Editing SSC/Structure/OtherAnalyticRamblings

__FORCETOC__
<!-- __NOTOC__ will force TOC off -->
=Ramblings Regarding the Stability of Other Analytically Definable, Spherical Equilibrium Models=

The material presented in this chapter was originally developed as a [[SSC/Structure/OtherAnalyticModels|subsection of a chapter that discusses "Other" Analytic Equilibrium Models."]]

{{ SGFworkInProgress }}

===Generic Setup===
Dividing the [[SSC/Structure/OtherAnalyticModels#Stabililty_2|above, 2<sup>nd</sup>-order ODE]] through by the quantity, <math>~[R^2 (P_0/P_c)]</math>, gives, 

<div align="center">
<math>
\frac{d^2x}{dr_0^2} 
+ \biggl[\frac{4}{r_0} - \frac{1}{R}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggl(\frac{P_c}{P_0}\biggr)\biggr] \frac{dx}{dr_0} 
- \biggl[\frac{1}{R}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{\alpha}{r_0} \biggr]  x 
= - \frac{1}{R^2}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \biggl( \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} \biggr) \biggr]  x \, ,
</math><br />
</div>


which matches [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad's (1949)] equation (1), namely,

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

<tr>
  <td align="right">
<math>~x^{' '} + \biggl[\frac{4}{r_0} - \frac{\mu(r_0) }{r_0}\biggr] x^{'} - \biggl[ \frac{\alpha \mu(r_0)}{r_0^2} \biggr] x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \frac{n^2\rho_c}{\gamma_g P_c} \biggr] x  \, ,</math>
  </td>
</tr>
</table>
</div>
where, primes indicate differentiation with respect to <math>~r_0</math>, and,
<div align="center">
<math>~\mu(r_0) \equiv \frac{r_0}{R} \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, .</math>
</div>

(Note that Prasad's equation has the awkward units of inverse length-squared.)  Regrouping terms in Prasad's governing equation, multiplying through by <math>~R^2</math> (to make the equation dimensionless), and now letting primes denote differentiation with respect to the ''dimensionless'' radial coordinate, <math>~\chi_0</math>, we quite generally can write the linear adiabatic wave equation as,

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

<tr>
  <td align="right">
<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[x^{' '} + \frac{4 x^'}{\chi_0}\biggr] - \frac{\mu(\chi_0)}{\chi_0} \biggl[ x^{'}   + \frac{\alpha x}{\chi_0} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) - \frac{\mu(\chi_0)}{\chi_0} \biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Defining, 

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

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{A\mu(\chi_0)}{\chi_0}  = \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
the governing equation becomes,

<div align="center" id="LAWE">
<table border="1" cellpadding="5"><tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{B}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) -\frac{A}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~B \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - A \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>

Notice that, because,
<div align="center">
<math>~g_0 = - \frac{1}{\rho_0} ~\frac{dP_0}{dr_0} \, ,</math>
</div>
at every radial location throughout the configuration, it must also be true that, for any equilibrium configuration,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1} \frac{d(P_0/P_c)}{d\chi_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{B}{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<span id="Examples">The following table shows that this relationship holds for a collection of analytically described equilibrium structures.</span>

<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="4"><font size="+1">Table 1a:  Properties of Analytically Defined Equilibrium Structures</font></th>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~\frac{\rho_0}{\rho_c}</math>
  <td align="center"><math>~\frac{P_0}{P_c}</math>
  <td align="center"><math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~1</math>
  <td align="center"><math>~1 - \chi_0^2</math>
  <td align="center"><math>~-2\chi_0</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1-\chi_0</math>
  <td align="center"><math>~\tfrac{1}{5} (5 -24 \chi_0^2 + 28\chi_0^3 - 9\chi_0^4)</math>
  <td align="center"><math>~\tfrac{1}{5}[- 48\chi_0 + 84\chi_0^2 - 36\chi_0^3]</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~1-\chi_0^2</math>
  <td align="center"><math>~\tfrac{1}{2} (2  - 5\chi_0^2 + 4\chi_0^4 - \chi_0^6)</math>
  <td align="center"><math>~- 5\chi_0 + 8\chi_0^3 - 3\chi_0^5</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  <td align="center"><math>~\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2</math>
  <td align="center"><math>~\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math>
</tr>
</table>


<div align="center" id="Table1b">
<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="5"><font size="+1">Table 1b:  Properties of Analytically Defined Equilibrium Structures</font></th>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~\frac{\rho_0}{\rho_c}</math>
  <td align="center"><math>~B\equiv \frac{g_0}{g_\mathrm{SSC}}</math>
  <td align="center"><math>~A \equiv \biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1}</math>
  <td align="center"><math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-2}</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~1</math>
  <td align="center"><math>~2\chi_0</math>
  <td align="center"><math>~1 - \chi_0^2</math>
  <td align="center"><math>~1 - \chi_0^2</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1-\chi_0</math>
  <td align="center"><math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math>
  <td align="center"><math>~\tfrac{1}{5} (1-\chi_0) (5 + 10\chi_0 - 9\chi_0^2)</math>
  <td align="center"><math>~(1 + 2\chi_0 - \tfrac{9}{5}\chi_0^2)</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~1-\chi_0^2</math>
  <td align="center"><math>~5\chi_0 - 3\chi_0^3</math>
  <td align="center"><math>~\tfrac{1}{2} (1-\chi_0^2) (2  - \chi_0^2)</math>
  <td align="center"><math>~(1  - \tfrac{1}{2} \chi_0^2)</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  <td align="center"><math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math>
  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  <td align="center"><math>~1</math>
</tr>
</table>
</div>

<span id="Generic">Leaning on this new expression for the ratio, <math>~B/A</math>, let's play with the form of the governing equation.</span>

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

<tr>
  <td align="right">
<math>~- \sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A \biggl\{ \frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A\biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl\{ \biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\}
</math>
  </td>
</tr>
</table>
</div>

===Polytropic Configurations===
Let's compare this presentation of the LAWE to the form of the LAWE that has been derived [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|specifically for polytropic equilibrium configurations]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V(\xi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{\xi}{(\theta/\theta_c)} \frac{d (\theta/\theta_c)}{d\xi} = \frac{g_0}{a_n}\biggl(\frac{a_n^2\rho_0}{P_0}\biggr)\frac{\xi}{(n+1)} \, .</math>
  </td>
</tr>
</table>
</div>
[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped.  This notation was introduced in our [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|separate discussion of the polytropic LAWE]] in order to make it clear how our derivations have overlapped earlier published work.]  Regrouping terms, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} - 
\biggl[ \frac{\alpha (n+1)V(x)}{\xi^2} \biggr]  x</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{d^2x}{d\xi^2} + \biggl(\frac{4}{\xi}\biggr)\frac{dx}{d\xi} \biggr] -\biggl[\frac{(n+1)V(\xi)}{\xi} \biggr] 
\biggl[\frac{dx}{d\xi} + \frac{\alpha x}{\xi} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) -\biggl[\frac{(n+1)V(\xi)}{\xi} \biggr] 
\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\xi} \biggr] 
\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Next we note that, written in terms of the traditional polytropic radial coordinate, <math>~\xi</math>, the fractional radius,
<div align="center">
<math>~\chi_0 \equiv \frac{r_0}{R} = \frac{\xi}{\xi_1} = \frac{a_n \xi}{R} \, .</math>
</div>
Hence, multiplying the polytropic LAWE through by the quantity, <math>~(R/a_n)^2</math>, gives,

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

<tr>
  <td align="right">
<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\chi_0} \biggr] 
\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[\omega^2 \biggl(\frac{R^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl(\frac{\theta_c}{\theta}\biggr)\sigma^2 x \, .</math>
  </td>
</tr>
</table>
</div>

Finally, noting that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\theta}{\theta_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{P_0}{P_c} \biggr)\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1} 
= \biggl( \frac{P_0}{P_c} \biggr)^{1/(n+1)} \, ,
</math>
  </td>
</tr>
</table>
</div>
we can rewrite the polytropic LAWE in the form,

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

<tr>
  <td align="right">
<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[\frac{d\ln(P_0/P_c)}{d\chi_0} \biggr] 
\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{P_0}{P_c} \biggr)^{-1}\biggl( \frac{\rho_0}{\rho_c} \biggr)\sigma^2 x \, ,</math>
  </td>
</tr>
</table>
</div>
which precisely matches the general expression for the LAWE presented at the end of our [[SSC/Structure/OtherAnalyticModels#Generic_Setup|generic setup, directly above]].  

