Editing Apps/MaclaurinSpheroids/GoogleBooks (section)

=Excerpts from A Treatise of Fluxions=
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! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like|<b>Maclaurin's<br />Original Text<br />&amp;<br />Analysis<br />(1742)</b>]]
|}
Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via [https://books.google.com/books/about/A_Treatise_of_Fluxions.html?id=dCAOAAAAQAAJ Google Books]. (Check out [http://mathworld.wolfram.com/Fluxion.html Wolfram's explanation of the term, ''fluxion''.]) In what follows, we provide an abbreviated table of contents for both volumes and present selected excerpts from these volumes.

<table border="1" align="center" width="50%">
<tr>
  <th align="center" colspan="2">
Title Pages with Google-Books Links
  </th>
</tr>
<tr>
  <td align="center" width="50%">
[[File:TitlePage1stEdition01.png|center|border|200px|Volume I (1742)]]
  </td>
  <td align="center">
[[File:TitlePage2ndEdition01.png|center|border|200px|Volume II (2<sup>nd</sup> Ed., 1801)]]
  </td>
</tr>
<tr>
  <td align="center">
[https://books.google.com/books?id=HqZeAAAAcAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume I (1<sup>st</sup> Ed., 1742)]
  </td>
  <td align="center">
[https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume II (2<sup>nd</sup> Ed., 1801)]
  </td>
</tr>
</table>


As is illustrated by the images displayed in the following two panels (click in one of the panels in order to view a higher-resolution image), each published "Figure Plate" was digitized (by Google Books) in segments that must be digitally pieced together in order to reconstruct some of the individual figures.  See, for example, the discussion that references [[#Paragraph630|Fig. 282, below]].

<table border="1" align="center">
<tr>
  <th align="center" colspan="2">
Example Digitized Figure Plates
  </th>
</tr>
<tr>
  <td align="center>
[[File:PlateXXXII_vol2.png|center|border|250px|Plate XXXII from Volume II (2<sup>nd</sup> Ed., 1801)]]
  </td>
  <td align="center>
[[File:PlateXXXIII_vol2.png|center|border|250px|Plate XXXIII from Volume II (2<sup>nd</sup> Ed., 1801)]]
  </td>
</tr>
<tr>
  <td align="center">
Plate XXXII
  </td>
  <td align="center">
Plate XXXIII
  </td>
</tr>
</table>


<div style="clear:both"></div>
==Volume I==
<table border="0" width="90%" align="center">
<tr>
  <td align="left">Dedication (an 18<sup>th</sup> Century Acknowledgment)</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">Preface</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">Introduction</td><td align="right">1</td>
</tr>
<tr>
  <td align="left">Book 1 (Of the Fluxions of Geometrical Magnitudes)</td><td align="right">51</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter I (Of the Grounds of this Method) &#8212; &sect;&sect; 1-77</td><td align="right">51</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter II (Of the Fluxions of plane rectilineal Figures) &#8212; &sect;&sect; 78-104</td><td align="right">109</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter III (Of the Fluxions of plane curvilineal Figures) &#8212; &sect;&sect; 105-123</td><td align="right">131</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter IV (Of the Fluxions of Solids, and of third Fluxions) &#8212; &sect;&sect; 124-139</td><td align="right">142</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter V (Of the Fluxions of Quantities that are in a continued Geometrical Progression, the first term of which is invariable) 
&#8212; &sect;&sect; 140-150</td><td align="right">152</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter VI (Of Logarithms, and of the Fluxions of logarithmic Quantities) 
&#8212; &sect;&sect; 151-179</td><td align="right">158</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter VII (Of the Tangents of curve Lines) 
&#8212; &sect;&sect; 180-214</td><td align="right">178</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter VIII (Of the Fluxions of curve Surfaces) 
&#8212; &sect;&sect; 215-237</td><td align="right">199</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter IX ([Identifying Extrema and Inflection Points] of Curves that are defined by a common or by a fluxional Equation) 
&#8212; &sect;&sect; 238-285</td><td align="right">214</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter X (Of the Asymptotes of curve Lines, the Areas bounded by them and &hellip;) 
&#8212; &sect;&sect; 286-362</td><td align="right">240</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter XI (Of the Curvature of Lines &hellip; different kinds of Contact [with other Curves] &hellip; Caustics &hellip; centripetal Forces &hellip;) 
&#8212; &sect;&sect; 363-</td><td align="right">304</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Parabola &#8212; &sect;371</td><td align="right">311</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Any Conic Section &#8212; &sect;373</td><td align="right">312</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> The Second Fluxion of a Curve &#8212; &sect;384</td><td align="right">324</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Refraction of Light &#8212; &sect;413</td><td align="right">344</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Centripetal and Centrifugal Forces &#8212; &sect;416</td><td align="right">346</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Gravity &#8212; &sect;419</td><td align="right">348</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Circular Motion &#8212; &sect;432</td><td align="right">356</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> When the Center of Forces is the Focus of a Conic Section &#8212; &sect;446</td><td align="right">370</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> When Gravity is Uniform or Varies as any Power of the Distance &#8212; &sect;458</td><td align="right">383</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Orbit of the Moon, taking into account the Gravity of both the Earth and the Sun &#8212; &sect;471</td><td align="right">391</td>
</tr>
<tr>
  <td align="center" colspan="2">[[File:MoonOrbitQuoteSect487.png|400px|thumb|center|Extracted directly from &sect;487 of Maclaurin's Book 1, as digitized by Google]]</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Prolate Spheroidal Fluid Figure &#8212; &sect;491</td><td align="right">409</td>
</tr>
<tr>
  <td align="center" colspan="2">[[File:ProlateSpheroidQuoteSect491.png|400px|thumb|center|Extracted directly from &sect;491 of Maclaurin's Book 1, as digitized by Google]]</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> The Earth's Equilibrium Shape &#8212; &sect;492</td><td align="right">410</td>
</tr>
<tr>
  <td align="center" colspan="2">[[File:EarthAsOblateSpheroidQuoteSect492.png|400px|thumb|center|Extracted directly from &sect;492 of Maclaurin's Book 1, as digitized by Google]]</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; End of Book I, Chapter XI, &sect; 494</td><td align="right">413</td>
</tr>
</table>


