Editing Appendix/Ramblings/ConcentricEllipsoidalCoordinates (section)

===Speculation1===

Building on our experience developing [[Appendix/Ramblings/T3Integrals#Integrals_of_Motion_in_T3_Coordinates|T3 Coordinates]] and, more recently, [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]], let's define the two "angles,"
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Zeta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\sinh^{-1}\biggl(\frac{qy}{x} \biggr)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\Upsilon</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\sinh^{-1}\biggl(\frac{pz}{x} \biggr) \, ,</math>
  </td>
</tr>
</table>
in which case we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^2(\cosh^2\Zeta + \sinh^2\Upsilon)\, .</math>
  </td>
</tr>
</table>
We speculate that the other two orthogonal coordinates may be defined by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x \biggl[ \sinh\Zeta \biggr]^{1/(1-q^2)} 
=
x \biggl[ \frac{qy}{x}\biggr]^{1/(1-q^2)}
=
x \biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)}
=
\biggl[ \frac{x^{q^2}}{qy}\biggr]^{1/(q^2-1)}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x \biggl[ \sinh\Upsilon \biggr]^{1/(1-p^2)} 
=
x \biggl[ \frac{pz}{x}\biggr]^{1/(1-p^2)} 
=
x \biggl[ \frac{x}{pz}\biggr]^{1/(p^2-1)} 
=
\biggl[ \frac{x^{p^2}}{pz}\biggr]^{1/(p^2-1)} 
\, .</math>
  </td>
</tr>
</table>

Some relevant partial derivatives are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{1}{qy}\biggr]^{1/(q^2-1)} \biggl[ \frac{q^2}{q^2-1} \biggr]x^{1/(q^2-1)}
=
\biggl[ \frac{q^2}{q^2-1} \biggr]\biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)} 
=
\biggl[ \frac{q^2}{q^2-1} \biggr]\frac{\lambda_2}{x} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{x^{q^2}}{q}\biggr]^{1/(q^2-1)} \biggl[ \frac{1}{1-q^2} \biggr] y^{q^2/(1-q^2)}
=
- \biggl[ \frac{1}{q^2-1} \biggr] \frac{\lambda_2}{y} 
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{p^2}{p^2-1} \biggr]\frac{\lambda_3}{x} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{1}{p^2-1} \biggr] \frac{\lambda_3}{z} 
\, .</math>
  </td>
</tr>
</table>
And the associated scale factors are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \biggl( \frac{q^2}{q^2-1} \biggr)\frac{\lambda_2}{x} \biggr]^2 + \biggl[ - \biggl( \frac{1}{q^2-1} \biggr) \frac{\lambda_2}{y} \biggr]^2 \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl( \frac{q^2}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{x^2}  + \biggl( \frac{1}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{y^2} \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{x^2 + q^4 y^2 \biggr\}^{-1}
\biggl[ \frac{(q^2 - 1)^2x^2 y^2}{\lambda_2^2} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~h_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{x^2 + p^4 z^2 \biggr\}^{-1}
\biggl[ \frac{(p^2 - 1)^2x^2 z^2}{\lambda_3^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
We can now fill in the rest of our direction-cosines table, namely,

<table border="1" cellpadding="8" align="center" width="60%">
<tr>
  <td align="center" colspan="4">
'''Direction Cosines for T6 Coordinates'''
<br />
<math>~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center" width="10%"><math>~n</math></td>
  <td align="center" colspan="3"><math>~i = x, y, z</math>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center">&nbsp;<br />
<math>~x\ell_{3D}</math><br />&nbsp;
  <td align="center"><math>~q^2 y \ell_{3D}</math>
  <td align="center"><math>~p^2 z \ell_{3D}</math>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center">
<math>~q^2 y \ell_q </math>
  <td align="center">
<math>~-x\ell_q</math>
  <td align="center">
<math>~0</math>
  </td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">
<math>~p^2 z \ell_p</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~-x\ell_p</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{21}
+ \hat\jmath \gamma_{22}
+\hat{k} \gamma_{23}
=
\hat\imath (q^2y\ell_q)
- \hat\jmath (x\ell_q) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \gamma_{31}
+ \hat\jmath \gamma_{32}
+\hat{k} \gamma_{33}
=
\hat\imath (p^2z\ell_p)
-\hat{k} (x\ell_p) \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\hat{e}_2 \cdot \hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(q^2y\ell_q)^2
+ (x\ell_q)^2
=
1 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_3 \cdot \hat{e}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(p^2z\ell_p)^2
+ (x\ell_p)^2
=
1 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_2 \cdot \hat{e}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(q^2y\ell_q)(p^2z\ell_p) \ne 0 \, .
</math>
  </td>
</tr>
</table>