User:Tohline/Apps/DysonPotential - Revision history

Tohline: /* Elliptic Integral Expressions */

2020-07-16T21:04:42Z

Elliptic Integral Expressions

← Older revision		Revision as of 21:04, 16 July 2020
Line 5,083:		Line 5,083:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\}		\biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} \, .
	</math>		</math>
	</td>		</td>
Line 5,174:		Line 5,174:
	<math>~		<math>~
	\frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr]		\frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr]
	\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\}		\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Elliptic Integral Expressions */

2020-07-16T21:02:23Z

Elliptic Integral Expressions

← Older revision		Revision as of 21:02, 16 July 2020
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	===Elliptic Integral Expressions===		===Elliptic Integral Expressions===
	Hence, drawing ~~upon the series expansion~~ for the ''complete elliptic integral of the first kind~~'' that can be found~~, ~~for example~~, ~~as eq. (8.113.3) in the Fourth Edition of [http~~://~~www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)],~~ we can write,		Hence, drawing from our set of [[User:Tohline/Appendix/Equation_templates#Complete_Elliptic_Integrals\|Key Expressions for the complete elliptic integral of the first kind]], specifically,
			{{User:Tohline/Math/EQ_EllipticIntegral03 }}

			we can write,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 5,027:		Line 5,030:
	<td align="right">		<td align="right">
	<math>~K(\mu)</math>		<math>~K(\mu)</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr){k'}^2~~
	~~+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} \biggr){k'}^4~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~+ \biggl( \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{2}{5\cdot 6} \biggr){k'}^6 + \cdots~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 5,166:		Line 5,140:
	-->		-->

	Also, drawing ~~upon the series expansion~~ for the ''complete elliptic integral of the second kind~~'' that can be found~~, ~~for example~~, ~~as eq. (8.114.3) in the Fourth Edition of [http~~://~~www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)],~~ we ~~can write~~,		Also, drawing from our set of [[User:Tohline/Appendix/Equation_templates#Complete_Elliptic_Integrals\|Key Expressions for the complete elliptic integral of the second kind]], specifically,
			{{User:Tohline/Math/EQ_EllipticIntegral04 }}

			we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 5,172:		Line 5,149:
	<td align="right">		<td align="right">
	<math>~E(\mu)</math>		<math>~E(\mu)</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	1
	~~~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{1\cdot 2}\biggr)(k')^2~~
	~~~+~ \frac{1^2\cdot 3}{2^2\cdot 4}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{1}{3\cdot 4}\biggr)(k')^4~~
	~~~+~ \frac{1^2\cdot 3^2\cdot 5}{2^2\cdot 4^2\cdot 6}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{1}{5\cdot 6}\biggr)(k')^6~~
	~~~+~ \cdots~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Our Attempt to Replicate */

2020-07-15T23:03:45Z

Our Attempt to Replicate

← Older revision		Revision as of 23:03, 15 July 2020
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	====Our Attempt to Replicate====		====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 [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|accompanying discussion]] of incomplete elliptic integrals. ~~According to, for example,~~ [~~https~~://~~en.wikipedia.org~~/~~wiki~~/~~Elliptic_integral~~#~~Complete_elliptic_integral_of_the_first_kind Wikipedia],~~ the relevant series-expansion expressions are:		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 [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|accompanying discussion]] of incomplete elliptic integrals. Drawing from our [[User:Tohline/Appendix/Equation_templates#Complete_Elliptic_Integrals\|accompanying set of Key mathematical relations]] — in which <math>~k</math>, rather than <math>~\mu</math>, represents the function modulus — the relevant series-expansion expressions are:
	<~~table border="0" cellpadding="5"~~ align="center">		<div align="center">
			{{ User:Tohline/Math/EQ_EllipticIntegral01 }}<br />
			{{ User:Tohline/Math/EQ_EllipticIntegral02 }}
			</div>