<div style="clear:both"></div>

==Volume II==
<table border="0" width="90%" align="center">
<tr>
  <td align="left">Table of Principal Contents (5 pp.)</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">Figure Plates (30 pp.)</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">Book 1 (continued)</td><td align="right">1</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter XII (Of the Methods of Infinitesimals &hellip;) &#8212; &sect;&sect; 495-570</td><td align="right">1</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Centre of Gravity &#8212; &sect;510</td><td align="right">13</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Collision of Bodies &#8212; &sect;511</td><td align="right">14</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Descent of Bodies that Act upon One Another &#8212; &sect;521</td><td align="right">27</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Centre of Oscillation &#8212; &sect;533</td><td align="right">40</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Motion of Water Issuing from a Cylindric Vessel &#8212; &sect;537</td><td align="right">44</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Catenaria &#8212; &sect;551</td><td align="right">59</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> General Observations Concerning the Angles of Contact, etc. &#8212; &sect;554</td><td align="right">61</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> General Observations Concerning centripetal Forces, etc. &#8212; &sect;563</td><td align="right">67</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter XIII (Determining the Lines of swiftest Descent in any Hypothesis of Gravity &hellip;) &#8212; &sect;&sect; 571-608</td><td align="right">74</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> When Gravity is Directed Towards a Given Centre &#8212; &sect;578</td><td align="right">80</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Isoperimetrical Problems &#8212; &sect;588</td><td align="right">88</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> The Solid of Least Resistance &#8212; &sect;606</td><td align="right">100</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; Chapter XIV (Of the Ellipse Considered as the Section of a Cylinder &hellip; Of the Figure of the Earth &hellip;) &#8212; &sect;&sect; 609-</td><td align="right">101</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Properties of the Ellipse &#8212; &sect;609</td><td align="right">101</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Gravitation towards Spheres and Spheroids &#8212; &sect;628</td><td align="right">110</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Of the Figures of Planets (including effects of rotation) &#8212; &sect;636</td><td align="right">116</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Key Concluding Theorem! &#8212; &sect;641</td><td align="right">119</td>
</tr>
<tr>
  <td align="center" colspan="2">[[File:Vol2Paragraph641.png|400px|thumb|center|Extracted directly from &sect;641 of Maclaurin's Book 1, as digitized by Google]]</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Using geometric relations, illustrate how to quantitatively determine gravitational acceleration &hellip;</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; [see related comment by [[#Todhunter1873|Todhunter (1873), below]]]</td><td align="right">&nbsp;</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
due to an arbitrarily shaped, uniform-density configuration &#8212; &sect;642</td><td align="right">120</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
outside or on the surface of a uniform-density sphere  &#8212; &sect;643</td><td align="right">121</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
at the pole of an oblate spheroid  &#8212; &sect;644</td><td align="right">122</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
at the equator of an oblate spheroid  &#8212; &sect;645</td><td align="right">123</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
at both the pole and the equator of an oblong (''i.e.,'' prolate) spheroid  &#8212; &sect;647</td><td align="right">125</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; <b>&#9656;</b> 
due to a pair of concentric, confocal spheroids  &#8212; &sect;648</td><td align="right">127</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Apply Specifically to the Earth &#8212; &sect;661</td><td align="right">136</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> What if the Earth's Density isn't Uniform but, instead, Varies Linearly with Distance? &#8212; &sect;670</td><td align="right">143</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Jupiter &#8212; &sect;682</td><td align="right">152</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Tides &#8212; &sect;686</td><td align="right">154</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <b>&middot;</b> Concluding Paragraph &#8212; &sect;696</td><td align="right">161</td>
</tr>
<tr>
  <td align="center" colspan="2">[[File:Vol2Paragraph696V2.png|400px|thumb|center|Extracted directly from &sect;696 of Maclaurin's Book 1, as digitized by Google]]</td>
</tr>
<tr>
  <td align="left">&nbsp; &nbsp; &nbsp; End of Book I</td><td align="right">162</td>
</tr>
</table>



