Editing StabilityVariationalPrinciple (section)

==Pressure-Truncated Configurations==

===Expectations===
For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index.  In this situation, then, we have,
<div align="center">
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">Free-energy expression:</td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mathcal{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">Virial equilibrium:</td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} +  \chi_\mathrm{eq}^{4 } \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">Stability indicator:</td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

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

<tr>
  <td align="right">
<math>~6g \chi_\mathrm{eq}^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
6g \chi_\mathrm{eq}^4 
+\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2e - \biggl[\frac{af(n+3)}{nb} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

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

<tr>
  <td align="right">
<math>~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
<!-- EARLIER DERIVATION

Hence, the equilibrium radius of the marginally unstable configuration is given by the expression,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e - 6g \chi_\mathrm{eq}^4 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e + 6g \biggl[ \frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n}  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[\frac{f(n+3)}{n^2} \biggr]  \chi_\mathrm{eq}^{(n-3)/n} + \biggl(\frac{2gb}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{3}\biggl(e + \frac{ag}{c} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\chi_\mathrm{eq}^{(n-3)/n}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2n^2}{3}\biggl[\frac{ec+ag}{cf(n+3) + 2ngb} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
EARLIER DERIVATION -->

===Energies and Structural Form Factors===
====Old Approach====

As has been developed in, for example, our [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]], we adopt the following normalizations:
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, , </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R^3_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>

Then, from separate summaries &#8212; both [[SSCpt1/Virial#Summary_of_Normalized_Expressions|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|here]] &#8212; we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)  \chi^3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx\biggr\}  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}} = \frac{U_\mathrm{int}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
  </td>
</tr>
</table>
</div>
where the [[SSCpt1/Virial#Structural_Form_Factors|structural form factors are defined]] as follows:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{f}_M </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^1  3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~  3\cdot 5 \int_0^1 \biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\}  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx \, .</math>
  </td>
</tr>
</table>
</div>

This gives, specifically for [[SSCpt1/Virial/FormFactors#PTtable|specifically for pressure-truncated polytropic configurations]],
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\tilde\mathfrak{f}_A
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

====New Approach====
In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions.  Basically, we will be repeating some [[SSCpt1/Virial#Expressions_for_Various_Energy_Terms|earlier derivations]].

=====Mass=====

Defining <math>~M_\mathrm{tot}</math> as the total mass of the ''isolated'' configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily ''equilibrium'') radius; and being careful to define the mean density of the truncated configuration such that,
<div align="center">
<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~M_r(r) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \int_0^r  4\pi r^2 \rho dr  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3}{4\pi} \int_0^r  4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr)  \frac{dr}{R_\mathrm{norm}}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^r  3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{dr}{R}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^\xi  3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{d\xi}{\tilde\xi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3  
\int_0^\xi  3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr)  \frac{d\xi}{\tilde\xi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}  
\int_0^\xi  3\xi^2 \theta^n  d\xi \, .  
</math>
  </td>
</tr>
</table>
</div>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi}  3\xi^2 \theta^n  d\xi  = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Now, from the,
<div align="center">
Polytropic Lane-Emden Equation<p></p>
{{ Math/EQ_SSLaneEmden01 }}
</div>
we realize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi^2 \theta^n \, .</math>
  </td>
</tr>
</table>
</div>
So these last two expressions may also be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr)  {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr]  
\, ; 
</math>
  </td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi  \, .</math>
  </td>
</tr>
</table>
</div>

=====Modified Internal Energy=====
Now we want to develop the appropriately scaled integral definition of a "variational" internal energy having the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{U_\Upsilon}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) } 
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{P}{P_\mathrm{norm}}\biggr) \biggl( \frac{dr}{R_\mathrm{norm}}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho}\biggr]^{\gamma_\mathrm{g}}
\biggl( \frac{M}{M_\mathrm{tot}}\biggr)^{\gamma_\mathrm{g}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma_\mathrm{g}}
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{P}{P_c}\biggr) \biggl( \frac{dr}{R}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}\chi^{-3/n}
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \, . </math>
  </td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~f</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f = \chi^{3/n}\biggl[ \frac{U_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \biggr\} \, ;</math>
  </td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_U(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f \rightarrow b = \chi^{3/n}\biggl[ \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} {\tilde\mathfrak{f}}_A
\, ;</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3  \xi^2 \theta^{n+1} d\xi \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>


When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala &amp; Horedt (1974)]], this integral over polytropic functions becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~{\tilde\mathfrak{f}}_A \equiv
{\tilde\xi}^{-3}\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[SSCpt1/Virial/FormFactors#PTtable|derived earlier]].

