Editing Appendix/Ramblings/T9Coordinates (section)

=Complete Orthogonality Check=
If the set of unit vectors is indeed orthogonal, then we must find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{mn}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_{mn} \, ,</math>
  </td>
</tr>
</table>
where the quantity <math>~M_{mn}</math> is the minor of <math>~\gamma_{mn}</math> in the determinant, <math>~|\gamma_{mn}|</math>.  (Note:  This last expression is true only for right-handed coordinate systems.  If the coordinate system is left-handed, we should find, <math>~\gamma_{mn} = - M_{mn}</math>.)  More specifically, for any right-handed, orthogonal curvilinear coordinate system we should find:
<table  border="1" align="center" cellpadding="10">
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="left">

<tr>
  <td align="right">
<math>~\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<table border="0" cellpadding="5" align="right">

<tr>
  <td align="right">
<math>~\gamma_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{33}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21} \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

and the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat\imath} x + 
\mathbf{\hat\jmath} y +
\mathbf{\hat{k}} z
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{e}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{e}}_3 (\gamma_{31} x + \gamma_{32} y + \gamma_{33} z) \, .
</math>
  </td>
</tr>
</table>

==For (&kappa;<sub>1</sub>, &kappa;<sub>2</sub>, &kappa;<sub>3</sub>)==

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

<tr>
  <td align="right">
<math>~\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ 0 \biggr]
+
\biggl[ \frac{( x^2 + q^4y^2 ) \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}  \biggr]
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
x\ell_{3D}
=
\gamma_{11} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bigg[ (x^2 + q^4y^2)^{1 / 2} \ell_{3D} \biggr]
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
-
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ 0 \biggr]
=
q^2 y \ell_{3D}
=
\gamma_{12} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
+
\biggl[ \frac{q^2 y p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x^2p^2 z \ell_{3D}}{(x^2 + q^4y^2)} \biggr]
+
\biggl[ \frac{q^4 y^2 p^2 z \ell_{3D}}{(x^2 + q^4y^2)} \biggr]
=
p^2 z \ell_{3D} 
=
\gamma_{13} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ p^2z \ell_{3D} \biggr]
-
\biggl[ 0\biggr]
\biggl[ q^2y \ell_{3D} \biggr]
=
\biggl[ \frac{xp^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
\gamma_{21} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 0\biggr]
\biggl[ x\ell_{3D} \biggr]
+
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ p^2z \ell_{3D} \biggr]
=
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
\gamma_{22} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ q^2y \ell_{3D} \biggr]
-
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ x\ell_{3D} \biggr]
=
-(x^2 + q^4y^2)^{1 / 2}\ell_{3D}
=
\gamma_{23} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ q^2y \ell_{3D} \biggr]
\biggl[ (x^2 + q^4y^2)^{1 / 2}\ell_{3D} \biggr]
-
\biggl[ p^2 z\ell_{3D} \biggr]
\biggl[ \frac{q^2y p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{ q^2 y \ell_{3D}^2 }{(x^2 + q^4y^2)^{1 / 2} } 
\biggl[ x^2 + q^4y^2 
+
p^4 z^2 \biggr] 
=
- \frac{ q^2 y }{(x^2 + q^4y^2)^{1 / 2} } 
=
\gamma_{31} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ p^2z \ell_{3D} \biggr]
\biggl[ \frac{x p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
+
\biggl[ x \ell_{3D} \biggr]
\biggl[ (x^2 + q^4y^2)^{1 / 2} \ell_{3D}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x \ell_{3D}^2 }{ (x^2 + q^4y^2)^{1 / 2} } \biggl[
x^2 + q^4y^2 + p^4 z^2 
\biggr]
=
\frac{x }{ (x^2 + q^4y^2)^{1 / 2} } 
=
\gamma_{32} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ x \ell_{3D} \biggr]
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
-
\biggl[ q^2y \ell_{3D} \biggr]
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
0
=
\gamma_{33} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

