4y^2 + p^4z^2 ) \biggr]^2
+
\biggl[ q^2 y(p^4z^2 + 2x^2 ) \biggr]^2
+
\biggl[ p^2z( x^2 - q^4y^2 ) \biggr]^2
=
\biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~h_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,
</math>
  </td>
</tr>
</table>
and the three associated direction cosines are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{51} = h_5 \biggl( \frac{\partial \lambda_5}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x(2 q^4y^2 + p^4z^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{52} = h_5 \biggl( \frac{\partial \lambda_5}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 y(p^4z^2 + 2x^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{53} = h_5 \biggl( \frac{\partial \lambda_5}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2z( x^2 - q^4y^2 )\biggl( \frac{\ell_{3D}}{\mathcal{D}} \biggr) \, .</math>
  </td>
</tr>
</table>
<span id="PartBCoordinatesT10">These direction cosines</span> exactly match what is required in order to ensure that the coordinate, <math>~\lambda_5</math>, is everywhere orthogonal to both <math>~\lambda_1</math> and <math>~\lambda_4</math>.  <font color="red">'''GREAT!'''</font>  The resulting summary table is, therefore:

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T10 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>~4</math></td>
  <td align="center"><math>~\frac{x y^{1/q^2}}{ z^{2/p^2}}</math></td>
  <td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\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>~-\frac{2\lambda_2}{p^2 z}</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{x}\biggr)</math></td>
  <td align="center"><math>~ \frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{q^2y}\biggr)</math></td>
  <td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(-\frac{2}{p^2z}\biggr)</math></td>
</tr>

<tr>
  <td align="center"><math>~5</math></td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}</math></td>
  <td align="center"><math>~-x(2 q^4y^2 + p^4z^2)</math></td>
  <td align="center"><math>~q^2 y(p^4z^2 + 2x^2 )</math></td>
  <td align="center"><math>~p^2z( x^2 - q^4y^2 )</math></td>
  <td align="center"><math>~-\frac{\ell_{3D}}{\mathcal{D}}\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]</math></td>
  <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]</math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<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>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>

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


Try &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^{-2q^4} \cdot y^{2q^2}
+
y^{q^2p^4} \cdot z^{-q^4p^2}
+
x^{-p^4} \cdot z^{p^2} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{x^{2q^4 + p^2} z^{q^4p^2} }\biggl\{
\mathfrak{F_5}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2q^4}{x}\biggl( \frac{ y^{2q^2} }{ x^{2q^4} } \biggr)
- \frac{p^4}{x}\biggl( \frac{ z^{p^2} }{ x^{p^2} } \biggr)
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~x^{2q^4 + p^2 + 1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{ z^{q^4p^2} }\biggl\{
\frac{\mathfrak{F_5}}{\lambda_5}
\biggr\} 
\, ,
</math>
  </td>
</tr>
</table>
we can also write,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{ z^{q^4p^2} }{x}\biggl\{
\frac{\lambda_5}{\mathfrak{F_5}}
\biggr\} 

\biggl[ 2q^4y^{2q^2} x^{p^2} + p^4 x^{2q^4}z^{p^2} \biggr] 
</math>
  </td>
</tr>
</table>
Similarly,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2q^2}{y} \biggl( \frac{ y^{2q^2} }{ x^{2q^4} } \biggr)
+
\frac{q^2p^4}{y} \biggl( \frac{ y^{q^2p^4} }{ z^{q^4p^2} } \biggr)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^{p^2}}{ y } \biggl\{ \frac{\lambda_5}{\mathfrak{F_5}} \biggr\}
\biggl[ 2q^2 \biggl( y^{2q^2} z^{q^4p^2} \biggr)
+
q^2p^4 \biggl( y^{q^2p^4} x^{2q^4} \biggr)\biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_5}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{q^4p^2}{z} \biggl( \frac{ y^{q^2p^4} }{ z^{q^4p^2} } \biggr)
+
\frac{p^4}{z} \biggl( \frac{ z^{p^4} }{ x^{p^2} } \biggr)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x^{2q^4} }{z} \biggl\{ \frac{\lambda_5}{\mathfrak{F_5}} \biggr\} \biggl[
p^4 \biggl( z^{p^4 + q^4p^2 }  \biggr)
- 
q^4p^2 \biggl( x^{p^2} y^{q^2p^4}  \biggr)\biggr]
</math>
  </td>
</tr>
</table>

===Understanding the Volume Element===

Let's see if the expression for the volume element makes sense; that is, does
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(h_1 h_4 h_5) d\lambda_1 d\lambda_4 d\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~dx dy dz \, ?</math>
  </td>
</tr>
</table>

First, let's make sure that we understand how to relate the components of the Cartesian line element with the components of our T10 coordinates. 

