Editing Aps/MaclaurinSpheroidFreeFall (section)

===Prolate Spheroids===
For a prolate-spheroidal mass distribution, by definition the "smallest" and "medium-length" semi-axes are equal to one another.  Hence, <math>a_m = a_s</math> and, according to [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#When_am_.3D_as|our associated derivation]],

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>A_\ell</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ (1-e^2)}{e^3} \cdot 
\ln \biggl[ \frac{1+e}{ 1-e }   \biggr]
- \frac{2(1-e^2)}{e^2 } 
\, ,
</math>
  </td>
</tr>
</table>

where, as above, <math>e \equiv (1 - a_s^2/a_\ell^2)^{1 / 2}</math>; and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_s = A_m</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2 } 
- \frac{ (1-e^2)}{2e^3} 
\cdot
\ln \biggl( \frac{1+e}{ 1-e }   \biggr) \, .
</math>
  </td>
</tr>
</table>
If the symmetry (in this case, longest) axis of this prolate-spheroidal mass distribution is aligned with the <math>z</math>-axis of the coordinate system, then we should set <math>A_1 = A_2 = A_s</math>, and <math>A_3 = A_\ell</math>.  This means that the expression for the gravitational potential is,

<div align="center">
<math>
\Phi(\vec{x}) = \pi G \rho \biggl[A_s \varpi^2 + A_\ell z^2 \biggr].
</math>
</div>
These same coefficient expressions may also be found in, for example, the second column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].


<font color="red">NOTE:</font>&nbsp; If, following [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we instead align the longest (and, in this case, symmetry) axis of the prolate mass distribution with the <math>x</math>-axis, then <math>A_2 = A_3 = A_s</math> and <math>A_1 = A_\ell</math>.  This matches the coefficient expressions presented in our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.7F.27.22.60UNIQ--postMath-0000004C-QINU.60.22.27.7F|parallel discussion]] of the potential inside and on the surface of a prolate-spheroidal mass distribution.