This seems to be a particularly insightful way to write the LAWE, as the only structural functions that appear explicitly are <math>~P_0(\chi_0)</math> and <math>~\rho_0(\chi_0)</math>.  It appears as though the eigenfunctions that describe ''adiabatic'' radial pulsations do not explicitly depend ''a priori'' on the radial dependence of the equilibrium gravitational acceleration.

===Examine Structural Pressure-Density Relation===

====Derivation====

One striking property exhibited by the [[SSC/Structure/OtherAnalyticModels#Examples|example configurations tabulated above]] is the ''structural'' relationship between the chosen function, <math>~\rho_0(\chi_0)</math>, and the corresponding radial pressure distribution, <math>~P_0(\chi_0)</math>, that is  [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|dictated by]], 

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} = - \frac{GM_r}{r^2} </math> ,
</div>

As has been detailed in the last column of [[SSC/Structure/OtherAnalyticModels#Table1b|Table 1b]], in all four cases, the ratio, <math>~(P_0/P_c)(\rho_0/\rho_c)^{-2}</math>, is an analytically prescribed polynomial expression.  That is, the pressure is "evenly divisible" by the square of the density.  Let's examine how broadly reliable this behavior is.  [Note that, for simplicity in typing, hereafter throughout this subsection we will drop the subscript zero and, rather than <math>~\chi_0</math>, we will use <math>~z \equiv r/R</math> to denote the dimensionless radial coordinate.] 


Assume a mass-distribution given by the general quadratic function,

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

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - a z - b z^2 \, ,</math>
  </td>
</tr>
</table>
</div>
where both coefficients, <math>~a</math> and <math>~b</math>, are positive.


<div align="center" id="Surface">
<table border="1" width="75%" align="center" cellpadding="5">
<tr>
<td align="center">ASIDE:  Surface Location</td>
</tr>
<tr><td align="left">
The surface of the configuration will be defined by the radial location, <math>~z_s</math>, at which the density first goes to zero.  If <math>~b = 0</math>, then the surface will be located at <math>~z_s = a^{-1}</math>; and if <math>~a = 0</math>, it will be located at <math>~z_s = b^{-1/2}</math>.  More generally, however, the roots of the quadratic equation that results from setting <math>~\rho/\rho_c</math> to zero are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{a}{2b}\biggl[1 \mp \biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Because only the <math>~z_+</math> solution provides positive roots, we conclude that, when both <math>~a</math> and <math>~b</math> are nonzero, the radial coordinate of the surface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_s = z_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{2b}\biggl[\biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  - 1\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

We acknowledge, as well, that the density profile can now be written in terms of these roots; specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b(z_+ - z)(z - z_-) \, .</math>
  </td>
</tr>
</table>
</div>

In the discussion, below, it may be advantageous to adopt the following notation:
<div align="center">
<math>~\ell^2 \equiv \frac{4b}{a^2} ~~~~~\Rightarrow ~~~~~ \ell = \frac{2b^{1/2}}{a} \, ,</math>
</div>
in which case,
<div align="center">
<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, .</math>
</div>

</td></tr>
</table>
</div>


This specified density profile implies a mass distribution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R^3 \rho_c \int_0^z (1 - a z - b z^2)z^2 dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

The hydrostatic balance condition therefore implies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{R\rho_c} \biggl( \frac{\rho}{\rho_c} \biggr)^{-1} \frac{dP}{dz}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{G}{R^2 z^2} \biggl[ 4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\biggl[ \frac{1}{4\pi G R^2 \rho_c^2 } \biggr] \frac{dP}{dz}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr)\biggl( \frac{\rho}{\rho_c} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
- \biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr) 
+ a \biggl( \frac{z^2}{3} - \frac{a z^3}{4} - \frac{b z^4}{5} \biggr) 
+ b \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{a}{4} + \frac{a}{3}  \biggr) + z^3 \biggl( \frac{b}{5} - \frac{a^2}{4} + \frac{b}{3}  \biggr) 
+ z^4 \biggl( - \frac{ab}{5} - \frac{ab}{4}  \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr) 
- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{P}{P_n} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int \biggl[ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr) 
- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr) \biggr] dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -\biggl(\frac{1}{6}\biggr)z^2 + z^3 \biggl( \frac{7a}{36} \biggr) + z^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) 
- z^5 \biggl( \frac{9ab}{100} \biggr) - z^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] + C \, .
</math>
  </td>
</tr>
</table>
</div>

where, <math>~C</math>, is an integration constant, and,
<div align="center">
<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>
</div>
The integration constant &#8212; which also proves to be the normalized central pressure &#8212; is determined by ensuring that the pressure goes to zero at [[SSC/Structure/OtherAnalyticModels#Surface|the surface of the configuration]], <math>~z_s</math>.  That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C = \frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -
\biggl[ -\biggl(\frac{1}{6}\biggr)z_s^2 + z_s^3 \biggl( \frac{7a}{36} \biggr) + z_s^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) 
- z_s^5 \biggl( \frac{9ab}{100} \biggr) - z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the pressure profile is,

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

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) - \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) (z_s^4 - z^4)
+ \biggl( \frac{9ab}{100} \biggr)(z_s^5 - z^5) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6) \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check this general expression against the specific cases described above.

====Example1====

First, let's set <math>~b=0</math>, but leave <math>~a</math> general.  As described in the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE]], this means that <math>~z_s=a^{-1}</math>.  So the pressure distribution is,

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

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) + \biggl( \frac{a^2}{16}\biggr) (z_s^4 - z^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(a^{-2} - z^2) - \biggl( \frac{7a}{36} \biggr)(a^{-3} - z^3) + \biggl( \frac{a^2}{16}\biggr) (a^{-4} - z^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^{-2} \biggl\{
\frac{1}{6}[1 - (az)^2] - \frac{7}{36} [1 - (az)^3] + \frac{1}{16} [1 - (az)^4]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And from the expression for the integration constant, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl(\frac{1}{6}\biggr)a^{-2}  - a^{-3} \biggl( \frac{7a}{36} \biggr) + a^{-4} \biggl( \frac{a^2}{16}\biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^{-2} \biggl[\frac{1}{6} - \frac{7}{36}  + \frac{1}{16} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2^4\cdot 3^2 a^2} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, dividing one expression by the other, we obtain,

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

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl\{ 24[1 - (az)^2] - 28 [1 - (az)^3] + 9 [1 - (az)^4] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check to see if this "general linear" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,

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

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - a z  \, .</math>
  </td>
</tr>
</table>
</div>
Strategically rewriting the expression for the pressure distribution gives,

