Editing Appendix/Ramblings/T9Coordinates (section)

===Unspecified Coefficients===

<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>~(ax^2 + by^2 + ez^2)^{-1 / 2} (ax^2 + by^2)^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(ax^2 + by^2)}{(ax^2 + by^2 + ez^2)}\biggr]^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-1 / 2} \, .</math>
  </td>
</tr>
</table>

<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}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial x}\biggl[ez^2 (ax^2 + by^2)^{-1 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez^2 }{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \biggl( -1\biggr) \biggl[(ax^2 + by^2)^{-2} \biggr]2ax
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a x e z^2 }{(ax^2 + by^2)^{2}}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \, ,
</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}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial y}\biggl[ez^2 (ax^2 + by^2)^{-1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez^2 }{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \biggl( -1\biggr) \biggl[(ax^2 + by^2)^{-2} \biggr]2by
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{by e z^2 }{(ax^2 + by^2)^{2}}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \, ,
</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}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial z}\biggl[ez^2 (ax^2 + by^2)^{-1 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)}  \, .
</math>
  </td>
</tr>
</table>

Hence,
<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{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}\biggr]^2
+
\biggl[ \frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \biggr]^2
+
\biggl[\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{e z ~\lambda_2^3}{(ax^2 + by^2)^{2}}\biggr]^2
\biggl[
(a x z)^2
+
(b y z )^2
+
(ax^2 + by^2)^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{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
\, .
</math>
  </td>
</tr>
</table>

The three direction-cosines are, then,
<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>~
\frac{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
axz
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</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>~
\frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
byz
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</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>~
-\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-(ax^2 + by^2)
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="9" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T9 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">(1)</td>
  <td align="center">(2)</td>
  <td align="center">(3)</td>
  <td align="center">(4)</td>
  <td align="center">(5)</td>
  <td align="center">(6)</td>
  <td align="center">(7)</td>
  <td align="center">(8)</td>
  <td align="center">(9)</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>~2A</math></td>
  <td align="center">---</td>
  <td align="center" bgcolor="pink"><math>~\ell_{3D}(x^2 + q^4 y^2)^{1 / 2}</math></td>
  <td align="center" bgcolor="white"><math>~\frac{xp^2z}{(x^2 + q^4y^2)} </math></td>
  <td align="center" bgcolor="white"><math>~\frac{q^2 y p^2z}{(x^2 + q^4y^2)}</math></td>
  <td align="center" bgcolor="white"><math>~-1</math></td>
  <td align="center"><math>~\frac{x p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~\frac{q^2 y p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~-\frac{(x^2 + q^4 y^2)\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
</tr>

<tr>
  <td align="center"><math>~2B</math></td>
  <td align="center" bgcolor="yellow"><math>~\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-1 / 2}</math></td>
  <td align="center"><math>~\biggl[\frac{(ax^2 + by^2)^2}{ez~\lambda_2^3} \biggr]\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~\frac{ax~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~\frac{by~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~- \frac{ez ~\lambda_2^3}{(ax^2 + by^2)} </math></td>
  <td align="center"><math>~axz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~byz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~-(ax^2 + by^2)~\mathfrak{J}_{2\mathrm{B}} </math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\biggl[ \frac{q^2y}{x^{q^2}} \biggr]^{1/(q^2-1)}</math></td>
  <td align="center"><math>~\frac{(q^2-1)x y }{\lambda_3(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~- \biggl( \frac{q^2}{q^2 - 1} \biggr) \frac{\lambda_3}{x}</math></td>
  <td align="center"><math>~\biggl( \frac{q^2}{q^2 - 1} \biggr) \frac{\lambda_3}{q^2y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~- \frac{q^2 y }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~\frac{x }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" 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^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}^{-2}_0 \equiv \frac{(x^2 + q^4 y^2)}{\ell_{3D}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[(xp^2 z)^2 + (q^2 y p^2 z )^2 + (x^2 + q^4y^2)^2\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}_{2\mathrm{B}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>


This table titled, "<i>Direction Cosine Components for T9 Coordinates</i>," contains the following information:
<ol>
<li>
The (first) row labeled, <math>~n = 1</math>, correctly details the scale-factor, <math>~h_1</math>, and the unit vector expression, <math>~\hat{e}_1 = (\hat\imath \gamma_{11} + \hat\jmath \gamma_{12} + \hat{k} \gamma_{13})</math>, that result from the given specification of the <math>~\lambda_1</math> coordinate.  By design, the unit vector, <math>~\hat{e}_1</math>, is everywhere normal to the "surface" of the ellipsoid.
</li>
<li>
The (fourth) row labeled, <math>~n = 3</math>, correctly details the scale-factor, <math>~h_3</math>, and the unit vector expression, <math>~\hat{e}_3 = (\hat\imath \gamma_{31} + \hat\jmath \gamma_{32} + \hat{k} \gamma_{33})</math>, that result from the given specification of the <math>~\lambda_2</math> coordinate.  By design, this unit vector, <math>~\hat{e}_3</math>, has no vertical component &#8212; that is, <math>~\gamma_{33} = 0</math> &#8212; and, by design, it is everywhere perpendicular to the "surface-normal" unit vector, <math>~\hat{e}_1</math>. 
</li>
<li>
We desire a unit vector, <math>~\hat{e}_2</math>, that is mutually orthogonal to the other two unit vectors; this has been accomplished by examining their cross-product, namely, we have set <math>~\hat{e}_{2\mathrm{A}} = \hat{e}_3 \times \hat{e}_1</math>.  Determined in this manner, the expressions for the three direction-cosine components of <math>~\hat{e}_{2\mathrm{A}}</math> have been written in the last three columns of the (second) row labeled, <math>~n=2\mathrm{A}</math>.  While we are confident that the correct specification of <math>~\hat{e}_2</math> is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_{2\mathrm{A}} = (\hat\imath \gamma_{21} + \hat\jmath \gamma_{22} + \hat{k} \gamma_{23})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (xp^2 z)\mathfrak{J}_0 + \hat\jmath (q^2y p^2 z)\mathfrak{J}_0 - \hat{k} (x^2 + q^4y^2) \mathfrak{J}_0 \, ,
</math>
  </td>
</tr>
</table>
as yet (18 February 2021), we have been unable to determine an expression for the coordinate, <math>~\lambda_{2\mathrm{A}}(x, y, z)</math>, from which all three of these direction-cosine expressions can be simultaneously derived &#8212; hence, the dashes in the second column of row 2A.  The expression for <math>~h_{2\mathrm{A}}</math> that has been presented in the third column of row 2A (and framed in pink) is a ''guess'' which, when divided into each of the three direction cosines, gives respectively the three ''guessed'' partial-derivative expressions shown in columns 4, 5, and 6 of row 2A.
</li>
<li>
The second column of row 2B contains a ''guess'' (framed in yellow) for the coordinate expression, <math>~\lambda_{2\mathrm{B}}</math>; this expression contains three unspecified scalar coefficients, <math>~a</math>, <math>~b</math> and <math>~e</math>.  The remaining columns of this row contain the three partial derivatives, the associated scale factor, and the three direction cosines that result from this ''guessed'' coordinate expression.  If we can find values of the three scalar coefficients that give (row 2B) expressions for the three direction cosines that perfectly match the direction cosines written in row 2A, then we will be able to state that <math>~\lambda_{2\mathrm{B}}</math> is &#8212; at least one form of &#8212; our sought-after third coordinate expression.
</li>
</ol>