Editing Apps/MaclaurinSpheroids/GoogleBooks (section)

==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>


-->