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

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl\{ 24[1 - (az)][1 + (az)] - 28 [1 - (az)][1 + (az) + (az)^2] + 9 [1 - (az)][1 + (az)][1 + (az)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{ 24[1 + (az)] - 28 [1 + (az) + (az)^2] + 9 [1 + (az)][1 + (az)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 24(az) - 28 [(az) + (az)^2] + 9 [(az) + (az)^2 + (az)^3] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 5(az) - 19 (az)^2 + 9 (az)^3 \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And, as luck would have it, the expression inside the curly braces can be "divided evenly" by the quantity, <math>~[1-(az)]</math>, one more time.  Specifically, the expression becomes,

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

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [5+ 10(az) - 9 (az)^2 ] \, .
</math>
  </td>
</tr>
</table>
</div>


====Example2====

First, let's set <math>~a=0</math>, but leave <math>~b</math> general.  As described in the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE]], this means that <math>~z_s=b^{-1/2}</math>.  So the pressure distribution is,

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

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{2b}{15} \biggr) (z_s^4 - z^4) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(b^{-1} - z^2) - \biggl( \frac{2b}{15} \biggr) (b^{-2} - z^4) + \biggl(\frac{b^2}{30}   \biggr)(b^{-3} - z^6) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{b}\biggl\{
\biggl(\frac{1}{6}\biggr)[1 - bz^2] - \biggl( \frac{2}{15} \biggr) [1 - (bz^2)^2] + \biggl(\frac{1}{30}   \biggr)[1 - (bz^2)^3] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And from the expression for the integration constant, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \biggl(\frac{1}{6}\biggr)z_s^2 - z_s^4 \biggl( \frac{2b}{15}\biggr) + z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{b}
\biggl[ \frac{1}{6} -  \frac{2}{15} + \frac{1}{30} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3 \cdot 5b} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, dividing one expression by the other, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl\{
5 [1 - bz^2] - 4  [1 - (bz^2)^2] +  [1 - (bz^2)^3] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check to see if this "general parabolic" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,

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

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - b z^2  \, .</math>
  </td>
</tr>
</table>
</div>
Strategically rewriting the expression for the pressure distribution gives,

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

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl\{ 5[1 - (bz^2)] - 4[1 - (bz^2)][1 + (bz^2)] + [1 - (bz^2)][1 + (bz^2) + (bz^2)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl\{ 5 - 4 [1 + (bz^2)] + [1 + (bz^2) + (bz^2)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl[ 2 - 3 (bz^2) + (bz^2)^2 \biggr] 
</math>
  </td>
</tr>
</table>
</div>

Again, as luck would have it, the expression inside the square brackets can be "divided evenly" by the quantity, <math>~[1-(bz^2)]</math>, one more time.  Specifically, the expression becomes,

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

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [2-(bz^2) ] \, .
</math>
  </td>
</tr>
</table>
</div>

====Example3====
In the most general quadratic case, we should rewrite the pressure distribution as,

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

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2^4\cdot 3^2\cdot 5^2}\biggl\{
2^3\cdot 3\cdot 5^2 (z_s^2 - z^2) - 2^2\cdot 5^2 \cdot 7 a(z_s^3 - z^3) - [2^5\cdot 3\cdot 5 b - 3^2\cdot 5^2 a^2] (z_s^4 - z^4)
+ 2^2\cdot 3^4 ab (z_s^5 - z^5) + 2^3\cdot 3\cdot 5b^2 (z_s^6 - z^6) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
600 (1 - \zeta^2) - 700 (a z_s) (1 - \zeta^3) - [480 (b z_s^2) - 225 (a z_s)^2] (1 - \zeta^4)
+ 324 (az_s)(b z_s^2) (1 - \zeta^5) + 120(b z_s^2)^2 (1 - \zeta^6) 
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>

where, <math>~\zeta \equiv z/z_s</math>.  Similarly, let's rewrite the integration constant as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
600 - 700 (a z_s)  - [480 (b z_s^2) - 225 (a z_s)^2] 
+ 324 (az_s)(b z_s^2)  + 120(b z_s^2)^2  
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

So the pressure can be written as,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P}{P_n} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~ \biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P_c}{P_n}\biggr]
- 600 \zeta^2 + 700 (a z_s) \zeta^3 + [480 (b z_s^2) - 225 (a z_s)^2] \zeta^4
- 324 (az_s)(b z_s^2) \zeta^5 - 120(b z_s^2)^2 \zeta^6  \, .
</math>
  </td>
</tr>
</table>
</div>

The question that remains to be answered is:  Is this expression for the pressure distribution "evenly divisible" by the square (or even the ''first'' power) of the normalized density distribution which, [[SSC/Structure/OtherAnalyticModels#Derivation|as defined above]] for the general quadratic case, is,
<div align="center">
<math>\frac{\rho}{\rho_c} = 1 - az - bz^2 = 1 - (az_s)\zeta - (bz_s^2)\zeta^2 \, .</math>
</div>

In attempting to answer this question, it may prove advantageous to refer back to the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE discussion]] of the roots of this quadratic function and, in particular, that,
<div align="center">
<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, ,</math>
</div>
where, <math>~\ell^2 \equiv 4b/a^2</math>.


<!--  HIDE TABLE 1C  
<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="5"><font size="+1">Table 1c:  Properties of Analytically Defined Equilibrium Structures</font></th>
</tr>
<tr>
  <td align="center" colspan="5">
<math>~\frac{\rho}{\rho_c} = 1 - a\chi_0 - b\chi_0^2</math>
  </td>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~a</math></td>
  <td align="center"><math>~b</math></td>
  <td align="center"><math>~C = \frac{P_c}{P_n}</math></td>
  <td align="center"><math>~\frac{P_0}{P_n}</math></td>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{6}</math>
  </td>
  <td align="center">
<math>C - \frac{1}{6}\chi_0^2 </math> 
  </td>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{6}\biggl[1 - \biggl( \frac{7}{6} \biggr) + \biggl( \frac{15}{40} \biggr) \biggr] = \frac{5}{2^4\cdot 3^2}</math></td>
  <td align="center">
<math>~
C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^3 \biggl( \frac{7}{36} \biggr) - \chi_0^4 \biggl( \frac{1}{16}\biggr) \biggr]
= C + \frac{1}{2^4\cdot 3^2}\biggl[ -24\chi_0^2 +  28\chi_0^3  - 9 \chi_0^4\biggr]
</math> 
  </td>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\frac{1}{6}\biggl[1 - \frac{32}{40}  + \frac{1}{5}  \biggr] = \frac{1}{15}</math></td>
  <td align="center">
<math>~
C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^4 \biggl( \frac{8}{60} \biggr) - \chi_0^6 \biggl(\frac{1}{30}   \biggr) \biggr]
= C + \frac{1}{30}\biggl[ -5\chi_0^2 + 4 \chi_0^4  - \chi_0^6 \biggr]
</math> 
  </td>
</tr>
<tr>
  <td align="center">General Linear</td>
  <td align="center"><math>~a</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{2^4\cdot 3^2}\biggl[
24 - 28a  + 9a^2  \biggr]
</math>
  </td>
  <td align="center">
<math>~
C + \frac{1}{2^4\cdot 3^2}\biggl[ -24 \chi_0^2 + 28a \chi_0^3 -9a^2 \chi_0^4  \biggr] 
</math> 
  </td>
</tr>
</table>
END HIDING of TABLE 1C -->

===Uniform Density===

In the case of a uniform-density configuration, the governing equation is,

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

<tr>
  <td align="right">
<math>~\sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\chi_0 \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - (1-\chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>
</div>

The following individual mode analyses should be compared with the results found in [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution|our discussion of Sterne's general solution]].

