Editing Appendix/Ramblings/ConcentricEllipsoidalT8Coordinates (section)

=Kappa (&kappa;8) Coordinates=
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2y^2 + p^2 z^2)^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\kappa_3</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{(qy)^{1/q^2}}{x} \biggr] \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial\kappa_3}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{(qy)^{2/q^2}}{x^2} \biggr]^{-1} \biggl[- \frac{(qy)^{1/q^2}}{x^2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\sin^2\kappa_3}{(qy)^{1/q^2}} = -\frac{\sin^2\kappa_3}{x\tan\kappa_3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\sin\kappa_3 \cos\kappa_3}{x}\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial\kappa_3}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{(qy)^{2/q^2}}{x^2} \biggr]^{-1} \biggl[\frac{q^{1/q^2} y^{1/q^2}}{q^2 x y}  \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \frac{1}{q^2y}\biggl[\frac{\tan\kappa_3}{1 + \tan^2\kappa_3 }\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \frac{\sin\kappa_3 \cos\kappa_3}{q^2y} \, .
</math>
  </td>
</tr>
</table>
Therefore,
<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>~
\biggl[-\frac{\sin\kappa_3 \cos\kappa_3}{x} \biggr]^2
+
\biggl[\frac{\sin\kappa_3 \cos\kappa_3}{q^2y} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2)
\biggl[\frac{\sin\kappa_3 \cos\kappa_3}{xq^2y} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ h_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2)^{-1 / 2}
\biggl[\frac{xq^2y}{\sin\kappa_3 \cos\kappa_3} \biggr] \, .
</math>
  </td>
</tr>
</table>


<span id="TableKappa8">&nbsp;</span>
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for &kappa;8 Coordinates'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\kappa_n</math></td>
  <td align="center"><math>~h_n</math></td>
  <td align="center"><math>~\frac{\partial \kappa_n}{\partial x}</math></td>
  <td align="center"><math>~\frac{\partial \kappa_n}{\partial y}</math></td>
  <td align="center"><math>~\frac{\partial \kappa_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>~\kappa_1 \ell_{3D}</math></td>
  <td align="center"><math>~\frac{x}{\kappa_1}</math></td>
  <td align="center"><math>~\frac{q^2 y}{\kappa_1}</math></td>
  <td align="center"><math>~\frac{p^2 z}{\kappa_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_{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="center"><math>~3</math></td>
  <td align="center"><math>~\tan^{-1}\biggl[ \frac{(qy)^{1/q^2}}{x} \biggr]</math></td>
  <td align="center"><math>~\frac{1}{\sin\kappa_3 \cos\kappa_3}\biggl[\frac{x q^2 y }{(x^2 + q^4y^2)^{1 / 2}}\biggr] </math></td>
  <td align="center"><math>~-\frac{\sin\kappa_3 \cos\kappa_3}{x}</math></td>
  <td align="center"><math>~\frac{\sin\kappa_3 \cos\kappa_3}{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="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>

</table>
  </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}{\kappa_4}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\biggr] </math></td>
  <td align="center"><math>~\frac{\kappa_4}{x}</math></td>
  <td align="center"><math>~\frac{\kappa_4}{q^2 y}</math></td>
  <td align="center"><math>~-\frac{2\kappa_4}{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">---</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>~\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>
Also note &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~AB \equiv \biggl( \frac{\mathcal{D}}{\ell_{3D}} \biggr)^{2} </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>
and the partial derivatives of <math>~A</math> and <math>~B</math> are detailed in [[Appendix/Ramblings/ConcentricEllipsodalCoordinates#ABderivatives|an accompanying discussion]].
  </td>
</tr>

</table>

The direction-cosines of the second unit vector &#8212; as has already been inserted into the "&kappa;8 coordinates" table &#8212; should be obtainable from the first and third unit vectors via the cross product,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_2 = \hat{e}_3 \times \hat{e}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl[ e_{3y}e_{1z} - e_{3z} e_{1y} \biggr]
+
\hat\jmath \biggl[ e_{3z}e_{1x} - e_{3x} e_{1z} \biggr]
+
\hat{k} \biggl[ e_{3x}e_{1y} - e_{3y} e_{1x} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl[ \frac{x (p^2z) \ell_{3D} }{(x^2 + q^4y^2)^{1 / 2}} \biggr]
+
\hat\jmath \biggl[  \frac{q^2 y  (p^2z) \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}\biggr]
-
\hat{k} \biggl[ \frac{(x^2 + q^4y^2) \ell_{3D} }{(x^2 + q^4y^2)^{1 / 2}} \biggr] \, .
</math>
  </td>
</tr>
</table>

The other boxes in the n = 2 row have been drawn from [[Appendix/Ramblings/ConcentricEllipsodalT8Coordinates#Associated_h3_Scale_Factor|our accompanying EUREKA! moment]] and the n = 3 row of the table that details "Direction Cosine Components for T8 Coordinates."

==Attempt 1==
Let's try &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{x(qy)^{1/q^2}}{(pz)^{1/p^2}} \biggr] \, ,</math>
  </td>
</tr>
</table>
which leads to,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial\kappa_3}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{x^2(qy)^{2/q^2}}{(pz)^{2/p^2}} \biggr]^{-1} \biggl[ \frac{(qy)^{1/q^2}}{(pz)^{1/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}\biggl[1+\tan^2\kappa_2\biggr]^{-1} \tan\kappa_2 = \frac{\sin\kappa_2 \cos\kappa_2}{x} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial\kappa_3}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \frac{x^2(qy)^{2/q^2}}{(pz)^{2/p^2}} \biggr]^{-1} \biggl[ \frac{xq^{1/q^2}(y)^{1/q^2}}{q^2 y(pz)^{1/p^2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{q^2y}\biggl[1+\tan^2\kappa_2\biggr]^{-1} \tan\kappa_2 = \frac{\sin\kappa_2 \cos\kappa_2}{q^2y} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial\kappa_3}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\sin\kappa_2 \cos\kappa_2}{p^2z} \, .
</math>
  </td>
</tr>

</table>
Hence,
<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>~\sin^2\kappa_2 \cos^2\kappa_2 \biggl[ \frac{1}{x^2} + \frac{1}{q^4y^2} + \frac{1}{p^4z^2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sin^2\kappa_2 \cos^2\kappa_2 \biggl[ \frac{x^2 + q^4 y^2 + p^4 z^2}{x^2 q^4y^2 p^4 z^2}  \biggr]
=
\biggl[ \frac{\sin^2\kappa_2 \cos^2\kappa_2 }{x^2 q^4y^2 p^4 z^2 \ell_{3D}^2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x q^2y p^2 z \ell_{3D}}{\sin\kappa_2 \cos\kappa_2 }  \biggr] \, .
</math>
  </td>
</tr>
</table>
The three direction-cosines are, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21} = h_2 \biggl(\frac{\partial \kappa_2}{\partial x}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x q^2y p^2 z \ell_{3D}}{\sin\kappa_2 \cos\kappa_2 }  \biggr] \frac{\sin\kappa_2 \cos\kappa_2}{x}
=
q^2y p^2z \ell_{3D} \, .
</math>
  </td>
</tr>
</table>