Difference between revisions of "User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks"

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(Begin quantitative assessment of Maclaurin's key theorem)
(→‎Interpreting Maclaurin's Key Concluding Theorem: Finished demonstration, but need to switch sign convention)
Line 320: Line 320:
<div align="center">
<div align="center">
<math>
<math>
  e \equiv \biggl[ 1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} .
  e^2 \equiv 1 - \biggl(\frac{a_3}{a_1}\biggr)^2 .
</math>  
</math>  
</div>
</div>
Line 395: Line 395:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{D} \equiv \frac{(\varpi \omega_0)^2}{\varpi}\biggr|_\mathrm{eq}</math>
<math>~\mathcal{V} \equiv \frac{(\varpi \omega_0)^2}{\varpi}\biggr|_\mathrm{eq}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 401: Line 401:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\varpi \omega_0^2 \, .</math>
<math>~a_1\omega_0^2 \, .</math>
   </td>
   </td>
</tr>
</tr>
Line 407: Line 407:
</div>
</div>


Maclaurin's theorem states that the rotating spheroidal configuration will be in equilibrium if,
<div align="center">
<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>
</div>
or, equivalently,
<div align="center">
<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>
</div>
We choose to rewrite this expression and label it as,
<div align="center">
<span id="MaclaurinTheorem"><font color="#770000">'''Maclaurin's Theorem'''</font></span><br />
<math>~\frac{\mathcal{V}}{a_1} = \frac{1}{a_1}\biggl[- \mathcal{A}\biggl( \frac{a_3}{a_1} \biggr) + \mathcal{D} \biggr] </math>
</div>
which, in our terminology, becomes
<div align="center">
<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_3\biggl( \frac{a_3}{a_1} \biggr)^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[ A_3(1-e^2) -A_1\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Now, as we have [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential|detailed elsewhere]], from an evaluation of the gravitational potential of uniform-density spheroids, we can write,
<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,
<div align="center">
<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>
</div>


=Related Discussions=
=Related Discussions=

Revision as of 20:58, 31 August 2015


Excerpts from A Treatise of Fluxions

Whitworth's (1981) Isothermal Free-Energy Surface
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Both Volume I and Volume II of Colin Maclaurin's "A Treatise of Fluxions" can now be accessed online via Google Books. (Check out 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.

Title Pages with Google-Books Links

Example Digitized Figure Plate

Volume I (1742)
Volume II (2nd Ed., 1801)
Plate XXXII from Volume II (2nd Ed., 1801)

Volume I (1st Ed., 1742)

Volume II (2nd Ed., 1801)

Plate XXXII

As is illustrated in the righthand panel of the above image trio (click in the panel 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 Fig. 282, below.

Volume I

Dedication (an 18th Century Acknowledgment) 
Preface 
Introduction1
Book 1 (Of the Fluxions of Geometrical Magnitudes)51
      Chapter I (Of the Grounds of this Method) — §§ 1-7751
      Chapter II (Of the Fluxions of plane rectilineal Figures) — §§ 78-104109
      Chapter III (Of the Fluxions of plane curvilineal Figures) — §§ 105-123131
      Chapter IV (Of the Fluxions of Solids, and of third Fluxions) — §§ 124-139142
      Chapter V (Of the Fluxions of Quantities that are in a continued Geometrical Progression, the first term of which is invariable) — §§ 140-150152
      Chapter VI (Of Logarithms, and of the Fluxions of logarithmic Quantities) — §§ 151-179158
      Chapter VII (Of the Tangents of curve Lines) — §§ 180-214178
      Chapter VIII (Of the Fluxions of curve Surfaces) — §§ 215-237199
      Chapter IX ([Identifying Extrema and Inflection Points] of Curves that are defined by a common or by a fluxional Equation) — §§ 238-285214
      Chapter X (Of the Asymptotes of curve Lines, the Areas bounded by them and …) — §§ 286-362240
      Chapter XI (Of the Curvature of Lines … different kinds of Contact [with other Curves] … Caustics … centripetal Forces …) — §§ 363-304
            · Parabola — §371311
            · Any Conic Section — §373312
            · The Second Fluxion of a Curve — §384324
            · Refraction of Light — §413344
            · Centripetal and Centrifugal Forces — §416346
            · Gravity — §419348
            · Circular Motion — §432356
            · When the Center of Forces is the Focus of a Conic Section — §446370
            · When Gravity is Uniform or Varies as any Power of the Distance — §458383
            · Orbit of the Moon, taking into account the Gravity of both the Earth and the Sun — §471391
Extracted directly from §487 of Maclaurin's Book 1, as digitized by Google
            · Prolate Spheroidal Fluid Figure — §491409
Extracted directly from §491 of Maclaurin's Book 1, as digitized by Google
            · The Earth's Equilibrium Shape — §492410
Extracted directly from §492 of Maclaurin's Book 1, as digitized by Google
      End of Book I, Chapter XI, § 494413