====Mode 0====
Try an eigenfunction of the form,
<div align="center">
<math>x = a_0\, ,</math>
</div>
in which case,
<div align="center">
<math>x^' = x^{' '} = 0 \, .</math>
</div>
In order for this to be a solution, we must have,

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

<tr>
  <td align="right">
<math>~\sigma^2  a_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\chi_0 \biggl[ \frac{\alpha a_0}{\chi_0} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\alpha = 2\biggl(3 - \frac{4}{\gamma_g}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_g - \frac{4}{3}\, .
</math>
  </td>
</tr>
</table>
</div>


====Mode 2====
Try an eigenfunction of the form,
<div align="center">
<math>x = a_0 + a_2\chi_0^2 \, ,</math>
</div>
in which case,
<div align="center">
<math>~x^' = 2 a_2\chi_0 </math>&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~x^{' '} = 2 a_2 \, . </math>
</div>
In order for this to be a solution, we must have,

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

<tr>
  <td align="right">
<math>~\sigma^2  \biggl(a_0 + a_2\chi_0^2\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \biggl[ \alpha (a_0 + a_2\chi_0^2) + \chi_0 (2 a_2\chi_0 )\biggr] - (1-\chi_0^2) \biggl[ \frac{4(2 a_2\chi_0 )}{\chi_0}+ 2a_2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\alpha a_0 + \chi_0^2[2a_2 (2+\alpha)] - 10a_2(1-\chi_0^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \sigma^2 a_0 - 2\alpha a_0 + 10a_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0^2 [-\sigma^2  + 2 (2+\alpha) + 10   ]a_2 \, .
</math>
  </td>
</tr>
</table>
</div>
Given that the coefficients on both sides of this expression must independently be zero, we have:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 (2+\alpha) + 10</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g}{6}\biggl[14 +2\biggl(3-\frac{4}{\gamma_g} \biggr)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g}{6}\biggl[20 -\frac{8}{\gamma_g}\biggr] = \frac{1}{3}\biggl(10\gamma_g - 4\biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
and
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{a_2}{a_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{10} \biggl[ 2\alpha - \sigma^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{10} \biggl\{ 2\alpha - [14+2\alpha) ] \biggr\} = - \frac{7}{5} \, .</math>
  </td>
</tr>
</table>
</div>

===Parabolic Density Distribution===

In the case of a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|parabolic density distribution]], the governing equation is,

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

<tr>
  <td align="right">
<math>~\sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5\chi_0 - 3\chi_0^3)\biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - \tfrac{1}{2} (1-\chi_0^2) (2  - \chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{15}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>
</div>


====First Trial====
Try an eigenfunction of the form,
<div align="center">
<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x^' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (2-\chi_0^2)^{-1} (2 b\chi_0 + 4c\chi_0^3) + 2\chi_0 (2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (2-\chi_0^2)^{-2}[(2-\chi_0^2) (2 b\chi_0 + 4c\chi_0^3) + 2\chi_0 (a + b\chi_0^2 + c\chi_0^4) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[(2-\chi_0^2) (b + 2c\chi_0^2) +  (a + b\chi_0^2 + c\chi_0^4) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[  (2b+4c\chi_0^2-b\chi_0^2 -2c\chi_0^4) +  (a + b\chi_0^2 + c\chi_0^4) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ] \, ;</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~x^{' '} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2-\chi_0^2)^{-1} (2 b + 12 c\chi_0^2) + 2\chi_0 (2-\chi_0^2)^{-2} (2 b\chi_0 + 4c\chi_0^3) 
+ 2(2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ 2\chi_0 (a + b\chi_0^2 + c\chi_0^4) [-2(2-\chi_0^2)^{-3}(-2\chi_0)] + 2\chi_0 (2-\chi_0^2)^{-2} (2 b\chi_0 + 4c\chi_0^3) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2-\chi_0^2)^{-3} [
2(4-4\chi_0^2 + \chi_0^4) (b + 6 c\chi_0^2) + 4\chi_0^2 (2-\chi_0^2) (b + 2c\chi_0^2) 
+ 2(2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ 8\chi_0^2 (a + b\chi_0^2 + c\chi_0^4)  + 4\chi_0^2 (2-\chi_0^2) (b + 2c\chi_0^2) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2(2-\chi_0^2)^{-3} \{
( 4b+24c\chi_0^2 - 4b\chi_0^2-24c\chi_0^4 + b\chi_0^4 + 6c\chi_0^6 ) +(8b \chi_0^2+16c\chi_0^4 -4b\chi_0^4 - 8c\chi_0^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+(2a+2b\chi_0^2 + 2c\chi_0^4 -a\chi_0^2 - b\chi_0^4 - c\chi_0^6)
+  (4a\chi_0^2 + 4b\chi_0^4 + 4c\chi_0^6)  \}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2(2-\chi_0^2)^{-3} \{
(4b+2a) + \chi_0^2(24c-4b+8b+2b-a+4a) + \chi_0^4(-24c+b+16c-4b+2c-b+4b) + \chi_0^6(6c-8c-c+4c)
\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 (2-\chi_0^2)^{-3} \{(2a+4b) + \chi_0^2(3a + 6b + 24c) + \chi_0^4(-6c) + \chi_0^6(c)\}
</math>
  </td>
</tr>
</table>
</div>


In order for this to be a solution, we must have,

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

<tr>
  <td align="right">
<math>~\sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5 - 3\chi_0^2)\biggl[ \alpha x + \chi_0 x^'\biggr] - (1-\chi_0^2) (2  - \chi_0^2) \biggl[ \frac{2x^'}{\chi_0}+ \frac{x^{' '}}{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\sigma^2  (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(5 - 3\chi_0^2)\biggl[ \alpha  (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) + 2\chi_0^2 (2-\chi_0^2)^{-2}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1-\chi_0^2) \biggl[ 4 (2-\chi_0^2)^{-1}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]+ (2-\chi_0^2)^{-2} \{(2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6\} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Multiplying through by <math>~(2-\chi_0^2)^2</math> gives,

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

<tr>
  <td align="right">
<math>~\sigma^2  (2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(5 - 3\chi_0^2)\biggl\{ \alpha  (2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) + 2\chi_0^2 [  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1-\chi_0^2) \biggl\{ 4 (2-\chi_0^2)[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]+ [(2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\sigma^2  [2a + \chi_0^2(2b-a) + \chi_0^4(2c -b) - c\chi_0^6]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(5 - 3\chi_0^2)\biggl\{ \alpha  [2a + \chi_0^2(2b-a) + \chi_0^4(2c -b) - c\chi_0^6]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (5 - 3\chi_0^2)\biggl\{   (2a + 4b)\chi_0^2 + 8c\chi_0^4 - 2c \chi_0^6  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1-\chi_0^2) \biggl\{ [  8(a + 2b)+ 32c\chi_0^2 - 8c \chi_0^4  ] - [  4\chi_0^2(a + 2b)+ 16c\chi_0^4 -4c \chi_0^6  ]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1-\chi_0^2) \biggl\{ (2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(5 - 3\chi_0^2)\biggl\{ 2a\alpha + \chi_0^2 [(2b-a)\alpha +  (2a + 4b) ] + \chi_0^4[ (2c -b)\alpha + 8c] - (2+\alpha)c\chi_0^6 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1-\chi_0^2) \biggl\{  (10a + 20b)+ \chi_0^2[56c - a - 2b] +\chi_0^4 [-30c  ]+ 5c \chi_0^6  \biggr\} 
</math>
  </td>
</tr>
</table>
</div>

