Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

==Think Again==

===Firm Relations===

In addition to the functions that are specified in our above [[#Table1DaringAttack|Daring Attack Table]], we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial x} \biggr)
=
\biggl(\lambda_1 \ell_{3D} \biggr)^2 \frac{x}{\lambda_1}
=
x \lambda_1 \ell_{3D}^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial y} \biggr)
=
\biggl(\lambda_1 \ell_{3D} \biggr)^2 \frac{q^2y}{\lambda_1}
=
q^2 y \lambda_1 \ell_{3D}^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_1^2 \biggl( \frac{\partial \lambda_1}{\partial z} \biggr)
=
\biggl(\lambda_1 \ell_{3D} \biggr)^2 \frac{p^2z}{\lambda_1}
=
p^2 z \lambda_1 \ell_{3D}^2 \, .
</math>
  </td>
</tr>
</table>

Check &hellip;
<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>~
\biggl( \frac{\partial x}{\partial \lambda_1} \biggr)^2
+
\biggl( \frac{\partial y}{\partial \lambda_1} \biggr)^2
+
\biggl( \frac{\partial z}{\partial \lambda_1} \biggr)^2
=
\lambda_1^2 \ell_{3D}^4 \biggl[
x^2 + q^4 y^2 + p^4z^2
\biggr]
=
\lambda_1^2 \ell_{3D}^2 \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">(Yes!)</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_3^2 \biggl( \frac{\partial \lambda_3}{\partial x} \biggr)
=
\biggl[ \frac{xq^2y \ell_q}{\lambda_3} \biggr]^2 \biggl( - \frac{\lambda_3}{x} \biggr)
=
- q^4 y^2 \ell_q^2 \biggl( \frac{x}{\lambda_3} \biggr) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_3^2 \biggl( \frac{\partial \lambda_3}{\partial y} \biggr)
=
\biggl[ \frac{xq^2y \ell_q}{\lambda_3} \biggr]^2 \biggl( + \frac{\lambda_3}{q^2y} \biggr)
=
x^2 \ell_q^2 \biggl( \frac{q^2y} {\lambda_3}\biggr) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_3^2 \biggl( \frac{\partial \lambda_3}{\partial z} \biggr)
=
0 \, .
</math>
  </td>
</tr>
</table>

Check &hellip;
<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{\partial x}{\partial \lambda_3} \biggr)^2
+
\biggl( \frac{\partial y}{\partial \lambda_3} \biggr)^2
+
\biggl( \frac{\partial z}{\partial \lambda_3} \biggr)^2
=
\frac{\ell_q^4}{\lambda_3^2} \biggl[x^2 q^8 y^4 + x^4 q^4y^2  \biggr]
=
\frac{x^2 q^4 y^2\ell_q^4}{\lambda_3^2} \biggl[q^4 y^2 + x^2  \biggr]
=
\frac{x^2 q^4 y^2\ell_q^2}{\lambda_3^2} \,  .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">(Yes!)</font>
  </td>
</tr>
</table>

And, last &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{21}
=
h_2 \ell_q \ell_{3D} (xp^2z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{22}
=
h_2 \ell_q \ell_{3D} (q^2 y p^2 z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{23}
=
- h_2 \ell_q \ell_{3D}(x^2 + q^4y^2) \, .
</math>
  </td>
</tr>
</table>

===Speculation===
====First====
From the direction-cosine expressions for <math>~\partial\lambda_2/\partial x_i</math> that have been summarized in our above [[#Table1DaringAttack|Daring Attack Table]], it seems reasonable to suggest 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>~(\ell_q \ell_{3D})^2 
=
\biggl[ (x^2 + q^4y^2)(x^2 + q^4y^2 + p^4z^2) \biggr]^{-1}
\, ,
</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>~xp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2yp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(x^2 + q^4y^2) \, ;</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_2} = h_2^2 \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\ell_q \ell_{3D})^2 xp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_2} = h_2^2 \biggl( \frac{\partial \lambda_2}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\ell_q \ell_{3D})^2 q^2yp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_2} = h_2^2 \biggl(\frac{\partial \lambda_2}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(\ell_q \ell_{3D})^2 (x^2 + q^4y^2) \, .</math>
  </td>
</tr>
</table>

