Editing ThreeDimensionalConfigurations/HomogeneousEllipsoids (section)

=Properties of Homogeneous Ellipsoids (1)=
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! style="height: 125px; width: 125px; background-color:white;" |[[H_BookTiledMenu#Equilibrium_Structures_2|<b>The<br />Gravitational<br />Potential]]<br />(A<sub>i</sub> coefficients)</b>
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==Gravitational Potential==

===The Defining Integral Expressions===

As has been shown in a separate discussion titled, [[PGE/PoissonOrigin#Origin_of_the_Poisson_Equation|"Origin of the Poisson Equation,"]] the acceleration due to the gravitational attraction of a distribution of mass {{Math/VAR_Density01}} <math>(\vec{x})</math> can be derived from the gradient of a scalar potential {{Math/VAR_NewtonianPotential01}} <math>(\vec{x})</math> defined as follows:

<div align="center">
<math>
\Phi(\vec{x}) \equiv - \int \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} d^3 x' .
</math>
</div>

As has been explicitly demonstrated in Chapter 3 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and summarized in Table 2-2 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form.  Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>(x,y,z) = (a_1,a_2,a_3)</math>,

<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 />

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

[[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,

<table align="center" border=0 cellpadding="3">
<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">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eqs. (18), (15 &amp; 22)</font><sup>1</sup><font color="#00CC00">, &amp; (8)</font>, respectively<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], <font color="#00CC00">Chapter 2, Table 2-2</font> 
</div>

This definite-integral definition of <math>A_i</math> may also be found in:
* [<b>[[Appendix/References#Lamb32|<font color="red">Lamb32</font>]]</b>]: as Eq. (6) in &sect;114 (p. 153); and as Eq. (5) in &sect;373 (p. 700).
* [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]: as Eq. (5) in &sect;10.2 (p. 234), but note that there is an error in the denominator of the right-hand-side &#8212; <math>a_1</math> appears instead of <math>a_i</math>.

===Evaluation of Coefficients===

As is [[#Derivation_of_Expressions_for_Ai|detailed below]], the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> can be evaluated in terms of the [http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind incomplete elliptic integral of the first kind],

<div align="center">
<math>
F(\theta,k) \equiv \int_0^\theta \frac{d\theta '}{\sqrt{1 - k^2 \sin^2\theta '}} \, ,
</math>
</div> 

and/or the [http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind incomplete elliptic integral of the second kind],

<div align="center">
<math>
E(\theta,k) \equiv \int_0^\theta {\sqrt{1 - k^2 \sin^2\theta '}}d\theta ' \, ,
</math>
</div> 

where, for our particular problem,

<div align="center">
<math>
\theta \equiv \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr) \, ,
</math><br />

<math>
k \equiv \biggl[\frac{a_1^2 - a_2^2}{a_1^2 - a_3^2} \biggr]^{1/2} = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1 / 2} \, ,
</math><br />

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (32)</font> 

</div>

or the integrals can be evaluated in terms of more elementary functions if either <math>a_2 = a_1</math> ([[#Oblate_Spheroids|oblate spheroids]]) or <math>a_3 = a_2</math> ([[#Prolate_Spheroids|prolate spheroids]]).

<span id="triaxial">&nbsp;</span>
====Triaxial Configurations (a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>)====

If the three principal axes of the configuration are unequal in length and related to one another such that <math>a_1 > a_2 > a_3 </math>, 

<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>
\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}\biggr]  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
I_\mathrm{BT}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] \, .
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eqs. (33), (34) &amp; (35)</font> 
</div>

Notice that there is no need to specify the actual value of <math>a_1</math> in any of these expressions, as they each can be written in terms of the pair of axis ''ratios'', <math>a_2/a_1</math> and <math>a_3/a_1</math>.  As a sanity check, let's see if these three expressions can be related to one another in the manner described by equation (108) in &sect;21 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], namely, 
<div align="center">
<math>\sum_{\ell=1}^3 A_\ell = 2 \, .</math>
</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{a_1^2}{2a_2 a_3} \biggl[A_1 + A_3 + A_2\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} 
+ \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>+ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{k^2(1-k^2)\sin^3\theta} \biggl\{(1-k^2)F(\theta,k) - (1-k^2)E(\theta,k) 
+ k^2(a_2/a_3) \sin\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>- k^2E(\theta,k)  + E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(1-k^2)\sin^2\theta} \biggl[ \frac{a_2}{a_3} - \frac{a_3}{a_2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{a_1^2}{a_2 a_3} \, .</math>
  </td>
</tr>
</table>

Q.E.D.