<div style="clear:both"></div>

==Maclaurin's Discussion of Self-Gravitating, Oblate-Spheroidal Configurations==

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" colspan="2">
Paragraph (and related figure) extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<br />
"''A Treatise of Fluxions''"<br />
Volume II, Chapter XIV, &sect;628
</td></tr>
<tr>
  <td align="center" rowspan="2">
[[File:Vol2Paragraph628.png|400px|center|Maclaurin (1742)]]
  </td>
  <td align="center" rowspan="1">
[[File:Vol2Figure283.png|300px|center|Maclaurin (1742)]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:frustums.jpg|link=http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum|300px|center|To be inserted: Diagram from MATHalino.com]]
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
<sup>&dagger;</sup>As displayed here (left panel), this paragraph has been pieced together from two text segments found on separate but sequential pages (pp. 110-111) of Google's digitized volume.  The diagram labeled Fig. 283 (top-right panel) has been extracted from Maclaurin's [http://www.vistrails.org/index.php/File:PlateXXXII_vol2.png Figure Plate XXXIII], which appears near the beginning of the same digitized file, and has been displayed here without modification.  The diagrams in the bottom-right panel have been reposted from [http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum Mathalino.com] in an effort to illustrate what Maclaurin means by "frustum of a cone."
  </td>
</tr>
</table>
</div>


<div align="center" id="Paragraph630">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" colspan="2">
Paragraph (and a pair of related diagrams) extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<p></p>
"''A Treatise of Fluxions''"<p></p>
Volume II, Chapter XIV, &sect;630
</td></tr>
<tr>
  <td align="center">
[[File:Vol2PP630and631.png|400px|center|Maclaurin (1742)]]
  </td>
  <td align="center">
[[File:Figure282Whole.png|400px|center|Maclaurin (1742)]]
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
<sup>&dagger;</sup>The paragraph (left panel) has been extracted from p. 111 of Google's digitized volume and displayed here without modification.  The pair of diagrams (right panel) has been extracted from [http://www.vistrails.org/index.php/File:PlateXXXII_vol2.png Figure Plate XXXIII], which appears near the beginning of the same digitized file.  Note that the diagram on the right has been (poorly) pieced together from segments that appear on two separate pages of Google's digitized volume, presumably because the figure plate, itself, is folded in the original print publication.  
  </td>
</tr>
</table>
</div>

==Interpreting Maclaurin's Key Concluding Theorem==

As noted above, in line with the [[#Volume_II|table of contents for Volume II]], the key theorem resulting from Maclaurin's analysis is &hellip;
[[File:Vol2Paragraph641.png|400px|thumb|center|Extracted directly from &sect;641 of Maclaurin's Book 1, as digitized by Google]]

Here we assess this ''geometrically'' formulated statement using today's more common differential operators and algebraic expressions.  In particular, we draw from the [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|discussion of Maclaurin spheroids that has been presented in an accompanying chapter]] of this H_Book.

<font color="maroon">'''CA'''</font>:  This is the semi-minor (polar) axis of the spheroid, which we have called, <math>a_3</math>. 

<font color="maroon">'''CD'''</font>:  This is the semi-major (equatorial) radius of the spheroid, which we have called, <math>a_1</math>. 