====Second====
Alternatively, after examining the direction-cosine expressions for <math>~\partial x_i/\partial \lambda_2</math> that we have just provided, one might suggest 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>~(\ell_q \ell_{3D})^{-2} 
=
(x^2 + q^4y^2)(x^2 + q^4y^2 + p^4z^2) 
=
p^4z^2(x^2 + q^4y^2) + (x^2 + q^4y^2)^2
\, ,
</math>
  </td>
</tr>
</table>
in which case, the expressions provided for <math>~\partial \lambda_2/\partial x_i</math> and  <math>~\partial x_i/\partial \lambda_2</math> must be swapped relative to our ''First'' speculation.

====Third====
Noticing that <math>~h_1^2</math> is proportional to <math>~\lambda_1^2</math> and that <math>~h_3^2</math> is inversely proportional to <math>~\lambda_3^2</math>, let's consider both as possible behaviors for the 2<sup>nd</sup> scale factor.  Let's try the first of these behaviors.  Specifically, what if we assume &hellip;

<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{xp^2 z}{\lambda_2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2y p^2z}{\lambda_2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{(x^2 + q^4y^2)}{\lambda_2} \, .</math>
  </td>
</tr>
</table>
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( \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
=
[\lambda_2 \ell_q \ell_{3D} ]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda_2 \ell_q \ell_{3D} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Primary implication:
<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>~(xp^2 z) \ell_q \ell_{3D} \ ,</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>~(q^2 y p^2 z) \ell_q \ell_{3D} \ ,</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>~-(x^2 + q^4 y^2) \ell_q \ell_{3D} \ .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
These perfectly match the direction-cosine expressions (<math>~\gamma_{2i}</math> for i = 1, 3)<br />that have been summarized in our above [[#Table1DaringAttack|Daring Attack Table]].
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{21}
=
\lambda_2 \ell_q^2 \ell_{3D}^2 (xp^2z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{22}
=
\lambda_2 \ell_q^2 \ell_{3D}^2 (q^2 y p^2 z) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial z}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \gamma_{23}
=
- \lambda_2 \ell_q^2 \ell_{3D}^2(x^2 + q^4y^2) \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>


Now, what specifically is the function, <math>~\lambda_2(x, y, z)</math> ?  Start by rewriting the three partial derivatives as,

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

<tr>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2^2)}{\partial x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~xp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2)^2}{\partial y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2y p^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{1}{2} \frac{\partial (\lambda_2)^2}{\partial z} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(x^2 + q^4y^2) \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\lambda_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2y^2)p^2z \, .</math>
  </td>
</tr>
</table>

Then we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2xp^2z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2q^2 yp^2z \, .</math>&nbsp; &nbsp; &nbsp; <font color="red">Great!</font>
  </td>
</tr>
</table>

But this cannot be the correct expression for <math>~\lambda_2^2</math> because,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2^2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2y^2)p^2 \, ,</math>
  </td>
</tr>
</table>
which does not match the desired partial derivative with respect to <math>~z</math>.

====Fourth====

Alternatively, if we assume &hellip;