<span id="oblate">&nbsp;</span>

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

If the longest axis, <math>a_1</math>, and the intermediate axis, <math>a_2</math>, of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius <math>a_1</math> and the object is referred to as an '''oblate spheroid'''.  For homogeneous oblate 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>
\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} ~~;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
A_1  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>A_3</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ,
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">&sect;4.5, Eqs. (48) &amp; (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1 / 2} \, .
</math>
</div>

<span id="prolate">&nbsp;</span>

====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>].

==Example Evaluations==

Here we adopt the notation mapping, <math>~(a_1, a_2, a_3) ~\leftrightarrow~ (a,b,c)</math>.  In general, for a given pair of axis ratios, <math>~(\tfrac{b}{a}, \tfrac{c}{a})</math>, a determination of the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, requires evaluation of elliptic integrals.  For practical applications, we have decided to evaluate these special functions using the fortran functions provided in association with the book, ''Numerical Recipes in Fortran'';  in order to obtain the results presented in our Table 2, below, we modified those default (single-precision) routines to generate results with double-precision accuracy.  Along the way (see results posted in our Table 1), we pulled cruder evaluations of both elliptic integrals, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math>, from the printed special-functions table found in a CRC handbook.

As we developed/debugged the numerical tool that would allow us to determine the values of these three coefficients for arbitrary choices of the pair of axis ratios, it was important that we compare the results of our calculations to those that have appeared in the published literature.  As a primary point of comparison, we chose to use ''The properties of the Jacobi ellipsoids'' as tabulated in &sect;39 (Chapter 6) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  In particular, for twenty-six separate axis-ratio pairs, Chandrasekhar's Table IV lists the values of the square of the angular velocity, <math>~\Omega^2</math>, and the total angular momentum, <math>~L</math>, of an equilibrium Jacobi ellipsoid that is associated with each axis-ratio pair.  We should be able to duplicate &#8212; or, via double-precision arithmetic, improve &#8212; Chandrasekhar's tabulated results using the expressions for "omega2",
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega^2}{\pi G\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B_{12}</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (5)</font> </td></tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[\frac{A_1 - (b/a)^2A_2}{1-(b/a)^2} \biggr] \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">using, [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;21, Eqs. (105) &amp; (107)</font></td></tr>
</table>
</div>
and, for "angmom",
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{L}{(GM^3)^{1/2}(abc)^{1/6}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{3}}{10}\biggl[ \frac{a^2 + b^2}{(abc)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2} </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (16)</font></td></tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{3}}{10}\biggl[ \frac{1 + (b/a)^2}{(b/a)^{2/3}(c/a)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Or, in connection with the free-energy discussion found in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, &amp; J. E. Tohline (1995, ApJ, 446, 472)],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{5L}{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 \biggr]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}\biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 \biggr]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}</math>
  </td>
</tr>
</table>
</div>

