User:Tohline/Appendix/Ramblings/Dyson1893Part1 - Revision history

Tohline: /* Try Again */

2018-10-07T22:22:09Z

Try Again

← Older revision		Revision as of 22:22, 7 October 2018
Line 3,026:		Line 3,026:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~R^2\sin^2\chi_R + [ R\cos\chi_R + a - c ]^2 \, ;</math>		<math>~R^2\sin^2\chi_R + [ R\cos\chi_R + (a - c) ]^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~R^2\sin^2\chi_R + [ R^2\cos^2\chi_R + 2(a-c)R\cos\chi_R + (a - c)^2 ] </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~R^2 + (a - c)^2 - 2(c-a)R\cos\chi_R \, ;</math>
	</td>		</td>
	</tr>		</tr>
Line 3,038:		Line 3,062:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{R\sin\chi_R}{R\cos\chi_R + a - c} \, .</math>		<math>~\frac{R\sin\chi_R}{R\cos\chi_R + (a - c)} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Try Again */

2018-10-07T22:15:10Z

Try Again

← Older revision		Revision as of 22:15, 7 October 2018
Line 2,906:		Line 2,906:

	==Try Again==		==Try Again==
			Let's transform from Wong's lengths and angles to the ones used by Dyson. (This will permit us to compare Wong's analytic solution to the approximate one presented by Dyson.) Given the coordinate-pair <math>~(\varpi, z)</math> and Dyson's length scale, <math>~c</math>, we have,

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

			<tr>
			<td align="right">
			<math>~\tan\chi_R</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{z}{c-\varpi} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~R^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 + (c-\varpi)^2 \, .</math>
			</td>
			</tr>
			</table>

			From this pair of expressions we can transform from <math>~(R,\chi_R)</math> to <math>~(\varpi,z)</math> as follows:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~c-\varpi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z \cot\chi_R</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ R^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 (1 + \cot\chi^2_R) =\biggl[ \frac{z}{\sin\chi_R} \biggr]^{2}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ z</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~R\sin\chi_R</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \varpi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~c - \cot\chi_R \cdot R\sin\chi_R = c - R\cos\chi_R \, .</math>
			</td>
			</tr>
			</table>

			Next, given the coordinate-pair <math>~(\varpi, z)</math> and Wong's length scale, <math>~a</math>, we have,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tan\chi_2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{z}{a-\varpi} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~r_2^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 + (a-\varpi)^2 \, ;</math>
			</td>
			</tr>
			</table>

			and, combining the two sets gives,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~r_2^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~R^2\sin^2\chi_R + [ R\cos\chi_R + a - c ]^2 \, ;</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\tan\chi_2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{R\sin\chi_R}{R\cos\chi_R + a - c} \, .</math>
			</td>
			</tr>
			</table>




	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Compare Notations Used by Dyson and Wong */

2018-10-07T21:40:12Z

Compare Notations Used by Dyson and Wong

← Older revision		Revision as of 21:40, 7 October 2018
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	</tr>		</tr>
	</table>		</table>

			==Try Again==



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Ratios */

2018-10-05T19:43:48Z

Ratios

← Older revision		Revision as of 19:43, 5 October 2018
Line 2,852:		Line 2,852:
	<math>~		<math>~
	\frac{ (\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 ) ~-~( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) }{ ( c - a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( a\lambda_2^2 - c )(\lambda_1^2 - 1) }		\frac{ (\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 ) ~-~( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) }{ ( c - a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( a\lambda_2^2 - c )(\lambda_1^2 - 1) }
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{ (\lambda_1^2 - \lambda_2^2)(c^2 - a^2) }{ (\lambda_1^2 + \lambda_2^2)(c+a) - 2c } \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Ratios */

2018-10-05T19:23:24Z

Ratios

← Older revision		Revision as of 19:23, 5 October 2018
Line 2,815:		Line 2,815:
	<td align="left">		<td align="left">
	<math>~ ( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) + \varpi( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) + \varpi^2( 1 - \lambda_1^2 )( \lambda_2^2 - 1) \, .</math>		<math>~ ( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) + \varpi( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) + \varpi^2( 1 - \lambda_1^2 )( \lambda_2^2 - 1) \, .</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			\varpi^2 [(1 - \lambda_2^2 )(\lambda_1^2 - 1) ~-~ ( 1 - \lambda_1^2 )( \lambda_2^2 - 1) ]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\varpi [ ( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( 2a\lambda_2^2 - 2c )(\lambda_1^2 - 1) ]
			~+~[ ( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) ~-~(\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 ) ]
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	~~That~~ is,		Now, because the coefficient of the <math>~\varpi^2</math> term (on the left-hand-side of this last expression) is zero, we find that,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,825:		Line 2,842:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0		<math>~
			2 \varpi
	</math>		</math>
	</td>		</td>
Line 2,833:		Line 2,851:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\~~varpi^2 [~~( ~~1 -~~ \lambda_1^2 )( \lambda_2^2 ~~- 1~~) ~-~ (~~1 -~~ \lambda_2^2 )(\lambda_1^2 ~~- 1~~) ]		\frac{ (\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 ) ~-~( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) }{ ( c - a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( a\lambda_2^2 - c )(\lambda_1^2 - 1) }
	~~~+~\varpi [~~ ( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( ~~2a\lambda_2^2 - 2c )(\lambda_1^2 - 1) ]~~
	~~~+~[ (~~ \lambda_2^2 - ~~1)(~~ c~~^2 -a^2\lambda_1^2~~ ) ~~~-~~~(\lambda_1^2 - 1)~~( c^2 - a^2\lambda_2^2 ) ]~~
	</math>		</math>
	</td>		</td>