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~K(\mu)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{\pi}{2} \biggl\{~~
	~~1 + \biggl[\frac{1}{2}\biggr]^2\mu^2~~
	~~+ \biggl[ \frac{1\cdot 3}{2\cdot 4}\biggr]^2\mu^4~~
	~~+ \biggl[ \frac{5\cdot 3\cdot 1}{6\cdot 4\cdot 2} \biggr]^2 \mu^6~~
	~~+ \cdots~~
	~~+ \biggl[ \frac{(2n-1)!!}{(2n)!!}\biggr]^2 \mu^{2n}~~
	~~+\cdots~~
	~~\biggr\} \, ;~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~E(\mu)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{\pi}{2} \biggl\{~~
	~~1 ~-~ \biggl[\frac{1}{2}\biggr]^2\frac{\mu^2}{1}~~
	~~~-~ \biggl[ \frac{1\cdot 3}{2\cdot 4}\biggr]^2 \frac{\mu^4}{3}~~
	~~~- ~\biggl[ \frac{5\cdot 3\cdot 1}{6\cdot 4\cdot 2} \biggr]^2 \frac{\mu^6}{5}~~
	~~~- ~\cdots~~
	~~~- ~\biggl[ \frac{(2n-1)!!}{(2n)!!}\biggr]^2 \frac{\mu^{2n}}{2n-1}~~
	~~~-~ \cdots~~
	~~\biggr\} \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	These expressions — up through <math>~\mathcal{O}(\mu^4)</math> — 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.		These expressions — up through <math>~\mathcal{O}(\mu^4)</math> — 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.

Tohline: /* Overview */

2020-07-15T19:11:49Z

Overview

← Older revision		Revision as of 19:11, 15 July 2020
Line 6:		Line 6:

	==Overview==		==Overview==
	~~In his~~ pioneering work, [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)] and [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D (1893b, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106)] used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] extended Dyson's work, using numerical techniques to obtain more accurate ~~— but still approximate —~~ equilibrium structures for incompressible tori having solid body rotation. Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613)] have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.
			Our focus, here, is on the pioneering work of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)] and [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D (1893b, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106)]. He used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] extended Dyson's work, using numerical techniques to obtain more accurate equilibrium structures for incompressible tori having solid body rotation. Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613)] have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.

			The most challenging aspect of each of these studies has been the development of an analytic and/or computational technique that can be used to accurately determine the gravitational potential of toroidal-shaped configurations. With this in mind, it should be appreciated that, in a paper that preceded his 1974 work, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)] derived an analytic expression for the ''exact'' potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, has heretofore been under-appreciated. In a [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|separate, accompanying discussion]], we detail how Wong accomplished this task.

	==External Potential==		==External Potential==

Tohline: /* Higher Order */

2018-09-19T22:37:11Z

Higher Order

← Older revision		Revision as of 22:37, 19 September 2018
Line 2,201:		Line 2,201:
	</table>		</table>

	Also, we have,		<span id="gammaInverse">Also, we have,</span>

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

Tohline: /* His Derived Expression */

2018-09-17T21:17:21Z

His Derived Expression

← Older revision		Revision as of 21:17, 17 September 2018
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	<td align="center">		<td align="center">
	[[File:DysonTorusIllustration03.png\|300px\|center\|Anchor Ring Schematic]]<br />		[[File:DysonTorusIllustration03.png\|300px\|center\|Anchor Ring Schematic]]<br />
	'''Caption:''' [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm Anchor ring] schematic, adapted from figure near the top of §2 (on p. 47) of Dyson (1893a)		'''Caption:''' [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm Anchor ring] schematic, adapted from figure near the top of §2 (on p. 47) of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)]
	</td>		</td>
	</tr>		</tr>

Tohline: /* His Derived Expression */

2018-09-17T02:06:55Z

His Derived Expression

← Older revision		Revision as of 02:06, 17 September 2018
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	==External Potential==		==External Potential==
	===His Derived Expression===		===His Derived Expression===
	(See an accompanying [[User:Tohline/Appendix/Ramblings/Dyson1893Part1\|''Ramblings Chapter'']] for additional derivation details.) On p. 62 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)], we find the following approximate expression for the potential at point "P", anywhere exterior to an [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]:		(See an accompanying [[User:Tohline/Appendix/Ramblings/Dyson1893Part1\|''Ramblings Chapter'']] for additional derivation details.) On p. 62, in §8 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)], we find the following approximate expression for the potential at point "P", anywhere exterior to an [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]:

	<table border="0" cellpadding="10" align="right" width="40%"><tr><td align="center">		<table border="0" cellpadding="10" align="right" width="40%"><tr><td align="center">

Tohline: /* His Derived Expression */

