Editing Appendix/Ramblings/T9Coordinates (section)

===Blue-Framed Guess 2C===
Let's try,
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<math>~\lambda_{2\mathrm{B}}</math>
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<math>~=</math>
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<math>~(cz)^{2e} (ax^2 + by^2)^{-f} \, .</math>
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  <td align="center" colspan="9">'''Direction Cosine Components for Additional T9 Coordinate Guesses'''</td>
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  <td align="center"><math>~n</math><br />&nbsp; <br />(1)</td>
  <td align="center"><math>~\lambda_n</math><br />&nbsp; <br />(2)</td>
  <td align="center"><math>~h_n</math><br />&nbsp; <br />(3)</td>
  <td align="center"><math>~\partial \lambda_n/\partial x</math><br />&nbsp; <br />(4)</td>
  <td align="center"><math>~\partial \lambda_n/\partial y</math><br />&nbsp; <br />(5)</td>
  <td align="center"><math>~\partial \lambda_n/\partial z</math><br />&nbsp; <br />(6)</td>
  <td align="center"><math>~\gamma_{n1}</math><br />&nbsp; <br />(7)</td>
  <td align="center"><math>~\gamma_{n2}</math><br />&nbsp; <br />(8)</td>
  <td align="center"><math>~\gamma_{n3}</math><br />&nbsp; <br />(9)</td>
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  <td align="center"><math>~2A</math></td>
  <td align="center">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center"><math>~x p^2 z ~\mathfrak{J}_0 </math></td>
  <td align="center"><math>~q^2 y p^2 z~\mathfrak{J}_0 </math></td>
  <td align="center"><math>~-(x^2 + q^4 y^2)~\mathfrak{J}_0 </math></td>
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  <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>
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  <td align="center"><math>~2C</math></td>
  <td align="center" bgcolor="lightblue"><math>~(cz)^{2e} (ax^2 + by^2)^{-f}</math></td>
  <td align="center"><math>~\biggl[\frac{z(ax^2 + by^2)}{2e~\lambda_{2\mathrm{C}}} \biggr]\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~- \frac{2afx~\lambda_{2\mathrm{C}} }{(ax^2 + by^2)}</math></td>
  <td align="center"><math>~- \frac{2bfy~\lambda_{2\mathrm{C}} }{(ax^2 + by^2)}</math></td>
  <td align="center"><math>~\frac{2e~\lambda_{2\mathrm{C}} }{z}</math></td>
  <td align="center"><math>~ -\biggl( \frac{af xz}{e} \biggr)~\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~- \biggl( \frac{bf yz}{e} \biggr)~\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~(ax^2 + by^2)~\mathfrak{J}_{2\mathrm{C}} </math></td>
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<math>~\ell_{3D}</math>
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<math>~\equiv</math>
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<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
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<math>~\mathfrak{J}^{-2}_0 \equiv \frac{(x^2 + q^4 y^2)}{\ell_{3D}^2}</math>
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<math>~=</math>
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<math>~
\biggl[(xp^2 z)^2 + (q^2 y p^2 z )^2 + (x^2 + q^4y^2)^2\biggr] \, ,
</math>
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<math>~\mathfrak{J}_{2\mathrm{B}}</math>
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<math>~\equiv</math>
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<math>~
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, ,
</math>
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<math>~\mathfrak{J}_{2\mathrm{C}}</math>
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<math>~\equiv</math>
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<math>~
\biggl[(x z)^2 \biggl( \frac{fa}{e} \biggr)^2 + (y z )^2\biggl( \frac{fb}{e} \biggr)^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, .
</math>
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Examining just the expression for <math>~\mathfrak{J}_{2\mathrm{C}}</math>, we see that we definitely need:  <math>~a = 1</math> and <math>~b=q^4</math>.  Also, we need,
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<math>~\frac{af}{e}</math>
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<math>~=</math>
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<math>~p^2</math>
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<td align="center">&nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; 
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<math>~\frac{f}{e}</math>
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<math>~=</math>
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<math>~p^2 \, ;</math>
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and, separately we need,

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<math>~\frac{fb}{e}</math>
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<math>~=</math>
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<math>~q^2 p^2</math>
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<td align="center">&nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; 
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<math>~\frac{f}{e}</math>
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<math>~=</math>
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<math>~\frac{p^2}{q^2} \, .</math>
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These cannot simultaneously be satisfied.    So the blue-framed <font color="red">guess 2C does not provide a correct second-coordinate expression</font>.  But we are very close!  We need one additional scalar coefficient degree of freedom.