====Line Element====

MF53 claim that the following relation gives the various expressions for the scale factors; we will go ahead and incorporate the expectation that, since our coordinate system is orthogonal, the off-diagonal elements are zero.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds^2 = dx^2 + dy^2 + dz^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{i=1,4,5} h_i^2 d\lambda_i^2 \, .
</math>
  </td>
</tr>
</table>
Let's see.  The first term on the RHS is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_1^2 d\lambda_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl[  
\biggl( \frac{\partial \lambda_1}{\partial x}\biggr)dx
+
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)dy
+
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)dz
\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl[  
\biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 dx^2
+
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 dy^2
+
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 dz^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)} dx~dy
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)} dx~dz
+
\cancel{2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)} dy~dz
\biggr] \, ;
</math>
  </td>
</tr>
</table>
the other two terms assume easily deduced, similar forms.  When put together and after regrouping terms, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\sum_{i=1,4,5} h_i^2 d\lambda_i^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2
\biggr] dx^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2
\biggr] dy^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[  
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
+
h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2
+
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggr] dz^2 \, .
</math>
  </td>
</tr>
</table>

Given that this summation should also equal the square of the Cartesian line element, <math>~(dx^2 + dy^2 + dz^2)</math>, we conclude that the three terms enclosed inside each of the pair of brackets must sum to unity.  Specifically, from the coefficient of <math>~dx^2</math>, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_1^2 \biggl( \frac{\partial \lambda_1}{\partial x}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>
Using this relation to replace <math>~h_1^2</math> in each of the other two bracketed expressions, we find for the coefficients of <math>~dy^2</math> and <math>~dz^2</math>, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ 
1 - h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 - h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 
\biggr]
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 
\biggr]\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ 
1 - h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 - h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 
\biggr]
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggr]\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \, .
</math>
  </td>
</tr>
</table>

We can use the first of these two expressions to solve for <math>~h_4^2</math> in terms of <math>~h_5^2</math>, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
- h_4^2\biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
- h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
h_4^2\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
h_4^2
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>
</table>
Analogously, the second of these two expressions gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
h_4^2 \biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
</math>
  </td>
</tr>
</table>
Eliminating <math>~h_4</math> between the two gives the desired overall expression for <math>~h_5</math>, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
\biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-
\biggl[
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  
+ h_5^2 \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
h_5^2 \biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} 
\biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_5^2 \biggl\{
\biggl[ \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
-
\biggl( \frac{\partial \lambda_5}{\partial y}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-
\biggl[ \biggl( \frac{\partial \lambda_5}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2
-
\biggl( \frac{\partial \lambda_5}{\partial z}\biggr)^2
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2} \biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\biggl[ \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 \biggr]
\biggl[
\biggl( \frac{\partial \lambda_4}{\partial y}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[ \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2  \biggr]
\biggl[ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
- \biggl( \frac{\partial \lambda_4}{\partial x}\biggr)^2 \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
\biggl[ \gamma_{51}^2 \gamma_{12}^2  
-
\gamma_{52}^2 \gamma_{11}^2 \biggr]
\biggl[ \gamma_{43}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{13}^2 
\biggr]
-
\biggl[ \gamma_{51}^2 \gamma_{13}^2
-
\gamma_{53}^2 \gamma_{11}^2 
\biggr]
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{12}^2 
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{h_4^2 h_1^2}\biggl\{
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[ \gamma_{43}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{13}^2 
\biggr]
-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggl[ \gamma_{43}^2\gamma_{11}^2
-~ \gamma_{41}^2 \gamma_{13}^2 
\biggr]
- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2 \gamma_{12}^2
\biggr]
+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggl[
\gamma_{42}^2 \gamma_{11}^2
- \gamma_{41}^2\gamma_{12}^2 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{h_5^2}{h_1^4 h_4^2 h_5^2}\biggl\{
\biggl[ \gamma_{51}^2 \gamma_{12}^2  
-
\gamma_{52}^2 \gamma_{11}^2 \biggr]

\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43} \gamma_{11}
- \gamma_{41} \gamma_{13} 
\biggr]

