Editing Appendix/Ramblings/ConcentricEllipsoidalCoordinates (section)

==Orthogonal Coordinates==

===Primary (''radial-like'') Coordinate===

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,
<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>
When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>.  (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)  

A vector, <math>~\bold{\hat{n}}</math>, that is normal to the <math>~\lambda_1</math> = constant surface is given by the gradient of the function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~F(x, y, z)</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} - \lambda_1 \, .</math>
  </td>
</tr>
</table>

In Cartesian coordinates, this means,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{\hat{n}}(x, y, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl( \frac{\partial F}{\partial x} \biggr)
+ \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr)
+ \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl[ x(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat{k}\biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \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}{\lambda_1} \biggr)
+ \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr)
+ \hat{k}\biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,
</math>
  </td>
</tr>
</table>
where it is understood that this expression is only to be evaluated at points, <math>~(x, y, z)</math>, that lie on the selected <math>~\lambda_1</math> surface &#8212; that is, at points for which the function, <math>~F(x,y,z) = 0</math>.  The length of this normal vector is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2
+ \biggl( \frac{q^2y}{\lambda_1} \biggr)^2
+ \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\lambda_1 \ell_{3D}}
</math>
  </td>
</tr>
</table>
where,
<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>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .</math>
  </td>
</tr>
</table>

It is therefore clear that the ''properly normalized'' normal unit vector that should be associated with any <math>~\lambda_1</math> = constant ellipsoidal surface is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} }
=
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat{k} (p^2 z \ell_{3D}) \, .
</math>
  </td>
</tr>
</table>
From our [[Appendix/Ramblings/DirectionCosines#Scale_Factors|accompanying discussion of direction cosines]], it is clear, as well, that the scale factor associated with the <math>~\lambda_1</math> coordinate is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1^2 \ell_{3D}^2 \, .</math>
  </td>
</tr>
</table>
We can also fill in the top line 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">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
</table>

===Other Coordinate Pair in the Tangent Plane===

Let's focus on a particular point on the <math>\lambda_1</math> = constant surface, <math>(x_0, y_0, z_0)</math>, that necessarily satisfies the function, <math>F(x_0, y_0, z_0) = 0</math>.  We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\hat{e}_1 </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, ,
</math>
  </td>
</tr>
</table>
where, for this specific point on the surface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\ell_{3D}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<div align="center">
'''Tangent Plane'''<br />
[See, for example, [http://math.furman.edu/~dcs/courses/math21/ Dan Sloughter's] ([https://www.furman.edu Furman University]) 2001 Calculus III  class lecture notes &#8212; specifically [http://math.furman.edu/~dcs/courses/math21/lectures/l-15.pdf Lecture 15]]
</div>

----

The two-dimensional plane that is tangent to the <math>\lambda_1</math> = constant surface ''at this point'' is given by the expression,
<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>
(x - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0 
+ (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0  
+ (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
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>
x_0^2  +  q^2 y_0^2 + p^2 z_0^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
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>
</td></tr></table>

Fix the value of <math>\lambda_1</math>.  This means that the relevant ellipsoidal surface is defined by the expression,
<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 + q^2y^2 + p^2z^2 \, .</math>
  </td>
</tr>
</table>
If <math>z = 0</math>, the semi-major axis of the relevant x-y ellipse is <math>\lambda_1</math>, and the square of the semi-minor axis is <math>\lambda_1^2/q^2</math>.  At any other value, <math>z = z_0 < c</math>, the square of the semi-major axis of the relevant x-y ellipse is, <math>~(\lambda_1^2 - p^2z_0^2)</math> and the square of the corresponding semi-minor axis is, <math>(\lambda_1^2 - p^2z_0^2)/q^2</math>.  Now, for any chosen <math>x_0^2 \le (\lambda_1^2 - p^2z_0^2)</math>, the y-coordinate of the point on the <math>~\lambda_1</math> surface is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>y_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{q^2}\biggl[ \lambda_1^2 - p^2 z_0 -x_0^2 \biggr] \, .</math>
  </td>
</tr>
</table>
The slope of the line that lies in the <math>z = z_0</math> plane and that is tangent to the ellipsoidal surface at <math>(x_0, y_0)</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>m \equiv \frac{dy}{dx}\biggr|_{z_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{x_0}{q^2y_0}</math>
  </td>
</tr>
</table>

