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Université de Bordeaux (Part 1)

Spheroid-Ring Systems

Through a research collaboration at the Université de Bordeaux, B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509 have published a paper titled, Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.

We discuss this topic in a separate, accompanying chapter.

Exterior Gravitational Potential of Toroids

J. -M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 5825-5838 have published a paper titled, The Exterior Gravitational Potential of Toroids. Here we examine how their work relates to the published work by C.-Y. Wong (1973, Annals of Physics, 77, 279), which we have separately discussed in detail.

Our Presentation of Wong's (1973) Result

Summary: First three terms in Wong's (1973) expression for the gravitational potential at any point, P(ϖ, z), outside of a uniform-density torus.

$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr\|_\mathrm{exterior}$	$~=$	$~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr) \Upsilon_{W0}(\eta_0) \biggl\{ \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) \biggr\}\, ,$
$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr\|_\mathrm{exterior}$	$~=$	$~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr) \Upsilon_{W1}(\eta_0) \times \cos\theta \biggl\{ \frac{a}{r_2} \cdot \boldsymbol{E}(k) \biggr\} \, ,$
$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr\|_\mathrm{exterior}$	$~=$	$~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0) \times \cos(2\theta) \biggl\{ \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) - \frac{a}{r_1} \cdot \boldsymbol{K}(k) \biggr\} \, ,$

where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z₀) — have been specified, we have,

$~a^2$	$~\equiv$	$~ R^2 - d^2$ and, $~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d} \, ,$
$~r_1^2$	$~\equiv$	$~(\varpi + a)^2 + (z - Z_0)^2 \, ,$
$~r_2^2$	$~\equiv$	$~(\varpi - a)^2 + (z - Z_0)^2 \, ,$
$~\cos\theta$	$~\equiv$	$~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,$
$~k$	$~\equiv$	$~ \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .$

		Leading Coefficient Expressions …	… evaluated for:	$~\frac{R}{d} = \cosh\eta_0 = 3$
$~\Upsilon_{W0}(\eta_0)$	$~\equiv$	$~ \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] \biggr\} \, ,$		7.134677
$~\Upsilon_{W1}(\eta_0)$	$~\equiv$	$~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] \biggr\} \, ,$		0.130324
$~\Upsilon_{W2}(\eta_0)$	$~\equiv$	$~ \frac{2^{3 / 2}}{3^2} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]$
		$~ -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] \biggr\} \, ,$		0.003153
where,
$~k_0$	$~\equiv$	$~ \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, .$		0.707106781

NOTE: In evaluating these "leading coefficient expressions" for the case, $~R/d = 3$ , we have used the complete elliptic integral evaluations, K(k₀) = 1.854074677 and E(k₀) = 1.350643881.

Setup

From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,

$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)$

$~=$

$~ -D_0 (\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, ,$

Wong (1973), §II.D, p. 294, Eqs. (2.59) & (2.61)

where,

$~D_0$	$~\equiv$	$~ \frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] = \frac{2^{3/2} }{3\pi^2} \biggl[\frac{(R^2 - d^2)^{3 / 2}}{d^2 R} \biggr] \, ,$
$~C_n(\cosh\eta_0)$	$~\equiv$	$~(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) - (n - \tfrac{3}{2}) Q_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \,$

Wong (1973), §II.D, p. 294, Eq. (2.63)

and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R,

$~\cosh\eta_0$	$~=$	$~\frac{R}{d} = \frac{1}{\epsilon} \, ,$
$~\sinh\eta_0$	$~=$	$~\frac{a}{d} = \frac{1}{d}\biggl[ R^2 - d^2 \biggr]^{1 / 2} = \frac{1}{\epsilon} \biggl[1 - \epsilon^2 \biggr]^{1 / 2} \, .$

These expressions incorporate a number of basic elements of a toroidal coordinate system. In what follows, we will also make use of the following relations:

Once the primary scale factor, $~a$ , has been specified, the illustration shown at the bottom of this inset box — see also our accompanying set of similar figures used by other researchers — helps in explaining how transformations can be made between any two of the referenced coordinate pairs: $~(\varpi, z)$ , $~(\eta, \theta)$ , $~(r_1, r_2)$ .

$~\varpi$	$~=$	$~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}$	$~\Rightarrow ~$	$~\cos\theta$	$~=$	$~\cosh\eta - \frac{a\sinh\eta}{\varpi}$
$~z - Z_0$	$~=$	$~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}$	$~\Rightarrow ~$	$~\sin\theta$	$~=$	$~\frac{(z - Z_0)}{\varpi} \cdot \sinh\eta$

Given that (sin²θ + cos²θ) = 1, we have,

$~1$	$~=$	$~ \biggl[ \frac{(z - Z_0)}{\varpi} \cdot \sinh\eta \biggr]^2 + \biggl[\cosh\eta - \frac{a\sinh\eta}{\varpi}\biggr]^2$
$~\Rightarrow ~~~ \coth\eta$	$~=$	$~ \frac{1}{2a\varpi}\biggl[\varpi^2 + a^2 + (z - Z_0)^2 \biggr] \, .$

We deduce as well that,

$~\frac{2}{\coth\eta + 1}$	$~=$	$~ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \, ,$ and,
$~\sinh\eta + \cosh\eta$	$~=$	$~ \frac{\varpi^2 + a^2 + (z - Z_0)^2}{(\varpi + a)^2 + (z - Z_0)^2} \, .$

Given the definitions,

$~r_1^2$	$~=$	$~(\varpi + a)^2 + (z - Z_0)^2 \, ,$
$~r_2^2$	$~=$	$~(\varpi - a)^2 + (z - Z_0)^2 \, ,$

we can use the transformations,

$~\varpi$	$~=$	$~\frac{(r_1^2 - r_2^2)}{4a}$ and,
$~(z - Z_0)^2$	$~=$	$~r_2^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 - 4a^2 \biggr]^2 \, ,$ or,
$~(z - Z_0)^2$	$~=$	$~r_1^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 + 4a^2 \biggr]^2 \, .$

Or we can use the transformations,

$~\eta$	$~=$	$~\ln \biggl(\frac{r_1}{r_2}\biggr) \, ,$
$~\cos\theta$	$~=$	$~\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .$

Additional potentially useful relations can be found in an accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system.

Leading (n = 0) Term

Wong's Expression

Now, from our separate derivation we have,

$~P_{-1 / 2}(\cosh\eta)$

$~=$

$~ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, .$

And if we make the function-argument substitution, $~z \rightarrow \coth\eta$ , in the "Key Equation,"

	$~Q_{-\frac{1}{2}}(z)$	$~=$	$~ \sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr)$
Abramowitz & Stegun (1995), p. 337, eq. (8.13.3)

we can write,

$~P_{-1 / 2}(\cosh\eta)$

$~=$

$~ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, ,$

where, from above, we recognize that,

$~ k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .$

So, the leading (n = 0) term gives,

$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)$	$~=$	$~ -D_0 (\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)P_{-\frac{1}{2}}(\cosh\eta)$
	$~=$	$~ -D_0~C_0(\cosh\eta_0) \biggl[ \frac{a \sinh\eta}{\varpi} \biggr]^{1 / 2} ~\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)$
	$~=$	$~ -\frac{D_0~C_0(\cosh\eta_0)}{\pi} \biggl[ \frac{2a }{\varpi} \biggr]^{1 / 2} ~ k \boldsymbol{K}(k)$
	$~=$	$~ - C_0(\cosh\eta_0) \cdot \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot \boldsymbol{K}(k) \, .$

Thin-Ring Evaluation of C₀

In an accompanying discussion of the thin-ring approximation, we showed that as $~\cosh\eta_0 \rightarrow \infty$

$~C_0(x)\biggr|_{x\rightarrow \infty}$

$~=$

$~\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0} \, .$

Hence, in this limit we can write,

$~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{thin-ring}$

$~=$

$~ - \frac{2 }{\pi} \cancelto{1}{\biggl[\frac{\sinh\eta_0}{\cosh\eta_0}\biggr]^3 } \frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot \boldsymbol{K}(k) \, .$

More General Evaluation of C₀

NOTE of CAUTION: In our above evaluation of the toroidal function, $~Q_{-\frac{1}{2}}(z)$ , we appropriately associated the functio

User:Tohline/Appendix/Ramblings/Bordeaux

From VisTrailsWiki

Current revision as of 17:05, 16 December 2020

Contents

Université de Bordeaux (Part 1)