With both <math>a_1</math> and <math>a_3</math> specified, the eccentricity of the oblate spheroid is also known via the expression,

<div align="center">
<math>
e^2 \equiv 1 - \biggl(\frac{a_3}{a_1}\biggr)^2  .
</math> 
</div>

<font color="maroon">'''Attraction of the Spheroid at the Pole'''</font>:  We interpret this phrase to mean the ''magnitude'' of the acceleration (force per unit mass) due to gravity at the pole, which is,

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

<tr>
  <td align="right">
<math>\mathcal{A} \equiv \biggl|- \frac{\partial \Phi}{\partial z}\biggr|_\mathrm{pole}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\pi G \rho  \biggl| \frac{\partial }{\partial z}
\biggl[  I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr]\biggr|_\mathrm{pole}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\pi G \rho A_3 a_3 \, .
</math>
  </td>
</tr>
</table>


<font color="maroon">'''Attraction at the Circumference of the Equator'''</font>:  We interpret this phrase to mean the ''magnitude'' of the acceleration (force per unit mass) due to gravity in the equatorial plane, at the surface of the configuration, which is,

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

<tr>
  <td align="right">
<math>\mathcal{D} \equiv \biggl|- \frac{\partial \Phi}{\partial \varpi}\biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\pi G \rho  \biggl| \frac{\partial }{\partial \varpi}
\biggl[  I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr]\biggr|_\mathrm{eq}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\pi G \rho A_1 a_1 \, .
</math>
  </td>
</tr>
</table>

<font color="maroon">'''Centrifugal force in the equatorial plane arising from rotation'''</font>:  We interpret this phrase to mean the centrifugal acceleration (force per unit mass) given by the expression,

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

<tr>
  <td align="right">
<math>\mathcal{V} \equiv \frac{(\varpi \omega_0)^2}{\varpi}\biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a_1\omega_0^2 \, .</math>
  </td>
</tr>
</table>

Maclaurin's theorem states that the rotating spheroidal configuration will be in equilibrium if,

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

<tr>
  <td align="right">
<math>\frac{\mathcal{D} - \mathcal{V}}{\mathcal{A}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{a_3}{a_1} \, ,</math>
  </td>
</tr>
</table>

or, equivalently,

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

<tr>
  <td align="right">
<math>\mathcal{D}-\mathcal{V}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mathcal{A}\biggl( \frac{a_3}{a_1} \biggr) \, .</math>
  </td>
</tr>
</table>

We choose to rewrite this expression and label it as,

<div align="center" id="MaclaurinTheorem">
<font color="#770000">'''Maclaurin's Theorem'''</font></span><br />

<math>\frac{\mathcal{V}}{a_1} = \frac{1}{a_1}\biggl[\mathcal{D}  - \mathcal{A}\biggl( \frac{a_3}{a_1} \biggr)\biggr] </math>
</div>

which, in our terminology, becomes

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

<tr>
  <td align="right">
<math>\omega_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\pi G \rho \biggl[ A_1 - A_3\biggl( \frac{a_3}{a_1} \biggr)^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\pi G \rho \biggl[A_1 -  A_3(1-e^2) \biggr] \, .</math>
  </td>
</tr>
</table>


This is precisely the equilibrium frequency that we have [[Apps/MaclaurinSpheroids#EquilibriumFrequency|derived elsewhere]], from an evaluation of the gravitational potential of uniform-density spheroids.

<!-- ALSO COULD WRITE THE FOLLOWING ...

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}  \biggr](1-e^2)^{1/2} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} = 2(1-A_1) \, .
</math>
  </td>
</tr>
</table>

Hence, Maclaurin's Theorem implies that, for a given spheroidal eccentricity, the equilibrium rotation frequency should be,

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

<tr>
  <td align="right">
<math>\omega_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi G \rho \biggl[ 2(1-A_1) (1-e^2) -A_1\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi G \rho \biggl[ 2(1-e^2 - A_1 + A_1e^2) - A_1 \biggr] 
</math>
  </td>
</tr>
</table>


-->

==Prolate Spheroid==
As one specific example, Maclaurin discusses the gravitational attraction at the pole of a uniform-density, prolate spheroid in his &sect;647 of Volume II (pp. 125-127).  While his discussion of this topic necessarily refers back to earlier sections &#8212; most importantly, &sect;643 (spheres) and &sect;644 (oblate spheroids) &#8212; it frequently refers to Figure 291, No. 2, which we have reproduced in the following figure.  