===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>

===Speculation2===

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

<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^{1/p^2}} \, ,
</math>
  </td>
</tr>
</table>
in which case,
<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>~
\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>~
\frac{x}{z^{1/p^2}} \biggl(-\frac{1}{q^2}\biggr) y^{-1/q^2 - 1} 
=
-\frac{\lambda_2}{q^2 y}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\lambda_2}{p^2 z}
\, .
</math>
  </td>
</tr>
</table>
The associated scale factor is, then,
<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( \frac{\partial \lambda_2}{\partial x} \biggr)^2
+ 
\biggl( \frac{\partial \lambda_2}{\partial y} \biggr)^2
+ 
\biggl( \frac{\partial \lambda_2}{\partial z} \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{ \lambda_2}{x} \biggr)^2
+ 
\biggl( -\frac{\lambda_2}{q^2y} \biggr)^2
+ 
\biggl( - \frac{\lambda_2}{p^2z} \biggr)^2
\biggr]^{-1}
</math>
  </td>
</tr>
</table>

===Speculation3===

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

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(x+p^2 z)^{1 / 2}}{y^{1/q^2} } \, ,
</math>
  </td>
</tr>
</table>
in which case,
<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>~
\frac{1}{2y^{1/q^2}}\biggl(x + p^2z\biggr)^{- 1 / 2}
=
\frac{\lambda_2}{2(x + p^2z) }
\, ,
</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>~
-\frac{\lambda_2}{q^2y}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~

\, .
</math>
  </td>
</tr>
</table>

===Speculation4===

====Development====

Here we stick with the [[#Primary_.28radial-like.29_Coordinate|primary (radial-like) coordinate as defined above]]; for example,
<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>~h_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_1 \ell_{3D} \, ,</math>
  </td>
</tr>

<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>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat{k} (p^2 z \ell_{3D}) \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
Note that, <math>~\hat{e}_1 \cdot \hat{e}_1 = 1</math>, which means that this is, indeed, a properly normalized ''unit'' vector.
</td></tr></table>

Then, drawing from our [https://www.phys.lsu.edu/astro/H_Book.current/Appendices/Mathematics/operators.tohline1.pdf earliest discussions of "T1 Coordinates"], we'll try defining the ''second'' coordinate as,
<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>~
\tan^{-1} u \, ,
</math>
&nbsp; &nbsp; &nbsp; where,
  </td>
</tr>

<tr>
  <td align="right">
<math>~u</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{y^{1/q^2}}{x} \, .</math>
  </td>
</tr>
</table>
The relevant partial derivatives are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{1 + u^2} \biggl[ - \frac{y^{1/q^2}}{x^2} \biggr]
=
- \biggl[ \frac{u}{1 + u^2}\biggr]\frac{1}{x} 
=
- \frac{\sin\lambda_3 \cos\lambda_3}{x} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_3}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{1 + u^2} \biggl[ \frac{y^{(1/q^2-1)}}{q^2x} \biggr]
=
\biggl[ \frac{u}{1 + u^2}\biggr]\frac{1}{q^2y} 
=
\frac{\sin\lambda_3 \cos\lambda_3}{q^2y} \, ,
</math>
  </td>
</tr>
</table>
which means that,
<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[ \biggl( \frac{\partial \lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_3}{\partial 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[ \frac{u}{1 + u^2}\biggr]^{-2} \biggl[ \frac{1}{x^2} + \frac{1}{q^4y^2} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1 + u^2}{u}\biggr]^{2} \biggl[ \frac{x^2 + q^4y^2}{x^2q^4y^2} \biggr]^{-1}
</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{1 + u^2}{u}\biggr]xq^2 y \ell_q = \frac{xq^2 y \ell_q}{\sin\lambda_3 \cos\lambda_3} \, ,
</math>
&nbsp; &nbsp; &nbsp; where, 
  </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>