Spheroid-Ring Systems

Exterior Gravitational Potential of Toroids

Our Presentation of Wong's (1973) Result

Setup

Leading (n = 0) Term

Wong's Expression

Thin-Ring Evaluation of C₀

More General Evaluation of C₀

@@ Line 1: / Line 1: @@
 <!-- __FORCETOC__ will force the creation of a Table of Contents -->
 <!-- __NOTOC__ will force TOC off -->
-=Universit&eacute; de Bordeaux=
+=Universit&eacute; de Bordeaux (Part 1)=
 {{LSU_HBook_header}}
@@ Line 8: / Line 8: @@
 Through a research collaboration at the [https://www.u-bordeaux.com Universit&eacute; de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais &amp; J. -M. Hur&eacute; (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems:  New Bifurcations, Critical Rotations, and Degenerate States.''
+We discuss this topic in a [[User:Tohline/Appendix/Ramblings/BordeauxSequences#Spheroid-Ring_Systems|separate, accompanying chapter]].
 ==Exterior Gravitational Potential of Toroids==
 [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Hur&eacute;, B. Basillais, V. Karas, A. Trova, &amp; O. Semer&aacute;k (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.''  Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]].
-===Their Presentation===
+===Our Presentation of Wong's (1973) Result===
-On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
+<table border="1" cellpadding="8" align="center" width="80%">
+<tr><td align="center">'''Summary:'''&nbsp; First three terms in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] expression for the gravitational potential at any point, P(&#x03D6;, z), outside of a uniform-density torus.</td></tr>
+<tr><td align="left">
+[[File:WongTorusIllustration02.png|500px|center|Wong diagram]]
+----
+<table border="0" cellpadding="5" align="center">
 <tr>
    <td align="right">
-<math>~\Psi_\mathrm{grav}(\vec{r})</math>
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math>
    </td>
    <td align="center">
-<math>~\approx</math>
+<math>~=</math>
    </td>
    <td align="left">
 <math>~
-\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
+- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
+\Upsilon_{W0}(\eta_0) \biggl\{
+\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k) \biggr\}\, ,
 </math>
    </td>
 </tr>
-</table>
-[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
-</div>
-where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
 <tr>
    <td align="right">
-<math>~\Psi_0 </math>
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
    </td>
    <td align="center">
@@ Line 47: / Line 49: @@
    <td align="left">
 <math>~
-- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}(k_0) \biggr]
+- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
+\Upsilon_{W1}(\eta_0) \times \cos\theta
+\biggl\{ \frac{a}{r_2} \cdot
+\boldsymbol{E}(k) \biggr\} \, ,
 </math>
    </td>
 </tr>
-</table>
-[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
-</div>
-and,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
 <tr>
    <td align="right">
-<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}</math>
    </td>
    <td align="center">
@@ Line 68: / Line 66: @@
    <td align="left">
 <math>~
-- \frac{G\pi \rho_0 R_c b^2}{4 (k')^2 \Delta_0^3} \biggl\{
+- \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0)
-[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k) - (k')^2 \Delta_0^2 \boldsymbol{K}(k)
+\times \cos(2\theta)
-\biggr\} \, .
+\biggl\{
+\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)
+-
+\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
+\biggr\} \, ,
 </math>
    </td>
@@ Line 76: / Line 78: @@
 </table>
-[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
+where, once the major ( R ) and minor ( d ) radii of the torus &#8212; as well as the vertical location of its equatorial plane (Z<sub>0</sub>) &#8212; have been specified, we have,
-</div>
-Note that the argument of the elliptic integral functions is,
-<div align="center">
 <table border="0" cellpadding="5" align="center">
 <tr>
    <td align="right">
-<math>~k</math>
+<math>~a^2 </math>
    </td>
    <td align="center">
@@ Line 91: / Line 90: @@
    <td align="left">
 <math>~
-\frac{2\sqrt{\varpi R}}{\Delta}
+R^2 - d^2</math>
+&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
+<math>~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d}
+\, ,
 </math>
    </td>
-<td align="center">&nbsp; &nbsp; where, &nbsp; &nbsp;</td>
+</tr>
+<tr>
    <td align="right">
-<math>~\Delta</math>
+<math>~r_1^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_2^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\cos\theta</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k</math>
    </td>
    <td align="center">
@@ Line 103: / Line 143: @@
    <td align="left">
 <math>~
-\biggl[ (R + \varpi)^2 + (Z-z)^2 \biggr]^{1 / 2} \, .
+\biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2}
+= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
+= \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}
+\, .
 </math>
    </td>
@@ Line 109: / Line 152: @@
 </table>
-[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (4) &amp; (5)
+----
-</div>
-===Our Presentation of Wong's (1973) Result===
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="left" colspan="2">&nbsp;</td>
+  <td align="left" colspan="1">Leading Coefficient Expressions &hellip;</td>
+  <td align="right" colspan="1" width="30%">&hellip; evaluated for:&nbsp; &nbsp;</td>
+  <td align="center" colspan="1"><math>~\frac{R}{d} = \cosh\eta_0 = 3</math>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~
+\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl\{
+K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
++ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
+- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
+\biggr\}  \, ,
+</math>
+  </td>
+  <td align="center"><font color="red">7.134677</font></td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
+\biggl\{
+K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
++~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
+-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
+\biggr\}
+\, ,
+</math>
+  </td>
+  <td align="center"><font color="red">0.130324</font></td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Upsilon_{W2}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~
+\frac{2^{3 / 2}}{3^2}  \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{
+K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
++  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]
+</math>
+  </td>
+  <td align="center">&nbsp;</td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left" colspan="2">
+<math>~
+-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ]
+\biggr\} \, ,
+</math>
+  </td>
+  <td align="center"><font color="red">0.003153</font></td>
+</tr>
+<tr><td align="left" colspan="5">where,</td></tr>
+<tr>
+  <td align="right">
+<math>~k_0</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~
+\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, .
+</math>
+  </td>
+  <td align="center"><font color="red">0.707106781</font></td>
+</tr>
+</table>
+NOTE:  In evaluating these "leading coefficient expressions" for the case, <math>~R/d = 3</math>, we have used the complete elliptic integral evaluations, '''K'''(k<sub>0</sub>) = <font color="red">1.854074677</font> and  '''E'''(k<sub>0</sub>) = <font color="red">1.350643881</font>.
+</td></tr>
+</table>
 ====Setup====
 From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression,
+<div align="center">
 <table border="0" cellpadding="5" align="center">
@@ Line 134: / Line 274: @@
 </tr>
 </table>
+[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eqs. (2.59) &amp; (2.61)
+</div>
 where,
+<div align="center">
 <table border="0" cellpadding="5" align="center">
@@ Line 168: / Line 310: @@
 </tr>
 </table>
+[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eq. (2.63)
+</div>
 and where, in terms of the major ( R ) and minor ( d ) radii of the torus &#8212; or their ratio, &epsilon; &equiv; d/R,
 <table border="0" cellpadding="5" align="center">
@@ Line 1,148: / Line 1,292: @@
 </table>
+<span id="Qrecurrence">&nbsp;</span>
 <table border="1" align="center" width="80%" cellpadding="10">
 <tr><td align="left">
@@ Line 1,252: / Line 1,396: @@
    <td align="left">
 <math>~
--~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 )
+-~\frac{1}{2^2}
+\biggl\{ z k_0~K ( k_0 )
 ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}
 </math>
@@ Line 1,294: / Line 1,439: @@
    <td align="left">
 <math>~
-Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0)  \, ,
+Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] \biggr\}
++ z \biggl\{ z k_0~K ( k_0 )
+~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} \biggl\{  [5 z]
+~-~z (z^2+3) \biggr\} E(k_0)
++ \biggl\{ z^2 k_0~
+- [(z-1)(z^2-1)]^{-1 / 2} [ 2^{-3 / 2} \cdot 5 (z-1)] \biggr\}K(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{-3 / 2}(z+1)^{-1 / 2} [4 z^2  -  5  ]K(k_0)
+-~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z  E(k_0)
 </math>
    </td>
@@ Line 1,576: / Line 1,769: @@
    <td align="left">
 <math>~
+\biggl[ \frac{2(3z^2 - 1)}{(z^2-1)}      \biggr]K(k_0)\cdot E(k_0)
 -~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0)
 -~\biggl[ \frac{ 5z}{(z-1)}  \biggr] E(k_0)\cdot E(k_0)
-+~\biggl[ \frac{2(3z^2 - 1)}{(z^2-1)}      \biggr]K(k_0)\cdot E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~(z_0^2-1)C_1(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+(3z^2 - 1) K(k_0)\cdot E(k_0)
+-~z_0(z_0-1) K(k_0)\cdot K(k_0)
+-~5z_0(z_0+1) E(k_0)\cdot E(k_0) \, .
 </math>
    </td>
 </tr>
 </table>
+Hence, we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]  C_1(\cosh\eta_0) \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2^2} \biggr]
+\boldsymbol{E}(k) \, .
+</math>
+  </td>
+</tr>
+</table>
+====Third (n = 2) Term====
+=====Part A=====
+The third (n = 2) term in [[User:Tohline/Apps/Wong1973Potential#D0andCn|Wong's (1973) expression for the exterior potential]] is,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-D_0
+(\cosh\eta - \cos\theta)^{1 / 2} \cdot 2 \cos(2\theta) \cdot  C_2(\cosh\eta_0)P_{+\frac{3}{2}}(\cosh\eta) \, ,
+</math>
+  </td>
+</tr>
+</table>
+where, <math>~D_0</math> is the same as [[#Setup|above]], and,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~C_2(\cosh\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~\tfrac{5}{2}Q_{+\frac{5}{2}}(\cosh \eta_0) Q_{+\frac{3}{2}}^2(\cosh \eta_0)
+- \tfrac{1}{2} Q_{+\frac{3}{2}}(\cosh \eta_0)~Q^2_{+ \frac{5}{2}}(\cosh \eta_0) \, .
+</math>
+  </td>
+</tr>
+</table>
+<table border="1" align="center" width="80%" cellpadding="10">
+<tr><td align="left">
+In order to evaluate <math>~C_2(z)</math>, we will need the following pair of expressions in addition to the ones already used:
