Editing ThreeDimensionalConfigurations/FerrersPotential

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=Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids=
{| class="Ferrers" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:#ffeeee;" |[[H_BookTiledMenu#Ellipsoidal_.26_Ellipsoidal-Like|<b>Ferrers<br />Potential<br />(1877)</b>]]
|}

In an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids|accompanying chapter]] titled, ''Properties of Homogeneous Ellipsoids (1),'' we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids.  In that discussion, we largely followed the derivations of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  In the latter part of the nineteenth-century, {{ Ferrers1877full }} showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions.  Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>\rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="90%" cellpadding="10">
<tr><td align="center">SUMMARY &#8212; copied from [[ThreeDimensionalConfigurations/Challenges#Trial_.232|accompanying, ''Trial #2'' Discussion]]</td></tr>

<tr><td align="left">
After studying the relevant sections of both [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] &#8212; this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription.  As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.

In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1|accompanying discussion]] we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{6}  \biggl(3A_{11}x^4 +  3A_{22}y^4 + 3A_{33}z^4  \biggr) 
\, ,
</math>
  </td>
</tr>
</table>

where,
<table border="0" align="center" cellpadding="10" width="80%">
<tr>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i \ne j</math></td></tr>
<tr>
  <td align="right">
<math>A_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (107)</font> ]</td></tr>
</table>

  </td>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i = j</math></td></tr>
<tr>
  <td align="right">
<math>2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{a_i^2} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (109)</font> ]</td></tr>
</table>

  </td>
</tr>
</table>
More specifically, in the three cases where the indices, <math>i=j</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3A_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_1^2} - (A_{12} + A_{13}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>3A_{22}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_2^2} - (A_{21} + A_{23}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>3A_{33}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_3^2} - (A_{31} + A_{32}) \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>


==Derivation==

<table border="1" cellpadding="8" width="80%" align="center">
<tr><td align="left">
Other references to Ferrers Potential:
<ul>
  <li>{{ Ferrers1877full }} &hellip; from [[Appendix/References#BT87|BT87]] References (p. 711)</li>
  <li>{{ Lebovitz79full }} </li>
  <li>[https://docs.galpy.org/en/v1.6.0/reference/potentialferrers.html Galpy methods]</li>
  <li>[https://www.mdpi.com/2075-4434/5/4/101 Lucas Antonio Carit&aacute;, Irapuan Rodriguez, &amp; I. Puerari (2017)] <i>Explicit Second Partial Derivatives of the Ferrers Potential</i></li>
  <li>[https://www.researchgate.net/publication/225744594_Anisotropic_and_inhomogeneous_S-type_Riemann_ellipsoids_inside_spheroidal_halos_II Martin G. Abrahamyan (2006, Astrophysics, 49(3), 306-319)], ''Anisotropic and inhomogeneous S-type Riemann ellipsoids inside spheroidal halos. II''</li>
</ul>
</td>
</tr>
</table>

Following &sect;2.3.2 (beginning on p. 60) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], let's examine ''inhomogeneous'' configurations whose isodensity surfaces (including the surface, itself) are defined by triaxial ellipsoids on which the Cartesian coordinates <math>(x_1, x_2, x_3)</math> satisfy the condition that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>m^2</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>a_1^2 \sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;20, p. 50, Eq. (75)</font> ]<br />
[ [[Appendix/References#BT87|BT87]], <font color="#00CC00">&sect;2.3.2, p. 61, Eq. (2-97)</font> ]
</td></tr>
</table>
be constant.  <span id="DensitySpecification">More specifically,</span> let's consider the case (related to the so-called ''Ferrers'' potentials) in which the configuration's density distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho(m^2)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_c \biggl[1 - \frac{m^2}{a_1^2}\biggr]^n </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_c \biggl[1 - 
\sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \biggr]^n 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]^n 
\, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<font color="red">'''NOTE:'''</font> &nbsp; &nbsp; In our [[ThreeDimensionalConfigurations/Challenges#Trial_.232|accompanying discussion]] of compressible analogues of Riemann S-type ellipsoids, we have discovered that &#8212; at least in the context of infinitesimally thin, nonaxisymmetric disks &#8212; this heterogeneous density profile can be nicely paired with an analytically expressible stream function, at least for the case where the integer exponent is, n = 1.
</td></tr></table>

According to Theorem 13 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; see his Chapter 3, &sect;20 (p. 53) &#8212; the potential at any point inside a triaxial ellipsoid with this specific density distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_\mathrm{grav}(\mathbf{x})</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \frac{\pi G \rho_c a_1 a_2 a_3}{(n+1)}  \int_0^\infty \frac{ du}{\Delta } Q^{n+1}  \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;20, p. 53, Eq. (101)</font> ]
</td></tr>
</table>
where, <math>\Delta</math> has the same definition as [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|above]], and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>Q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \, .
</math>
  </td>
</tr>
</table>

For purposes of illustration, in what follows we will assume that, <math>a_1 > a_2 > a_3</math>.