Tohline: /* Ratios */

2018-10-05T19:04:37Z

Ratios

← Older revision		Revision as of 19:04, 5 October 2018
Line 2,793:		Line 2,793:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[c~~^2 - 2c\varpi + \varpi~~^2 - a^2\lambda_2^2 + 2a~~\varpi~~\lambda_2^2 - \varpi^2\lambda_2^2 ] </math>		<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[ ( c^2 - a^2\lambda_2^2 ) + ( 2a\lambda_2^2 - 2c )\varpi + \varpi^2(1 - \lambda_2^2 ) ] </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 2,799:		Line 2,799:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~( \lambda_2^2 - 1) [ c^2 + 2c\varpi + \varpi^2 -a^2\lambda_1^2 - 2a \varpi\lambda_1^2 - \varpi^2\lambda_1^2 ]</math>		<math>~( \lambda_2^2 - 1) [ ( c^2 -a^2\lambda_1^2 ) + ( 2c - 2a \lambda_1^2)\varpi + \varpi^2( 1 - \lambda_1^2 ) ]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			(\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 )
			+ \varpi^2(1 - \lambda_2^2 )(\lambda_1^2 - 1) + \varpi( 2a\lambda_2^2 - 2c )(\lambda_1^2 - 1)
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ ( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) + \varpi( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) + \varpi^2( 1 - \lambda_1^2 )( \lambda_2^2 - 1) \, .</math>
			</td>
			</tr>
			</table>

			That is,

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

			<tr>
			<td align="right">
			<math>~0
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\varpi^2 [( 1 - \lambda_1^2 )( \lambda_2^2 - 1) ~-~ (1 - \lambda_2^2 )(\lambda_1^2 - 1) ]
			~+~\varpi [ ( 2c - 2a \lambda_1^2)( \lambda_2^2 - 1) ~-~ ( 2a\lambda_2^2 - 2c )(\lambda_1^2 - 1) ]
			~+~[ ( \lambda_2^2 - 1)( c^2 -a^2\lambda_1^2 ) ~-~(\lambda_1^2 - 1)( c^2 - a^2\lambda_2^2 ) ]
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Relationship Between Lengths */

2018-10-05T18:47:42Z

Relationship Between Lengths

← Older revision		Revision as of 18:47, 5 October 2018
Line 2,626:		Line 2,626:
	</table>		</table>

			===Ratios===

			Let's work with the following (dimensionless) length ratios:

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

			<tr>
			<td align="right">
			<math>~\lambda_2^2 \equiv \frac{R^2}{r_2^2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{z^2 + (c-\varpi)^2}{z^2 + (a-\varpi)^2} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\lambda_1^2 \equiv \frac{R_1^2}{r_1^2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{z^2 + (c+\varpi)^2}{z^2 + (a+\varpi)^2} \, .</math>
			</td>
			</tr>
			</table>

			Solving for <math>~z^2</math> in the first case, we find,

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

			<tr>
			<td align="right">
			<math>~\lambda_2^2 [z^2 + (a-\varpi)^2]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 + (c-\varpi)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~z^2( \lambda_2^2 - 1) </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(c-\varpi)^2 - \lambda_2^2(a-\varpi)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~z^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(c-\varpi)^2 - \lambda_2^2(a-\varpi)^2}{( \lambda_2^2 - 1)} \, ;</math>
			</td>
			</tr>
			</table>

			and in the second case, we find,

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

			<tr>
			<td align="right">
			<math>~\lambda_1^2 [z^2 + (a+\varpi)^2]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 + (c+\varpi)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ z^2 (\lambda_1^2 - 1)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(c+\varpi)^2 - \lambda_1^2 (a+\varpi)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ z^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(c+\varpi)^2 - \lambda_1^2(a+\varpi)^2}{(\lambda_1^2 - 1)} \, .</math>
			</td>
			</tr>
			</table>