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~Q^0_{m-\tfrac{1}{2}}</math> [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|recurrence]] with m = 3, gives:  &nbsp; &nbsp;
+<math>~Q_{+\tfrac{5}{2}}(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{8}{5} z~Q_{+\tfrac{3}{2}}(z_0) - \frac{3}{5} Q_{+\tfrac{1}{2}}(z_0)</math>
+  </td>
+</tr>
+</table>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Rightarrow~~~15Q_{+\tfrac{5}{2}}(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~8 z~\biggl[ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr]
+-
+\biggl[ z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)  \biggr]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z~k_0 K(k_0) \biggl[ 8(4z^2 - 1 ) - 9 \biggr]
++
+[2(z+1)]^{1 / 2} E(k_0) \biggl[-32z^2 + 9  \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z~k_0 K(k_0) [ 32z^2 - 17 ]
++
+[2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] \, .
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+Hence, &nbsp; &nbsp; <math>~Q_{+\frac{5}{2}}(3)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~0.002080867 \, .</math>
+  </td>
+</tr>
+</table>
+And, setting m = 2 in the [[#Qrecurrence|above recurrence relation for]] <math>~Q^2_{m+\frac{1}{2}}(z)</math> gives,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl[ (m - \tfrac{3}{2})Q^{2}_{m+\tfrac{1}{2}} (z) \biggr]_{m=2}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ (2m)z Q_{m-\tfrac{1}{2}}^{2}(z) - (m + \tfrac{3}{2})Q^{2}_{m - \tfrac{3}{2}}(z) \biggr]_{m=2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~  Q^{2}_{+\tfrac{5}{2}} (z) </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z Q_{+\tfrac{3}{2}}^{2}(z) - 7 Q^{2}_{+ \tfrac{1}{2}}(z)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z \biggl[ 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \biggr] - 7 Q^{2}_{+ \tfrac{1}{2}}(z)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z Q^{2}_{- \tfrac{1}{2}}(z_0)
+- [32z^2 +7]Q_{+\tfrac{1}{2}}^{2}(z_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z \biggl\{
+[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ \frac{[32z^2 +7]}{4} \biggl\{
+ z k_0~K ( k_0 )
+~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+</table>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~  4Q^{2}_{+\tfrac{5}{2}} (z) </math>  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^5\cdot 5z \biggl\{ 2^{1 / 2}
+[(z-1)(z^2-1)]^{-1 / 2} [zE(k_0) ]
+-
+^{-3 / 2}[(z-1)(z^2-1)]^{-1 / 2} [(z-1)K(k_0)]
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ [32z^2 +7] \biggl\{
+ z k_0~K ( k_0 )
+~-~2^{1 / 2}(z^2+3) [ (z-1)(z^2-1) ]^{-1 / 2} E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+^{11 / 2}\cdot 5  [z^2 ] - 2^{1 / 2} [32z^2 +7]  (z^2+3)
+\biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~2^{7 / 2}\cdot 5 \biggl\{ [(z-1)(z^2-1)]^{-1 / 2} (z-1) \biggr\} z K(k_0)
++ [32z^2 +7] \biggl\{ 2^{1 / 2}[z + 1]^{-1 / 2} \biggr\}  z K ( k_0 )
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{1 / 2}\biggl\{ 32z^2 - 33 \biggr\}  z [z + 1]^{-1 / 2}  K ( k_0 )
+-~2^{1 / 2}
+\biggl\{
+z^4 - 57  z^2   + 21
+\biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33)   z (z-1)  K ( k_0 )
+-~(32z^4 - 57  z^2   + 21)E(k_0) \biggr]
+\, .
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+Hence, &nbsp; &nbsp; <math>~Q^2_{+\frac{5}{2}}(3)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~0.03377378 \, .</math>
+  </td>
+</tr>
+</table>
+</td></tr></table>
+=====Part B=====
+Let's evaluate <math>~C_2(z)</math> specifically for the case where <math>~z = \cosh\eta_0 = 3</math>, using the already separately evaluated values of the four relevant toroidal functions.  We find,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~2C_2(3)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~5Q_{+\frac{5}{2}}(3) Q_{+\frac{3}{2}}^2(3)
+- Q_{+\frac{3}{2}}(3)~Q^2_{+ \frac{5}{2}}(3)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\cdot ( 0.002080867 ) \times ( 0.132453829 ) -  ( 0.014544576 ) \times (0.03377378 )
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+.868687\times 10^{-4} \, .
+</math>
+  </td>
+</tr>
+</table>
+Next, let's develop a consolidated expression for <math>~C_2(z_0)</math> that replaces all the toroidal functions with complete elliptic integrals of the first and second kind.
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~2C_2(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~5Q_{+\frac{5}{2}}(z_0) Q_{+\frac{3}{2}}^2(z_0)
+- Q_{+\frac{3}{2}}(z_0)~Q^2_{+ \frac{5}{2}}(z_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{1}{3}\biggl\{
+z~k_0 K(k_0) [ 32z^2 - 17 ]
++
+[2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ]
+\biggr\}
+\times \biggl\{
+^{-3 / 2}(z+1)^{-1 / 2} [4 z^2  -  5  ]K(k_0)
+-~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z  E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2^2\cdot 3}
+\biggl\{ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0)  \biggr\}
+\times \biggl\{
+^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33)   z (z-1)  K ( k_0 )
+-~(32z^4 - 57  z^2   + 21)E(k_0) \biggr]
+\biggr\}
+</math>
+  </td>
+</tr>
+</table>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ 2^{2} \cdot 3 (z^2-1) C_2(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+K(k_0) z[ 32z^2 - 17 ]
++
+(z+1) E(k_0) [9 -32z^2 ]
+\biggr\}
+\times \biggl\{
+(z-1) [4 z^2  -  5  ]K(k_0)
+-~4 (z^2 - 2)z  E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- ~
+\biggl\{ (4z^2 - 1 )  K(k_0) - 4 z(z+1) E(k_0)  \biggr\}
+\times \biggl\{
+(32z^2 - 33)   z (z-1)  K ( k_0 )
+-~(32z^4 - 57  z^2   + 21)E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+(z-1)[ 32z^2 - 17 ] [4 z^2  -  5  ]z K(k_0) \cdot K(k_0)
+-~4 (z^2 - 2)z^2 [ 32z^2 - 17 ]  K(k_0) \cdot E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++  \biggl\{
+(z-1) (z+1) [9 -32z^2 ] [4 z^2  -  5  ]K(k_0) \cdot E(k_0)
+-~4 (z^2 - 2)z (z+1) [9 -32z^2 ] E(k_0) \cdot E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ ~
+\biggl\{
+(32z^4 - 57  z^2   + 21)(4z^2 - 1 ) K(k_0) \cdot E(k_0)
+-~(32z^2 - 33)   z (z-1)(4z^2 - 1 )  K ( k_0 ) \cdot K(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ ~
+\biggl\{
+z(z+1)(32z^2 - 33)   z (z-1)  K ( k_0 ) \cdot E(k_0)
+-~4 z(z+1)(32z^4 - 57  z^2   + 21)E(k_0) \cdot E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~(z-1)\biggl\{
+\biggl[
+( 32z^2 - 17 ) (4 z^2  -  5  )z \biggr]
+-~\biggl[ (32z^2 - 33)   z (4z^2 - 1 ) \biggr] \biggr\} K ( k_0 ) \cdot K(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++  \biggl\{ \biggl[
+(z-1) (z+1) (9 -32z^2 ) (4 z^2  -  5  )\biggr]
+-~\biggl[ 4 (z^2 - 2)z^2 ( 32z^2 - 17 ) \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ ~
+\biggl[
+(32z^4 - 57  z^2   + 21)(4z^2 - 1 ) \biggr]
++ ~   \biggl[ 4 z(z+1)(32z^2 - 33)   z (z-1)\biggr]\biggr\}  K ( k_0 ) \cdot E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~2z(z+1) \biggl\{ \biggl[ 2 (32z^4 - 57  z^2   + 21) \biggr]
++~2\biggl[  (z^2 - 2) (9 -32z^2 )\biggr] \biggr\} E(k_0) \cdot E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z(z-1)\biggl\{[ 52 - 64z^2 ] \biggr\} K ( k_0 ) \cdot K(k_0)
+-~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++  \biggl\{
+\biggl[ 5(32z^4 - 41z^2 + 9 ) \biggr]
+- \biggl[ (32z^4 - 57  z^2   + 21)\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ ~
+z^2\biggl[ (32z^4 - 57  z^2   + 21)
++  (32z^4 - 65z^2 + 33)  + (-32z^4 + 41z^2 -9 )  +~( -32z^4 + 81z^2 - 34 )
+ \biggr]\biggr\}  K ( k_0 ) \cdot E(k_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+z(z-1)\biggl\{ 13 - 16z^2  \biggr\} K ( k_0 ) \cdot K(k_0)
+-~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0)
++  8\biggl\{
+z^4 -13z^2 + 3 \biggr\}  K ( k_0 ) \cdot E(k_0) \, .
+</math>
+  </td>
+</tr>
+</table>
+Finally, let's evaluate this consolidated expression for the specific case of <math>~z_0 = \cosh\eta_0 = 3</math>, remembering that in this specific case <math>~k_0  = 2^{-1 / 2}</math>, <math>~K(k_0) = 1.854074677</math>, and <math>~E(k_0) = 1.350643881</math>.  We find,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~2C_2(z_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+[2 \cdot 3 (z^2-1) ]^{-1} \biggl\{
+z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0)
+-~4z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0)
++  8[
+z^4 -13z^2 + 3 ]  K ( k_0 ) \cdot E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+[48 ]^{-1} \biggl\{
+-24[ 131 ] K ( k_0 ) \cdot K(k_0)
+-~48 [ 147] E(k_0) \cdot E(k_0)
++  8[ 1182 ]  K ( k_0 ) \cdot E(k_0)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+.8708 \times 10^{-4} \, .
+</math>
+  </td>
+</tr>
+</table>
+<font color="red">This matches the numerically evaluated expression, from above (6/30/2020)</font>.  There is a tremendous amount of cancellation between the three key terms in this expression, so the match  is only to three significant digits.</tr>
+=====Part C=====
+Next &hellip;
+<table border="1" cellpadding="8" align="center" width="60%">
+<tr><td align="left">
+<div align="center">'''Useful Relations from Above'''</div>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\cosh\eta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{r_1^2 + r_2^2}{2r_1 r_2} \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\sinh\eta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{r_1^2 - r_2^2}{2r_1 r_2} \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\varpi</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{r_1^2 - r_2^2}{2a} \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\cosh\eta - \cos\theta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{2a^2}{r_1 r_2} \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~ \cos\theta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{r_1^2 + r_2^2-4a^2}{2r_1 r_2} \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\frac{2}{\coth\eta + 1}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{4a\varpi}{r_1^2} \, .</math>
+  </td>
+</tr>
+</table>
+</td></tr>
+</table>
+Now, from our tabulation of [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|example recurrence relations]], we see that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~ P_{+\frac{3}{2}}(\cosh\eta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{4}{3} \cdot \cosh\eta ~P_{+\frac{1}{2}}(\cosh\eta) - \frac{1}{3} ~ P_{-\frac{1}{2}}(\cosh\eta) </math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{4}{3} \cdot \cosh\eta \biggl[ \frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) \biggr]
+-
+\frac{1}{3} ~ \biggl[ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{2^{1 / 2}}{3\pi}
+\biggl[ 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k)
+-
+(\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]
+\, ,</math>
+  </td>
+</tr>
+</table>
+where, as above,
+<div align="center">
+<math>~
+k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}
+=
+\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}
+=
+\biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} \, .