So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~2a\sigma^2 - 10a\alpha +10a +20b =~a(2\sigma^2 - 10\alpha+10) + 20b</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~(2b-a)\sigma^2 -5[(2b-a)\alpha +  (2a + 4b) ] +6a\alpha+[56c - a - 2b] - (10a + 20b)</math><p>
<math>=~a[-\sigma^2 -5(-\alpha + 2) +6\alpha-1-10] + b[2\sigma^2-10\alpha-20-2-20 ] + c[ 56]</math></p><p>
<math>=~a[-\sigma^2 + 11\alpha - 21] + b[2\sigma^2-10\alpha-42 ] + 56 c</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~(2c-b)\sigma^2 - 5[ (2c -b)\alpha + 8c] - 30c + 3[(2b-a)\alpha +  (2a + 4b)] - [56c - a - 2b ] </math><p>
<math>=~a[-3\alpha+7] + b[-\sigma^2+5\alpha+6\alpha+12+2] + c[2\sigma^2-10\alpha-40-30-56]</math></p><p>
<math>=~a[7-3\alpha] + b[11\alpha-\sigma^2+14] + c[2\sigma^2-10\alpha-126]</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~- c\sigma^2 +(10+5\alpha)c +3[ (2c -b)\alpha + 8c] +5c +30c</math><p>
<math>=~b[-3\alpha] + c[-\sigma^2 + 10 + 5\alpha+6\alpha+24+35]</math></p><p>
<math>=~b[-3\alpha] + c[11\alpha -\sigma^2 + 69]</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~- 3(2+\alpha)c -5c =~c[-11-3\alpha]</math>
  </td>
</tr>
</table>
</div>

===Independent Investigation of Parabolic Distribution===
In the specific case of a parabolic density distribution, the leading factor on the LHS is,

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

<tr>
  <td align="right">
<math>~\frac{1}{A_\mathrm{parab}} \equiv \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} 
= \frac{1}{(1-\chi_0^2) (1-\tfrac{1}{2}\chi_0^2)}
= \frac{2}{2 - 3\chi_0^2 + \chi_0^4}
\, ,</math>
  </td>
</tr>
</table>
</div>

and the function appearing on the RHS is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu(\chi_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\chi_0^2(1-\chi_0^2)(5-3\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} 
= \frac{\chi_0^2 (5-3\chi_0^2)}{A_\mathrm{parab}}
\, .</math>
  </td>
</tr>
</table>
</div>
Multiplying the linear adiabatic wave equation through by <math>~A_\mathrm{parab}</math>, gives,

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

<tr>
  <td align="right">
<math>~- \sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
- \frac{B_\mathrm{parab} }{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)   \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>
~B_\mathrm{parab} \equiv \chi_0 (5-3\chi_0^2) \, .
</math>
</div>

Now we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr] + \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \frac{\chi_0^4 x^'}{2} \frac{d}{d\chi_0}\biggl[ 1 - \frac{3}{\chi_0^2} + \frac{2}{\chi_0^4}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \biggl[ \frac{3}{\chi_0^3} - \frac{4}{\chi_0^5}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 -  3\chi_0^2 ) \, .
</math>
  </td>
</tr>
</table>
</div>

Similarly we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr] + \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 
\frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ 5\chi_0^{1-\alpha} -3 \chi_0^{3-\alpha}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \biggl[ 5(1-\alpha)\chi_0^{-\alpha} -3 (3-\alpha)\chi_0^{2-\alpha}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the LAWE can be rewritten as,

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

<tr>
  <td align="right">
<math>~- \sigma^2  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 -  3\chi_0^2 ) 
- \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] + x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr]  \, ;
</math>
  </td>
</tr>
</table>
</div>
then multiplying through by <math>~\chi_0</math>, and rearranging terms gives,

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

<tr>
  <td align="right">
<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' - B_\mathrm{parab}~x\biggr] + x^'(4 -  3\chi_0^2 ) \, .
</math>
  </td>
</tr>
</table>
</div>
Next, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
A_\mathrm{parab}~x^'
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - x \biggl[\frac{d}{d\chi_0}\biggl(A_\mathrm{parab}\biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - \frac{x}{2} \biggl[\frac{d}{d\chi_0}\biggl(2 - 3\chi_0^2 + \chi_0^4\biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + x (3\chi_0 - 2\chi_0^3 )</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~A_\mathrm{parab}~x^' - B_\mathrm{parab}~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + \biggl[(3\chi_0 - 2\chi_0^3 ) - ( 5\chi_0 - 3\chi_0^3 )\biggr] x</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x \, .</math>
  </td>
</tr>
</table>
</div>
So, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ \frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x\biggr] + x^'(4 -  3\chi_0^2 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) -
\chi_0 ~\frac{d}{d\chi_0}\biggl[ (2\chi_0 - \chi_0^3 ) x\biggr] 
+ x^'(4 -  3\chi_0^2 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) - \biggl\{ \frac{d}{d\chi_0}\biggl[ (2\chi_0^2 - \chi_0^4)x\biggr] - (2\chi_0 - \chi_0^3)x \biggr\}
+ \biggl\{ \frac{d}{d\chi_0}\biggl[ (4-3\chi_0^2)x\biggr] + 6\chi_0 x \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-3\chi_0^2)x -(2\chi_0^2 - \chi_0^4)x\biggr] 
+ (2\chi_0 - \chi_0^3)x + 6\chi_0 x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-5\chi_0^2 + \chi_0^4)x \biggr] 
+ (8\chi_0 - \chi_0^3)x \, .
</math>
  </td>
</tr>
</table>
</div>
Moving the last term on the RHS of this expression to the LHS, and factoring the polynomial coefficients of the terms inside of the first and second derivatives gives,

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

<tr>
  <td align="right">
<math>~x \biggl\{ (5\alpha - 13 - \sigma^2)\chi_0 + (10 - 3\alpha)\chi_0^{3} \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ (1-\chi_0^2)(2-\chi_0^2)x \biggr] + \frac{d}{d\chi_0}\biggl[(1-\chi_0^2)(4-\chi_0^2)x \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>




====First Trial (Same as Above)====
Try an eigenfunction of the form,
<div align="center">
<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
</div>
in which case,

<!--  LHS -->

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

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2}\biggl\{ (a + b\chi_0^2 + c\chi_0^4) (2-\chi_0^2) [ (5\alpha - 13 - \sigma^2) + (10 - 3\alpha)\chi_0^{2}] \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) \biggl\{
(10\alpha - 26 - 2\sigma^2) + (\sigma^2 -11\alpha + 33  )\chi_0^2 + (3\alpha -10)\chi_0^{4}
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2}\biggl\{  
a(10\alpha - 26 - 2\sigma^2) + \chi_0^2[a(\sigma^2 -11\alpha + 33  ) + b(10\alpha - 26 - 2\sigma^2)] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\chi_0^{4}[ a (3\alpha -10) + b(\sigma^2 -11\alpha + 33  ) + c(10\alpha - 26 - 2\sigma^2)] 
+ \chi_0^{6}[ b (3\alpha -10) + c(\sigma^2 -11\alpha + 33  )] + c(3\alpha -10)\chi_0^{8}
\biggr\}  
</math>
  </td>
</tr>
</table>
</div>

