Jet53man at 22:57, 12 April 2022

2022-04-12T22:57:38Z

← Older revision		Revision as of 18:57, 12 April 2022
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	=Dyson (1893a) Part I: Some Details=		=Dyson (1893a) Part I: Some Details=
	This chapter provides some derivation details relevant to our [[~~User:Tohline/~~Apps/DysonPotential#Dyson_.281893.29\|accompanying discussion]] of Dyson's analysis of the gravitational potential exterior to an ''anchor ring.''		This chapter provides some derivation details relevant to our [[Apps/DysonPotential#Dyson_.281893.29\|accompanying discussion]] of Dyson's analysis of the gravitational potential exterior to an ''anchor ring.''

	==Overview==		==Overview==
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	===Step 2===		===Step 2===
	Taking a queue from our [[~~User:Tohline/~~2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System\|accompanying discussion of toroidal coordinates]], if we adopt the variable notation,		Taking a queue from our [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System\|accompanying discussion of toroidal coordinates]], if we adopt the variable notation,
	<div align="center">		<div align="center">
	<math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math>		<math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math>
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	</table>		</table>

	In preparation for a [[~~User:Tohline/~~Apps/Wong1973Potential#Compare_With_Dyson_.281893.29\|comparison between Dyson's expression and the expression derived by Wong (1973)]], let's regroup terms according to the various trigonometric functions. Keep in mind that,		In preparation for a [[Apps/Wong1973Potential#Compare_With_Dyson_.281893.29\|comparison between Dyson's expression and the expression derived by Wong (1973)]], let's regroup terms according to the various trigonometric functions. Keep in mind that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	Now, drawing from our [[~~User:Tohline/~~2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|accompanying discussion of relationships between complete elliptic integrals and relationships between their parameter arguments]] and, in particular, associating Dyson's parameter, <math>~\mu = (R_1-R)/(R_1+R)</math>, with <math>~k_1</math>, we understand that,		Now, drawing from our [[2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|accompanying discussion of relationships between complete elliptic integrals and relationships between their parameter arguments]] and, in particular, associating Dyson's parameter, <math>~\mu = (R_1-R)/(R_1+R)</math>, with <math>~k_1</math>, we understand that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	From an [[~~User:Tohline/~~Apps/DysonPotential#gammaInverse\|accompanying discussion]] that builds upon the law of cosines, we also may write,		From an [[Apps/DysonPotential#gammaInverse\|accompanying discussion]] that builds upon the law of cosines, we also may write,

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

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2021-07-24T17:36:13Z

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Appendix/Ramblings/Dyson1893Part1 - Revision history

Jet53man at 22:57, 12 April 2022

Jet53man: Created page with "__FORCETOC__ =Dyson (1893a) Part I: Some Details= This chapter provides some der..."

Jet53man: Created page with "FORCETOC =Dyson (1893a) Part I: Some Details= This chapter provides some der..."