<div align="center" id="Paragraph630">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" colspan="2">
Figure extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<br />
"''A Treatise of Fluxions''"<br />
Volume II, Figure Plate XXXIII
</td></tr>
<tr>
  <td align="center">
[[File:Figure291Pt2.png|300px|center|Maclaurin (1742)]]
  </td>
  <td align="center">
<!-- [[File:MyFig291Pt2C.png|380px|center|Our Construction]] -->
[[File:Maclaurin01.gif|380px|center|Our Construction]]
  </td>
</tr>
<tr>
  <td align="left">
<sup>&dagger;</sup>The .  
  </td>
  <td align="left">
In this animation sequence, the blue ellipse has an eccentricity, <math>e = 0.55</math>, and the red line-segment is drawn at an angle to the horizontal axis that varies over the range,  <math>\tfrac{3\pi}{8} \ge \alpha \ge \tfrac{\pi}{8}</math>.  As is described in the accompanying text, the orange curve &#8212; Maclaurin's "AKG" curve &#8212; is mapped out as the angle, <math>\alpha</math>, is varied.
  </td>
</tr>
</table>
</div>

While trying to decipher Maclaurin's geometrically based derivation, we have found it useful to re-draw key elements of this figure, deriving relevant algebraic or trigonometric relations along the way.  Because the "oblong" ellipse in Maclaurin's figure is oriented such that the axis of revolution &#8212; the symmetry axis &#8212; is horizontal, and because we prefer to associate the <math>z</math> axis with the symmetry axis of the configuration, we will rather unconventionally refer to the horizontal axis as the <math>z</math> (symmetry) axis.  Adopting a cylindrical coordinate system furthermore means that, in our notation, the vertical axis will be the <math>\varpi</math> axis.

The blue ellipse centered on the origin of our plot represents Maclaurin's "ADBE" ellipse; in the expressions that follow, we will use <math>a_1</math> and <math>a_3</math> to refer, respectively, to the length of the semi-major axis (Maclaurin's line-segment "CA") and semi-minor axis (Maclaurin's line-segment "CD") of the ellipse; and, following Maclaurin, we will refer to the leftmost point on the ellipse (Maclaurin's point "A") as the "pole" which is positioned at coordinate <math>(z, \varpi) = (-a_1, 0)</math>.   (Note that in our plot, we have set <math>a_1 = 1</math>, but otherwise we will leave its specification arbitrary.)   In our plot, Maclaurin's "CNH" circle is represented by the green quarter circle centered on the pole.  The blue ellipse is defined by the relation,

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

<tr>
  <td align="right">
<math>\frac{z^2}{a_1^2} + \frac{\varpi^2}{a_3^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{z^2}{a_1^2} + (1-e^2)^{-1}\frac{\varpi^2}{a_1^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \, ,</math>
  </td>
</tr>
</table>

and, because it is [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|off-center]], the green quarter circle is defined by the relation,

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

<tr>
  <td align="right">
<math>(a_1 + z)^2 + \varpi^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a_1^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl(1 + \frac{z}{a_1} \biggr)^2 + \frac{\varpi^2}{a_1^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \, .</math>
  </td>
</tr>
</table>

The red line-segment in our plot is what Maclaurin refers to as "AM" which, in his words, is ''any right line from the pole A.''  We will use the variable, <math>\alpha</math>, to identify the angle between this line-segment and the horizontal axis; in determining the gravitational attraction at pole A, Maclaurin ultimately sweeps this angle through the range, <math>0 \le \alpha \le \tfrac{\pi}{2}</math>.  Points lying along this line are defined by the relation,

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

<tr>
  <td align="right">
<math>\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(a_1+z)\tan\alpha</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{\varpi}{a_1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(1+\frac{z}{a_1}\biggr)\tan\alpha \, .</math>
  </td>
</tr>
</table>


===Intersection With Circle===

Hence, the red line-segment intersects the green quarter-circle when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1- \biggl(1 + \frac{z}{a_1} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+\frac{z}{a_1}\biggr)^2\tan^2\alpha </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\biggl(1+\frac{z}{a_1}\biggr)^2(1+\tan^2\alpha)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{z}{a_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1+\tan^2\alpha)^{-1/2} -1 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos\alpha -1 \, .</math>
  </td>
</tr>
</table>
</div>

Correspondingly,
<div align="center">
<math>\frac{\varpi}{a_1} = (\cos\alpha)\tan\alpha = \sin\alpha \, .</math>
</div>