The third row of direction cosines can now be filled in to give,

<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">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
<tr>
  <td align="center"><math>~3</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>
which means that the associated unit vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\hat\imath (q^2 y \ell_{q}) + \hat\jmath (x \ell_{q})  \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
Note that, <math>~\hat{e}_3 \cdot \hat{e}_3 = 1</math>, which means that this also is a properly normalized ''unit'' vector.  Note, as well, that the dot product between our "first" and "third" unit vectors should be zero if they are indeed orthogonal to each other.  Let's see &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_3 \cdot \hat{e}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(- q^2y \ell_q)x\ell_{3D} + (x\ell_q) q^2y\ell_{3D} = 0 \, .
</math>
  </td>
</tr>
</table>
Q.E.D.
</td></tr></table>

Now, even though we have not yet determined the proper expression for the "second" orthogonal coordinate, <math>~\lambda_2</math>, we should be able to obtain an expression for its associated unit vector from the cross product of the "third" and "first" unit vectors.  Specifically we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_2 \equiv \hat{e}_3 \times \hat{e}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl[ (e_3)_2 (e_1)_3 - (e_3)_3(e_1)_2 \biggr]
+
\hat\jmath \biggl[ (e_3)_3 (e_1)_1 - (e_3)_1(e_1)_3 \biggr]
+
\hat{k} \biggl[ (e_3)_1 (e_1)_2 - (e_3)_2(e_1)_1 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath \biggl[ (x \ell_q) (p^2 z \ell_{3D}) - 0 \biggr]
+
\hat\jmath \biggl[ 0 - (-q^2y \ell_q)(p^2z \ell_{3D}) \biggr]
+
\hat{k} \biggl[ (-q^2y \ell_q) (q^2 y \ell_{3D}) - (x\ell_q)(x\ell_{3D}) \biggr]
</math>
  </td>
</tr>

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

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

<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
Note that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_3 \cdot \hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ell_q^2 \ell_{3D} \biggl[
(- q^2y )x p^2 z  
+ (x) q^2y p^2 z  
\biggr] = 0 \, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_1 \cdot \hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(x\ell_{3D})xp^2z \ell_q \ell_{3D}
+ (q^2y \ell_{3D}) q^2yp^2 z \ell_q \ell_{3D}
- (x^2 + q^4 y^2)\ell_q \ell_{3D} (p^2 z \ell_{3D} )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \ell_q \ell_{3D}^2 \biggl[
x^2p^2z 
+ (q^4y^2 ) p^2 z 
- (x^2 + q^4 y^2) (p^2 z )
\biggr] = 0 \, .
</math>
  </td>
</tr>
</table>
We conclude, therefore, that <math>~\hat{e}_2</math> is perpendicular to both of the other unit vectors. <font color="red">'''Hooray!'''</font>
</td></tr></table>


Filling in the second row of the direction cosines table gives,

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

<tr>
  <td align="center"><math>~3</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>

====Analysis====

Let's break down each direction cosine into its components.

<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>
</table>

Try,
<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>~
\tan^{-1} w \, ,
</math>
&nbsp; &nbsp; &nbsp; where,
  </td>
</tr>