+</math>
+</div>
+So we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{5/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
+C_2(\cosh\eta_0)\cos(2\theta)
+\biggl\{ (\cosh\eta - \cos\theta)^{1 / 2}P_{+\frac{3}{2}}(\cosh\eta) \biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
+C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2}
+\biggl\{
+\cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k)
+-
+(\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
+C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times
+\biggl\{
+\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \biggl[\frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{+1 / 2} \biggl[ \frac{4a}{r_1^2} \biggl( \frac{r_1^2 - r_2^2}{2a} \biggr)\biggr]^{-1 / 2}  \boldsymbol{E}(k)
+-
+\biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2}  \biggr]^{-1 / 2} ~\biggl[ \frac{4a}{r_1^2}\biggl( \frac{r_1^2 - r_2^2}{2a} \biggr) \biggr]^{1 / 2} \boldsymbol{K}(k)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{9/2} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
+C_2(\cosh\eta_0)\cos(2\theta)
+\times
+\biggl\{
+\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)
+-
+\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+Finally, inserting the expression for <math>~\sinh^2\eta_0~ C_2(\cosh\eta_0) = (z_0^2-1)C_2(z_0)</math> that we have derived, above, gives,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{9/2} }{3^3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
+\times \cos(2\theta)
+\biggl\{
+\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)
+-
+\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times \biggl\{
+z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0)
+-~z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0)
++  2 [ 16z^4 -13z^2 + 3 ]  K ( k_0 ) \cdot E(k_0)
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+====Summary====
+Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely,
+<div align="center">
+<math>~a^2 \equiv R^2 - d^2\, ,</math> &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; <math>~\cosh\eta_0 \equiv \frac{R}{d} \, ,</math>
+</div>
+in which case also, <math>~\sinh\eta_0 = a/d \, .</math>  Once the mass-density ( &rho;<sub>0</sub> ) of the torus has been specified, the torus mass is given by the expression,
+<div align="center">
+<math>~M = 2\pi^2 \rho_0 d^2 R \, .</math>
+</div>
+In addition to the principal pair of meridional-plane coordinates, <math>~(\varpi, z)</math>, it is useful to define the pair of distances,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~r_1^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_2^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
+  </td>
+</tr>
+</table>
+where, the equatorial plane of the torus is located at <math>~z = Z_0</math>.  As we have shown above, the leading (n = 0) term in Wong's (1973) series-expression for the gravitational potential anywhere outside the torus is,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{2^{3} }{3\pi^3}
+\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
+\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times \biggl\{
+K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
++ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
+- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
+\biggr\}  \, .
+</math>
+  </td>
+</tr>
+</table>
+where, the two distinctly different arguments &#8212; one with, and one without a zero subscript &#8212; of the complete elliptic-integral functions are,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}
+=
+\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}
+= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
+= \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k_0</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2}  \, .
+</math>
+  </td>
+</tr>
+</table>
+As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot
+\boldsymbol{E}(k)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~\times
+\biggl\{
+K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
++~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
+-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
+\biggr\}
+\, .
+</math>
+  </td>
+</tr>
+</table>
+Note that a transformation from the <math>~(r_1, r_2)</math> coordinate pair to the toroidal-coordinate pair <math>~(\eta, \theta)</math> includes the expression,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\cos\theta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .
+</math>
+  </td>
+</tr>
+</table>
+So this (n = 1) term's explicit dependence on "cos(n&theta;)" is clear.  Finally,  the third (n = 2) term in Wong's (1973) series-expression for the exterior gravitational potential is,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
+\times \cos(2\theta)
+\biggl\{
+\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)
+-
+\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times \frac{2^{3 / 2}}{3^2}\biggl\{
+K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
++  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ]
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+===The Hur&eacute;, ''et al'' (2020) Presentation===
+{{LSU_WorkInProgress}}
+====Notation====
+In [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], the major and minor radii of the torus surface ("shell") are labeled, respectively, R<sub>c</sub> and b, and their ratio is denoted,
+<div align="center">
+<math>~e \equiv \frac{b}{R_c} \, .</math>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (1)
+</div>
+The authors work in cylindrical coordinates, <math>~(R, Z)</math>, whereas we refer to this same coordinate-pair as, <math>~(\varpi_W, z_W)</math>.  The quantity,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Delta^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+[R + (R_c + b\cos\theta_H)]^2 + [Z - b\sin\theta_H]^2 \, .
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (5) &amp; (7)
+</div>
+We have affixed the subscript "H" to their meridional-plane angle, &theta;, to clarify that it has a different coordinate-base definition from the meridional-plane angle, &theta;, that appears in our above discussion of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop <math>~(b \rightarrow 0)</math>, that is to say,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Delta_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+[R + R_c]^2 + Z^2 \, .
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (13)
+</div>
+Generally, the argument (modulus) of the complete elliptic integral functions is,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k_H</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{2}{\Delta}\biggl[ R (R_c + b\cos\theta_H) \biggr]^{1 / 2}  \, ,
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (4)
+</div>
+and, as stated in the first sentence of their &sect;3, reference may also be made to the ''complementary modulus'',
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k'_H</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~[1 - k_H^2]^{1 / 2} \, .</math>
+  </td>
+</tr>
+</table>
+(Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].)  And in the case of an infinitesimally thin hoop <math>~(b\rightarrow 0)</math>,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~[k^2_H]_0</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{4R R_c}{\Delta_0^2}  \, .
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (12)
+</div>
+====Key Finding====
+On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Psi_\mathrm{grav}(\vec{r})</math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
+</div>
+where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Psi_0 </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}([k_H]_0) \biggr]
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
+</div>
+and,
+<div align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{G\pi \rho_0 R_c b^2}{4 (k'_H)^2 \Delta_0^3} \biggl\{
+[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
+</div>
+Rewriting this last expression in a form that can more readily be compared with Wong's work, we obtain,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^3\pi}{e^2} \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{ 1 }{(k'_H)^2 \Delta_0^3}\biggl\{
+[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} +  \frac{\boldsymbol{K}(k_H)}{\Delta_0} \, .
+</math>
+  </td>
+</tr>
+</table>
+<span id="Step01">Hence, also,</span>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~ \frac{ \Psi_0 }{GM}  + \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~- \frac{2}{\pi}\biggl\{
+\frac{\boldsymbol{K}([k_H]_0)}{\Delta_0}
+\biggr\} +
+\frac{e^2}{2^3\pi}\biggl\{
+\frac{\boldsymbol{K}(k_H)}{\Delta_0}
+- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~- \frac{2}{\pi \Delta_0}\biggl\{
+\boldsymbol{K}([k_H]_0)
+-
+\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) \biggr\}
+- \frac{2}{\pi\Delta_0} \cdot \frac{e^2}{2^4}\biggl\{\biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]
+\biggr\} \boldsymbol{E}(k_H)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\boldsymbol{K}([k_H]_0)
+-
+\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H)
++  \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]  \boldsymbol{E}(k_H) \, .
+</math>
+  </td>
+</tr>
+</table>
+===Compare First Terms===
+Rewriting the first term in the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] series expression for the potential, we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{\Psi_0}{GM} </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{2}{\pi} \biggl\{ \frac{\boldsymbol{K}([k_H]_0) }{[ (\varpi_W + R_c)^2 + z_W^2]^{1 / 2}} \biggr\} \, ,
+</math>
+  </td>
+</tr>
+</table>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~[k_H]_0</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{4\varpi_W R_c}{\Delta_0^2}  \biggr]^{1 / 2}
+=
+\biggl\{ \frac{4\varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2} \biggr\}^{1 / 2} \, .
+</math>
+  </td>
+</tr>
+</table>
+For comparison, the first term in Wong's expression is,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{\Phi_\mathrm{W0}}{GM} </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
+\Upsilon_{W0}(\eta_0) \biggl\{
+\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, ,
+</math>
+  </td>
+</tr>
+</table>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~a^2 </math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+R^2 - d^2 ~~~\Rightarrow ~~~ a = R_c(1 - e^2)^{1 / 2} \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_1^2</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \varpi + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + \biggl[z - Z_0 \biggr]^2 \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \frac{4\varpi R_c(1-e^2)^{1 / 2}}{[\varpi + R_c(1-e^2)^{1 / 2}]^2 + [z - Z_0]^2} \biggr\}^{1 / 2}
