Editing 3Dconfigurations/RiemannEllipsoids (section)

==Gravitational Potential==
For the gravitational potential, <math>V</math>, (see p. 178 of [http://www.kendrickpress.com/Riemann.htm BCO2004]), Riemann adopts the expression,

<table border="0" align="center" cellpadding="2" width="90%">

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  <td align="right" width="10%">
<math>V</math>
  </td>
  <td align="center" width="3%">=</td>
  <td align="left">
<math>
H - A\xi^2 -B\eta^2 - C\zeta^2
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="center" width="3%">=</td>
  <td align="left">
<math>
\pi \int_0^\infty \frac{ds}{\Delta_R} \biggl[ 1 
- \frac{\xi^2}{(a^2 + s)} - \frac{\eta^2}{(b^2+s)} - \frac{\zeta^2}{(c^2 + s)} \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,

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  <td align="right" width="10%">
<math>\Delta_R</math>
  </td>
  <td align="center" width="3%"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[ \biggl( 1 + \frac{s}{a^2} \biggr) \biggl( 1 + \frac{s}{b^2} \biggr)\biggl( 1 + \frac{s}{c^2} \biggr)\biggr]^{1 / 2}\, .
</math>
  </td>
</tr>
</table>
From our [https://www.vistrails.org/index.php/User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential separate discussion of the gravitational potential of homogeneous ellipsoids], which closely follows the notation used in EFE, we find,

<div align="center">
<math>
~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr],
</math><br />

[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, Eq. (40)</font><sup>1,2</sup> ]<br />
[ [[User:Tohline/Appendix/References#BT87|BT87]], <font color="#00CC00">Chapter 2, Table 2-2</font> ]
</div>

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[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 15 August 2020:  This integral definition of A_i also appears as Eq. (5) of &sect;10.2 (p. 234) of T78, but it contains an error &#8212; in the denominator on the right-hand-side, a_1 appears instead of a_i.]]
-->
where,

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<tr>
  <td align="right">
<math>
~A_i
</math>
  </td>
  <td align="center">
<math>
~\equiv
</math>
  </td>
  <td align="left">
<math>
~a_1 a_2 a_3 \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~I_\mathrm{BT}
</math>
  </td>
  <td align="center">
<math>
~\equiv
</math>
  </td>
  <td align="left">
<math>
~\frac{a_2 a_3}{a_1} \int_0^\infty \frac{du}{\Delta} = A_1 + A_2\biggl(\frac{a_2}{a_1}\biggr)^2+ A_3\biggl(\frac{a_3}{a_1}\biggr)^2 ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\Delta
</math>
  </td>
  <td align="center">
<math>
~\equiv
</math>
  </td>
  <td align="left">
<math>
~\biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u)  \biggr]^{1/2} .
</math>
  </td>
</tr>
</table>

<div align="center">
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, Eqs. (18), (15 &amp; 22)</font><sup>1</sup><font color="#00CC00">, &amp; (8)</font>, respectively ]<br />
[ [[User:Tohline/Appendix/References#BT87|BT87]], <font color="#00CC00">Chapter 2, Table 2-2</font> ]
</div>
How do these two equations for the potential relate to one another?  Multiplying Riemann's expression through by <math>(-G\rho)</math>, we have,

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  <td align="right" width="10%">
<math>(-G\rho)V</math>
  </td>
  <td align="center" width="3%">=</td>
  <td align="left">
<math>
- \pi G\rho \int_0^\infty \frac{(abc)ds}{\Delta} \biggl[ 1 
- \frac{\xi^2}{(a^2 + s)} - \frac{\eta^2}{(b^2+s)} - \frac{\zeta^2}{(c^2 + s)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="center" width="3%">=</td>
  <td align="left">
<math>
- \pi G\rho \biggl\{ 
(abc)\int_0^\infty \frac{ds}{\Delta} 
-
(abc)\xi^2\int_0^\infty \frac{ds}{\Delta} \biggl[  
\frac{1}{(a^2 + s)} \biggr]
-
(abc)\eta^2\int_0^\infty \frac{ds}{\Delta} \biggl[  
\frac{1}{(b^2+s)} \biggr]
-
(abc)\zeta^2\int_0^\infty \frac{ds}{\Delta} \biggl[ 
\frac{1}{(c^2 + s)} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="center" width="3%">=</td>
  <td align="left">
<math>
- \pi G\rho \biggl\{ 
I_\mathrm{BT}a^2 
- A_1\xi^2 - A_2\eta^2 - A_3 \zeta^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
We see, therefore, that 

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<tr>
  <td align="right">
<math>(-G\rho)V</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\Phi(\vec{x}) \, ,
</math>
  </td>
</tr>
</table>
and we recognize the following notation associations:

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  <td align="right">
<math>(A, B, C)~~ \leftrightarrow ~~ (A_1, A_2, A_3)</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="left">
<math>
H ~~ \leftrightarrow ~~ a_1^2 I_\mathrm{BT} =a_1^2 A_1^2 + a_2^2 A_2^2 + a_3^2 A_3^2 \, .
</math>
  </td>
</tr>
</table>