ThreeDimensionalConfigurations/HomogeneousEllipsoids - Revision history

Joel2: /* Index Symbols of the 2nd Order */

2024-08-04T23:50:14Z

Index Symbols of the 2nd Order

@@ Line 2,161: / Line 2,161: @@
 <table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
 Now, according to the relations stated in [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|an accompanying discussion]], we should find that, for <math>i = j</math>,
@@ Line 2,292: / Line 2,294: @@
 <math>
 \int_{a_\ell^2}^\infty \frac{dx}{x z^{5 / 2} }
 </math>
    </td>
@@ Line 2,298: / Line 2,341: @@
 <table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
 Again, according to the relations stated in [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|an accompanying discussion]], we should find that, for <math>i = j</math>,
@@ Line 2,410: / Line 2,455: @@
 </tr>
 </table>
 </td></tr></table>

2024-08-04T23:29:10Z

Index Symbols of the 2nd Order

@@ Line 2,357: / Line 2,357: @@
 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
 \biggr\}
 </math>
    </td>

2024-08-04T22:13:30Z

When am = aℓ

@@ Line 2,339: / Line 2,339: @@
    <td align="left">
 <math>\frac{1}{(1-e^2)} - a_\ell^2 A_{s \ell} </math>
    </td>
 </tr>

2024-08-04T22:05:46Z

Index Symbols of the 2nd Order

@@ Line 1,951: / Line 1,951: @@
 \biggr] \biggr\}
 \, .
 </math>
    </td>
@@ Line 2,294: / Line 2,271: @@
 </table>
 Q. E. D. &nbsp; &nbsp; &nbsp; <font color="red">Yeah!</font>
 </td></tr></table>

2024-08-03T18:17:16Z

Index Symbols of the 2nd Order

2024-08-03T17:38:53Z

Index Symbols of the 2nd Order

2024-08-03T17:22:37Z

Index Symbols of the 2nd Order

← Older revision		Revision as of 13:22, 3 August 2024
Line 1,873:		Line 1,873:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[2a_s^2 - ~~\frac{~~5a_\ell^2 }{4(a_\ell^2 - a_s^2)^2 a_\ell^4} \biggr] a_s		\biggl[\frac{2a_s^2 - 5a_\ell^2 }{4(a_\ell^2 - a_s^2)^2 a_\ell^4} \biggr] a_s
	+		+
	\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}		\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}
Line 1,900:		Line 1,900:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{~~5 -~~ 2(1-e^2)}{4e^4 } \biggr] \frac{a_s^2}{a_\ell^4}		\biggl[ \frac{2(1-e^2) - 5}{4e^4 } \biggr] \frac{a_s^2}{a_\ell^4}
	+		+
	\frac{3 }{4 e^{5}}		\frac{3 }{4 e^{5}}
Line 1,922:		Line 1,922:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{5 - 2(1-~~e^2)~~}{4e^4 } \biggr] (1-e^2)		\biggl[ \frac{-2e^2-3}{4e^4 } \biggr] (1-e^2)
	+		+
	\frac{3 (1 - e^2)^{1 / 2}}{4 e^{5}}		\frac{3 (1 - e^2)^{1 / 2}}{4 e^{5}}
Line 1,944:		Line 1,944:
	<math>		<math>
	\frac{1}{4e^4}\biggl\{		\frac{1}{4e^4}\biggl\{
	~~\biggl[ 5~~ - 2(~~1-e~~^2) ~~\biggr]~~ (1-e^2)		- (3 + 2e^2) (1-e^2)
	+		+
	3 (1 - e^2)^{1 / 2}		3 (1 - e^2)^{1 / 2}

2024-08-03T17:15:43Z

Index Symbols of the 2nd Order

← Older revision		Revision as of 13:15, 3 August 2024
Line 1,832:		Line 1,832:
	<math>		<math>
	\biggl[		\biggl[
	\biggl( \frac{3}{4a^2 a_\ell^2} - \frac{1}{2a a_\ell^4} \biggr) [(a_s^2 - a_\ell^2) + a_\ell^2]^{1 / 2}		\biggl( - \frac{3}{4a^2 a_\ell^2} + \frac{1}{2a a_\ell^4} \biggr) [(a_s^2 - a_\ell^2) + a_\ell^2]^{1 / 2}
	\biggr]		\biggr]
	+		+
Line 1,853:		Line 1,853:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl( \frac{3a_\ell^2 ~~- 2a~~}{4a^2 a_\ell^4} \biggr) a_s		\biggl( \frac{2a - 3a_\ell^2 }{4a^2 a_\ell^4} \biggr) a_s
	+		+
	\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}		\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}
Line 1,873:		Line 1,873:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{5a_\ell~~^2 - 2a_s~~^2}{4(a_\ell^2 - a_s^2)^2 a_\ell^4} \biggr] a_s		\biggl[2a_s^2 - \frac{5a_\ell^2 }{4(a_\ell^2 - a_s^2)^2 a_\ell^4} \biggr] a_s
	+		+
	\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}		\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}}

2024-08-02T23:26:27Z

Index Symbols of the 2nd Order

2024-08-02T23:24:26Z

Index Symbols of the 2nd Order

@@ Line 2,182: / Line 2,182: @@
 </tr>
 </table>
 ==When a<sub>m</sub> = a<sub>s</sub>==