Editing SSC/Virial/PolytropesEmbeddedOutline (section)

===Third Effort===
In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically]], specifically, for indexes <math>~n=5</math> and <math>~n=1</math>.  The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> ''do not match'' the general form-factor expressions that were rather cavalierly "derived" during our first effort.  Oddly enough, as we discovered while [[SSCpt1/Virial/FormFactors#Fiddling_Around|fiddling around with the new results]], the ''ratio'' of these form factors appears to be the same as before, namely, 
<div align="center">
<table border="0" cellpadding="5" align="center">

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

</table>
</div>
It is worth noting that, as a result of this more thorough "third effort" examination, we have confirmed that the third key form factor, 
<div align="center">
<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>
</div>
is the same as before and the same as for isolated polytropes.  We also have determined that,
<div align="center">
<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} = 
\biggl(\frac{\tilde\xi^2 \tilde\theta^'}{\xi_1 \theta^'_1} \biggr)\biggl[- \frac{\tilde\xi}{3\tilde\theta^'}\biggr] 
= - \frac{\tilde\xi^3 }{3\xi_1 \theta^'_1} \, ,
</math>
</div>
except in the case of <math>~n=5</math> structures, for which we have determined,
<div align="center">
<math>~\biggl[\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_{n=5} = 
\ell^3 = \biggl( \frac{\tilde\xi^2}{3} \biggr)^{3/2} \, .
</math>
</div>