Editing ThreeDimensionalConfigurations/HomogeneousEllipsoids (section)

====Prolate Spheroids (a<sub>1</sub> > a<sub>2</sub> = a<sub>3</sub>)====

If the shortest axis <math>(a_3)</math> and the intermediate axis <math>(a_2)</math> of the ellipsoid are equal to one another &#8212; and the symmetry (longest, <math>a_1</math>) axis is aligned with the <math>x</math>-axis &#8212; then a cross-section in the <math>y-z</math> plane of the object presents a circle of radius <math>a_3</math> and the object is referred to as a '''prolate spheroid'''.  For homogeneous prolate spheroids, evaluation of the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> gives, 

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{1}{e^2} - \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{2e^3}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
A_2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
I_\mathrm{BT}
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>~
A_1 + 2(1-e^2)A_2 = \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{e} \, ,
</math>
  </td>
</tr>
</table>
<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (38)</font>
</div>

where, again, the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1/2} \, .
</math>
</div>


<font color="red">NOTE:</font>&nbsp; If, instead, the longest (and, in this case, symmetry) axis of the prolate mass distribution is aligned with the <math>z</math>-axis &#8212; in which case, <math>a_1 = a_2 < a_3</math> &#8212; then, <math>e = (1 - a_1^2/a_3^2)^{1 / 2}</math> and the mathematical expressions for the <math>A_i</math> coefficients must be altered; they are essentially "swapped."  This modified set of coefficient expressions can be found in a [[Aps/MaclaurinSpheroidFreeFall#Prolate_Spheroids|parallel discussion]] of the potential inside and on the surface of prolate-spheroidal mass distributions, as well as in the second column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].