+\, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\sinh\eta_0}{\cosh\eta_0}\biggl\{
+K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
++ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
+- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
+\biggr\}  \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{(1-e^2)^{1 / 2}}{e^2} \biggl\{
+- K(k_0)\cdot K(k_0) (1-e)
++ 2K(k_0)\cdot E(k_0) (1+e^2)
+- E(k_0)\cdot E(k_0) (1+e)
+\biggr\}  \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k_0</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{2}{1+1/e} \biggr]^{1 / 2}
+=
+\biggl[ \frac{2e}{1+e} \biggr]^{1 / 2} \, .
+</math>
+  </td>
+</tr>
+</table>
+This expression is correct for any value of the aspect ratio, <math>~e</math>.  But let's set <math>~Z_0 = 0</math> &#8212; as Hur&eacute;, et al. (2020) have done &#8212; then see how the expression simplifies for an infinitesimally thin hoop, that is, if we let <math>~e \rightarrow 0</math>.  First we note that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k\biggr|_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \frac{4\varpi R_c}{[\varpi + R_c]^2 + z^2} \biggr\}^{1 / 2}
+\, ,
+</math>
+  </td>
+</tr>
+</table>
+so in this limit the modulus of the complete elliptic integral of the first kind becomes identical to the modulus used by Hur&eacute;, et al. (2020), <math>~[k_H]_0</math>.  Next, we note that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~r_1\biggr|_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~[( \varpi + R_c )^2 + z^2]^{1 / 2} \, .</math>
+  </td>
+</tr>
+</table>
+As a result, we can write,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\Psi_0}{GM}  \cdot
+\biggl[
+\biggl( \frac{2^{2} }{3\pi^2} \biggr)
+\Upsilon_{W0}(\eta_0)
+\biggr]_{e\rightarrow 0} \, .
+</math>
+  </td>
+</tr>
+</table>
+Now let's evaluate the coefficient, <math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>.
+<table align="center"  border="1" width="100%" cellpadding="8"><tr><td align="left">
+<div align="center">
+<math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>.
+</div>
+First, drawing from our [[User:Tohline/Apps/Wong1973Potential#Phase_0C|separate examination of the behavior of complete elliptic integral functions]], we appreciate that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  E(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+~+~\frac{1}{2^5} ~k_0^4
+~+~\frac{1}{2^5} ~ k_0^6
++ \mathcal{O}(k_0^{8}) \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  K(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \frac{1}{2} k_0^2
++ \frac{11}{2^5} ~k_0^4
++ \frac{17}{2^6} ~ k_0^6
++ \mathcal{O}(k_0^{8}) \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot  E(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~-~\frac{1}{2^6} ~ k_0^6
++ \mathcal{O}(k_0^{8}) \, .
+</math>
+  </td>
+</tr>
+</table>
+Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2e(1+e)^{-1}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2e(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math>
+  </td>
+</tr>
+</table>
+and,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k_0^4</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~4e^2(1+e)^{-2}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~4e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, .</math>
+  </td>
+</tr>
+</table>
+Hence, we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
++ \frac{11}{2^5} ~k_0^4
++ \mathcal{O}(k_0^{6})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
++ \mathcal{O}(k_0^{6})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~
++ \mathcal{O}(k_0^{6})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + e (  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots )
++ \frac{11}{2^3} \cdot ~e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots )
++ \mathcal{O}(e^{3})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots )
++ \mathcal{O}(e^{3})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- (1+e) \biggl[ 1 - e ~(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) ~
+-~ \frac{1}{2^3} \cdot~ e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) ~
++ \mathcal{O}(e^{3})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + e (  1 - e  )
++ \frac{11}{2^3} \cdot ~e^2
+ \biggr]
++ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2
+ \biggr]
+- (1+e) \biggl[ 1 - e ~(  1 - e   ) ~
+-~ \frac{1}{2^3} \cdot~ e^2 ~
+\biggr]
++ \mathcal{O}(e^{3})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \biggl[ 1 + e (  1 - e  )
++ \frac{11}{2^3} \cdot ~e^2
+ \biggr]
++e \biggl[ 1 + e
+\biggr]
++ 2 \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2
+ \biggr]
++ 2 e^2
+- \biggl[ 1 - e ~(  1 - e   ) ~
+-~ \frac{1}{2^3} \cdot~ e^2 ~
+\biggr]
+- e \biggl[ 1 - e
+\biggr]
++ \mathcal{O}(e^{3})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-1 - e + e^2
+- \frac{11}{2^3} \cdot ~e^2
++ e + e^2
++ 2 ~+~\frac{1}{2^2} \cdot~e^2
++ 2 e^2
+-1 + e - e^2  ~
++~ \frac{1}{2^3} \cdot~ e^2 ~
+- e +e^2
++ \mathcal{O}(e^{3})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{e^2}{2^3}
+\biggl[ 2^5
+- 11~
+~+~3 \biggr]~
++ \mathcal{O}(e^{3})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+e^2
++ \mathcal{O}(e^{3})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+[1 + \mathcal{O}(e^{1})]\cdot (1 - e^2)^{1 / 2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\, .</math>
+  </td>
+</tr>
+</table>
+</td></tr></table>
+Given that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\, ,</math>
+  </td>
+</tr>
+</table>
+we conclude that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\Psi_0}{GM}  \, ,
+</math>
+  </td>
+</tr>
+</table>
+that is, we conclude that <math>~\Psi_0</math> matches <math>~\Phi_{W0}</math> in the limit of, <math>~e\rightarrow 0</math>.
+===Go to Higher Order===
+Let's keep higher order terms in Wong's n = 0 component, and let's examine contributions to the same order that come from Wong's n = 1 and (if necessary) n = 2 components.
+<span id="Step02">First, note that,</span>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr)
+\Upsilon_{W0}(\eta_0) \biggl\{
+\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, .
+</math>
+  </td>
+</tr>
+</table>
+====Keeping Higher Order in Wong's First Component====
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  E(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+~+~\frac{1}{2^5} ~k_0^4
+~+~\frac{1}{2^5} ~ k_0^6
+~+~\frac{231}{2^{13}} ~ k_0^8
++ \mathcal{O}(k_0^{10}) \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  K(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \frac{1}{2} k_0^2
++ \frac{11}{2^5} ~k_0^4
++ \frac{17}{2^6} ~ k_0^6
++ \frac{1787}{2^{13}} ~k^8
++ \mathcal{O}(k_0^{10})
+\, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot  E(k_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4
+~-~\frac{1}{2^6} ~ k_0^6
+~-~\frac{77}{2^{13}} ~ k_0^8
++ \mathcal{O}(k_0^{10}) \, .
+</math>
+  </td>
+</tr>
+</table>
+<table border="1" width="80%" cellpadding="5" align="center">
+<tr><td align="center">'''Add One Additional Term'''</td></tr>
+<tr><td align="center">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  E(k) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
++ \biggl( \frac{1}{2} \biggr)^2k^2
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
++ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
++ \cdots
++ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
++ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
+- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
+~-~ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
+~-~ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
+- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
+\biggr\}
+~+~
+\biggl\{
+\biggl( \frac{1}{2} \biggr)^2k^2
+\biggr\}
+\times~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~+~
+\biggl\{
+\biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
+\biggr\}
+\times~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+\biggr\}
+~+~
+\biggl\{
+\biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
+\biggr\}\times
+\biggl\{1 - \frac{1}{2^2} ~k^2
+\biggr\}
++ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \biggl(\frac{5}{2^8}\biggr)~k^6
+- \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)~k^8
+\biggr\}
+~+~
+\biggl\{
+\biggl( \frac{1}{2^2} \biggr)k^2
+- \frac{1}{2^4} ~k^4
+- \frac{3}{2^8}~ k^6
+-\frac{5}{2^{10}} ~k^8
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+~+~
+\biggl( \frac{3^2}{2^6}\biggr) k^4
+~-~
+\biggl( \frac{3^2}{2^8}\biggr) k^6
+~-~ \biggl(\frac{3^3}{2^{12}}\biggr) ~k^8
+~+~
+\biggl( \frac{5^2}{2^8}\biggr) k^6
+~-~\biggl(\frac{5^2}{2^{10}}\biggr) ~k^8
+~+~\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) ~k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+~+~\biggl[ \frac{1}{2^2}
+- \frac{1}{2^2} \biggr] ~k^2
+~+~\biggl[ \frac{3^2}{2^6}
+- \frac{3}{2^6}
+- \frac{1}{2^4} \biggr]~k^4
+~+~\biggl[ \frac{5^2}{2^8}
+- \frac{5}{2^8}
+- \frac{3}{2^8}
+~-~\frac{3^2}{2^8} \biggr]~ k^6
++ \biggl[
+\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) - \biggl(\frac{5^2}{2^{10}}\biggr) - \biggl(\frac{3^3}{2^{12}}\biggr) - \frac{5}{2^{10}} - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)
+\biggr]~k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+~+~\frac{1}{2^5} ~k^4
+~+~\frac{1}{2^5} ~ k^6
+~+~\frac{231}{2^{13}} ~ k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+</table>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  K(k) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
++ \biggl( \frac{1}{2} \biggr)^2k^2
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
++ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
++ \cdots
++ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
++ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times~
+\biggl\{
++ \biggl( \frac{1}{2} \biggr)^2k^2
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
++ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
++ \cdots
++ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
++ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
++ \frac{5^2}{2^8} k^6
++ \frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
+\times~
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
++ \frac{5^2}{2^8} k^6
++ \frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
++ \frac{5^2}{2^8} k^6
++ \frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
+~+~\frac{1}{2^2} k^2
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
++ \frac{5^2}{2^8} k^6
+\biggr\}
+~+~\frac{3^2}{2^6} k^4
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
+\biggr\}
+~+~
+\biggl\{
+\frac{5^2}{2^8} k^6
+\biggr\}
+\biggl\{
++ \frac{1}{2^2} k^2
+\biggr\}
+~+~
+\biggl\{
+\frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
++ \frac{1}{2^2} k^2
++ \frac{3^2}{2^6} k^4
++ \frac{5^2}{2^8} k^6
++ \frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
+~+~
+\biggl\{
+\frac{1}{2^2} k^2
++ \frac{1}{2^4} k^4
++ \frac{3^2}{2^8} k^6