<span id="Table1">&nbsp;</span>
<table align="center" cellpadding="5" border="1">
<tr>
  <th align="center" colspan="12"><font size="+1">Table 1:&nbsp; Example Evaluations</font></th>
</tr>
<tr>
  <th align="center" colspan="2">Given</th>
  <th align="center" colspan="10">Determined using calculator and (crude) CRC tables of elliptic integrals</th>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>~\frac{a_3}{a_1}</math></td>
  <td align="center" colspan="2"><math>~\theta</math></td>
  <td align="center" rowspan="2"><math>~k</math></td>
  <td align="center" colspan="2"><math>~\sin^{-1}k</math></td>
  <td align="center" rowspan="2"><math>~F(\theta,k)</math></td>
  <td align="center" rowspan="2"><math>~E(\theta,k)</math></td>
  <td align="center" rowspan="2"><math>~A_1</math></td>
  <td align="center" rowspan="2"><math>~A_2</math></td>
  <td align="center" rowspan="2"><math>~A_3</math></td>
</tr>
<tr>
  <td align="center">radians</td>
  <td align="center">degrees</td>
  <td align="center">radians</td>
  <td align="center">degrees</td>
</tr>
<tr>
  <td align="right">1.00</td>
  <td align="right">0.582724</td>
  <td align="right">0.94871973</td>
  <td align="right">54.3576</td>
  <td align="right">0.00000000</td>
  <td align="right">0.00000000</td>
  <td align="right">0.000000</td>
  <td align="right">0.94871973</td>
  <td align="right">0.94871973</td>
  <td align="right">0.51589042</td>
  <td align="right">0.51589042</td>
  <td align="right">0.96821916</td>
</tr>
<tr>
  <td align="right">0.96</td>
  <td align="right">0.570801</td>
  <td align="right">0.96331527</td>
  <td align="right">55.1939</td>
  <td align="right">0.34101077</td>
  <td align="right">0.34799191</td>
  <td align="right">19.9385</td>
  <td align="right">0.975</td>
  <td align="right">0.946</td>
  <td align="right">+0.4937</td>
  <td align="right">+0.5319</td>
  <td align="right">+0.9744</td>
</tr>
<tr>
  <td align="right">0.60</td>
  <td align="right">0.433781</td>
  <td align="right">1.12211141</td>
  <td align="right">64.292</td>
  <td align="right">0.88788426</td>
  <td align="right">1.09272580</td>
  <td align="right">62.609</td>
  <td align="right">1.3375</td>
  <td align="right">0.9547</td>
  <td align="right">0.3455</td>
  <td align="right">0.6741</td>
  <td align="right">0.9803</td>
</tr>
</table>


<b>With regard to our Table 1 (immediately above):</b> To begin with, we picked three axis-ratio pairs from Table IV of EFE, and considered them to be "given."  For each pair, we used a hand-held calculator to calculate the corresponding values of the two arguments of the elliptic integrals, namely, <math>~\theta</math> and <math>~k</math>, as [[#Evaluation_of_Coefficients|defined above]].  By default, each determined value of <math>~\theta</math> is in radians.  Because the published CRC special-functions tables quantify both arguments of the special functions in angular ''degrees'', we converted <math>~\theta</math> from radians to degrees (see column 4 of Table 1) and, similarly, we converted <math>~\sin^{-1}k</math> to degrees (see column 7 of Table 1).  For the axisymmetric configuration &#8212; the first row of numbers in Table 1, for which <math>~a_2/a_1 = 1</math> &#8212; the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, were determined to eight digits of precision using the [[#Oblate_Spheroids|appropriate expressions for oblate spheroids]].  Note that, in this axisymmetric case, <math>~F(\theta,0) = E(\theta,0) = \theta</math>, but these function values are irrelevant with respect to the determination of the <math>~A_\ell</math> coefficients.

<div align="center" id="Table2">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="1">
<font size="+1">Table 2:&nbsp; Double-Precision Evaluations</font><p></p>
Related to Table IV in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>
</th>
</tr>
<tr><td align="left">
<pre>
                                                                                                                                 precision
         b/a      c/a              F                   E                  A1                  A2                  A3          [2-(A1+A2+A3)]/2

        1.00   0.582724          -----               -----          5.158904180D-01     5.158904180D-01     9.682191640D-01        0.0D+00
        0.96   0.570801     9.782631357D-01     9.487496699D-01     5.024584655D-01     5.292952683D-01     9.682462661D-01        4.4D-16
        0.92   0.558330     1.009516282D+00     9.489290273D-01     4.884500698D-01     5.432292722D-01     9.683206580D-01        0.0D+00
        0.88   0.545263     1.042655826D+00     9.492826127D-01     4.738278227D-01     5.577100115D-01     9.684621658D-01        2.2D-16
        0.84   0.531574     1.077849658D+00     9.498068890D-01     4.585648648D-01     5.727687434D-01     9.686663918D-01        2.2D-16