-
\biggl[ \gamma_{51}^2 \gamma_{13}^2
-
\gamma_{53}^2 \gamma_{11}^2 
\biggr]

\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12} 
\biggr] 
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41} \gamma_{12} 
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{h_4^2 h_1^2}\biggl\{
\biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}

\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43} \gamma_{11}
- \gamma_{41} \gamma_{13} 
\biggr]

-
\biggl( \frac{\partial \lambda_1}{\partial y}\biggr)^2 
\biggl[ \gamma_{43}\gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\biggl[ \gamma_{43}\gamma_{11}
-~ \gamma_{41} \gamma_{13} 
\biggr]

- \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)^{2}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12}
\biggr]
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41} \gamma_{12}
\biggr]


+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr)^2 
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} 
\biggr]
\biggl[
\gamma_{42} \gamma_{11}
- \gamma_{41}\gamma_{12} 
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{h_1^4 h_4^2 }\biggl\{
-
\biggl[ \gamma_{51} \gamma_{12}  
+
\gamma_{52} \gamma_{11} \biggr]
\gamma_{43}
\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]
\gamma_{52}

+
\biggl[ \gamma_{51} \gamma_{13}
+
\gamma_{53} \gamma_{11} 
\biggr]
\gamma_{42}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12} 
\biggr] \gamma_{53}

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{ h_1^4 h_4^2}\biggl\{
\gamma_{12}^2 
\biggl[ \gamma_{43}\gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]\gamma_{52}
-\gamma_{11}^{2}
\biggl[ \gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} 
\biggr]\gamma_{52}


- \gamma_{11}^{2}
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41} \gamma_{12}
\biggr]\gamma_{53}


+ \gamma_{13}^2 
\biggl[
\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} 
\biggr]\gamma_{53} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{h_1^4 h_4^2 }\biggl\{
\biggl[
(- \gamma_{51} \gamma_{12}  
-
\gamma_{52} \gamma_{11} )
\gamma_{43}
+ 
\gamma_{12}^2 

-\gamma_{11}^{2}
\biggr]
(\gamma_{43} \gamma_{11}
+ \gamma_{41} \gamma_{13} )
\gamma_{52}
+
\biggl[
(\gamma_{51} \gamma_{13}
+
\gamma_{53} \gamma_{11} )
\gamma_{42}

- \gamma_{11}^{2}

+ \gamma_{13}^2 
\biggr]
(\gamma_{42} \gamma_{11}
+ \gamma_{41}\gamma_{12} )
\gamma_{53} 
\biggr\}
</math>
  </td>
</tr>
</table>
&hellip; Not sure this is headed anywhere useful!

====Volume Element====

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

<tr>
  <td align="right">
<math>~(h_1 h_4 h_5) d\lambda_1 d\lambda_4 d\lambda_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(h_1 h_4 h_5)
\biggl[ 
\biggl( \frac{\partial \lambda_1}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_1}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_1}{\partial z}\biggr) dz
\biggr]
\biggl[ 
\biggl( \frac{\partial \lambda_4}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_4}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_4}{\partial z}\biggr) dz
\biggr]
\biggl[ 
\biggl( \frac{\partial \lambda_5}{\partial x}\biggr) dx
+ \biggl( \frac{\partial \lambda_5}{\partial y}\biggr) dy
+ \biggl( \frac{\partial \lambda_5}{\partial z}\biggr) dz
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
\biggl( \gamma_{11}\biggr) dx
+ \biggl( \gamma_{12}\biggr) dy
+ \biggl( \gamma_{13}\biggr) dz
\biggr]
\biggl[ 
\biggl( \gamma_{41}\biggr) dx
+ \biggl( \gamma_{42}\biggr) dy
+ \biggl( \gamma_{43}\biggr) dz
\biggr]
\biggl[ 
\biggl( \gamma_{51} \biggr) dx
+ \biggl( \gamma_{52} \biggr) dy
+ \biggl( \gamma_{53} \biggr) dz
\biggr]
</math>
  </td>
</tr>
</table>

=COLLADA=
Here we try to use the 3D-visualization capabilities of COLLADA to test whether or not the three coordinates associated with the T6 Coordinate system are indeed orthogonal to one another.  We begin by making a copy of the '''Inertial17.dae''' text file, which we obtain from [[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS#The_COLLADA_Code_.26_Initial_3D_Scene|an accompanying discussion]].  When viewed with the Mac's '''Preview''' application, this group of COLLADA-based instructions displays a purple ellipsoid with axis ratios, (b/a, c/a) = (0.41, 0.385).  This means that we are dealing with an ellipsoid for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>q \equiv \frac{a}{b}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.44</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~p \equiv \frac{a}{c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.60 \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\lambda_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x y^{1/q^2}}{ z^{2/p^2}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>
</table>