Volume II

Table of Principal Contents (5 pp.) 
Figure Plates (30 pp.) 
Book 1 (continued)1
      Chapter XII (Of the Methods of Infinitesimals …) — §§ 495-5701
            · Centre of Gravity — §51013
            · Of the Collision of Bodies — §51114
            · Of the Descent of Bodies that Act upon One Another — §52127
            · Of the Centre of Oscillation — §53340
            · Of the Motion of Water Issuing from a Cylindric Vessel — §53744
            · Of the Catenaria — §55159
            · General Observations Concerning the Angles of Contact, etc. — §55461
            · General Observations Concerning centripetal Forces, etc. — §56367
      Chapter XIII (Determining the Lines of swiftest Descent in any Hypothesis of Gravity …) — §§ 571-60874
            · When Gravity is Directed Towards a Given Centre — §57880
            · Isoperimetrical Problems — §58888
            · The Solid of Least Resistance — §606100
      Chapter XIV (Of the Ellipse Considered as the Section of a Cylinder … Of the Figure of the Earth …) — §§ 609-101
            · Properties of the Ellipse — §609101
            · Of the Gravitation towards Spheres and Spheroids — §628110
            · Of the Figures of Planets (including effects of rotation) — §636116
            · Key Concluding Theorem! — §641119
Extracted directly from §641 of Maclaurin's Book 1, as digitized by Google
            · Itemize Numerous Implications for Equilibrium Spheroids — §642120
            · Apply Specifically to the Earth — §661136
            · What if the Earth's Density isn't Uniform but, instead, Varies Linearly with Distance? — §670143
            · Jupiter — §682152
            · Tides — §686154
            · Concluding Paragraph — §696161
Extracted directly from §696 of Maclaurin's Book 1, as digitized by Google
      End of Book I162


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

Paragraph (and related figure) extracted from Colin Maclaurin (1742)

"A Treatise of Fluxions"

Volume II, Chapter XIV, §628

Maclaurin (1742)
Maclaurin (1742)
Diagram reposted from MATHalino.com

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 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 Mathalino.com in an effort to illustrate what Maclaurin means by "frustum of a cone."


Paragraph (and a pair of related diagrams) extracted from Colin Maclaurin (1742)

"A Treatise of Fluxions"

Volume II, Chapter XIV, §630

Maclaurin (1742)
Maclaurin (1742)

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 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.


Interpreting Maclaurin's Key Concluding Theorem

As noted above, in line with the table of contents for Volume II, the key theorem resulting from Maclaurin's analysis is …

Extracted directly from §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 discussion of Maclaurin spheroids that has been presented in an accompanying chapter of this H_Book.

CA: This is the semi-minor (polar) axis of the spheroid, which we have called, <math>~a_3</math>.

CD: 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,

<math>

e^2 \equiv 1 - \biggl(\frac{a_3}{a_1}\biggr)^2  .

</math>

Attraction of the Spheroid at the Pole: We interpret this phrase to mean the acceleration (force per unit mass) due to gravity at the pole, directed toward the equatorial plane, which is,

<math>~\mathcal{A} \equiv - \frac{\partial \Phi}{\partial z}\biggr|_\mathrm{pole}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<math>~ -2\pi G \rho A_3 a_3 \, . </math>


Attraction at the Circumference of the Equator: We interpret this phrase to mean the acceleration (force per unit mass) due to gravity in the equatorial plane, at the surface of the configuration, and directed radially toward the center of the spheroid, which is,

<math>~\mathcal{D} \equiv - \frac{\partial \Phi}{\partial \varpi}\biggr|_\mathrm{eq}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<math>~ -2\pi G \rho A_1 a_1 \, . </math>

Centrifugal force in the equatorial plane arising from rotation: We interpret this phrase to mean the centrifugal acceleration (force per unit mass) given by the expression,

<math>~\mathcal{V} \equiv \frac{(\varpi \omega_0)^2}{\varpi}\biggr|_\mathrm{eq}</math>

<math>~=</math>

<math>~a_1\omega_0^2 \, .</math>

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

<math>~\frac{\mathcal{D} - \mathcal{V}}{\mathcal{A}}</math>

<math>~=</math>

<math>~\frac{a_3}{a_1} \, ,</math>

or, equivalently,

<math>~\mathcal{D}-\mathcal{V}</math>

<math>~=</math>

<math>~\mathcal{A}\biggl( \frac{a_3}{a_1} \biggr) \, .</math>

We choose to rewrite this expression and label it as,

Maclaurin's Theorem

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

which, in our terminology, becomes

<math>~\omega_0^2</math>

<math>~=</math>

<math>~2\pi G \rho \biggl[ A_3\biggl( \frac{a_3}{a_1} \biggr)^2 -A_1\biggr] </math>

 

<math>~=</math>

<math>~2\pi G \rho \biggl[ A_3(1-e^2) -A_1\biggr] \, .</math>

Now, as we have detailed elsewhere, from an evaluation of the gravitational potential of uniform-density spheroids, we can write,

<math> ~A_1 </math>

<math> = </math>

<math> \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr](1-e^2)^{1/2} \, , </math>

<math> ~A_3 </math>

<math> = </math>

<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>

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

<math>~\omega_0^2</math>

<math>~=</math>

<math>~ 2\pi G \rho \biggl[ 2(1-A_1) (1-e^2) -A_1\biggr] </math>

 

<math>~=</math>

<math>~ 2\pi G \rho \biggl[ 2(1-e^2 - A_1 + A_1e^2) - A_1 \biggr] </math>

Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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