===Intersection With Ellipse===

Similarly, the red line-segment intersects the blue ellipse when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1-e^2)\biggl[1- \biggl(\frac{z}{a_1} \biggr)^2\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+\frac{z}{a_1}\biggr)^2\tan^2\alpha </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ (1-e^2) - (1-e^2)\biggl(\frac{z}{a_1} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1+2\biggl(\frac{z}{a_1}\biggr)+\biggl(\frac{z}{a_1}\biggr)^2\biggr]\tan^2\alpha </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\tan^2\alpha - (1-e^2)\biggr]+2\biggl(\frac{z}{a_1}\biggr)\tan^2\alpha
+\biggl(\frac{z}{a_1}\biggr)^2\biggl[\tan^2\alpha+ (1-e^2) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

The roots of this quadratic equation are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{z}{a_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b}{2a}\biggl\{ -1 \pm \biggl[ 1 - \frac{4ac}{b^2} \biggr]^{1/2} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan^2\alpha}{(\tan^2\alpha+ 1-e^2 )}\biggl\{ -1 
\pm \biggl[ 1 - \frac{[\tan^2\alpha - (1-e^2)][ \tan^2\alpha+ (1-e^2) ]}{\tan^4\alpha} \biggr]^{1/2} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan^2\alpha}{(\tan^2\alpha+ 1-e^2 )}\biggl\{ -1 
\pm \frac{1}{\tan^2\alpha}\biggl[ \tan^4\alpha - [\tan^4\alpha - (1-e^2)^2] \biggr]^{1/2} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{[\tan^2\alpha \mp (1-e^2) ]}{[\tan^2\alpha+ (1-e^2 )]}  \, .</math>
  </td>
</tr>
</table>
</div>
The "inferior" root gives, 

<div align="center">
<math>~\frac{z}{a_1}\biggr|_- = -1 \, ,</math>
</div>
while the "superior" root gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{z}{a_1}\biggr|_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- ~\frac{[\tan^2\alpha - (1-e^2) ]}{[\tan^2\alpha+ (1-e^2) ]} \, .</math>
  </td>
</tr>
</table>
</div>

Correspondingly,
<div align="center">
<math>\frac{\varpi}{a_1}\biggr|_- = 0 \, ,</math>
</div>
and,
<div align="center">
<math>\frac{\varpi}{a_1}\biggr|_+ = \tan\alpha\biggl\{  1
- ~\frac{[\tan^2\alpha - (1-e^2) ]}{[\tan^2\alpha+ (1-e^2) ]}
\biggr\} \, .</math>
</div>

The "inferior" root clearly identifies the intersection of the red line-segment with the pole of the prolate spheroid.  The "superior" root identifies the point at which the red line-segment intersects and crosses the blue ellipse, away from the pole.

===Generate Curve AKG===

In &sect;644 (p. 122 of Volume II) of his Treatise, Maclaurin states that "the gravity at the pole A towards the spheroid ADBE will be measured by [the ratio],"
<div align="center">
<math>\frac{\mathrm{AKGC}}{AC} \, ,</math>
</div>
where the numerator, "AKGC," is the area under the curve, "AKG," and the denominator is the length of the semi-major axis, <math>~a_1</math>.  So a key element of Maclaurin's derivation is the proper construction of the curve, "AKG."  The key sentence from his &sect;644 appears to be the following:

[[File:Vol2Par644KeyPhrase.png|400px|thumb|center|Extracted directly from &sect;644 of Maclaurin's Book 1, as digitized by Google]]

We interpret this phrase to mean that, as the angle, <math>~\alpha</math>, varies from <math>~\tfrac{\pi}{2}</math> to <math>~0</math>, the curve "AKG" is generated by plotting, as the ordinate, the length of the line-segment "AQ" and, as the abscissa, the <math>~z</math>-coordinate location of the vertical line, "RN" &#8212; we'll label this, <math>~z_\mathrm{RN}</math>.  In our terminology, as [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Circle|derived above]], <math>~z_\mathrm{RN}</math> is the <math>~z</math>-coordinate of the intersection of the red line-segment with the green quarter-circle, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\mathrm{RN}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos\alpha - 1 \, .</math>
  </td>
</tr>
</table>
</div>
And the magnitude of "AQ" is obtained from the <math>~z</math>-coordinate location of the vertical line, "MQ" &#8212; that is, the <math>~z</math>-coordinate of the intersection of the red line-segment with the blue ellipse &#8212; as [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse|derived above]].  Specifically, we interpret Maclaurin's phrase to mean that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathrm{AQ}}{a_1} \equiv 1 + \frac{z}{a_1}\biggr|_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - ~\frac{[\tan^2\alpha - (1-e^2) ]}{[\tan^2\alpha+ (1-e^2) ]} \, .</math>
  </td>
</tr>
</table>
</div>