== First Trial==
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''First Trial'''<br />(specified variable values have bgcolor="pink")</td>
</tr>
<tr>
  <td align="center">x</td>
  <td align="center">y</td>
  <td align="center">z</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.5</td>
  <td align="center">0.35493</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.46052</td>
  <td align="center">2.11310</td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat{k} (p^2 z_0 \ell_{3D})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (0.23026) + \hat\jmath (0.97313)  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x_0 q^2 y_0 p^2 z_0}{\mathcal{D}} \biggl\{
\hat{\imath} \biggl( \frac{1}{x_0} \biggr)
+ \hat{\jmath}  \biggl( \frac{1}{q^2 y_0} \biggr)
+ \hat{k} \biggl( -\frac{2}{p^2 z_0} \biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{k} ~\biggl(  \frac{2x_0 q^2 y_0  }{\mathcal{D}} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{k} ~\biggl(  1 \biggr)
\ ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_{3D}}{\mathcal{D}} 
\biggl\{
-\hat\imath \biggl[ x_0(2 q^4y_0^2 + p^4z_0^2 ) \biggl]
+ \hat\jmath \biggl[ q^2 y_0(p^4z_0^2 + 2x_0^2 )  \biggl]
+ \hat{k} \biggl[ p^2z_0( x_0^2 - q^4y_0^2 ) \biggl]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2q^2 x_0 y_0\ell_{3D}}{\mathcal{D}} 
\biggl\{
-\hat\imath ( q^2y_0 ) 
+ \hat\jmath  (x_0) 
\biggr\}
=
\biggl(1\biggr)\ell_{3D}\biggl\{ -\hat\imath ( q^2y_0 ) + \hat\jmath  (x_0) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\hat\imath (0.97313 ) + \hat\jmath  (0.23026) \, .
</math>
  </td>
</tr>
</table>

===Tangent Plane===

From our [[Appendix/Ramblings/ConcentricEllipsoidalCoordinates#Other_Coordinate_Pair_in_the_Tangent_Plane|above derivation]], the plane that is tangent to the ellipsoid's surface at <math>~(x_0, y_0, z_0)</math> is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x x_0  + q^2 y y_0  + p^2 z z_0 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\lambda_1^2)_0 \, .
</math>
  </td>
</tr>
</table>
For this ''First Trial,'' we have (for all values of <math>~z</math>, given that <math>~z_0 = 0</math>) &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
(0.5)x   + (2.11310)y   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(1 - 0.5x)}{2.11310} \, .
</math>
  </td>
</tr>
</table>
So let's plot a segment of the tangent plane whose four corners are given by the coordinates,
<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">Corner</td>
  <td align="center">x</td>
  <td align="center">y</td>
  <td align="center">z</td>
</tr>
<tr>
  <td align="center">A</td>
  <td align="center" bgcolor="pink">x_0 - 0.25 = +0.25</td>
  <td align="center">0.41408</td>
  <td align="center" bgcolor="pink">-0.25</td>
</tr>
<tr>
  <td align="center">B</td>
  <td align="center" bgcolor="pink">x_0 + 0.25 = +0.75</td>
  <td align="center">0.29577</td>
  <td align="center" bgcolor="pink">-0.25</td>
</tr>
<tr>
  <td align="center">C</td>
  <td align="center" bgcolor="pink">x_0 - 0.25 = +0.25</td>
  <td align="center">0.41408</td>
  <td align="center" bgcolor="pink">+0.25</td>
</tr>
<tr>
  <td align="center">D</td>
  <td align="center" bgcolor="pink">x_0 + 0.25 = +0.75</td>
  <td align="center">0.29577</td>
  <td align="center" bgcolor="pink">+0.25</td>
</tr>
</table>
Now, in order to give some thickness to this tangent-plane, let's adjust the four corner locations by a distance of <math>~\pm 0.1</math> in the <math>~\hat{e}_1</math> direction.

