Editing Appendix/Ramblings/ToHadleyAndImamura (section)

====For n = 3 Configurations====
Following &sect;1.3 of Blaes85, let's specifically consider the case of slim PP-tori that have a polytropic index, <math>~n=3</math>.  In this case, we have,

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~2^2[2^5\cos^2\theta - 3]\eta^2  - 13 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^{13}\cdot 3\eta^2\cos^2\theta ]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

<!-- OMIT
Hence, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 - 13 
+ 2^2[2^5\cos^2\theta - 3]\eta^2 }{  2^{6}(2\cdot 3)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

END OMIT -->

Now, rather than examining the structural behavior of the amplitude and phase of the function, <math>~[W/W_0 -1]</math>, as we have done, above, Blaes evaluated the amplitude (only) of the function, 
<div align="center">
<math>~\frac{W}{W_0} = 1 + \mathcal{A}_\mathrm{Blaes}(\eta,\theta) + i \mathcal{B}_\mathrm{Blaes}(\eta,\theta) \, .</math>
</div>

For the specific case of <math>~n=3</math>, the square of the amplitude of ''this'' function is,

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

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2a_3}{2^5}  
+ \frac{ (\beta m)^4 }{2^{12}} (a_3^2 + b_3^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^5}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>

Therefore, to lowest order in the "slimness" parameter, <math>~\beta</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^6}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>

This (almost exactly) matches the amplitude expression derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] and evaluated for the specific case of <math>~\beta = (101)^{-1/2}</math> &#8212; the relevant equation (1.12) from Blaes85 is digitally reproduced in the table that follows.  

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation (1.12) extracted without modification from p. 556 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq1.12.png|400px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

Our expression differs from the one published in Blaes85 in one detail:  In Blaes85, the pre-factor of the second term inside the curly braces is, evidently, <math>~(\beta m)^2/2^5</math>, which is a factor of two larger than the corresponding pre-factor found in our expression.  In our attempt to re-derive the Blaes85 expression (1.12), this extra factor of two disappears when we take the square-root of both sides to obtain the modulus, rather than the square of the modulus.  Hopefully further reflection will resolve this discrepancy between our approximate expression for <math>~|W/W_0|_{n=3}</math> and the analogous one presented in Blaes85.