Given that the prescribed interrelationships between all nine direction cosines are satisfied, we conclude that the <math>~(\kappa_1, \kappa_2, \kappa_3)</math> coordinate system is an orthogonal one.  Accordingly, the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
x^2 \ell_{3D} 
+ q^2y^2 \ell_{3D} 
+ p^2z^2 \ell_{3D}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_2 \biggl\{
x^2 + q^2y^2  - \frac{1}{p^2}(x^2 + q^4 y^2) 
\biggr\} \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\mathbf{\hat{e}}_3 \biggl\{ 
- \frac{xq^2y}{(x^2 + q^4y^2)^{1 / 2}} 
+ \frac{xy}{(x^2 + q^4 y^2)^{1 / 2}}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ \kappa_1^2 \ell_{3D} \biggr]
~+~
\mathbf{\hat{e}}_2 \biggl[
x^2 + q^2y^2  - \frac{1}{p^2}(x^2 + q^4 y^2) 
\biggr] \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
~+~
\mathbf{\hat{e}}_3 
\biggl[ 
\frac{(1-q^2)xy}{(x^2 + q^4 y^2)^{1 / 2}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ \kappa_1^2 \ell_{3D} \biggr]
~+~
\mathbf{\hat{e}}_2 \biggl[
x^2\biggl(1 - \frac{1}{p^2}\biggr) - q^2y^2 \biggl(q^2 - 1 \biggr) 
\biggr] \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
~-~
\mathbf{\hat{e}}_3 
\biggl[ 
\frac{(q^2 - 1)xy}{(x^2 + q^4 y^2)^{1 / 2}} \biggr] \, .
</math>
  </td>
</tr>
</table>

==For (&kappa;<sub>1</sub>, &kappa;<sub>4</sub>, &kappa;<sub>5</sub>)==

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

<tr>
  <td align="right">
<math>~\gamma_{42}\gamma_{53} - \gamma_{43}\gamma_{52}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{1}{q^2y} \biggr) \biggr]
\biggl[ \frac{p^2 z \ell_{3D}}{\mathcal{D}} \biggl(x^2 - q^4y^2 \biggr)  \biggr]
+
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{2}{p^2z} \biggr) \biggr]
\biggl[ \frac{q^2y \ell_{3D}}{\mathcal{D}}\biggl( p^4 z^2 + 2x^2 \biggr) \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x\ell_{3D}}{\mathcal{D}^2} \biggl[
x^2p^4z^2  
+
4x^2 q^4y^2 
+
q^4y^2 p^4 z^2 \biggr]
=
x\ell_{3D} = \gamma_{11} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{43}\gamma_{51} - \gamma_{41}\gamma_{53}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{2}{p^2 z} \biggr) \biggr]
\biggl[ \frac{x \ell_{3D}}{\mathcal{D}} \biggl(2q^4y^2 +p^4z^2\biggr)  \biggr]
-
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{1}{x} \biggr) \biggr]
\biggl[ \frac{p^2 z \ell_{3D}}{\mathcal{D}}\biggl( x^2 - q^4y^2 \biggr) \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2 y \ell_{3D}}{\mathcal{D}^2}  \biggl[
4x^2 q^4y^2 + x^2 p^4z^2 + q^4y^2 p^4 z^2 
\biggr]
=
q^2y \ell_{3D}
= \gamma_{12} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{41}\gamma_{52} - \gamma_{42}\gamma_{51}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{1}{x} \biggr) \biggr]
\biggl[ \frac{q^2 y\ell_{3D}}{\mathcal{D}} \biggl(p^4z^2 + 2x^2\biggr)  \biggr]
+
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{1}{q^2 y} \biggr) \biggr]
\biggl[ \frac{x \ell_{3D}}{\mathcal{D}}\biggl( 2q^4y^2 + p^4z^2 \biggr) \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2 z \ell_{3D}}{\mathcal{D}^2} \biggl[
q^4y^2 p^4z^2 + 4x^2 q^4 y^2
+ x^2p^4z^2  
\biggr]
=
p^2 z \ell_{3D}
=
\gamma_{13} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{52}\gamma_{13} - \gamma_{53}\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( p^4z^2 + 2x^2 \biggr) \biggr]
-
\biggl[ \frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( x^2 - q^4y^2 \biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl[ x^2 + q^4y^2 + p^4z^2 \biggr]
=
\frac{q^2 y p^2 z }{\mathcal{D}} 
=
\gamma_{41} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{53}\gamma_{11} - \gamma_{51}\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( x^2 - q^4y^2 \biggr)  \biggr]
+
\biggl[ \frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( 2q^4y^2 +p^4z^2\biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl[ x^2 + q^4y^2 +p^4z^2  \biggr]
=
\frac{xp^2 z}{\mathcal{D}}
=
\gamma_{42} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{51}\gamma_{12} - \gamma_{52}\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl[ \frac{x q^2y \ell_{3D}^2}{\mathcal{D}} \biggl( 2q^4y^2 +p^4z^2\biggr)  \biggr]
-
\biggl[ \frac{x q^2y \ell_{3D}^2}{\mathcal{D}} \biggl( p^4z^2 + 2x^2\biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2x q^2y \ell_{3D}^2}{\mathcal{D}}\biggl[ x^2 + q^4y^2 +p^4z^2  \biggr]
=
-\frac{2x q^2y }{\mathcal{D}}
=
\gamma_{43} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{12}\gamma_{43} - \gamma_{13}\gamma_{42}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2x q^4 y^2 \ell_{3D} }{\mathcal{D}} \biggr]
- \biggl[ \frac{x p^4 z^2 \ell_{3D} }{\mathcal{D}} \biggr]
=
- \biggl[ \frac{x \ell_{3D} }{\mathcal{D}} \biggr] (2q^4 y^2 + p^4z^2)
= \gamma_{51} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}\gamma_{41} - \gamma_{11}\gamma_{43}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2 y p^4z^2 \ell_{3D} }{\mathcal{D}} \biggr]
+ 
\biggl[ \frac{2x^2 q^2 y \ell_{3D} }{\mathcal{D}} \biggr]
=
\biggl[ \frac{q^2 y  \ell_{3D} }{\mathcal{D}} \biggr](p^4z^2 + 2x^2)
=
\gamma_{52} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{11}\gamma_{42} - \gamma_{12}\gamma_{41}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x^2 p^2z \ell_{3D}}{\mathcal{D}} \biggr]
- 
\biggl[ \frac{q^4 y^2 p^2 z\ell_{3D} }{\mathcal{D}} \biggr]
=
\biggl[ \frac{p^2z \ell_{3D}}{\mathcal{D}} \biggr](x^2 - q^4y^2)
=
\gamma_{53} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