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

The 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>~
\frac{1}{1 + w^2} \biggl[ \frac{x}{(x^2 + q^2y^2)^{1 / 2}~z^{1/p^2}} \biggr]
=
\frac{w}{1 + w^2} \biggl[ \frac{x}{(x^2 + q^2y^2)} \biggr]
\, ,
</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>~
\frac{1}{1 + w^2} \biggl[ \frac{q^2y}{(x^2 + q^2y^2)^{1 / 2}~z^{1/p^2}} \biggr]
=
\frac{w}{1 + w^2} \biggl[ \frac{q^2y}{(x^2 + q^2y^2)} \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{w}{1 + w^2} \biggl[- \frac{1}{p^2 z} \biggr]
\, ,
</math>
  </td>
</tr>
</table>

which means that,
<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( \frac{\partial \lambda_2}{\partial x} \biggr)^2
+
\biggl( \frac{\partial \lambda_2}{\partial y} \biggr)^2
+
\biggl( \frac{\partial \lambda_2}{\partial z} \biggr)^2
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{w}{1 + w^2}\biggr)^2 
\biggl[ \frac{(x^2 + q^4y^2)(p^4 z^2) + (x^2 + q^2y^2)^2}{(x^2 + q^2y^2)^2~p^4 z^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{1 + w^2}{w}\biggr) \biggl\{ \frac{(x^2 + q^2y^2)~p^2 z}{ \mathcal{D}}  \biggr\} 
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + q^4y^2)(p^4 z^2) + (x^2 + q^2y^2)^2]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Hence, the trio of associated direction cosines are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21} = h_2 \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1 + w^2}{w}\biggr) \biggl\{ \frac{(x^2 + q^2y^2)~p^2 z}{ \mathcal{D}}  \biggr\}\frac{w}{1 + w^2} \biggl[ \frac{x}{(x^2 + q^2y^2)} \biggr]
=
\biggl\{ \frac{x~p^2 z}{ \mathcal{D}}  \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22} = h_2 \biggl( \frac{\partial \lambda_2}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1 + w^2}{w}\biggr) \biggl\{ \frac{(x^2 + q^2y^2)~p^2 z}{ \mathcal{D}}  \biggr\} \frac{w}{1 + w^2} \biggl[ \frac{q^2y}{(x^2 + q^2y^2)} \biggr]
=
\biggl\{ \frac{q^2 y~p^2 z}{ \mathcal{D}}  \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23} = h_2 \biggl( \frac{\partial \lambda_2}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1 + w^2}{w}\biggr) \biggl\{ \frac{(x^2 + q^2y^2)~p^2 z}{ \mathcal{D}}  \biggr\}\frac{w}{1 + w^2} \biggl[- \frac{1}{p^2 z} \biggr]
=
\biggl\{- \frac{(x^2 + q^2y^2)}{ \mathcal{D}}  \biggr\} \, .
</math>
  </td>
</tr>
</table>

<font color="red">'''VERY close!'''</font>

Let's examine the function, <math>~\mathcal{D}^2</math>.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\ell_{3D}^2 \ell_d^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(x^2 + q^4 y^2)(x^2 + q^4 y + p^4 z)
=
(x^2 + q^4 y^2)p^4 z + (x^2 + q^4 y^2)^2 \, .
</math>
  </td>
</tr>
</table>

===Eureka (NOT!)===


Try,
<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>~
\tan^{-1} w \, ,
</math>
&nbsp; &nbsp; &nbsp; where,
  </td>
</tr>

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

The 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>~
\frac{1}{1 + w^2} \biggl[ \frac{x}{(x^2 + q^2y^2)^{1 / 2}~p^2 z} \biggr]
=
\frac{w}{1 + w^2} \biggl[ \frac{x}{(x^2 + q^2y^2)} \biggr]
\, ,
</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>~
\frac{1}{1 + w^2} \biggl[ \frac{q^2y}{(x^2 + q^2y^2)^{1 / 2}~p^2z} \biggr]
=
\frac{w}{1 + w^2} \biggl[ \frac{q^2y}{(x^2 + q^2y^2)} \biggr]
\, ,
</math>
  </td>
</tr>

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

which means that,
<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( \frac{\partial \lambda_2}{\partial x} \biggr)^2
+
\biggl( \frac{\partial \lambda_2}{\partial y} \biggr)^2
+
\biggl( \frac{\partial \lambda_2}{\partial z} \biggr)^2
</math>
  </td>
</tr>

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

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


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