Using the first of these two expressions, we can rewrite the tangent function directly in terms of the ordinate, <math>~z_\mathrm{RN}</math>; specifically,
<div align="center">
<math>\tan^2\alpha = \frac{1- \cos^2\alpha}{\cos^2\alpha} = \frac{1- (1+z_\mathrm{RN})^2}{(1+z_\mathrm{RN})^2}  \, .</math>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathrm{AQ}}{a_1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - ~\frac{(1+z_\mathrm{RN})^2[\tan^2\alpha - (1-e^2) ]}{(1+z_\mathrm{RN})^2[\tan^2\alpha+ (1-e^2) ]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - ~\frac{[1-(1+z_\mathrm{RN})^2 - (1-e^2)(1+z_\mathrm{RN})^2 ]}{[1-(1+z_\mathrm{RN})^2+ (1-e^2) (1+z_\mathrm{RN})^2]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - ~\frac{[1+(1+z_\mathrm{RN})^2 (-1 - (1-e^2) ]}{[1+(1+z_\mathrm{RN})^2(-1+1-e^2)]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - ~\frac{[1-(2-e^2)(1+z_\mathrm{RN})^2 ]}{[1-e^2(1+z_\mathrm{RN})^2]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[1-e^2(1+z_\mathrm{RN})^2]-[1-(2-e^2)(1+z_\mathrm{RN})^2 ]}{[1-e^2(1+z_\mathrm{RN})^2]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[2(1-e^2)(1+z_\mathrm{RN})^2 ]}{[1-e^2(1+z_\mathrm{RN})^2]} \, .</math>
  </td>
</tr>
</table>
</div>

In our above figure, this function, <math>~\mathrm{AQ}(z_\mathrm{RN})</math> normalized to <math>~a_1</math>, is delineated by the orange triangles over the coordinate range, <math>~-1 \le z_\mathrm{RN} \le 0</math>.  If our interpretation of Maclaurin's discussion is correct, this is precisely the curve in Maclauin's Figure 291, No. 2 that is labeled, "AKG."

===Area Under the Curve===
The area, "AKGC" &#8212; which, according to Maclaurin, gives the gravitational acceleration at the pole A &#8212; can be determined by carrying out the integral,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathrm{AKGC}}{a_1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_{-1}^0 \biggl[\frac{\mathrm{AQ}}{a_1}\biggr]dz_\mathrm{RN} \, .</math>
  </td>
</tr>
</table>
</div>

This integral can be more cleanly evaluated if we make the variable substitution, 
<div align="center">
<math>u_\mathrm{RN} \equiv 1 + z_\mathrm{RN} = \cos\alpha \, ,</math>
</div> 
with a corresponding shift in the limits of integration from <math>~(-1 \rightarrow 0)</math>, for <math>~z_\mathrm{RN}</math>, to <math>~(0 \rightarrow 1)</math>, for <math>~u_\mathrm{RN}</math>.  Specifically, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathrm{AKGC}}{a_1}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2(1-e^2) \int_{0}^1 \frac{u_\mathrm{RN}^2 }{1-e^2 u_\mathrm{RN}^2} ~du_\mathrm{RN}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2(1-e^2) \biggl[ \frac{u_\mathrm{RN}}{e^2} \biggr]_{0}^1 + \frac{1}{e^4} \int_{0}^1 \frac{du_\mathrm{RN}}{e^{-2}- u_\mathrm{RN}^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{2(1-e^2)}{e^2}  + \frac{2(1-e^2)}{e^4} \biggl\{ \frac{e}{2} \ln \biggl[ \frac{e^{-2} +u_\mathrm{RN} }{e^{-2} -u_\mathrm{RN} } \biggr] \biggr\}_{0}^1 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-e^2)}{e^3} \ln \biggl[ \frac{1 + e^{2} }{1-e^{2}} \biggr]-\frac{2(1-e^2)}{e^2}   \, .</math>
  </td>
</tr>
</table>
</div>
This is precisely the expression for the coefficient, <math>~A_1</math>, that is [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.28a1_.3E_a2_.3D_a3.29|traditionally used in the expression for the gravitational potential inside homogeneous, prolate spheroids]].  


