Editing Appendix/Ramblings/PPToriPt1A (section)

===Additional Notation===

<div align="center">
<math>~(ky)_\mathrm{GGN} = \biggl( \frac{my}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ (m\phi)_\mathrm{Blaes}</math>
</div>

<div align="center">
<math>~\beta_\mathrm{GGN} \equiv \biggl( \frac{ma}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ m\beta_\mathrm{Blaes}</math>
</div>

From equation (5.16) of GGN86 we obtain "the lowest order [complex] expression for the [perturbed] velocity potential," namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} \, .</math>
  </td>
</tr>
</table>
</div>

Working on the imaginary part of this expression to put it in the terminology of Blaes85, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}(\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} \biggl(\frac{m}{\varpi_0}\biggr) [\varpi_0 (\eta\beta_\mathrm{Blaes})\cos\theta ](m\beta_\mathrm{Blaes}) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>
  </td>
</tr>
</table>
</div>

which exactly matches <math>~\mathrm{Im}(f_m)</math> as derived by Blaes85 and [[#Incompressible_Slim_Tori|summarized above]].  Similarly,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}(\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\frac{1}{4} \biggl(\frac{m}{\varpi_0}\biggr)^2[\varpi_0^2 r^2(5\cos^2\theta - 3\sin^2\theta)] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\frac{1}{4} \eta^2 m^2 \beta^2_\mathrm{Blaes}[8\cos^2\theta - 3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

This exactly matches <math>~\mathrm{Re}(f_m)</math> as derived by Blaes85 and [[#Incompressible_Slim_Tori|summarized above]].  This is in line with the following statement that appears in the acknowledgement section of GGN86:  "We note that Omar Blaes &hellip; [has] independently derived many of the results reported in this paper."