        0.80   0.517216     1.115314984D+00     9.505192815D-01     4.426242197D-01     5.884274351D-01     9.689483451D-01       -4.4D-16
        0.76   0.502147     1.155290552D+00     9.514282210D-01     4.259717080D-01     6.047127268D-01     9.693155652D-01        2.2D-16
        0.72   0.486322     1.198053140D+00     9.525420558D-01     4.085724682D-01     6.216515450D-01     9.697759868D-01       -4.4D-16
        0.68   0.469689     1.243931393D+00     9.538724717D-01     3.903895871D-01     6.392680107D-01     9.703424022D-01        2.2D-16
        0.64   0.452194     1.293310292D+00     9.554288569D-01     3.713872890D-01     6.575860416D-01     9.710266694D-01        4.4D-16

        0.60   0.433781     1.346645618D+00     9.572180643D-01     3.515319835D-01     6.766289416D-01     9.718390749D-01       -3.3D-16
        0.56   0.414386     1.404492405D+00     9.592491501D-01     3.307908374D-01     6.964136019D-01     9.727955606D-01       -6.7D-16
        0.52   0.393944     1.467522473D+00     9.615263122D-01     3.091371405D-01     7.169543256D-01     9.739085339D-01        4.4D-16
        0.48   0.372384     1.536570313D+00     9.640523748D-01     2.865506903D-01     7.382563770D-01     9.751929327D-01       -2.2D-16
        0.44   0.349632     1.612684395D+00     9.668252052D-01     2.630231082D-01     7.603153245D-01     9.766615673D-01        8.9D-16

        0.40   0.325609     1.697213059D+00     9.698379297D-01     2.385623719D-01     7.831101146D-01     9.783275135D-01        0.0D+00
        0.36   0.300232     1.791930117D+00     9.730763540D-01     2.132011181D-01     8.065964525D-01     9.802024294D-01        2.2D-15
        0.32   0.273419     1.899227853D+00     9.765135895D-01     1.870102340D-01     8.307027033D-01     9.822870627D-01       -1.3D-15
        0.28   0.245083     2.022466812D+00     9.801112910D-01     1.601127311D-01     8.553054155D-01     9.845818534D-01       -2.4D-15
        0.24   0.215143     2.166555572D+00     9.838093161D-01     1.327137129D-01     8.802197538D-01     9.870665333D-01        1.4D-14

        0.20   0.183524     2.339102805D+00     9.875217566D-01     1.051389104D-01     9.051602520D-01     9.897008376D-01       -1.6D-14
        0.16   0.150166     2.552849055D+00     9.911267582D-01     7.790060179D-02     9.296886827D-01     9.924107155D-01       -3.4D-14
        0.12   0.115038     2.831664019D+00     9.944537935D-01     5.180880535D-02     9.531203882D-01     9.950708065D-01        1.4D-13
        0.08   0.078166     3.229072310D+00     9.972669475D-01     2.817821170D-02     9.743504218D-01     9.974713665D-01        3.9D-13
        0.04   0.039688     3.915557866D+00     9.992484565D-01     9.281550546D-03     9.914470033D-01     9.992714461D-01        9.8D-13
</pre>
</td></tr>
</table>
</div>

<b>With regard to our Table 2 (immediately above):</b> Next, given each pair of axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> &#8212; copied from Table IV of EFE (see columns 1 and 2 of our Table 2)  &#8212; we used some fortran routines from [http://numerical.recipes/ Numerical Recipes] to calculate <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> (see columns 3 and 4 of our Table 2); we converted the routines to accommodate double-precision arithmetic.  We subsequently evaluated the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, (columns 5, 6, &amp; 7 of Table 2) using the expressions given above, then demonstrated that, in each case, the three coefficients sum to 2.0 to better than twelve digits accuracy.