Editing SSC/Stability/InstabilityOnsetOverview (section)

=====D&eacute;j&agrave; Vu=====
It should not be a surprise, but is nevertheless comforting to see, that our general expression for the polytropic displacement function, 

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<math>~x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, ,</math>
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morphs into the analytically defined, polytropic displacement functions that had previously been identified, namely, for n = 3 and n = 5 configurations.  The first of these transformations is trivial to demonstrate because, when n = 3, the leading coefficient of the radially dependent portion of the expression for <math>~x_P</math> is zero.  Hence,
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<math>~x_P\biggr|_{n=3} \rightarrow \biggl[\frac{3(n-1)}{2n}\biggl]_{n=3} =  1 \, .</math>
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In the second case, drawing on the definition of <math>~\theta(\xi)</math> for n = 5 polytropes, as given [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|in an accompanying chapter]], we have,
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<math>~x_P\biggr|_{n=5}</math>
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<math>~=</math>
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<math>~
\frac{6}{5}\biggl[1 + \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=5} 
</math>
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</tr>

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&nbsp;
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<math>~=</math>
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<math>~
\frac{6}{5} - \frac{3}{5\xi} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{5/2} \frac{\xi}{3} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} 
</math>
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</tr>

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&nbsp;
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<math>~=</math>
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<math>~
\frac{6}{5} - \frac{1}{5} \biggl( 1 + \frac{\xi^2}{3} \biggr)
</math>
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&nbsp;
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<math>~=</math>
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<math>~
1 - \frac{\xi^2}{15} \, .
</math>
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</tr>
</table>
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