			Combined, this gives,

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

			<tr>
			<td align="right">
			<math>~\frac{(c-\varpi)^2 - \lambda_2^2(a-\varpi)^2}{( \lambda_2^2 - 1)} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(c+\varpi)^2 - \lambda_1^2(a+\varpi)^2}{(\lambda_1^2 - 1)} </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[(c-\varpi)^2 - \lambda_2^2(a-\varpi)^2] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~( \lambda_2^2 - 1) [ (c+\varpi)^2 - \lambda_1^2(a+\varpi)^2 ]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[(c^2 - 2c\varpi + \varpi^2) - \lambda_2^2(a^2 - 2a\varpi +\varpi^2 )] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~( \lambda_2^2 - 1) [ (c^2 + 2c\varpi + \varpi^2) - \lambda_1^2(a^2+ 2a \varpi + \varpi^2) ]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[c^2 - 2c\varpi + \varpi^2 - a^2\lambda_2^2 + 2a\varpi\lambda_2^2 - \varpi^2\lambda_2^2 ] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~( \lambda_2^2 - 1) [ c^2 + 2c\varpi + \varpi^2 -a^2\lambda_1^2 - 2a \varpi\lambda_1^2 - \varpi^2\lambda_1^2 ]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ (\lambda_1^2 - 1)[c^2 - 2c\varpi + \varpi^2 - a^2\lambda_2^2 + 2a\varpi\lambda_2^2 - \varpi^2\lambda_2^2 ] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~( \lambda_2^2 - 1) [ c^2 + 2c\varpi + \varpi^2 -a^2\lambda_1^2 - 2a \varpi\lambda_1^2 - \varpi^2\lambda_1^2 ]</math>
			</td>
			</tr>
			</table>

	==Relationship Between Key Angles==		==Relationship Between Key Angles==

Tohline: /* Compare Notations Used by Dyson and Wong */

2018-10-05T18:14:14Z

Compare Notations Used by Dyson and Wong

Show changes

Tohline: /* Relationship Between Lengths */

2018-10-05T02:23:43Z

Relationship Between Lengths

← Older revision		Revision as of 02:23, 5 October 2018
Line 2,565:		Line 2,565:
	</table>		</table>

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

			<tr>
			<td align="right">
			<math>~r_2^2- (a-\varpi)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~r_1^2 - (a+\varpi)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ r_1^2 - (a+\varpi)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~r_2^2- (a-\varpi)^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ r_1^2 - r_2^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(a+\varpi)^2 - (a-\varpi)^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(a^2 + 2a \varpi + \varpi^2) - (a^2 - 2a\varpi + \varpi^2) </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4a \varpi </math>
			</td>
			</tr>
			</table>

	==Relationship Between Key Angles==		==Relationship Between Key Angles==

Tohline: /* Compare Notations Used by Dyson and Wong */

2018-10-04T20:38:45Z

Compare Notations Used by Dyson and Wong

← Older revision		Revision as of 20:38, 4 October 2018
Line 2,289:		Line 2,289:
	</table>		</table>

	~~Law of Cosines:~~		Key relationship, with understanding that, <math>~\epsilon \equiv d/c</math>:
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~(2c)^2</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~R_1^2 + R^2 - 2RR_1\cos\psi~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~(2a)^2</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~r_1^2 + r_2^2 - 2r_1 r_2 \cos\theta~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~Relationship~~, with understanding that, <math>~\epsilon \equiv d/c</math>:
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 2,336:		Line 2,305:
	</table>		</table>

			==Relationship Between Lengths==

			===Subtraction===
	In both cases, lengths are referenced to the same "external" point whose meridional-plane coordinates are, <math>~(\varpi,z)</math>. Hence,		In both cases, lengths are referenced to the same "external" point whose meridional-plane coordinates are, <math>~(\varpi,z)</math>. Hence,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,536:		Line 2,508:
	<td align="left">		<td align="left">
	<math>~2c^2\epsilon^2</math>		<math>~2c^2\epsilon^2</math>
			</td>
			</tr>
			</table>

			===Multiplication===

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

			<tr>
			<td align="right">
			<math>~R^2R_1^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~[z^2 + (c-\varpi)^2][z^2 + (c+\varpi)^2]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^4 + z^2[(c-\varpi)^2 + (c+\varpi)^2] + (c-\varpi)^2(c+\varpi)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~r_1^2r_2^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~[z^2 + (a-\varpi)^2][z^2 + (a+\varpi)^2]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z^4 + z^2[(a-\varpi)^2 + (a+\varpi)^2 ] + (a-\varpi)^2(a+\varpi)^2</math>
			</td>
			</tr>
			</table>


			==Relationship Between Key Angles==

			Law of Cosines:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(2c)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			R_1^2 + R^2 - 2RR_1\cos\psi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~(2a)^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			r_1^2 + r_2^2 - 2r_1 r_2 \cos\theta
			</math>
	</td>		</td>
	</tr>		</tr>