<!--  RHS (1st part) -->


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

<tr>
  <td align="right">
RHS (1<sup>st</sup> term)
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ (1-\chi_0^2)(a + b\chi_0^2 + c\chi_0^4) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ a + (b-a)\chi_0^2 + (c-b)\chi_0^4 - c\chi_0^6\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\chi_0}{2} \frac{d}{d\chi_0} \biggl[ 2(b-a)\chi_0 + 4(c-b)\chi_0^3 - 6c\chi_0^5\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\chi_0}{2} [ 2(b-a) + 12(c-b)\chi_0^2 - 30c\chi_0^4]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(b-a)\chi_0 + 6(c-b)\chi_0^3 - 15c\chi_0^5 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{ (4-4\chi_0^2 + \chi_0^4)
[(b-a) + 6(c-b)\chi_0^2 - 15c\chi_0^4 ] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{ 
[(4b-4a) + (24c- 24b)\chi_0^2 - 60c\chi_0^4 ] 
+ [(-4b + 4a)\chi_0^2 + (-24c + 24b)\chi_0^4 + 60c\chi_0^6 ] 
+ [(b-a)\chi_0^4 + 6(c-b)\chi_0^6 - 15c\chi_0^8 ] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{ 
(4b-4a) + [(24c- 24b) + (-4b + 4a)]\chi_0^2 +[  - 60c  + (-24c + 24b) + (b-a)]\chi_0^4 
+ [60c  + 6(c-b)]\chi_0^6 - 15c\chi_0^8  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{ 
(4b-4a) + [ 24c- 28b + 4a]\chi_0^2 +[  - 84c  + 25b  -a ]\chi_0^4 
+ [66c  -6b ]\chi_0^6 - 15c\chi_0^8  
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

<!--  RHS (2nd part) -->


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

<tr>
  <td align="right">
RHS (2<sup>nd</sup> term)
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi_0}\biggl[(1-\chi_0^2)(4-\chi_0^2)(2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2-\chi_0^2)^{-1} \frac{d}{d\chi_0}\biggl[(4-5\chi_0^2+\chi_0^4)(a + b\chi_0^2 + c\chi_0^4) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
(4-5\chi_0^2+\chi_0^4)(a + b\chi_0^2 + c\chi_0^4) \frac{d}{d\chi_0}\biggl[(2-\chi_0^2)^{-1} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2-\chi_0^2)^{-1} \frac{d}{d\chi_0}\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]  \frac{d}{d\chi_0}\biggl[(2-\chi_0^2)^{-1} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2-\chi_0^2)^{-2}\biggl\{(2-\chi_0^2) \biggl[\chi_0 (8b-10a) + \chi_0^3 (16c-20b+4a) + \chi_0^5 (6b-30c) + 8c\chi_0^7 \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+2\chi_0
\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]    
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2}\biggl\{\biggl[(16b-20a) + \chi_0^2 (32c-40b+8a) + \chi_0^4 (12b-60c) + 16c\chi_0^6 \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[\chi_0^2 (8b-10a) + \chi_0^4 (16c-20b+4a) + \chi_0^6 (6b-30c) + 8c\chi_0^8 \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
\biggl[8a + \chi_0^2 (8b-10a) + \chi_0^4 (8c-10b+2a) + \chi_0^6 (2b-10c) + 2c\chi_0^8 \biggr]    
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2}\biggl\{[(16b-20a) + 8a] + \chi_0^2 [(32c-40b+8a) + (10a-8b) + (8b-10a)] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \chi_0^4 [(12b-60c)+ (20b-4a-16c) + (8c-10b+2a)] + \chi_0^6[16c + (30c-6b) + (2b-10c) ] - 6c\chi_0^8 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi_0 (2-\chi_0^2)^{-2}\biggl\{[16b-12a] + \chi_0^2 [(32c-40b+8a) ] + \chi_0^4 [-2a + 22b -68c] + \chi_0^6[36c -4b ] - 6c\chi_0^8 
\biggr\}
</math>
  </td>
</tr>
</table>
</div>


So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="SecondTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a(10\alpha - 26 - 2\sigma^2) - (4b-4a) - [16b-12a] ~=a(10\alpha - 10 - 2\sigma^2) - 20b</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~[a(\sigma^2 -11\alpha + 33  ) + b(10\alpha - 26 - 2\sigma^2)] - [ 24c- 28b + 4a]- [(32c-40b+8a) ]</math><p>
<math>= ~[a(\sigma^2 -11\alpha + 21  ) + b(10\alpha + 42 - 2\sigma^2)] -  56c</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~[ a (3\alpha -10) + b(\sigma^2 -11\alpha + 33  ) + c(10\alpha - 26 - 2\sigma^2)] - [  - 84c  + 25b  -a ] - [-2a + 22b -68c]</math><p>
<math>~=~[ a (3\alpha -7) + b(\sigma^2 -11\alpha -14  ) + c(10\alpha + 126 - 2\sigma^2)]</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~[ b (3\alpha -10) + c(\sigma^2 -11\alpha + 33  )] - [66c  -6b ]-[36c -4b ]</math><p>
<math>~=~[ b (3\alpha ) + c(\sigma^2 -11\alpha - 69  )] </math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~c(3\alpha -10) + 15c + 6c = c[3\alpha+11]</math>
  </td>
</tr>
</table>
</div>

The expressions for the coefficients presented in [[SSC/Structure/OtherAnalyticModels#SecondTable|this table]] exactly match the entire set of expressions [[SSC/Structure/OtherAnalyticModels#FirstTable|derived earlier]], except for the adopted sign convention &#8212; every term in this second derivation has the opposite sign to the corresponding term in the earlier derivation.

===Conjecture===

Returning to the [[SSC/Structure/OtherAnalyticModels#Generic|generic formulation derived earlier]], we have,

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

<tr>
  <td align="right">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

Now, ''suppose'' that the expression on the RHS is of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{RHS}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~u dv + v du \, ,</math>
  </td>
</tr>
</table>
</div>
where <math>~u \equiv (P_0/P_c)</math>?  Then the function,
<div align="center">
<math>~v \equiv \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, ,</math>
</div>
and the eigenvector, <math>~x</math>, must satisfy ''both'' of the relations:

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

<tr>
  <td align="right">
<font size="+1">Relation I</font>
  </td>
  <td align="center">
<math>~:</math>
  </td>
  <td align="left">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x  
= \frac{d}{d\chi_0}\biggl[ \biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<font size="+1">Relation II</font>
  </td>
  <td align="center">
<math>~:</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0}\biggl[ \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)\biggr]
= \frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
</math>
  </td>
</tr>
</table>
</div>

The validity (or not) of this conjecture can be tested against both configurations whose  --- ABANDON!