=====Modified Gravitational Potential Energy=====

Similarly, we have,

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

<tr>
  <td align="right">
<math>~\frac{W_\Upsilon}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\int_0^R \Upsilon_W(r) \biggl(\frac{GM_r}{r}\biggr) 4\pi r^2 \rho dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \frac{R_\mathrm{norm}\rho_c R^2}{M_\mathrm{tot}}\int_0^R 4\pi \Upsilon_W(r) \biggl(\frac{M_r}{M_\mathrm{tot}}\biggr)  \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\chi^{-1}
\int_0^R 3\Upsilon_W(r) \biggl[\frac{M_r}{M_\mathrm{tot}}\biggr]  \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 3\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~e</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e = -\chi \biggl[ \frac{W_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, ;
</math>
  </td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_W(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e \rightarrow a = -\chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{5}\biggl[\frac{1}{{\tilde\mathfrak{f}}_M} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 ~{\tilde\mathfrak{f}}_W \, ;
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_W</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Now, according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala &amp; Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

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

<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
As we have [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
{\tilde\xi}^{-5}
\int_0^\tilde\xi 5 \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

===Test Virial Equilibrium Condition===

If the correct normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is specified, our [[#Expectations|expectation regarding virial equilibrium]] is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~3nc\chi_\mathrm{eq}^{4 } - 3b\chi_\mathrm{eq}^{(n-3)/n} + an</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0\, .</math>
  </td>
</tr>
</table>
</div>
Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair &#8212; <math>~R_\mathrm{eq}</math> and <math>~P_e</math> &#8212; that has been given by [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt (1970)]], namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\chi_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}} =  \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} 
\, ,
</math>
  </td>
</tr>
</table>
</div>

where we have taken into account the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|shift in normalization factors]],
<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">Switch from [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
  <td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}\biggr]^{1/(4-3\gamma)}
= \biggl[(n+1)^{-1} \biggl( 4\pi \biggr)^{1/n}\biggr]^{n/(n-3)}
= \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)}
\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl\{ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3} \biggl( \frac{1}{4\pi} \biggr) \biggr\}^{\gamma/(4-3\gamma)}
= \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>

We therefore have:
====First Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~3n\biggl[\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr]\chi_\mathrm{eq}^{4 }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
4\pi n \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
 \biggr\}^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{4(n-1)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi n \biggl[(n+1)^{[3(n+1)-4n]} ( 4\pi )^{[4-(n+1)]} \biggr]^{1/(n-3)}
{\tilde\xi}^4  \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{[2(n+1)+ 4(1-n)]/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{[4(n-1)-2(n+1)]/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{n}{(n+1) }\biggr]
{\tilde\xi}^4  \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^2 \, .
</math>
  </td>
</tr>
</table>
</div>

====Second Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~3b\chi_\mathrm{eq}^{(n-3)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
4\pi ~n \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} 
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\biggr\}^{(n-3)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4\pi ~n}{(5-n)} \biggl[ \frac{1}{4\pi} \biggl( - \frac{\tilde\xi}{\tilde\theta^'}\biggr) \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} 
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
\times (n+1)^{-1} ( 4\pi )^{1/n}
{\tilde\xi}^{(n-3)/n} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2} 
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
\times {\tilde\xi}^{[(n-3)/n + 3(n+1)/n]} ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)/n - (n+1)/n]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2} 
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
  </td>
</tr>
</table>
</div>


====Third Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~an</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{3}{5}\biggl[\biggl( - \frac{\tilde\xi}{3\tilde\theta^'} \biggr) \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 
\frac{3\cdot 5~n}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr)  \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^2 
\frac{n \tilde\xi^4}{(5-n)} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr)  \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} \, .
</math>
  </td>
</tr>
</table>
</div>

====Combined====
Combining the three terms, the virial expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ (5-n)(n+1)\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^{-2} \biggl[ 3nc\chi_\mathrm{eq}^{4 } + an - 3b\chi_\mathrm{eq}^{(n-3)/n} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
n(5-n)
{\tilde\xi}^4  \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} 
+ n(n+1)\tilde\xi^4 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-n \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4  \biggl\{
(5-n)\tilde\theta_n^{n+1} 
+ (n+1) \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr)  \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(n+1) ( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4  \biggl\{ 0 \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Q. E. D.