===Eight Corners of Tangent Plane===

Corner 1:  Shift surface-point location <math>~(x_0, y_0, z_0)</math> by <math>~(+\Delta e_1)</math> in the <math>~\hat{e}_1</math> direction,  by <math>~(+\Delta e_2)</math> in the <math>~\hat{e}_2</math> direction, and by  by <math>~(+\Delta e_3)</math> in the <math>~\hat{e}_3</math> direction.  This gives &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_0 + (\Delta e_1)0.23026 - (\Delta e_2)0.97313</math>
  </td>
</tr>
</table>

==Second Trial==
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''Second Trial''' &hellip; <math>~(q = 2.44, p = 2.60)</math><br />[specified variable values have bgcolor="pink"]</td>
</tr>
<tr>
  <td align="center">x_0</td>
  <td align="center">y_0</td>
  <td align="center">z_0</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.5</td>
  <td align="center">0.35493</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.46052</td>
  <td align="center">2.11310</td>
</tr>
</table>

===Generic Unit Vector Expressions===

Let's adopt the notation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath ~[e_{ix}] + \hat\jmath ~[e_{iy}] + \hat{k} ~[e_{iz}]</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; &nbsp;</td>
  <td align="center"><math>~i = 1,3 \, .</math></td>
</tr>
</table>

Then, for the T6 Coordinate system, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e_{1x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2y_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2 z_0 \ell_{3D} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{2x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{q^2 y_0 p^2 z_0}{\mathcal{D}}\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x_0 p^2 z_0}{\mathcal{D}} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 q^2 y_0}{\mathcal{D}}\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{3x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-x_0(2 q^4y_0^2 + p^4z_0^2 )\frac{\ell_{3D}}{\mathcal{D}}  \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2 y_0(p^4z_0^2 + 2x_0^2 )\frac{\ell_{3D}}{\mathcal{D}}   \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2z_0( x_0^2 - q^4y_0^2 )\frac{\ell_{3D}}{\mathcal{D}} \, .</math>
  </td>
</tr>

</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="4">'''Second Trial'''</td>
</tr>
<tr>
  <td align="center"> &nbsp;</td>
  <td align="center"> x</td>
  <td align="center"> y</td>
  <td align="center"> z</td>
</tr>
<tr>
  <td align="center"> <math>~e_1</math></td>
  <td align="center">0.23026</td>
  <td align="center">0.97313</td>
  <td align="center"> 0.0</td>
</tr>
<tr>
  <td align="center"> <math>~e_2</math></td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">-1.0</td>
</tr>
<tr>
  <td align="center"> <math>~e_3</math></td>
  <td align="center">- 0.97313</td>
  <td align="center">0.23026</td>
  <td align="center"> 0.0</td>
</tr>
</table>


What are the coordinates of the eight corners of a thin tangent-plane?  Let's say that we want the plane to extend &hellip;
<ul>
  <li>From <math>~(-\Delta_1)</math> to <math>~(+\Delta_1)</math> in the <math>~\hat{e}_1</math> direction &hellip; here we set <math>~\Delta_1 = 0.05</math>;</li>
  <li>From <math>~(-\Delta_2)</math> to <math>~(+\Delta_2)</math> in the <math>~\hat{e}_2</math> direction &hellip; here we set <math>~\Delta_2 = 0.25</math>;</li>
  <li>From <math>~(-\Delta_3)</math> to <math>~(+\Delta_3)</math> in the <math>~\hat{e}_3</math> direction &hellip; here we set <math>~\Delta_3 = 0.25</math>.</li>
</ul>

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

<tr>
  <td align="right">
<math>~\Delta_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1x} + \Delta_2 e_{2x} + \Delta_3 e_{3x} = -0.23177 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1y} + \Delta_2 e_{2y} + \Delta_3 e_{3y} = +0.10622 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1z} + \Delta_2 e_{2z} + \Delta_3 e_{3z} = -0.25000 \, .</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="left">[[File:TangentPlaneSchematic.png|Tangent Plane Schematic]]</td>
  <td align="left">
<table border="1" cellpadding="5" align="center">
  <tr>
    <td align="center">vertex</td>
    <td align="center">x</td>
    <td align="center">y</td>
    <td align="center">z</td>
    <td align="center" rowspan="9" bgcolor="lightgray">&nbsp;</td>
    <td align="center">x</td>
    <td align="center">y</td>
    <td align="center">z</td>
  </tr>
  <tr>
    <td align="center">0<sup>&dagger;</sup></td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center">0.24871</td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">1</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center">0.46115</td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">2</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center"> 0.24871 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">3</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center"> 0.46115 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">4</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.73177</td>
    <td align="center"> 0.24871 </td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">5</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.46115 </td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">6</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.24871 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">7</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.46115 </td>
    <td align="center">0.25</td>
  </tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
<sup>&dagger;</sup>In the figure on the left, vertex 0 is hidden from view behind the (yellow) solid rectangle.
  </td>
</tr>
</table>

