User:Tohline/SSC/Structure/Other Analytic Ramblings - Revision history

Tohline: /* Third Guess */ Finished cleaning up simplification of 3rd guess table

2015-08-19T17:51:10Z

Third Guess: Finished cleaning up simplification of 3rd guess table

← Older revision		Revision as of 17:51, 19 August 2015
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - ~~27n~~ a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2		\alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 47n a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2
	</math><p>		</math><p>
	<math>		<math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	~~a_0[3a_2~~\alpha +~~6a_2~~ (n + m ) ~~] +~~ a_2~~[\alpha(3a_0~~ - ~~11a_2) + (-22n~~ a_2 ~~+6m a_0~~ - ~~10a_2~~ m~~) +~~ a_2~~\sigma~~^2]		\alpha(6a_0a_2 - 11a_2^2) + \sigma^2(a_2^2) + 11 n a_0a_2 +22 m a_0a_2 - 47n a_2^2 - 25m a_2^2 -12 n m a_2^2 + 4n m a_0a_2
	</math><p>		</math><p>
	<math>~		<math>~
	~~a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - 25na_2~~ -12 n ~~m a_2 + 5ma_0 ]~~		- 10 n(n-1)a_2^2 -2m(m-1)a_2^2 +4m(m-1)a_0 a_2
	~~</math></p><p>~~
	~~<math>~~~
	~~-[ 2n~~(n-1)a_2^2 +2m(m-1)a_2^2 ~~] + [ -8n(n-1)a_2^2~~ +4m(m-1)a_0 a_2 ]
	</math></p>		</math></p>
	</td>		</td>

Tohline: /* Second Guess */

2015-08-19T17:36:21Z

Second Guess

← Older revision		Revision as of 17:36, 19 August 2015
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ \alpha (-~~a_2~~^2) +\sigma^2 a_2^2 -47n a_2^2 + 11na_0a_2+ 22 m a_0a_2 -25ma_2^2 -12n m a_2^2 + 4n m a_0a_2		<math>~ \alpha (-11a_2^2 + 6a_0 a_2) +\sigma^2 a_2^2 -47n a_2^2 + 11na_0a_2+ 22 m a_0a_2 -25ma_2^2 -12n m a_2^2 + 4n m a_0a_2
	</math><p>		</math><p>
	<math>~-10n(n-1) a_2^2 - 2m(m-1)a_2^2 + 4m(m-1)a_0 a_2		<math>~-10n(n-1) a_2^2 - 2m(m-1)a_2^2 + 4m(m-1)a_0 a_2

Tohline: /* Second Guess */

2015-08-19T16:14:29Z

Second Guess

← Older revision		Revision as of 16:14, 19 August 2015
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + ~~5na_2~~ + 5m a_2 ] + [ 2n(n-1) a_2^2 + 2m(m-1)a_2^2 ]		<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + 11na_2 + 11m a_2 ] + [ 2n(n-1) a_2^2 + 2m(m-1)a_2^2 ]
	</math>		</math>
	</td>		</td>
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ \{ 3\alpha + [ 4n m + 5n + 5m ] + [ 2n(n-1) + 2m(m-1) ]\}a_2^2		<math>~ \{ 3\alpha + [ 4n m + 11n + 11m ] + [ 2n(n-1) + 2m(m-1) ]\}a_2^2
	</math>		</math>
	</td>		</td>

