Editing Apps/Ostriker64 (section)

===Basics===
In &sect;IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\cos\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>
As Ostriker states, <font color="darkgreen">"The coordinate <math>~r</math> is the distance from a reference circle of radius <math>~R</math> (later chosen to be the major radius of the ring) &hellip;"</font>  The angle, <math>~\theta</math>, plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, <math>~\phi</math> is a meridional-plane polar angle measured counterclockwise from the equatorial plane.  For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are, 

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\varpi \equiv (x^2+y^2)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R+r\cos\phi \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 1 extracted without modification from p. 1077 of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. P. Ostriker (1964; Paper II)]<p></p>
"''The Equilibrium of Self-Gravitating Rings''"<p></p>
ApJ, vol. 140, pp. 1067-1087 &copy; American Astronomical Society
</td>
</tr>

<tr>
<td align="center">
[[File:Ostriker64PaperIIFig1.png|600px|Figure 1 from Ostriker (1964) Paper II]]
</td>
</tr>
</table>
</div>
For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta s^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta r^2 + r^2 \delta\phi^2 + (R+r\cos\phi)^2\delta\theta^2 \, ,
</math>
  </td>
</tr>
</table>
which means that the relevant scale factors for the adopted coordinate system, <math>~(r,\phi,\theta)</math>, are
<div align="center">
<math>~h_1 = 1 \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_2 = r \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_3 = (R+r\cos\phi) \, ,</math>
</div>
and the relevant differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_1 h_2 h_3 dr d\phi d\theta
= r(R+r\cos\phi) dr d\phi d\theta\, .
</math>
  </td>
</tr>
</table>