===Exploration===

====Compare LAWE to Hydrostatic Balance Condition====

Returning to the [[SSC/Structure/OtherAnalyticModels#Generic|generic formulation derived earlier]], we have,

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

<tr>
  <td align="right">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

Dividing this entire expression through by <math>~(P_0/P_c)x</math> gives,

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

<tr>
  <td align="right">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x \chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl[ \ln \biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{1}{x\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
+ \frac{d}{d\chi_0}\biggl[ \ln \biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Now, let's step aside from the LAWE and look directly at the differential relationship between the mass-density and the pressure, as dictated by combining the [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|two principal governing relations]], the

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
</div>
and,
<div align="center">
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr)  = 4\pi G \rho </math> .<br />
</div>
In combination, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-4\pi G \rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{r_0^2}\frac{d}{dr_0}\biggl[ \frac{r_0^2}{\rho_0} \frac{dP_0}{dr_0}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -4\pi G \rho_0 \biggl(\frac{R^2\rho_c}{P_c}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \frac{\chi_0^2}{(\rho_0/\rho_c)}\cdot \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1}\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{d}{d\chi_0}\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\chi_0^2 (P_0/P_c)}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] 
+ \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{p^'}{p}\biggr)
\frac{1}{\chi_0^2 p^'}\frac{d}{d\chi_0}\biggl[ \chi_0^2 p^'\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d\ln(p)}{d\chi_0}\cdot
\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p^'</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>

Let's compare the form of this "equilibrium" relation with the form of the LAWE just constructed:
<div align="center" id="Compare">
<table border="1" align="center" width="75%">
<tr><td align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~- [4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{p^'}{p} \biggr) \cdot 
\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<font size="+1">''versus''</font>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
+ \frac{d\ln(p)}{d\chi_0}\cdot  \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] 
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

I like this layout because it unveils similarities in the way the differential operators interact with the functions that describe the radial profiles of variables &#8212; specifically, the mass-density, the pressure, and the fractional radial displacement, <math>~x</math>, during pulsations.  However, it is not yet obvious how best to translate between the two differential equations in order to aid in solving for the unknown variable, <math>~x(\chi_0)</math>.

====Dabbling with Equilibrium Condition====
In the meantime, I've found it instructive to play with the first of these two expressions to see how it might be restructured in order to most directly confirm that it is satisfied by the expressions presented in [[SSC/Structure/OtherAnalyticModels#Examples|Table 1]].  Adopting the shorthand notation,
<div align="center">
<math>~\Gamma \equiv 4\pi G\rho_c \tau_\mathrm{SSC}^2</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\varpi \equiv \frac{\rho_0}{\rho_c} \, ,</math>
</div>
and multiplying the "equilibrium" relation through by <math>~(-\varpi p)</math>, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\Gamma \varpi^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \varpi p^'\biggl\{
\frac{1}{\chi_0^2 p^'} \frac{d}{d\chi_0} (\chi_0^2 p^') - \frac{1}{\varpi}\frac{d\varpi}{d\chi_0}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^' \varpi^' - \varpi \frac{dp^'}{d\chi_0}  -\frac{2\varpi p^'}{\chi_0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \varpi \frac{dp^'}{d\chi_0}  \, ;
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\Gamma \varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}  \, .
</math>
  </td>
</tr>
</table>
</div>

=====Specific Cases=====

<font color="red">'''Case 1''' (Parabolic)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = 1 -\chi_0^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = -2\chi_0 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = -5\chi_0 + 8\chi_0^3 - 3\chi_0^5 = \chi_0(1-\chi_0^2)(-5+3\chi_0^2)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = -5 +3\chi_0^2  \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
Also, note: 
  </td>
  <td align="left">
<math>~\frac{d(p^')}{d\chi_0} = -5 +24\chi_0^2 -15\chi_0^4 \, .</math>
  </td>
</tr>
</table>
</div>

For the parabolic case, therefore, the RHS of the "equilibrium" expression is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<font size="+1">RHS</font>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-5+3\chi_0^2)\biggl[ -2\chi_0^2 - 2(1-\chi_0^2) \biggr] - (-5 +24\chi_0^2 -15\chi_0^4)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(10 - 6\chi_0^2) + (5 -24\chi_0^2 +15\chi_0^4)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~15(1-2\chi_0^2+\chi_0^4) \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the LHS of the "equilibrium" relation, if,
<div align="center">
<math>~\Gamma = 15</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{15}{4\pi G \rho_c} \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Stabililty_2|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the parabolic density distribution.


<font color="red">'''Case 2''' (Linear)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = 1 -\chi_0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = -1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = \tfrac{12}{5}[- 4\chi_0 + 7\chi_0^2 - 3\chi_0^3]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = \tfrac{12}{5}(-4 +3\chi_0)  \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
Also, note: 
  </td>
  <td align="left">
<math>~\frac{d(p^')}{d\chi_0} = \tfrac{12}{5}[- 4 + 14\chi_0 - 9\chi_0^2] \, .</math>
  </td>
</tr>
</table>
</div>

For the linear case, therefore, the RHS of the "equilibrium" expression is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<font size="+1">RHS</font>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}(-4 +3\chi_0)\biggl[ -\chi_0  - 2(1-\chi_0) \biggr] - \tfrac{12}{5}(- 4 + 14\chi_0 - 9\chi_0^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}\biggl[ (4 -3\chi_0)( 2-\chi_0 ) + (4 - 14\chi_0 + 9\chi_0^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}(12-24\chi_0 + 12\chi_0^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{2^4\cdot 3^2}{5}(1-2\chi_0 + \chi_0^2) \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the LHS of the "equilibrium" relation, if,
<div align="center">
<math>~\Gamma = \frac{2^4\cdot 3^2}{5}</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{2^2\cdot 3^2}{5\pi G \rho_c} \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Lagrangian_Approach|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the linear density distribution.



<font color="red">'''Case 3''' (n = 1 polytrope)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = \frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = \frac{\cos(\pi\chi_0)}{\chi_0} -  \frac{\sin(\pi\chi_0)}{\pi\chi_0^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = \frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = \frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]  \, .</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">

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

<tr>
  <td align="right">
Also, note: &nbsp; &nbsp; &nbsp;<math>~\frac{d(p^')}{d\chi_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2\pi \cdot \cos(\pi\chi_0)}{(\pi^2\chi_0^3)} - \frac{6\sin(\pi\chi_0)}{(\pi^2\chi_0^4)}  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] 
+\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\cos(\pi\chi_0) - \pi^2\chi_0 \sin(\pi\chi_0) - \pi \cos(\pi\chi_0) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
\biggl[ 2\pi \chi_0\cdot \cos(\pi\chi_0) - 6\sin(\pi\chi_0)  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] 
+2 \chi_0 \sin(\pi\chi_0) \biggl[ - \pi^2\chi_0 \sin(\pi\chi_0) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
2\pi^2\chi_0^2\cos^2(\pi\chi_0) -8\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+6\sin^2(\pi\chi_0)
- 2 \pi^2 \chi_0^2 \sin^2(\pi\chi_0)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
-4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
</table>
</div>

For the case of an n = 1 polytropic configuration, therefore, the equilibrium requirement is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\Gamma \varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[ \cos(\pi\chi_0) -  \frac{\sin(\pi\chi_0)}{\pi\chi_0} - \frac{2\sin(\pi\chi_0)}{\pi\chi_0} \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
-4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{ \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[\pi\chi_0 \cos(\pi\chi_0) -  3\sin(\pi\chi_0) \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{3\sin^2(\pi\chi_0) - 4\pi\chi_0\sin(\pi\chi_0)\cos(\pi\chi_0) + (\pi\chi_0)^2 \biggl[1-\sin^2(\pi\chi_0) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{ (\pi\chi_0)^2 \sin^2(\pi\chi_0)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 \biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}  \biggr]^2
</math>
  </td>
</tr>
</table>
</div>
So, the equilibrium condition is satisfied if,
<div align="center">
<math>~\Gamma = 2\pi^2</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{2\pi^2}{4\pi G \rho_c} = \frac{\pi}{2G \rho_c}  \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Stability/Polytropes#Setup|as presented in a separate chapter discussion]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of an n = 1 polytropic configuration.