++ \frac{5^2}{2^{10}} k^8
+\biggr\}
+~+~
+\biggl\{
+\frac{3^2}{2^6} k^4
+~+~\frac{3^2}{2^8} k^6
+~+~\frac{3^4}{2^{12}} k^8
+\biggr\}
+~+~
+\biggl\{
+\frac{5^2}{2^8} k^6
+~+~\frac{5^2}{2^{10}} k^8
+\biggr\}
+~+~
+\biggl\{
+\frac{5^2 \cdot 7^2}{2^{14}} k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \biggl[ \frac{1}{2^2}
+~+~ \frac{1}{2^2} \biggr] ~k^2
++ \biggl[ \frac{3^2}{2^6}
++ \frac{1}{2^4}
+~+~\frac{3^2}{2^6} \biggr]~k^4
++ \biggl[ \frac{5^2}{2^8}
++ \frac{3^2}{2^8}
+~+~\frac{3^2}{2^8}
+~+~\frac{5^2}{2^8} \biggr]~ k^6
+~+~\biggl[
+\frac{5^2 \cdot 7^2}{2^{14}}+ \frac{5^2}{2^{10}} ~+~\frac{3^4}{2^{12}} ~+~\frac{5^2}{2^{10}} ~+~\frac{5^2 \cdot 7^2}{2^{14}}
+\biggr]k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \frac{1}{2} k^2
++ \frac{11}{2^5} ~k^4
++ \frac{17}{2^6} ~ k^6
++ \frac{1787}{2^{13}} ~k^{8}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+</table>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[E(k) \cdot  E(k) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
+- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
+~-~ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
+~-~ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+\times~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
+- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
+- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
+~-~ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
+~-~ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \frac{5}{2^8}~ k^6
+- \frac{5^2\cdot 7}{2^{14}} ~k^8
+\biggr\}
+\times
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \frac{5}{2^8}~ k^6
+- \frac{5^2\cdot 7}{2^{14}} ~k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \frac{5}{2^8}~ k^6
+- \frac{5^2\cdot 7}{2^{14}} ~k^8
+\biggr\}
+~+~
+\biggl\{- \frac{1}{2^2} ~k^2
+\biggr\}
+\times
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \frac{5}{2^8}~ k^6
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+~+~
+\biggl\{
+- \frac{3}{2^6}~ k^4
+\biggr\}
+\times
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+\biggr\}
+~+~
+\biggl\{
+- \frac{5}{2^8}~ k^6
+\biggr\}
+\times
+\biggl\{
+- \frac{1}{2^2} ~k^2
+\biggr\}
+~+~
+\biggl\{
+- \frac{5^2\cdot 7}{2^{14}}k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+- \frac{1}{2^2} ~k^2
+- \frac{3}{2^6}~ k^4
+- \frac{5}{2^8}~ k^6
+- \frac{5^2\cdot 7}{2^{14}} ~k^8
+\biggr\}
+~+~
+\biggl\{
+- \frac{1}{2^2} ~k^2  + \frac{1}{2^4} ~k^4
++ \frac{3}{2^8}~ k^6
++ \frac{5}{2^{10}}~k^8
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+~+~
+\biggl\{
+~-~ \frac{3}{2^6}~ k^4
+~+~ \frac{3}{2^8}~ k^6
++ \frac{3^2}{2^{12}}~k^8
+\biggr\}
+~+~
+\biggl\{
+~-~\frac{5}{2^8}~ k^6
++ \frac{5}{2^{10}}~k^8
+\biggr\}
+~+~
+\biggl\{
+~-~ \frac{5^2\cdot 7}{2^{14}}k^8
+\biggr\}
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \biggl[ - \frac{1}{2^2}- \frac{1}{2^2} \biggr]k^2
++ \biggl[ - \frac{3}{2^6}+ \frac{1}{2^4}~-~ \frac{3}{2^6} \biggr] k^4
++ \biggl[ - \frac{5}{2^8}+ \frac{3}{2^8}~+~ \frac{3}{2^8}~-~\frac{5}{2^8} \biggr] k^6
++ \biggl[ - \frac{5^2\cdot 7}{2^{14}} + \frac{5}{2^{10}}+ \frac{3^2}{2^{12}}+ \frac{5}{2^{10}}~-~ \frac{5^2\cdot 7}{2^{14}} \biggr] k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \biggl[ - \frac{2}{2^2} \biggr]k^2
++ \biggl[ - \frac{3}{2^5}+ \frac{2}{2^5} \biggr] k^4
++ \biggl[ - \frac{5}{2^7}+ \frac{3}{2^7} \biggr] k^6
++ \biggl[ - \frac{5^2\cdot 7}{2^{13}} + \frac{5\cdot 2^4}{2^{13}}+ \frac{2 \cdot 3^2}{2^{13}}\biggr] k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 ~-~\frac{77}{2^{13}} ~ k^8
++ \mathcal{O}(k^{10})
+</math>
+  </td>
+</tr>
+</table>
+</td></tr>
+</table>
+Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~k_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2e(1+e)^{-1}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2e(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k_0^4</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~4e^2(1+e)^{-2}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~4e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k_0^6</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2^3e^3(1+e)^{-3}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2^3e^3(  1 - 3e + 6e^2 - 10e^3 + 15e^4 - 21e^5 + \cdots ) \, ;</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~k_0^8</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2^4e^4(1+e)^{-4}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2^4e^4(1 - 4e + 10 e^2 - 20e^3 + 35e^4 - 56e^5 + \cdots) \, .</math>
+  </td>
+</tr>
+</table>
+<span id="Step03">Hence, we have,</span>
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
++ \frac{11}{2^5} ~k_0^4
++ \frac{17}{2^6} ~ k_0^6
++ \frac{1787}{2^{13}} ~k_0^8
++ \mathcal{O}(k_0^{10})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
+~+~\frac{1}{2^5} ~ k_0^6
+~+~ \frac{231}{2^{13}}~k_0^8
++ \mathcal{O}(k_0^{10})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~
+~-~\frac{1}{2^6} ~ k_0^6
+~-~\frac{77}{2^{13}}~k_0^8
++ \mathcal{O}(k_0^{10})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + e (  1 - e +e^2 -e^3 )
++ \frac{11}{2^3} \cdot ~e^2(  1 - 2e  + 3e^2)
++ \frac{17}{2^6} ~\cdot 2^3 e^3 ( 1 - 3e)
++ \frac{1787}{2^{13}} \cdot 2^4e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} \cdot~4e^2(  1 - 2e  + 3e^2)
+~+~\frac{1}{2^5} ~ 2^3e^3 (1-3e)
+~+~\frac{231}{2^{13}} \cdot 2^4e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- (1+e) \biggl[ 1 - e ~(  1 - e +e^2 -e^3) ~
+-~ \frac{1}{2^5} \cdot~ 4e^2(  1 - 2e  +3e^2) ~
+~-~\frac{1}{2^6} ~ 2^3e^3(1 - 3e)
+~-~\frac{77}{2^{13}}~2^4e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1-e) \biggl[ 512 (1+  e - e^2 +e^3 -e^4 )
++ 704  \cdot ~(  e^2 - 2e^3  + 3e^4)
++ 1088 ~\cdot ( e^3 - 3e^4)
++ 1787 \cdot e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ (1+e^2) 2^{-9}\biggl[ 1024
+~+~128 \cdot~(  e^2 - 2e^3  + 3e^4)
+~+~256 ~ (e^3-3e^4)
+~+~462 \cdot e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1+e) \biggl[ 512 ~(1  -e + e^2 -e^3 + e^4) ~
+-~ 64 \cdot~ (  e^2 - 2e^3  +3e^4) ~
+~-~64 ~ (e^3 - 3e^4)
+~-~77~e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1-e) \biggl[
++ 512e + 192e^2 + e^3(512 - 1408 + 1088) + e^4(704-512 - 3264 + 1787)
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2^{-9}(1+e^2) \biggl[
++ 128e^2 + e^3(-256 + 256  ) + e^4(384 -768 + 462)
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1+e) \biggl[
+- 512e + e^2(512 - 64  ) + e^3(-512 +128 -64  ) + e^4(512 - 192 - 192 - 77)
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1-e) \biggl[
++ 512e + 192e^2 + 192e^3 - 1285 e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2^{-9}(1+e^2) \biggl[
++ 128e^2 + 78e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1+e) \biggl[
+- 512e + 448e^2 - 448 e^3 + 51e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{-9}\biggl[
+(- 192+ 128 - 448)e^2
++ (- 192 + 448) e^3
++ (1285 + 78- 51)e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2^{-9} e \biggl[
+e + (192- 448)e^2
++ (192+ 448) e^3
++ \mathcal{O}(e^{4})
+\biggr]
++ 2^{-9} e^2 \biggl[
++ 128e^2
++ \mathcal{O}(e^{3})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{-9} \biggl[
+(- 192+ 128 - 448)e^2  + 2048 e^2
++ (1285 + 78- 51)e^4
++ (192+ 448) e^4
++ 128e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^{-9} \biggl[
+e^2
++ 2080e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+e^2
++ \biggl[ \frac{5\cdot 13}{2^4}\biggr] e^4
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
+</math>
+  </td>
+</tr>
+</table>
+Hence,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+[1 + \mathcal{O}(e^{2})]\cdot (1 - e^2)^{1 / 2}
+\biggl\{
+\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, ,
+</math>
+  </td>
+</tr>
+</table>
+or, more precisely,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
+\biggl\{
+\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, .
+</math>
+  </td>
+</tr>
+</table>
+====Next Factors====
+Now,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Delta_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+(\varpi_W + R_c)^2 + z_W^2 \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_1^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  - \biggl[ (\varpi_W + R_c)^2 + z_W^2 \biggr]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\varpi_W^2 + 2\varpi_W R_c (1 - e^2 )^{1 / 2} + R_c^2 (1 - e^2 ) + z_W^2  - [\varpi_W^2 +  2\varpi_W R_c + R_c^2 + z_W^2 ]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~2\varpi_W R_c [(1 - e^2 )^{1 / 2}   - 1]  -e^2 R_c^2 \, .  </math>
+  </td>
+</tr>
+</table>
+----
+Again, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~(1 -e^2)^{1 / 2}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2}e^2 + \biggl[\frac{ \tfrac{1}{2}(-\tfrac{1}{2}) }{ 2 } \biggr]e^4 - \biggl[ \frac{ \tfrac{1}{2}(-\tfrac{1}{2} )(-\tfrac{3}{2} ) }{ 3! } \biggr]e^6
++ \biggl[ \frac{ \tfrac{1}{2} (-\tfrac{1}{2})(-\tfrac{3}{2})(-\tfrac{5}{2})  }{ 4! } \biggr]e^8 + \cdots
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{1}{2}e^2 - \frac{1}{2^3} e^4 - \frac{1}{2^4}e^6 - \frac{5}{2^7} e^8 - \mathcal{O}(e^{10}) \, .
+</math>
+  </td>
+</tr>
+</table>
+----
+<div align="center" id="Step04">
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\varpi_W R_c \biggl[- e^2 - \frac{1}{2^2} e^4 - \frac{1}{2^3}e^6 - \frac{5}{2^6} e^8 - \mathcal{O}(e^{10})  \biggr]  -e^2 R_c^2   </math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~\frac{ r_1^2}{\Delta_0^2} </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~1
+ -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr]
+- \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10})  \biggr] </math>
+  </td>
+</tr>
+</table>
+</div>
+====Now Work on Elliptic Integral Expressions====
+From a [[User:Tohline/2DStructure/ToroidalGreenFunction#Series_Expansions|separate discussion]] we can draw the series expansion of <math>~\boldsymbol{K}(k)</math>, specifically,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2K(k)}{\pi}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \biggl( \frac{1}{2} \biggr)^2k^2
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
++ \cdots
+</math>
+  </td>
+</tr>
+</table>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{k^2}{4}</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{1}{4} \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]
+= \biggl[ \frac{a\varpi}{r_1^2} \biggr]
+= \biggl[ \frac{a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2}  \, .
+</math>
+  </td>
+</tr>
+</table>
+Also,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2K(k_H)}{\pi}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
++ \biggl( \frac{1}{2} \biggr)^2k_H^2
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
++ \cdots
+</math>
+  </td>
+</tr>
+</table>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{k_H^2}{4}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{1}{\Delta^2}\biggl[ R (R_c + b\cos\theta_H) \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2}  \, .