==Third Trial==
===GoodPlane01===
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''Third Trial''' &hellip; <math>~(q = 2.44, p = 2.60)</math><br />[specified variable values have bgcolor="pink"]</td>
</tr>
<tr>
  <td align="center">x_0</td>
  <td align="center">y_0</td>
  <td align="center">z_0</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.8</td>
  <td align="center">0.24600</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.59959</td>
  <td align="center">2.34146</td>
</tr>
</table>

Again, for the T6 Coordinate system, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e_{1x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2y_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2 z_0 \ell_{3D} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{2x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{q^2 y_0 p^2 z_0}{\mathcal{D}}\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x_0 p^2 z_0}{\mathcal{D}} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 q^2 y_0}{\mathcal{D}}\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{3x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-x_0(2 q^4y_0^2 + p^4z_0^2 )\frac{\ell_{3D}}{\mathcal{D}}  \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2 y_0(p^4z_0^2 + 2x_0^2 )\frac{\ell_{3D}}{\mathcal{D}}   \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2z_0( x_0^2 - q^4y_0^2 )\frac{\ell_{3D}}{\mathcal{D}} \, .</math>
  </td>
</tr>

</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="5">'''Third Trial'''</td>
</tr>
<tr>
  <td align="center"> &nbsp;</td>
  <td align="center"> x</td>
  <td align="center"> y</td>
  <td align="center"> z</td>
  <td align="center"> <math>~\Delta_\mathrm{TP}</math></td>
</tr>
<tr>
  <td align="center"> <math>~e_1</math></td>
  <td align="center">0.47967</td>
  <td align="center">0.87745</td>
  <td align="center"> 0.0</td>
  <td align="center"> 0.02</td>
</tr>
<tr>
  <td align="center"> <math>~e_2</math></td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">-1.0</td>
  <td align="center"> 0.25</td>
</tr>
<tr>
  <td align="center"> <math>~e_3</math></td>
  <td align="center">- 0.87753</td>
  <td align="center"> 0.47952 </td>
  <td align="center">0.0</td>
  <td align="center"> 0.25</td>
</tr>
</table>

In constructing the Tangent-Plane (TP) for a 3D COLLADA display, we first move from the point that is on the surface of the ellipsoid, <math>~\vec{x}_0 = (x_0, y_0, z_0) = (0.8, 0.246, 0.0)</math>, to

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" rowspan="2">vertex <br />"m"</td>
  <td align="center" rowspan="2"><math>~\vec{P}_m</math></td>
  <td align="center" colspan="3">Components</td>
</tr>
<tr>
  <td align="center"><math>~x_m = \hat\imath \cdot \vec{P}_m</math></td>
  <td align="center"><math>~y_m = \hat\jmath \cdot \vec{P}_m</math></td>
  <td align="center"><math>~z_m = \hat{k} \cdot \vec{P}_m</math></td>
</tr>

<tr>
  <td align="center">0</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 - \Delta_2\hat{e}_2 - \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} - \Delta_2 e_{2x} - \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} - \Delta_2 e_{2y} - \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} - \Delta_2 e_{2z} - \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  - 0.25 (-0.87752) = 1.00979  
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  - 0.25 (0.47952) = 0.10847
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">1</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 + \Delta_2\hat{e}_2 - \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} + \Delta_2 e_{2x} - \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} + \Delta_2 e_{2y} - \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} + \Delta_2 e_{2z} - \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  - 0.25 (-0.87752) =  1.00979
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  - 0.25 (0.47952) = 0.10847
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">2</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 - \Delta_2\hat{e}_2 + \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} - \Delta_2 e_{2x} + \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} - \Delta_2 e_{2y} + \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} - \Delta_2 e_{2z} + \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  + 0.25 (-0.87752) =   0.56307
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  + 0.25 (0.47952) = 0.34823
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">3</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 + \Delta_2\hat{e}_2 + \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} + \Delta_2 e_{2x} + \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} + \Delta_2 e_{2y} + \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} + \Delta_2 e_{2z} + \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  + 0.25 (-0.87752) = 0.57103
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  + 0.25 (0.47952) = 0.34823
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">4</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  - 0.25 (-0.87752) =   1.0290
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  - 0.25 (0.47952) = 0.1436
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">5</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  - 0.25 (-0.87752) =  1.0290
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  - 0.25 (0.47952) = 0.1436
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">6</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  + 0.25 (-0.87752) =   0.59021
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  + 0.25 (0.47952) = 0.38333
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">7</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  + 0.25 (-0.87752) = 0.59021
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  + 0.25 (0.47952) = 0.38333
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="left">[[File:TangentPlaneSchematic.png|Tangent Plane Schematic]]</td>
  <td align="left">[[File:ExcelVertices080.png|Vertex Locations via Excel]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~x_0 = 0.8, z_0 = 0.0, y_0 = 0.246, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgray">[[File:GoodPlane01.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
</tr>
</table>