Given that the prescribed interrelationships between all nine direction cosines are satisfied, we conclude that the <math>~(\kappa_1, \kappa_4, \kappa_5)</math> coordinate system is an orthogonal one.  Accordingly, the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{e}}_2 (\gamma_{41} x + \gamma_{42} y + \gamma_{43} z)
+
\mathbf{\hat{e}}_3 (\gamma_{51} x + \gamma_{52} y + \gamma_{53} z) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
x^2\ell_{3D} 
+ q^2y^2 \ell_{3D} 
+ p^2z^2\ell_{3D}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_2 \biggl\{
\frac{xq^2yp^2z}{\mathcal{D}} 
+ \frac{x y p^2z}{\mathcal{D}}
- \frac{2 xq^2y z}{\mathcal{D}}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_3 \biggl\{
- \frac{\ell_{3D} x^2}{\mathcal{D}} (2q^4y^2 + p^4z^2)
+ \frac{\ell_{3D} q^2 y^2}{\mathcal{D}} (p^4z^2 + 2x^2) 
+ \frac{\ell_{3D} p^2z^2}{\mathcal{D}} (x^2 - q^4y^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
\kappa_1^2 \ell_{3D}
\biggr\}
+
\mathbf{\hat{e}}_2 \biggl\{
q^2p^2 + p^2 - 2 q^2 
\biggr\} \frac{xyz}{\mathcal{D}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_3 \biggl\{
- x^2 (2q^4y^2 + p^4z^2)
+ q^2 y^2 (p^4z^2 + 2x^2) 
+ p^2z^2 (x^2 - q^4y^2) 
\biggr\} \frac{\ell_{3D}}{\mathcal{D}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
\kappa_1^2 \ell_{3D}
\biggr\}
~+~
\mathbf{\hat{e}}_2 \biggl\{
q^2p^2 + p^2 - 2 q^2 
\biggr\} \frac{xyz}{\mathcal{D}}
~-~
\mathbf{\hat{e}}_3 \biggl\{ 2 x^2 q^2y^2(q^2-1) + x^2 p^2z^2(p^2-1) + q^2 y^2 p^2 z^2(q^2 - p^2)
\biggr\} \frac{\ell_{3D}}{\mathcal{D}} \, .
</math>
  </td>
</tr>
</table>