Editing Appendix/Ramblings/T9Coordinates (section)

===Yellow-Framed Guess 2B===

Referencing the 2B table row, above, we are looking for coefficient values that map,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~axz \rightarrow xp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~byz \rightarrow q^2 yp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~(ax^2 + by^2) \rightarrow (x^2 + q^4y^2) \, ,</math>
  </td>
</tr>
</table>

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

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

into the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{J}_0 \equiv \frac{\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
(x^2 + q^4y^2 + p^4 z^2)(x^2 + q^4 y^2)
\biggr]^{-1 / 2}
</math>
  </td>
</tr>

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

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

A portion of these mappings are accomplished by setting <math>~a = p^2</math> and <math>~b = q^2p^2</math>, but this pair of specified coefficient values does not satisfy other mappings.  Alternatively, a separate subset of mappings &#8212; but, again, not all mappings &#8212; is satisfied by setting <math>~a = 1</math> and <math>~b = q^4</math>.  So the yellow-framed <font color="red">guess 2B does not provide a correct second-coordinate expression</font>.