===The Case Where n = 0===

When <math>n = 0</math>, we have a uniform-density configuration, and the "interior" potential will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_\mathrm{grav}(\mathbf{x})</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>- \pi G \rho_c a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{x^2}{ a_1^2 + u } \biggr)
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{y^2}{ a_2^2 + u } \biggr) 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{z^2}{ a_3^2 + u }  \biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
- ~x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u) } 
- ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u) }  
- ~ \int_0^\infty \frac{ du}{\Delta (a_3^2 + u) } 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math> = </math>
  </td>
  <td align="left">
<math>
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>

As a check, let's see if this scalar potential satisfies the differential form of the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
</div>
<span id="SumTo2">Given that,</span>
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\sum_{\ell = 1}^3 A_\ell</math>
  </td>
  <td align="center">
<math> = </math>
  </td>
  <td align="left">
<math>2 \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (108)</font> ]</td></tr>
</table>

we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla^2\Phi_\mathrm{grav} = \biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr]\Phi_\mathrm{grav}</math>
  </td>
  <td align="center">
<math> = </math>
  </td>
  <td align="left">
<math>
+ 2\pi G \rho_c (A_1 + A_2 + A_3) = 4\pi G\rho_c \, .
</math>
  </td>
</tr>
</table>
Q.E.D.

===The Case Where n = 1===

When <math>n = 1</math>, we have a specific heterogeneous density configuration, and the "interior" potential will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} a_1 a_2 a_3  \biggl\{  
\int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
- ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
- ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

The first definite-integral expression inside the curly braces is, to within a leading factor of <math>\tfrac{1}{2}</math>, identical to the entire expression for the normalized potential that was derived in the case where n = 0. That is, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
- \frac{1}{2} a_1 a_2 a_3  \biggl\{ ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
+ ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Then, from &sect;22, p. 56 of [[Appendix/References#EFE|EFE]], we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta (a_i^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( A_i - \sum_{\ell=1}^3 A_{i\ell} x_\ell^2 \biggr) \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, p. 53, Eq. (125)</font> ]
</td></tr>
</table>

<span id="GravFor1">Applying this result to each of the other three definite integrals gives us,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
- \frac{x^2}{2}  \biggl( A_1 - \sum_{\ell=1}^3 A_{1\ell} x_\ell^2 \biggr)
- \frac{y^2}{2}  \biggl( A_2 - \sum_{\ell=1}^3 A_{2\ell} x_\ell^2 \biggr)
- \frac{z^2}{2}  \biggl( A_3 - \sum_{\ell=1}^3 A_{3\ell} x_\ell^2 \biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
- \frac{x^2}{2}  \biggl[ A_1 - \biggl( A_{11}x^2 + A_{12}y^2 + A_{13}z^2 \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- \frac{y^2}{2}  \biggl[ A_2 - \biggl( A_{21}x^2 + A_{22}y^2 + A_{23}z^2 \biggr) \biggr]
- \frac{z^2}{2}  \biggl[ A_3 - \biggl( A_{31}x^2 + A_{32}y^2 + A_{33}z^2 \biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\, ,
</math>
  </td>
</tr>
</table>

where,
<table border="1" align="center" cellpadding="10" width="80%">
<tr>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i \ne j</math></td></tr>
<tr>
  <td align="right">
<math>A_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (107)</font> ]</td></tr>
</table>

  </td>
  <td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>i = j</math></td></tr>
<tr>
  <td align="right">
<math>2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{a_i^2} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (109)</font> ]</td></tr>
</table>

  </td>
</tr>
</table>


and we have made use of the symmetry relation, <math>A_{ij} = A_{ji}</math>.  Again, as a check, let's see if this scalar potential satisfies the differential form of the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
</div>
We find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr] </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2}\biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr] 
\biggl[- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial}{\partial x}   \biggl[- A_1 x  ~+ A_{12} x y^2 + A_{13} x z^2 ~+ A_{11}x^3  \biggr]
+\frac{\partial}{\partial y}  \biggl[- A_2 y   ~+  A_{12} x^2y  + A_{23} y z^2~+ A_{22}y^3  \biggr]
+ \frac{\partial}{\partial z}\biggl[- A_3 z  ~+ A_{13} x^2z + A_{23} y^2z~+ A_{33}z^3  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[- A_1  + A_{12} y^2 + A_{13} z^2 ~+ 3A_{11}x^2  \biggr]
+ \biggl[- A_2 +  A_{12} x^2  + A_{23} z^2~+ 3A_{22}y^2  \biggr]
+ \biggl[- A_3  + A_{13} x^2 + A_{23} y^2~+ 3A_{33}z^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- (A_1 + A_2 + A_3) + x^2(3A_{11} + A_{12} + A_{13}) + y^2( 3A_{22} + A_{12} + A_{23}) + z^2( 3A_{33} + A_{13} + A_{23})\, .
</math>
  </td>
</tr>
</table>

In addition to recognizing, as [[#SumTo2|stated above]], that <math>(A_1 + A_2 + A_3) = 2</math>, and making explicit use of the relation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{a_i^2} \, ,</math>
  </td>
</tr>
</table>
this last expression can be simplified to discover that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr] </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- (2) + \frac{2x^2}{a_1^2} + \frac{2y^2}{a_2^2} + \frac{2z^2}{a_3^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \nabla^2 \Phi_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G \rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
This does indeed demonstrate that the derived gravitational potential is consistent with [[#DensitySpecification|our selected mass distribution in the case where n = 1]], namely,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>

Q.E.D.

=See Also=
<ul>
<li>Our [[Appendix/Ramblings/T6CoordinatesPt2#Speculation6|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>
<li>[[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Challenges Constructing Ellipsoidal-Like Configurations]]</li>
<li>[[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids_.281.29|Properties of Homogeneous Ellipsoids (1)]] &#8212; The Gravitational Potential (A<sub>i</sub> Coefficients)</li>
</ul>


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