2018-09-17T01:58:55Z

His Derived Expression

← Older revision		Revision as of 01:58, 17 September 2018
Line 10:		Line 10:
	==External Potential==		==External Potential==
	===His Derived Expression===		===His Derived Expression===
	On p. 62 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)], we find the following approximate expression for the potential at point "P", anywhere exterior to an [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]:		(See an accompanying [[User:Tohline/Appendix/Ramblings/Dyson1893Part1\|''Ramblings Chapter'']] for additional derivation details.) On p. 62 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)], we find the following approximate expression for the potential at point "P", anywhere exterior to an [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]:

	<table border="0" cellpadding="10" align="right" width="40%"><tr><td align="center">		<table border="0" cellpadding="10" align="right" width="40%"><tr><td align="center">

Tohline: /* Insert Expressions for K and E */

2018-09-15T15:42:36Z

Insert Expressions for K and E

← Older revision		Revision as of 15:42, 15 September 2018
Line 7,687:		Line 7,687:
	\frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi		\frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi
	\biggr\} \, .		\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

			This expression differs from the 2<sup>nd</sup>-order term in Dyson's equation (6) by the amount,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Delta \biggr\|_{\mathcal{O}(a^2/c^2)} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{a}{c}\biggr)^2\biggl\{
			\frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi
			\biggr\}
			-
			\biggl(\frac{a}{c}\biggr)^2 \biggl\{ \frac{\lambda - \frac{1}{2}}{16} + \frac{3(\lambda + \frac{17}{36})}{16}\cos2\chi \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
			60\lambda - (48\lambda + 12)~\cos\chi +(36\lambda+23)\cos 2\chi
			\biggr\}
			-
			\frac{1}{16\cdot 12}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 12\lambda - 6 + (36\lambda + 17) \cos2\chi \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
			48\lambda + 6 - (48\lambda + 12)~\cos\chi +(6)\cos 2\chi
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^2\biggl\{
			(\cos 2\chi -1)- (8\lambda + 2)~(1+\cos\chi)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2\biggl\{
			(\cos\chi - 1)- (4\lambda + 1)~
			\biggr\} (1+\cos\chi)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2 (\cos\chi - 2- 4\lambda ) (1+\cos\chi)
	</math>		</math>
	</td>		</td>

Tohline: /* To Second Order */

2018-09-14T21:14:16Z

To Second Order

← Older revision		Revision as of 21:14, 14 September 2018
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	\ln\biggl( \frac{2^3c}{a}\biggr)		\ln\biggl( \frac{2^3c}{a}\biggr)
	+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi		+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi
	~~</math>~~		~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	~~</td>~~		-71 ~+~84\cos\chi -98\cos^2\chi
	~~</tr>~~		~+~24\ln\biggl( \frac{2^3c}{a}\biggr)(1 - 2\cos\chi + 3\cos^2\chi )

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	-~98\cos^2\chi ~~~+~ 75~~\cos\chi ~~~-~71~~
	~+~(24 ~~- 39\cos\chi + 72\cos^2\chi ) \biggl[~~ \ln\biggl( \frac{2^3c}{a}\biggr)\~~biggr]~~
	\biggr\}		\biggr\}
	~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) ~~\, .~~		~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{		<math>~
	-~49(1+\cos2\chi) ~+~ 75\~~cos\chi ~-~71~~		~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	~+~(24 - 39\cos\chi) ~~\biggl[ \lambda + 2\biggr]~~		-71 ~+~84\cos\chi -49(1+\cos2\chi )
	~+~36(1 + \cos2\chi) ~~\biggl[ \lambda + 2\biggr]~~		~+~24(\lambda + 2)(1 - 2\cos\chi )
			~+~36(\lambda+2)(1 + \cos2\chi )
	\biggr\}		\biggr\}
	</math>		</math>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{		<math>~
	-~~~49\cos2\chi~~ ~+~ 75\cos\chi ~-~~~120~~		~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	~+~(24 - 39\cos\chi) \~~biggl[~~ \~~lambda + 2\biggr]~~		-71 ~+~84\cos\chi -49 -49 \cos2\chi
	~+~(36 + 36\~~cos2~~\chi) \~~biggl[ \lambda +~~ 2\~~biggr]~~		~+~24(\lambda + 2 -2\lambda \cos\chi - 4\cos\chi)
			~+~36(\lambda+2 +\lambda\cos 2\chi + 2\cos 2\chi)
	\biggr\}		\biggr\}
	</math>		</math>
Line 7,659:		Line 7,649:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{		<math>~
	~+~ 75\cos\chi ~~~-~120~~		~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	~+~24 ~~\biggl[~~ \lambda + 2\~~biggr]~~		-71 ~+~84\cos\chi -49 -49 \cos2\chi
	~~~- 39~~\cos\chi \~~biggl[~~ \~~lambda + 2\biggr]~~		~+~24\lambda + 48 -48\lambda \cos\chi - 96\cos\chi
	~+~~~\biggl[~~ 36\lambda + 72\~~biggr]~~		~+~36\lambda+72 +36\lambda\cos 2\chi + 72\cos 2\chi
	~~~+~~~\~~cos2~~\chi \~~biggl[ 36~~\~~lambda + 23\biggr]~~
	\biggr\}		\biggr\}
	</math>		</math>
Line 7,678:		Line 7,667:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~+~ \frac{1}{16} \biggl( \frac{a}{c}\biggr)^2\biggl\{		<math>~
	5\lambda		~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
	~-~ \~~biggl[~~\~~frac{1}{4}~~		60\lambda -48\lambda \cos\chi - 12\cos\chi +36\lambda\cos 2\chi + 23\cos 2\chi
	~+ \~~frac{39}{~~12~~}\lambda\biggr]~~ \cos\chi
	~+~3\~~biggl[~~ \~~lambda~~ +~~\frac{~~23~~}{36}~~\~~biggr]\cos2~~\chi
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			~+~ \biggl( \frac{a}{c}\biggr)^2\biggl\{
			\frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi
			\biggr\} \, .
	</math>		</math>
	</td>		</td>