Tohline: /* Second Guess */ Fix some algebra errors

2015-08-19T16:11:13Z

Second Guess: Fix some algebra errors

← Older revision		Revision as of 16:11, 19 August 2015
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - x^4 [(2na_2 + 2ma_2 ) - a_2\alpha ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4]		\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4]
	-\sigma^2 [ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ] (a_0+a_2x^2)		-\sigma^2 [ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ] (a_0+a_2x^2)
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - x^4 [(2na_2 + 2ma_2 ) - a_2\alpha ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4]		\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4]
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	-\sigma^2\biggl\{ 2a_0^2 + [2a_0a_2 + a0(2a_2-a_0)]x^2 +[a_2(2a_2-a_0) -a_0a_2]x^4 - a_2^2 x^6 \biggr\}		-\sigma^2\biggl\{ 2a_0^2 + [2a_0a_2 + a_0(2a_2-a_0)]x^2 +[a_2(2a_2-a_0) -a_0a_2]x^4 - a_2^2 x^6 \biggr\}
	</math>		</math>
	</td>		</td>
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~5a_0[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] + \alpha [2a_0 ](5a_2-3a_0)-\sigma^2[2a_0a_2 + a0(2a_2-a_0)]		<math>~5a_0[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] + \alpha [2a_0 ](5a_2-3a_0)-\sigma^2[2a_0a_2 + a_0(2a_2-a_0)]
	</math><p>		</math><p>
	<math>~+ 2a_0[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ] + (2a_2-a_0)[ ( 5m a_0 - 10n a_2) ] - [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]		<math>~+ 2a_0[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ] + (2a_2-a_0)[ ( 5m a_0 - 10n a_2) ] - [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ - 5a_0[(2na_2 + 2ma_2 ) - a_2\alpha ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - 6a_0 a_2\alpha -\sigma^2[a_2(2a_2-a_0) -a_0a_2]		<math>~ - 5a_0[(2na_2 + 2ma_2 ) + a_2\alpha ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - 6a_0 a_2\alpha -\sigma^2[a_2(2a_2-a_0) -a_0a_2]
	</math><p>		</math><p>
	<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2 ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ] - a_2(5ma_0 - 10na_2)		<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2 ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ] - a_2(5ma_0 - 10na_2)
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ -3a_2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - (5a_2-3a_0)[(2na_2 + 2ma_2 ) - a_2\alpha ] +\sigma^2 a_2^2		<math>~ -3a_2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - (5a_2-3a_0)[(2na_2 + 2ma_2 ) + a_2\alpha ] +\sigma^2 a_2^2
	</math><p>		</math><p>
	<math>~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2 ] - a_2[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ]		<math>~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2 ] - a_2[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ]
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ \alpha (10a_2^2 -~~6a_0a_2~~+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_2)		<math>~ \alpha (10a_2^2 - 22a_0a_2+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_2)
	-47n a_0a_2 + ~~50n~~ a_2^2 - ~~45ma_0a_2~~ + 11m a_0^2 + 10m a_2^2-12n m a_0a_2 + 8n m a_2^2		-47n a_0a_2 + 60n a_2^2 - 50ma_0a_2 + 11m a_0^2 + 10m a_2^2-12n m a_0a_2 + 8n m a_2^2
	</math><p>		</math><p>
	<math>~+ 16n(n-1) a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2		<math>~+ 16n(n-1) a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2

Tohline: /* Third Guess */ Correct an algebra error

2015-08-19T15:59:30Z

Third Guess: Correct an algebra error

← Older revision		Revision as of 15:59, 19 August 2015
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	<td align="left">		<td align="left">
	<math>~		<math>~
	+ [ -15ma_2 - ~~5na_2~~ -12 n m a_2 + 5ma_0 ]x^4 + [5ma_2 + 5na_2 + 4n m a_2 ]x^6		+ [ -15ma_2 - 25na_2 -12 n m a_2 + 5ma_0 ]x^4 + [5ma_2 + 5na_2 + 4n m a_2 ]x^6
	\biggr\} (a_0 + a_2x^2)		\biggr\} (a_0 + a_2x^2)
	</math>		</math>
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	</math><p>		</math><p>
	<math>~		<math>~
	+ a_0 [ -15ma_2 - ~~5na_2~~ -12 n m a_2 + 5ma_0 ] + a_2[10ma_2 + 40na_2 + 8n m a_2 -15ma_0 ]		+ a_0 [ -15ma_2 - 25na_2 -12 n m a_2 + 5ma_0 ] + a_2[10ma_2 + 40na_2 + 8n m a_2 -15ma_0 ]
	</math></p><p>		</math></p><p>
	<math>		<math>
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	</math><p>		</math><p>
	<math>~		<math>~
	a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - ~~5na_2~~ -12 n m a_2 + 5ma_0 ]		a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - 25na_2 -12 n m a_2 + 5ma_0 ]
	</math></p><p>		</math></p><p>
	<math>~		<math>~
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	</math><p>		</math><p>
	<math>~		<math>~
	a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - ~~5na_2~~ -12 n m a_2 + 5ma_0 ]		a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - 25na_2 -12 n m a_2 + 5ma_0 ]
	</math></p><p>		</math></p><p>
	<math>~		<math>~

