Editing Apps/DysonPotential (section)

====Our Attempt to Replicate====
First, let's test the accuracy of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson's (1893a)] "series expansion" expression for the elliptic integrals, <math>~K(\mu)</math> and <math>~E(\mu)</math>; in the following table, the high-precision evaluations labeled "''Numerical Recipes''" have been drawn from the tabulated data that is provided in our [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals|accompanying discussion]] of incomplete elliptic integrals.  Drawing from our [[Appendix/SpecialFunctions#Complete_Elliptic_Integrals|accompanying set of Key mathematical relations]] &#8212; in which <math>~k</math>, rather than <math>~\mu</math>, represents the function modulus &#8212; the relevant series-expansion expressions are:
<div align="center">
{{ Math/EQ_EllipticIntegral01 }}<br />
{{ Math/EQ_EllipticIntegral02 }}
</div>

These expressions &#8212; up through <math>~\mathcal{O}(\mu^4)</math> &#8212; can be found in the middle of p. 58 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)].  We strongly suspect that, in constructing the equipotential contours shown in his figures 1-6, Dyson used expressions for <math>~K(\mu)</math> and <math>~E(\mu)</math> that were more accurate than this.  For example, we found it necessary to include terms up through <math>~\mathcal{O}(\mu^{10})</math> in order to match to three digits accuracy the potential contour values and coordinate locations reported by Dyson.

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" rowspan="2"><math>~\mu</math></td>
  <td align="center" colspan="2">Numerical Recipes</td>
  <td align="center" colspan="2">Series expansion up through <math>~\mathcal{O}(\mu^4)</math></td>
  <td align="center" colspan="2">Series expansion up through <math>~\mathcal{O}(\mu^{10})</math></td>
</tr>
<tr>
  <td align="center"><math>~K(\mu)</math></td>
  <td align="center"><math>~E(\mu)</math></td>
  <td align="center"><math>~K(\mu)</math></td>
  <td align="center"><math>~E(\mu)</math></td>
  <td align="center"><sup>&dagger;</sup><math>K(\mu)~</math></td>
  <td align="center"><math>~E(\mu)</math></td>
</tr>
<tr>
  <td align="right">0.34202014</td>
  <td align="right">1.62002589</td>
  <td align="right">1.52379921</td>
  <td align="right">1.6198</td>
  <td align="right">1.5239</td>
  <td align="right">1.6200263</td>
  <td align="right">1.5237989</td>
</tr>
<tr>
  <td align="right">0.57357644</td>
  <td align="right">1.73124518</td>
  <td align="right">1.43229097</td>
  <td align="right">1.7239</td>
  <td align="right">1.4336</td>
  <td align="right">1.73124518</td>
  <td align="right">1.43230</td>
</tr>
<tr>
  <td align="right">0.76604444</td>
  <td align="right">1.93558110</td>
  <td align="right">1.30553909</td>
  <td align="right">1.8773</td>
  <td align="right">1.3150</td>
  <td align="right">1.93558109</td>
  <td align="right">1.3061</td>
</tr>
<tr>
  <td align="right">0.90630779</td>
  <td align="right">2.30878680</td>
  <td align="right">1.16382796</td>
  <td align="right">2.042</td>
  <td align="right">1.199</td>
  <td align="right">2.308784</td>
  <td align="right">1.1700</td>
</tr>
<tr>
  <td align="right">0.98480775</td>
  <td align="right">3.15338525</td>
  <td align="right">1.04011440</td>
  <td align="right">2.16</td>
  <td align="right">1.12</td>
  <td align="right">3.150</td>
  <td align="right">1.069</td>
</tr>
<tr>
  <td align="left" colspan="7">
<sup>&dagger;</sup>We actually used the "descending Landen transformation" to evaluate <math>~K(\mu)</math> through <math>~\mathcal{O}(\mu^{10})</math>.
  </td>
</tr>
</table>


For <math>~c=1</math> and a specification of the ratio, <math>~a/c</math>, take the following steps to map out an equipotential curve that has <math>~V_2 = V_0</math>:
* Choose a value of <math>~R \ge a</math>
** ''Guess'' a value of <math>~(c-R) \le R_1 \le (c+R) ~~~\Rightarrow ~~~ \varpi = (R_1^2 - R^2)/(4c)</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~z = \pm \sqrt{ R_1^2 - (c+\varpi)^2}</math>
** Set <math>~ \cos\psi = (R_1^2 + R^2 - 4c^2)/(2RR_1)</math>
** Evaluate the function, <math>~V_2</math>
** If <math>~V_2 \ne V_0</math> to the desired accuracy, loop back up and guess another value of <math>~R_1</math>
* If <math>~V_2 = V_0</math> to the desired accuracy, save the coordinate location, <math>~(\varpi,z)</math>, and loop back up to pick another value of <math>~R</math>


<table border="1" align="center" cellpadding="8">
<tr><td align="center" colspan="1" bgcolor="lightgreen">
''Top Panel''<br />(as above) '''Figure 3 extracted without modification from p. 65 of <br />[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<br />
''II.&nbsp; The Potential of an Anchor Ring''<br /> Phil. Trans. Royal Soc. London. A., Vol. 184<br />[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]<br />
[[File:PermissionsRectYellow.png|75px|link=Appendix/Permissions#Dyson1893]]
</td></tr>
<tr>
<td rowspan="1" align="left" valign="bottom">
[[File:RoverD5over2.png|800px|center|The Potential Exterior to an Anchor Ring; R/d = 2.5]]
</td>
</tr>
<tr>
  <td align="left" valign="bottom">&nbsp; &nbsp;[[File:DysonCompare01.png|625px|Compare with Dyson]]</td>
</tr>
</table>