{{SGFworkInProgress}}

===Comparison With More Traditional Derivation===

In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole|separate derivation of the gravitational acceleration at the pole of a prolate spheroid]] that uses a more traditional approach, we obtained this key coefficient by integrating over a different function, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{Z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(1-\zeta)}{ [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{1/2}} -1 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-\zeta)^{1/2}}{ [ 2-e^2 - e^2\zeta  ]^{1/2}} -1 \, .</math>
  </td>
</tr>
</table>
</div>

By comparing Maclaurin's approach to the more traditional one, we conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_{-1}^0 \biggl[\frac{\mathrm{AQ}}{a_1}\biggr]dz_\mathrm{RN} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_{-1}^1 \mathcal{Z} d\zeta \, .</math>
  </td>
</tr>
</table>
</div>

We should be able to demonstrate explicitly that these two integrands are the same.  We begin by recognizing that the relationship between the dimensionless coordinate, <math>~z/a_1</math>, that we have used to decipher Maclaurin's presentation and the dimensionless coordinate, <math>~\zeta</math>, that was [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole|used in our separate analysis]], is,
<div align="center">
<math>\frac{z}{a_1} = -\zeta \, .</math>
</div>

Returning to the first expression in the above subsection that details the [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse|intersection of the red line-segment with the blue ellipse]], we can therefore write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1-e^2) (1-\zeta^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\zeta)^2\tan^2\alpha </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ \tan^2\alpha </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-e^2) (1+\zeta)}{(1-\zeta)} \, .</math>
  </td>
</tr>
</table>
</div>

Recalling that,
<div align="center">
<math>u_\mathrm{RN} = \cos\alpha = [1+\tan^2\alpha]^{-1/2} \, ,</math>
</div>
we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathrm{AQ}}{a_1} = \frac{2(1-e^2)u_\mathrm{RN}^2}{1-e^2 u_\mathrm{RN}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(1-e^2)[1+\tan^2\alpha]^{-1}}{1-e^2[1+\tan^2\alpha]^{-1}}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(1-e^2)}{1-e^2 +\tan^2\alpha}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(1-e^2)(1-\zeta)}{(1-e^2)(1-\zeta) +(1-e^2)(1+\zeta)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\zeta) \, .</math>
  </td>
</tr>
</table>
</div>

Furthermore, we see that the transformation from <math>~u_\mathrm{RN}</math> to <math>~\zeta</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u_\mathrm{RN} = [1+\tan^2\alpha ]^{-1/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{(1-e^2) (1+\zeta)}{(1-\zeta)}  \biggr]^{-1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(1-\zeta) + (1-e^2) (1+\zeta)}{(1-\zeta)}  \biggr]^{-1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\zeta)^{1/2} (2-e^2 -e^2\zeta)^{-1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ du_\mathrm{RN}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[
-2^{-1}(1-\zeta)^{-1/2} (2-e^2 -e^2\zeta)^{-1/2}
-2^{-1}(1-\zeta)^{1/2} (2-e^2 -e^2\zeta)^{-3/2}(-e^2)]d\zeta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2^{-1}(1-\zeta)^{-1/2}  (2-e^2 -e^2\zeta)^{-3/2}[
(2-e^2 -e^2\zeta)-e^2(1-\zeta) ]d\zeta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(1-e^2)(1-\zeta)^{-1/2}  (2-e^2 -e^2\zeta)^{-3/2} d\zeta \, .</math>
  </td>
</tr>
</table>
</div>

Putting these two expressions together, we conclude that the integrand is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{\mathrm{AQ}}{a_1}\biggr] du_\mathrm{RN}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[\frac{(1-e^2)(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{3/2}}\biggr] d\zeta </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)}  \biggr]\biggl[\frac{(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{1/2}}\biggr] d\zeta \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)}  \biggr]\biggl[1 + \mathcal{Z}\biggr] d\zeta \, .</math>
  </td>
</tr>
</table>
</div>

This does not appear to match the other integrand.  I must be misinterpreting something!

To be done:  I suspect that Maclaurin's approach will look more familiar if I redo the "traditional" derivation from the point of view of a spherical coordinate system (positioning pole A at the origin), in which an axisymmetric volume element takes the form,
<div align="center">
<math>~dV = 2\pi r^2 \sin\alpha dr d\alpha = 2\pi r^2 dr du_\mathrm{RN} \, ,</math>
</div>
where, from above, <math>u_\mathrm{RN} \equiv \cos\alpha \, .</math>