Tohline: /* Second Guess */ Corrected a couple of errors in x^4 coefficient, but still missing something

2015-08-18T21:27:05Z

Second Guess: Corrected a couple of errors in x^4 coefficient, but still missing something

← Older revision		Revision as of 21:27, 18 August 2015
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	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ - 5a_0[(2na_2 + 2ma_2 ) - a_2\alpha ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ]-\sigma^2[a_2(2a_2-a_0) -a_0a_2]		<math>~ - 5a_0[(2na_2 + 2ma_2 ) - a_2\alpha ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - 6a_0 a_2\alpha -\sigma^2[a_2(2a_2-a_0) -a_0a_2]
	</math><p>		</math><p>
	<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2 ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ]		<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2 ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ] - a_2(5ma_0 - 10na_2)
	</math></p><p>		</math></p><p>
	<math>~-[ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ] + [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]		<math>~-[ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ] + [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]

Tohline: /* Third Guess */

2015-08-18T21:10:46Z

Third Guess

← Older revision		Revision as of 21:10, 18 August 2015
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	<td align="left">		<td align="left">
	<math>~		<math>~
	~~a_0 [~~\alpha(3a_0 ~~- 11a_2~~) ~~+ (~~-~~22n a_2 +6m a_0 - 10a_2 m) + a_2~~\sigma^2~~] + a_2[\alpha~~(- ~~11a_0 +10a_2~~) + ~~(20n~~ a_2 - ~~10m~~ a_0) + ~~(a_0~~ - ~~2a_2)\sigma~~^2 ]		\alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 27n a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2
	</math><p>		</math><p>
	~~<math>~~~
	~~+ a_0 [ -15ma_2 - 5na_2 -12 n m a_2 + 5ma_0 ] + a_2[10ma_2 + 40na_2 + 8n m a_2 -15ma_0 ]~~
	~~</math></p><p>~~
	<math>		<math>
	~ ~~- [ -8n~~(n-1)a_2^2 +4m(m-1)a_0 a_2 ~~] + [ 8n(n-1)a_2^2~~ +2m(m-1)a_0^2 ]		~ + 16n(n-1)a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2
	</math></p>		</math></p>
	</td>		</td>

Tohline: /* Third Guess */ Finished simplifying x^2 coefficients

2015-08-18T20:20:22Z

Third Guess: Finished simplifying x^2 coefficients

← Older revision		Revision as of 20:20, 18 August 2015
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	</div>		</div>


			After simplification:
			<div align="center" id="FirstTable">
			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="right"><math>~x^0</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~ \alpha(10a_0^2) + \sigma^2(- 2a_0^2) -20n a_0a_2 + 10ma_0^2 </math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^2</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~\alpha(- 11a_0^2 +20 a_0a_2) + \sigma^2(a_0^2 - 4 a_0a_2) + 60n a_0a_2 -20na_2^2- 25m a_0^2
			+ 20m a_0a_2 + 8n m a_0a_2
			</math><p><math>
			- [ 8n(n-1)a_2^2 + 2m(m-1)a_0^2 ]
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^4</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~
			a_0 [\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m) + a_2\sigma^2] + a_2[\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0) + (a_0 - 2a_2)\sigma^2 ]
			</math><p>
			<math>~
			+ a_0 [ -15ma_2 - 5na_2 -12 n m a_2 + 5ma_0 ] + a_2[10ma_2 + 40na_2 + 8n m a_2 -15ma_0 ]
			</math></p><p>
			<math>
			~ - [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ] + [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^6</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~
			a_0[3a_2\alpha +6a_2 (n + m ) ] + a_2[\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m) + a_2\sigma^2]
			</math><p>
			<math>~
			a_0[5ma_2 + 5na_2 + 4n m a_2 ] + a_2[ -15ma_2 - 5na_2 -12 n m a_2 + 5ma_0 ]
			</math></p><p>
			<math>~
			-[ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ] + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^8</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~
			a_2[3a_2\alpha +6a_2 (n + m ) ] + a_2[5ma_2 + 5na_2 + 4n m a_2 ] + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]
			</math>
			</td>
			</tr>