+</math>
+  </td>
+</tr>
+</table>
+What we want to do is write <math>~K(k)</math> in terms of <math>~K(k_H)</math>.  Let's try &hellip;
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2K(k)}{\pi}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\frac{2K(k_H)}{\pi} + \delta_K \, ,</math>
+  </td>
+</tr>
+</table>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\delta_K</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left">
+<math>~\frac{2K(k)}{\pi} - \frac{2K(k_H)}{\pi} </math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{1 + \frac{k^2}{4}
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
++ \cdots
+\biggr\}
+- \biggl\{1 + \frac{k_H^2}{4}
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
++ \cdots
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{1 + \frac{k^2}{4}
+\biggr\}
+- \biggl\{1 + \frac{k_H^2}{4}
+\biggr\}
+= \frac{k^2}{4} - \frac{k_H^2}{4}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2}
+-
+\frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \varpi_W R_c(1-e^2)^{1 / 2} \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2  + 2\varpi_W R_c(1-e^2)^{1 / 2} - R_c^2 e^2 \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-
+\biggl\{ \varpi_W R_c (1 + e\cos\theta_H) \biggr\} \biggl\{[\varpi_W + R_c(1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \varpi_W R_c \biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2  + 2\varpi_W R_c\biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] - R_c^2 e^2 \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-
+\biggl\{ \varpi_W R_c \biggl[ 1 + e\cos\theta_H \biggr] \biggr\}
+\biggl\{\varpi_W^2 + 2\varpi_W R_c(1+e\cos\theta_H)
++ R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H)  + z_W^2 - 2 z_W R_c e\sin\theta_H + R_c^2 e^2 \sin\theta^2_H  \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{\varpi_W R_c  -  e^2  \biggl[ \frac{\varpi_W R_c}{2} \biggr]  + \mathcal{O}(e^4) \biggr\}
+\biggl\{ [ (\varpi_W +R_c)^2 + z_W^2 ] - e^2 \biggl[  \varpi_W R_c + R_c^2 \biggr]  + \mathcal{O}(e^4) \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-
+\biggl\{ \varpi_W R_c  +  e \biggl[ \varpi_W R_c \cos\theta_H \biggr]\biggr\}
+\biggl\{ [ (\varpi_W^2 + R_c)^2 + z_W^2 ]
++ 2R_c e(R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H) + R_c^2 e^2  \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr]  \biggr\}
+\biggl\{ 1 - e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~
+\biggl\{ \frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  } \biggl[ 1 + e\cos\theta_H \biggr] \biggr\}
+\biggl\{ 1
++ e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
++ e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr]  \biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr]
+\biggl\{ 1 + e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~\frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  } \biggl[ 1 + e\cos\theta_H \biggr]
+\biggl\{ 1
+- e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
+- e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr]  \biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+-~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  }
+\biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~\frac{e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] }
+\biggl\{ \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } - \frac{1}{2} \biggr]
+- \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr]
+- \cos\theta_H \biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+-~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  [ (\varpi_W^2 + R_c)^2 + z_W^2 ] }
+\biggl[ \cos\theta_H [(\varpi_W^2 + R_c)^2 + z_W^2 ]  -2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H) \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~\frac{\tfrac{1}{2}e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ][ (\varpi_W^2 + R_c)^2 + z_W^2 ] }
+\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- 2R_c^2
+- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+-~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 }
+\biggl[ \cos\theta_H   - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 }
+\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- 2R_c^2
+- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+Let's subtract <math>~K([k_H]_0)</math> from the potential expression.  But first, let's adopt the shorthand notation &hellip;
+<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
+Given that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{
+\boldsymbol{K}(k)  \biggr\} \frac{\Delta_0}{r_1}~\biggl( \frac{2^2}{3\pi^2} \biggr) \Upsilon_{W0}(\eta_0)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl[\frac{r_1^2 }{ \Delta_0^2 } \biggr]^{-1 / 2}
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\}
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2  - \mathcal{O}(e^{4}) \biggr]
+</math>
+  </td>
+</tr>
+</table>
+let's define the variable, <math>~\mathcal{A}</math>, such that,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{\boldsymbol{K}(k)\biggr\} \{1 + e^2\mathcal{A}\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \{ 1 + e^2 \cdot \mathcal{A} \}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\}
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2  - \mathcal{O}(e^{4}) \biggr]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] \biggr\}
+\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2  \biggr] \biggl[ 1 - \frac{1}{2}e^2   \biggr]</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2  - \frac{1}{2}e^2 \biggr\}</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ 2\mathcal{A}</math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{41}{2^3\cdot 3}\biggr) \, . </math>
+  </td>
+</tr>
+</table>
+</td></tr></table>
+We can therefore write,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~- K([k_H]_0) +
+\biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\}
+\{ 1 + e^2 \cdot \mathcal{A} \}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~- K([k_H]_0)
++
+K(k_H)
+\{ 1 + e^2 \cdot \mathcal{A} \}
++
+\frac{\pi}{2} \cdot \delta_K \, ,
+</math>
+  </td>
+</tr>
+</table>
+where we should keep in mind that <math>~\delta_k</math> is <math>~\mathcal{O}(e^1)</math>.  So, let's examine the piece,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+ \biggl\{1 + \frac{k_H^2}{4}
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
++ \cdots
+\biggr\} -
+ \biggl\{1 + \frac{k_H^2}{4}
++ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
++ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
++ \cdots
+\biggr\}_{e\rightarrow 0}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{k_H^2}{4} - \biggl[ \frac{k_H^2}{4} \biggr]_{e\rightarrow 0}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]
+-
+\biggl[  \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]_{e\rightarrow 0}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\varpi_W R_c (1 + e\cos\theta_H)
+\biggl\{ [\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1}
+-
+\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\varpi_W R_c (1 + e\cos\theta_H)
+\biggl\{
+\varpi_W^2 + 2\varpi_W R_c(1 + e\cos\theta_H) + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 -2z_W R_c e\sin\theta_H + R_c^2 e^2\sin^2\theta_H
+\biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-
+\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+-\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
++ \biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
+\biggl\{1
++ \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
++ \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
+\biggr\}^{-1}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+-\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
++\biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
+\biggl\{1
+- \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
++ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{ [(\varpi_W + R_c)^2 + z_W^2]^2}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
+\biggl\{
+- \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
++ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{ [(\varpi_W + R_c)^2 + z_W^2]^2}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
+\biggl\{ \frac{e\cos\theta_H [(\varpi_W + R_c)^2 + z_W^2] -e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr]
+\biggl\{
+e\cos\theta_H
+- \frac{ eR_c\cos\theta_H [2\varpi_W  + 2 R_c - 2z_W  \tan\theta_H]  }{ \Delta_0^2}
+- \frac{e^2 R_c^2 }{ \Delta_0^2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{ \Delta_0^2}
++ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^4}
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+</table>
+Now we have,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~- K([k_H]_0)
++
+K(k_H)
+\{ 1 + e^2 \cdot \mathcal{A} \}
++
+\frac{\pi}{2} \cdot \delta_K
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ - \Delta_0 \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\frac{2}{\pi} K([k_H]_0)
++ \frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]
++
+\delta_K
++
+\frac{2}{\pi} K(k_H) e^2 \cdot \mathcal{A} \, .
+</math>
+  </td>
+</tr>
+</table>
+But, as we have just demonstrated,
+ <table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]+ \delta_K
+</math>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr]
+\biggl\{
+e\cos\theta_H
+- \frac{ eR_c\cos\theta_H [2\varpi_W  + 2 R_c - 2z_W  \tan\theta_H]  }{ \Delta_0^2}
+- \frac{e^2 R_c^2 }{ \Delta_0^2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{ \Delta_0^2}
++ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^4}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+-~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 }
+\biggl[ \cos\theta_H   - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 }
+\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- 2R_c^2
+- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+\biggr\} \, .
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
+\biggl\{
+- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- R_c^2
+- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr><td align="center" colspan="3"><font color="red">TEMPORARY BREAK HERE</font></td></tr>
+</table>
+Hence,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~ - \frac{\pi  \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+K([k_H]_0)
++
+K(k_H) e^2 \cdot \mathcal{A}
++~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
+\biggl\{
+- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- R_c^2
+- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
+\biggr\}
+</math>
+  </td>
+</tr>
+</table>
+===Include Second Wong Term===
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
+\Upsilon_{W1}(\eta_0) \times \cos\theta
+\biggl\{ \frac{a}{r_2} \cdot
+\boldsymbol{E}(k) \biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl( \frac{2^{2} }{3\pi^2} \biggr)
+\Upsilon_{W1}(\eta_0) \times \cos\theta
+\biggl\{ \frac{\Delta_0}{r_2} \cdot