<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}{xp^2 z} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <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 p^2z} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <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}{(x^2 + q^4y^2)} \, ,</math>
  </td>
</tr>
</table>
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( \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{\lambda_2}{xp^2 z} \biggr)^2
+ \biggl( \frac{\lambda_2}{q^2y p^2z} \biggr)^2
+\biggl( \frac{\lambda_2}{x^2 + q^4y^2} \biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (h_2 \lambda_2)^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ q^4y^2p^4z^2 (x^2 + q^4y^2)^2 + x^2p^4z^2  (x^2 + q^4y^2)^2 + x^2 q^4y^2 p^8z^4}{x^2 q^4y^2p^8z^4(x^2 + q^4y^2)^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ h_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\lambda_2} \biggl[
\frac{x^2 q^4y^2p^8z^4(x^2 + q^4y^2)^2}{ q^4y^2p^4z^2 (x^2 + q^4y^2)^2 + x^2p^4z^2  (x^2 + q^4y^2)^2 + x^2 q^4y^2 p^8z^4}
\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_2} \biggl\{
\frac{x q^2y p^2z(x^2 + q^4y^2)}{ [ q^4y^2 (x^2 + q^4y^2)^2 + x^2  (x^2 + q^4y^2)^2 + x^2 q^4y^2 p^4z^2 ]^{1 / 2}}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\lambda_2} \biggl\{
\frac{x q^2y p^2z(x^2 + q^4y^2)}{ [ (x^2 + q^4y^2)^3 + x^2 q^4y^2 p^4z^2 ]^{1 / 2}}
\biggr\}
</math>
  </td>
</tr>
</table>
Let's check for consistency with one of the direction-cosines.
<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{q^2y (x^2 + q^4y^2)}{ [ (x^2 + q^4y^2)^3 + x^2 q^4y^2 p^4z^2 ]^{1 / 2}}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{ \gamma_{21} }{\ell_q(xp^2z) }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(x^2 + q^4y^2)^{1 / 2}}{xp^2z} \biggl\{
\frac{q^2y (x^2 + q^4y^2)}{ [ (x^2 + q^4y^2)^3 + x^2 q^4y^2 p^4z^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^2y}{xp^2z} \biggl\{
\frac{(x^2 + q^4y^2)^{3 / 2}}{ [ (x^2 + q^4y^2)^3 + x^2 q^4y^2 p^4z^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^2y}{xp^2z} 
\biggl[1 + \frac{x^2q^4y^2p^4z^2}{(x^2 + q^4y^2)^3}  \biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>
This does not match the term in the expression for <math>~\gamma_{21}</math> &#8212; namely, <math>~\ell_{3D}</math> &#8212; that is expected from the original tabulation.

====Better Organized====

From our above [[#Table1DaringAttack|Daring Attack Table]], we appreciate that the three direction cosines associated with the (as yet unknown) second curvilinear coordinate are,
<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>~\ell_q \ell_{3D} (xp^2z) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ell_q \ell_{3D} (q^2 y p^2z) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\ell_q \ell_{3D} (x^2 + q^4 y^2) \, .</math>
  </td>
</tr>
</table>

It is easy to see that the desired ''orthogonality'' relationship,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sum_{i=1}^3 (\gamma_{2i})^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ,</math>
  </td>
</tr>
</table>
is satisfied because,
<table border="0" cellpadding="5" align="center">

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

Now, as we attempt to determine the functional form of the second curvilinear coordinate, <math>~\lambda_2(x, y, z)</math>, a seemingly useful intermediate step is to determine the functional form of each of the three partial derivatives of this key coordinate function, namely, <math>~\partial \lambda_2/\partial x_i</math>, for i = 1, 3.  Here, we will accomplish this intermediate step by ''guessing'' the functional form of the second scale factor, <math>~h_2(x, y, z)</math>, then applying the relation,

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial x_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_{2i}}{h_2} \, .</math>
  </td>
</tr>
</table>
Notice that, without violating the above-state ''orthogonality'' relationship, we can adopt virtually any functional form for <math>~h_2(x, y, z)</math> and deduce that,
<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>~A(x, y, z) (xp^2z) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A(x, y, z) (q^2 y p^2z) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-A(x, y, z) (x^2 + q^4 y^2) \, ,</math>
  </td>
</tr>
</table>
as long as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(x, y, z)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{ \ell_q \ell_{3D} }{h_2} \, .
</math>
  </td>
</tr>
</table>

This key, leading coefficient function is unity &#8212; and, hence, is independent of position &#8212; if, as in our [[#First|''First'' speculation]] above, we ''guess'' that <math>~h_2^2 = (\ell_q \ell_{3D})^2</math>.  If, as in our [[#Second|''Second'' speculation]] above, we ''guess'' that <math>~h_2^2 = (\ell_q \ell_{3D})^{-2}</math>, we find that, <math>~A = (\ell_q \ell_{3D})^2</math>.  Our above [[#Third|''Third'' speculation]] is replicated if we ''guess'' that <math>~h_2^2 = (\lambda_2 \ell_q \ell_{3D})^2</math>; we immediately see that, in this ''Third'' 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{xp^2 z}{\lambda_2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2y p^2z}{\lambda_2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{(x^2 + q^4y^2)}{\lambda_2} \, .</math>
  </td>
</tr>
</table>