Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

==Background==
Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosines|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of the so-called [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Analysis|T6]] (concentric elliptic) coordinate system, here we take a somewhat daring attack on this problem, mixing our approach to identifying the expression for the third curvilinear coordinate.  Broadly speaking, this entire study is motivated by our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].

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

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\tan^{-1}\biggl( \frac{y^{1/q^2}}{x} \biggr)</math></td>
  <td align="center"><math>~\frac{xq^2 y \ell_q}{\sin\lambda_3 \cos\lambda_3}</math></td>
  <td align="center"><math>~-\frac{\sin\lambda_3 \cos\lambda_3}{x}</math></td>
  <td align="center"><math>~+\frac{\sin\lambda_3 \cos\lambda_3}{q^2y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~-q^2 y \ell_q</math></td>
  <td align="center"><math>~x\ell_q</math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="center" colspan="9">
<table border="0" cellpadding="5" 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^4 y^2 + p^4 z^2]^{- 1/ 2 }</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\ell_q</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[x^2 + q^4 y^2 ]^{- 1/ 2 }</math>
  </td>
</tr>
</table>

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

As before, let's adopt the first-coordinate expression,
<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>
</table>
but for the third-coordinate expression we will abandon the trigonometric expression and instead simply use,
<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>~\frac{y^{1/q^2}}{x} \, .</math>
  </td>
</tr>
</table>
<span id="Table1DaringAttack">This modified third-coordinate expression means that the last row of the above table changes, as follows.</span>

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">'''Daring Attack'''</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">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~\ell_q \ell_{3D} (xp^2z)</math></td>
  <td align="center"><math>~\ell_q \ell_{3D} (q^2 y p^2z) </math></td>
  <td align="center"><math>~- (x^2 + q^4y^2)\ell_q \ell_{3D}</math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\frac{y^{1/q^2}}{x} </math></td>
  <td align="center"><math>~\frac{xq^2 y \ell_q}{\lambda_3}</math></td>
  <td align="center"><math>~-\frac{\lambda_3}{x}</math></td>
  <td align="center"><math>~+\frac{\lambda_3}{q^2y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~-q^2 y \ell_q</math></td>
  <td align="center"><math>~x\ell_q</math></td>
  <td align="center"><math>~0</math></td>
</tr>
</table>

Notice that the direction cosine functions for the (as yet, unknown) second-coordinate function remain the same.  This is because the direction-cosine functions associated with both <math>~\lambda_1</math> and <math>~\lambda_3</math> remain unchanged, so it must be true that the cross product of the first and third unit vectors leads to the same components for the second unit vector.