+\boldsymbol{E}(k) \biggr\} \, ;
+</math>
+  </td>
+</tr>
+</table>
+====Leading (Upsilon) Coefficient====
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~\equiv</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
+\biggl\{
+K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
++~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
+-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
+\biggr\}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow~~~ \biggl[ \frac{e^2}{ (1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left" colspan="2">
+<math>~
+- (1-e)K(k_0)\cdot K(k_0)
++~2(3-e^2)K(k_0)\cdot E(k_0)
+-~5(1+e) E(k_0)\cdot E(k_0)  \, .
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
++ \frac{11}{2^5} ~k_0^4
++ \frac{17}{2^6} ~ k_0^6
++ \frac{1787}{2^{13}} ~k_0^8
++ \mathcal{O}(k_0^{10})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
+~+~\frac{1}{2^5} ~ k_0^6
+~+~ \frac{231}{2^{13}}~k_0^8
++ \mathcal{O}(k_0^{10})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 5(1+e) \biggl[ 1
+- ~\frac{1}{2} ~k_0^2 ~
+-~ \frac{1}{2^5} ~ k_0^4 ~
+~-~\frac{1}{2^6} ~ k_0^6
+~-~\frac{77}{2^{13}}~k_0^8
++ \mathcal{O}(k_0^{10})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- (1-e) \biggl[ 1 + \frac{1}{2} \cdot 2e(  1 - e +e^2 - e^3 )
++ \frac{11}{2^5} ~\cdot 4e^2(  1 - 2e + 3e^2 )
++ \frac{17}{2^6} ~ \cdot 2^3e^3(  1 - 3e  )
++ \frac{1787}{2^{13}} ~\cdot 2^4e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~\cdot 4e^2(  1 - 2e + 3e^2 )
+~+~\frac{1}{2^5} ~ \cdot 2^3e^3(  1 - 3e  )
+~+~ \frac{231}{2^{13}}~\cdot 2^4e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 5(1+e) \biggl[ 1
+- ~\frac{1}{2} ~\cdot 2e(  1 - e +e^2 - e^3 ) ~
+-~ \frac{1}{2^5} ~ \cdot 4e^2(  1 - 2e + 3e^2 )
+~-~\frac{1}{2^6} ~ \cdot 2^3e^3(  1 - 3e  )
+~-~\frac{77}{2^{13}}~\cdot 2^4e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(1-e) \biggl[ 2^9 + 2^9e(  1 - e +e^2 - e^3 )
++ 2^6 \cdot 11 ~\cdot e^2(  1 - 2e + 3e^2 )
++ 2^6\cdot 17 ~ \cdot e^3(  1 - 3e  )
++ 1787 ~\cdot e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2^{-9} (6- 2e^2) \biggl[ 2^9 ~+~2^6~\cdot e^2(  1 - 2e + 3e^2 )
+~+~2^7 \cdot e^3(  1 - 3e  )
+~+~ 231~\cdot e^4
++ \mathcal{O}(e^{5})
+ \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
+- 2^{-9}(5+ 5e) \biggl[ 2^9
+- ~2^9 \cdot e(  1 - e +e^2 - e^3 ) ~
+-~ 2^6 \cdot e^2(  1 - 2e + 3e^2 )
+~-~2^6 \cdot e^3(  1 - 3e  )
+~-~77~\cdot e^4
++ \mathcal{O}(e^{5})
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~\frac{2^{11}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^9 - 2^9e(  1 - e +e^2 - e^3 )
+- 2^6 \cdot 11 ~\cdot e^2(  1 - 2e + 3e^2 )
+- 2^6\cdot 17 ~ \cdot e^3(  1 - 3e  )
+- 1787 ~\cdot e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 2^9e + 2^9e^2(  1 - e +e^2 )
++ 2^6 \cdot 11 ~\cdot e^3(  1 - 2e )
++ 2^6\cdot 17 ~ \cdot e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 6\biggl[ 2^9 ~+~2^6~\cdot e^2(  1 - 2e + 3e^2 )
+~+~2^7 \cdot e^3(  1 - 3e  )
+~+~ 231~\cdot e^4
+ \biggr]
+-2^{10}e^2 ~-~2^7~\cdot e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 5 \biggl[ -2^9
++ ~2^9 \cdot e(  1 - e +e^2 - e^3 ) ~
++~ 2^6 \cdot e^2(  1 - 2e + 3e^2 )
+~+~2^6 \cdot e^3(  1 - 3e  )
+~+~77~\cdot e^4
+\biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~5 \biggl[ -2^9e
++ ~2^9 \cdot e^2(  1 - e +e^2 ) ~
++~ 2^6 \cdot e^3(  1 - 2e )
+~+~2^6 \cdot e^4
+\biggr]
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^9( e^2 - e^3 + e^4 )
+- 2^6 \cdot 11 ~(  e^2 - 2e^3 + 3e^4 )
+- 2^6\cdot 17 ~(  e^3 - 3e^4  )
+- 1787 ~e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++2^9 (  e^2 - e^3 +e^4 )
++ 2^6 \cdot 11 (  e^3 - 2e^4 )
++ 2^6\cdot 17 ~ e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~ 3\cdot 2^7~(  e^2 - 2e^3 + 3e^4 )
+~+~3\cdot 2^8 (  e^3 - 3e^4  )
+~+~ 2\cdot 3^2 \cdot 7\cdot 11~\cdot e^4
+-2^{10}e^2 ~-~2^7~\cdot e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ 5\cdot 2^9 (  - e^2 +e^3 - e^4 ) ~
++~ 5\cdot 2^6 (  e^2 - 2e^3 + 3e^4 )
+~+~5\cdot 2^6 (  e^3 - 3e^4  )
+~+~5\cdot 77~e^4
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++5\cdot 2^9 (  e^2 - e^3 +e^4 ) ~
++~ 5\cdot 2^6 (  e^3 - 2e^4 )
+~+~5\cdot 2^6 e^4
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+e^2 [ 2^9 - 2^6\cdot 11 + 2^9 + 3\cdot 2^7 -2^{10} -5\cdot 2^9 + 5\cdot 2^6 + 5\cdot 2^9 ]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ e^3 [ -2^9 - 2^7\cdot 11 - 2^6\cdot 17 - 2^9 +2^6\cdot 11 - 3\cdot 2^8+3\cdot 2^8  + 5\cdot 2^9-5\cdot 2^7 +5\cdot 2^6 -5\cdot 2^9 + 5\cdot 2^6]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ e^4 [ 2^9 - 3\cdot 11\cdot 2^6+3\cdot 17\cdot 2^6 - 1787 + 2^9 - 11\cdot 2^7 + 17\cdot 2^6 + 3^2\cdot 2^7 - 3^2\cdot 2^8+ 2\cdot 3^2\cdot 7\cdot 11 - 2^7
+- 5\cdot 2^9 + 3\cdot 5\cdot 2^6 - 3\cdot 5\cdot 2^6 + 5\cdot 7\cdot 11  + 5\cdot 2^9 - 5\cdot 2^7 +5\cdot 2^6]
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+e^2 [ - 2^6\cdot 11  + 3\cdot 2^7  + 5\cdot 2^6  ]
++ e^3 [ -2^{10} - 2^7\cdot 11 - 2^6\cdot 17 +2^6\cdot 11  ]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ e^4 [ 2^{10}~-~ 1787~+~ 2\cdot 3^2\cdot 7\cdot 11~+~ 5\cdot 7\cdot 11 ~+~ 2^6\cdot (3\cdot 17 - 3\cdot 11 - 22 + 17 + 18 - 36 - 2 - 10 + 5)  ]
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+^6 e^2 [ 0  ] ~- 2^8 \cdot 11 e^3 + e^4 [ 2^{10}~-~ 1787~+~ 7\cdot 11\cdot 23 ~-~ 2^8\cdot 3  ]
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- 2^8 \cdot 11 e^3 + 2^4\cdot 3\cdot 5e^4
++ \mathcal{O}(e^{5})
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~\frac{2^{2}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+- \frac{11}{2} e^3 + \frac{3\cdot 5}{2^5} e^4
++ \mathcal{O}(e^{5}) \, .
+</math>
+  </td>
+</tr>
+</table>
+<table border="1" cellpadding="8" align="center" width="90%">
+<tr>
+  <td align="center" bgcolor="black"><font color="white">'''Floating Comparison Summary'''</font></td>
+</tr>
+<tr><td align="left">
+As [[#Step01|shown above]], the first three terms of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] series expression may be written as,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\boldsymbol{K}([k_H]_0)
+-
+\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H)
++  \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]  \boldsymbol{E}(k_H) \, .
+</math>
+  </td>
+</tr>
+</table>
+Let's see how it compares to the first term of Wong's (1973) expression which, as [[#Step02|shown separately above]], can be written in the form,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr)
+\Upsilon_{W0}(\eta_0) \biggl\{
+\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, .
+</math>
+  </td>
+</tr>
+</table>
+----
+First, as [[#Step03|shown above]],
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
+</math>
+  </td>
+</tr>
+</table>
+Note that, in order to determine the functional form of the <math>~\mathcal{O}(e^{2})</math> term in this expression, we will have to include <math>~k_0^8</math> terms in the various expressions for products of elliptic integrals.  Second, [[#Step04|we have shown that]],
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\frac{ r_1^2}{\Delta_0^2} </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~1
+ -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr]
+- \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10})  \biggr] </math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\Rightarrow ~~~ \frac{\Delta_0}{r_1} </math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+  + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr]
+ \, ,</math> &nbsp; &nbsp; &nbsp; and we are defining <math>~\delta_K</math> such that,
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~K(k)</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~K(k_H) + \frac{\pi}{2} \cdot \delta_K \, .</math>
+  </td>
+</tr>
+</table>
+----
+Hence,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\}
+\biggl\{ 1  + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] \biggr\}\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~
+\boldsymbol{K}([k_H]_0)
++
+\boldsymbol{K}(k_H) e^2 \cdot \mathcal{A}
++~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
+\biggl\{
+- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ]
+- R_c^2
+- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+&nbsp;
+  </td>
+  <td align="left">
+<math>~
++ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
+\biggr\} \, ,
+</math>
+  </td>
+</tr>
+</table>
+and,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\mathcal{A}</math>
+  </td>
+  <td align="center">
+<math>~\approx</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{2\Delta_0^2}\biggr] + \biggl( \frac{41}{2^4\cdot 3}\biggr) \, . </math>
+  </td>
+</tr>
+</table>
+----
+Second,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\boldsymbol{E}(k)
+\biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\}
+\biggl[ \biggl( \frac{2^{2} }{3\pi^2} \biggr) \Upsilon_{W1}(\eta_0) \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+\biggl\{ \boldsymbol{E}(k_H) + \frac{\pi}{2}\cdot \delta_E \biggr\}
+\biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\}
+\biggl[- \biggl(\frac{11}{2\cdot 3} \biggr) e + \biggl(\frac{5}{2^5}\biggr) e^2 + \mathcal{O}(e^{5}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
+</math>
+  </td>
+</tr>
+</table>
+</td></tr>
+</table>
+====Geometric Factor====
+By definition,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\Delta_0^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~
+(\varpi_W + R_c)^2 + z_W^2 \, ,
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_1^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~r_2^2</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \varpi_W - R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+<math>~\cos^2\theta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~\biggl[ \frac{r_1^2 + r_2^2 - 4R_c^2(1-e^2)}{2r_1 r_2} \biggr]^2  \, .</math>
+  </td>
+</tr>
+</table>
+Hence,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~r_2^2 - \Delta_0^2 \cdot \cos^2\theta</math>
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~</math>
+  </td>
+</tr>
+</table>
+=See Also=
+<ul>
+<li>Universit&eacute; de Bordeaux (Part 2): &nbsp;[[User:Tohline/Appendix/Ramblings/BordeauxSequences|Spheroid-Ring Sequences]]</li>
+<li>Universit&eacute; de Bordeaux (Part 3): &nbsp;[[User:Tohline/Appendix/Ramblings/BordeauxPostDefense|Discussions Following Dissertation Defense]]</li>
+</ul>
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User:Tohline/Appendix/Ramblings/Bordeaux

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Current revision as of 17:05, 16 December 2020

Contents

Université de Bordeaux (Part 1)

Spheroid-Ring Systems

Exterior Gravitational Potential of Toroids

Our Presentation of Wong's (1973) Result

Setup

Leading (n = 0) Term

Wong's Expression

Thin-Ring Evaluation of C0

More General Evaluation of C0

Thin-Ring Evaluation of C₀

More General Evaluation of C₀