===GoodPlane02===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center"><math>~x_0 = 0.075, z_0 = 0.0, y_0 = 0.4089, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane02.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
</tr>
</table>

===GoodPlane03===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="2"><math>~x_0 = 0.25, z_0 = 0.20, y_0 = 0.33501, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane03.png|400px|Tangent Plane Schematic]]</td>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane03B.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.10, \Delta_3 = 0.25</math></td>
</tr>
<tr>
  <td align="left" colspan="2">
CAPTION: &nbsp; The image on the right differs from the image on the left in only one way &#8212; <math>~\Delta_2</math> = 0.1 instead of 0.25.  It illustrates more clearly that the <math>~\hat{e}_3</math> (longest) coordinate axis is not parallel to the z-axis when <math>~z_0 \ne 0.</math>
  </td>
</tr>
</table>

===GoodPlane04===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="1"><math>~x_0 = 0.25, z_0 = 1/3, y_0 = 0.1777, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane04A.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.10, \Delta_3 = 0.25</math></td>
</tr>
</table>

=Further Exploration=

Let's set:  <math>~x_0 = 0.25, y_0 = 0.33501, z_0 = 0.2 ~~~\Rightarrow ~~~\lambda_1 = 1.00000, \lambda_2 = 0.33521 \, .</math>


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

<tr>
  <td align="right">
<math>~q \equiv \frac{a}{b}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.43972</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~p \equiv \frac{a}{c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.5974 \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\lambda_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x y^{1/q^2}}{ z^{2/p^2}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>
  </td>
</tr>
</table>

Next, let's examine the curve that results from varying <math>~z</math> while <math>~\lambda_1</math> and <math>~\lambda_2</math> are held fixed.  From the expression for <math>~\lambda_2</math> we appreciate that,

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

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda_2 z^{2/p^2}}{y^{1/q^2}} \, ;</math>
  </td>
</tr>
</table>
and from the expression for <math>~\lambda_1</math> we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 - q^2y^2 - p^2z^2 \, .</math>
  </td>
</tr>
</table>
Hence, the relationship between <math>~y</math> and <math>~z</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\lambda_2 z^{2/p^2}}{y^{1/q^2}} \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 - q^2y^2 - p^2z^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \lambda_2^2 z^{4/p^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y^{2/q^2} \biggl[ \lambda_1^2 - q^2y^2 - p^2z^2\biggr] \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8" width="80%">
<tr><td align="left">
Alternatively, 

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

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

Hence, the relationship between <math>~x</math> and <math>~z</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 - p^2z^2 - q^2\biggl[ \frac{\lambda_2 z^{2/p^2}}{x}\biggr]^{2q^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x^{2(q^2+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^{2q^2} \biggl[\lambda_1^2 - p^2z^2\biggr] - q^2\biggl[ \lambda_2 z^{2/p^2}\biggr]^{2q^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ x^{2q^2} \biggl[\lambda_1^2 - p^2z^2\biggr] - q^2\biggl[ \lambda_2 z^{2/p^2}\biggr]^{2q^2} \biggr\}^{1/(q^2+1)}</math>
  </td>
</tr>
</table>

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

Here are some example values &hellip;