			</table>
			</div>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Second Guess */ More simplification

2015-08-18T20:01:15Z

Second Guess: More simplification

← Older revision		Revision as of 20:01, 18 August 2015
Line 6,010:		Line 6,010:
	<td align="center">  :  </td>		<td align="center">  :  </td>
	<td align="left">		<td align="left">
	<math>~ ~~-3a_2[(4n a_2 - 2m a_0) +~~ \alpha (~~2a_2~~-~~a_0~~) ~~] - (5a_2-3a_0)[(2na_2~~ + ~~2ma_2 )~~ - a_2~~\alpha ]~~ +~~\sigma~~^2 a_2^2		<math>~ \alpha (-a_2^2) +\sigma^2 a_2^2 -47n a_2^2 + 11na_0a_2+ 22 m a_0a_2 -25ma_2^2 -12n m a_2^2 + 4n m a_0a_2
	</math><p>		</math><p>
	<math>~~~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2 ] - a_2[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0~~ - 10n ~~a_2) ]~~		<math>~-10n(n-1) a_2^2 - 2m(m-1)a_2^2 + 4m(m-1)a_0 a_2
	~~</math></p><p>~~
	~~<math>~- [ 2n~~(n-1) a_2^2 + 2m(m-1)a_2^2 ~~] + [ -8n(n-1) a_2^2~~ + 4m(m-1)a_0 a_2 ]
	</math></p>		</math></p>
	</td>		</td>

Tohline: /* Second Guess */ Some simplification

2015-08-18T19:33:46Z

Second Guess: Some simplification

← Older revision		Revision as of 19:33, 18 August 2015
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	<td align="left">		<td align="left">
	<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + 5na_2 + 5m a_2 ] + [ 2n(n-1) a_2^2 + 2m(m-1)a_2^2 ]		<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + 5na_2 + 5m a_2 ] + [ 2n(n-1) a_2^2 + 2m(m-1)a_2^2 ]
			</math>
			</td>
			</tr>
			</table>
			</div>


			After simplification:
			<div align="center" id="FirstTable">
			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="right"><math>~x^0</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~
			10a_0^2 \alpha - 2a_0^2\sigma^2 + 10m a_0^2 - 20n a_0a_2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^2</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~\alpha (20a_0a_2-11a_0^2) -\sigma^2[4a_0a_2 -a_0^2]
			</math><p>
			<math>~+ 60na_0a_2 -20na_2^2 + 20m a_0a_2 -25m a_0^2 + 8n m a_0a_2
			- [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^4</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~ \alpha (10a_2^2 -6a_0a_2+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_2)
			-47n a_0a_2 + 50n a_2^2 - 45ma_0a_2 + 11m a_0^2 + 10m a_2^2-12n m a_0a_2 + 8n m a_2^2
			</math><p>
			<math>~+ 16n(n-1) a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^6</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~ -3a_2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0) ] - (5a_2-3a_0)[(2na_2 + 2ma_2 ) - a_2\alpha ] +\sigma^2 a_2^2
			</math><p>
			<math>~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2 ] - a_2[(4n m a_2 + 5na_2 + 5m a_2)- ( 5m a_0 - 10n a_2) ]
			</math></p><p>
			<math>~- [ 2n(n-1) a_2^2 + 2m(m-1)a_2^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]
			</math></p>
			</td>
			</tr>

			<tr>
			<td align="right"><math>~x^8</math></td>
			<td align="center">  :  </td>
			<td align="left">
			<math>~ \{ 3\alpha + [ 4n m + 5n + 5m ] + [ 2n(n-1) + 2m(m-1) ]\}a_2^2
	</math>		</math>
	</td>		</td>