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="7"><math>~\lambda_1 = 1</math>&nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\lambda_2 = 0.33521</math></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~z</math></td>
  <td align="center" colspan="2">1<sup>st</sup> Solution</td>
  <td align="center" rowspan="20" bgcolor="lightgray">&nbsp;</td>
  <td align="center" colspan="2">2<sup>nd</sup> Solution</td>
  <td align="center" rowspan="20" bgcolor="white">[[File:Lambda3ImageA.png|450px|lambda_3 coordinate]]</td>
</tr>
<tr>
  <td align="center"><math>~y_1</math></td>
  <td align="center"><math>~x_1</math></td>
  <td align="center"><math>~y_2</math></td>
  <td align="center"><math>~x_2</math></td>
</tr>
<tr>
  <td align="center">0.01</td>
  <td align="center">0.407825695</td>
  <td align="center">0.0995168</td>
  <td align="center">-</td>
  <td align="center">-</td>
</tr>
<tr>
  <td align="center">0.03</td>
  <td align="center">0.40481851</td>
  <td align="center">0.138</td>
  <td align="center">-</td>
  <td align="center">-</td>
</tr>
<tr>
  <td align="center">0.04</td>
  <td align="center">0.40309223</td>
  <td align="center">0.1503934</td>
  <td align="center">-</td>
  <td align="center">-</td>
</tr>
<tr>
  <td align="center">0.08</td>
  <td align="center">0.393779065</td>
  <td align="center">0.1854283</td>
  <td align="center">-</td>
  <td align="center">-</td>
</tr>
<tr>
  <td align="center">0.12</td>
  <td align="center">0.37990705</td>
  <td align="center">0.2103761</td>
  <td align="center">-</td>
  <td align="center">-</td>
</tr>
<tr>
  <td align="center">0.16</td>
  <td align="center">0.36067787</td>
  <td align="center">0.23111</td>
  <td align="center">1.04123&times;10<sup>-4</sup></td>
  <td align="center">0.9095546</td>
</tr>
<tr>
  <td align="center">0.2</td>
  <td align="center">0.33500747</td>
  <td align="center">0.2500033</td>
  <td align="center">2.23778&times;10<sup>-4</sup></td>
  <td align="center">0.85448</td>
</tr>
<tr>
  <td align="center">0.22</td>
  <td align="center">0.31923525</td>
  <td align="center">0.2592611</td>
  <td align="center">3.36653 &times;10<sup>-4</sup></td>
  <td align="center">0.82065</td>
</tr>
<tr>
  <td align="center">0.24</td>
  <td align="center">0.30106924</td>
  <td align="center">0.2686685</td>
  <td align="center">5.2327 &times;10<sup>-4</sup></td>
  <td align="center">0.78192</td>
</tr>
<tr>
  <td align="center">0.26</td>
  <td align="center">0.2799962</td>
  <td align="center">0.2784963</td>
  <td align="center">8.53243 &times;10<sup>-4</sup></td>
  <td align="center">0.73752</td>
</tr>
<tr>
  <td align="center">0.28</td>
  <td align="center">0.25521147</td>
  <td align="center">0.2891526</td>
  <td align="center">1.491545 &times;10<sup>-3</sup></td>
  <td align="center">0.68634</td>
</tr>
<tr>
  <td align="center">0.3</td>
  <td align="center">0.22530908</td>
  <td align="center">0.3013752</td>
  <td align="center">2.89262 &times;10<sup>-3</sup></td>
  <td align="center">0.62671</td>
</tr>
<tr>
  <td align="center">0.32</td>
  <td align="center">0.1873233</td>
  <td align="center">0.3168808</td>
  <td align="center">6.6223 &times;10<sup>-3</sup></td>
  <td align="center">0.55579</td>
</tr>
<tr>
  <td align="center">0.34</td>
  <td align="center">0.13149897</td>
  <td align="center">0.3423994</td>
  <td align="center">2.09221 &times;10<sup>-2</sup></td>
  <td align="center">0.46637</td>
</tr>
<tr>
  <td align="center">0.343</td>
  <td align="center">0.1191543</td>
  <td align="center">0.3490285</td>
  <td align="center">0.026458</td>
  <td align="center">0.4496</td>
</tr>
<tr>
  <td align="center">0.344</td>
  <td align="center">0.1145</td>
  <td align="center">0.3517</td>
  <td align="center">0.02880</td>
  <td align="center">0.4435</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="center">0.1093972</td>
  <td align="center">0.354688</td>
  <td align="center">0.03155965</td>
  <td align="center">0.4371186</td>
</tr>
<tr>
  <td align="center">0.3485</td>
  <td align="center">0.0847372</td>
  <td align="center">0.3713588</td>
  <td align="center">0.0480478</td>
  <td align="center">0.4085204</td>
</tr>
</table>

=See Also=

<ul>
  <li>
[[User:Tohline/Appendix/Ramblings/DirectionCosines|Direction Cosines]]
  </li>
</ul>


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