====Dabbling with LAWE====
Now, let's experiment with the [[SSC/Structure/OtherAnalyticModels#Compare|LAWE as presented above]], that is,

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

<tr>
  <td align="right">
<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
+ \frac{d\ln(p)}{d\chi_0}\cdot  \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>
</div> 

After multiplying though by <math>~(-p)</math>, this expression may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\sigma^2 \varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p}{x}\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr]
+ \frac{\alpha p^'}{\chi_0}\biggl[1 + \frac{\chi_0 x^'}{\alpha x}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{p}{x}\biggr) x^{' '} + p\biggl[ \frac{4}{\chi_0}\biggr]\frac{x^'}{x}
+ p^'\biggr[\frac{x^'}{x}\biggr] + \frac{\alpha p^'}{\chi_0} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
px^{' '} + \biggl[ \frac{4p}{\chi_0} + p^'\biggr]x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 0</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
px^{' '} + \biggl[ 4p + \chi_0 p^'\biggr]\frac{x^'}{\chi_0} + \biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x \, .
</math>
  </td>
</tr>
</table>
</div>

(We could have, perhaps, obtained this expression in a more direct fashion had we started directly from the form of the [[SSC/Structure/OtherAnalyticModels#LAWE|LAWE derived earlier]].)

=====Specific Case Attempts=====
======Uniform Density======
<font color="red">'''Case 0''' (Uniform density)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~1-\chi_0^2 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\alpha p^'}{\chi_0}</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~- 2\alpha \, .</math>
  </td>
</tr>
</table>
</div>

For the uniform-density case, therefore, the the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(1-\chi_0^2)}{x}\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr]
-2\alpha \biggl[1 + \frac{\chi_0 x^'}{\alpha x}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ (2\alpha -\sigma^2)x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-\chi_0^2)\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr] - 2\chi_0 x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-\chi_0^2)\frac{dx^'}{d\chi_0}  + (1-\chi_0^2)\biggl[ \frac{4x^'}{\chi_0}\biggr] - 2\chi_0 x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-\chi_0^2)\frac{dx^'}{d\chi_0}  + \frac{2x^'}{\chi_0}\biggl( 2 - 3\chi_0^2 \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>
where, [[SSC/Structure/OtherAnalyticModels#Stabililty_2|as defined above]],
<div align="center">
<math>~\alpha \equiv 3 - \frac{4}{\gamma_g} \, .</math>
</div>

'''Mode 3'''

Try an eigenfunction of the form,
<div align="center">
<math>x = a + b\chi_0^2 + c\chi_0^4 \, ,</math>
</div>
in which case,
<div align="center">
<math>~\frac{2x^'}{\chi_0} = \frac{2}{\chi_0}(2 b\chi_0 +4c\chi_0^3) = 4b+8c\chi_0^2</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~x^{' '} = 2 b + 12c\chi_0^2 \, . </math>
</div>
In order for this to be a solution, we must have,

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

<tr>
  <td align="right">
<math>~(2\alpha -\sigma^2)( a + b\chi_0^2 + c\chi_0^4) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-\chi_0^2)(2 b + 12c\chi_0^2 )  + ( 2 - 3\chi_0^2 )(4b+8c\chi_0^2 ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2 b + 12c\chi_0^2)  - \chi_0^2(2 b + 12c\chi_0^2 )  + 2(4b+8c\chi_0^2 ) - 3\chi_0^2 (4b+8c\chi_0^2 )  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~10b + \chi_0^2(12c-2b+16c-12b) - \chi_0^4(12c + 24c)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~10b + \chi_0^2(28c-14b) - \chi_0^4(36c) \, .
</math>
  </td>
</tr>
</table>
</div>


So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a\mathfrak{F} +10b</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~b\mathfrak{F} -14b + 28c</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~c[\mathfrak{F} -36]</math>
  </td>
</tr>
</table>
</div>
where, following [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29|Sterne's (1937) presentation]],
<div align="center">
<math>~\mathfrak{F}  \equiv \sigma^2 - 2 \alpha \, .</math>
</div>

In order for all three of the coefficients to be zero, we must have:

First: &nbsp;&nbsp;&nbsp; <math>~\mathfrak{F} = 36 \, ;</math>

Second: &nbsp;&nbsp;&nbsp; <math>~22b = -28c ~~~~~\Rightarrow ~~~~~ c = - (11/14)b \, ;</math>

Third: &nbsp;&nbsp;&nbsp; <math>~36a = -10b ~~~~~\Rightarrow ~~~~~ b = -(18/5)a \, .</math>

Hence, choosing <math>~a = 1</math> implies:  <math>~b = -18/5</math> &nbsp; &nbsp; &nbsp; &nbsp;and  &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~ c = (11/7)(9/5) = +99/35 \, .</math>  This precisely matches the "j = 2" mode identified by Sterne.


======Parabolic======

<font color="red">'''Case 1''' (Parabolic)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = 1 -\chi_0^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = -2\chi_0 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p = \tfrac{1}{2}(1 -\chi_0^2)^2 (2-\chi_0^2)</math>
  </td>
  <td align="center">
&nbsp;  
  </td>
  <td align="left">
&nbsp;  
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = -5\chi_0 + 8\chi_0^3 - 3\chi_0^5 = \chi_0(1-\chi_0^2)(-5+3\chi_0^2)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{\alpha p^'}{\chi_0} = \alpha (1-\chi_0^2)(-5+3\chi_0^2)  \, .</math>
  </td>
</tr>
</table>
</div>


For the parabolic case, therefore, the the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
px^{' '} + \biggl[ 4p + \chi_0 p^'\biggr]\frac{x^'}{\chi_0} + \biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{1}{2}\varpi^2 (2-\chi_0^2)x^{' '} 
+ \biggl[ 2\varpi^2 (2-\chi_0^2) + \chi_0^2\varpi(-5+3\chi_0^2)\biggr]\frac{x^'}{\chi_0} 
+ \biggl[\sigma^2 \varpi + \alpha \varpi(-5+3\chi_0^2)  \biggr]x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{2}\biggl\{
\varpi (2-\chi_0^2)x^{' '} 
+ \biggl[ 4\varpi (2-\chi_0^2) + \chi_0^2(-10+6\chi_0^2)\biggr]\frac{x^'}{\chi_0} 
+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr]x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{2}\biggl\{
(2-3\chi_0^2 + \chi_0^4)x^{' '} 
+ \biggl[ 8-22\chi_0^2 + 10\chi_0^4\biggr]\frac{x^'}{\chi_0} 
+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr]x 
\biggr\}
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\mathfrak{K} \equiv 2(\sigma^2 - 5\alpha) \, .</math>
</div>

'''Mode Inverse'''

Try an eigenfunction of the form,
<div align="center">
<math>x = (1 + a\chi_0^2)^{-\beta} = (1 + a\chi_0^2)^{-(\beta+2)} (1 + 2a\chi_0^2 + a^2\chi_0^4)\, ,</math>
</div>
in which case,

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

<tr>
  <td align="right">
<math>~x^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta \chi_0 (1 + a\chi_0^2)^{-\beta-1} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta (\chi_0 + a\chi_0^3)(1 + a\chi_0^2)^{-(\beta+2)} \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~x^{' '}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta (1 + a\chi_0^2)^{-\beta-1} - 2a\beta \chi_0\biggl[ -2a(\beta+1) \chi_0(1 + a\chi_0^2)^{-\beta-2}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta (1 + a\chi_0^2)^{-(\beta+2)}\biggl[(1 + a\chi_0^2)  -2a(\beta+1) \chi_0^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta (1 + a\chi_0^2)^{-(\beta+2)}\biggl[1 - a(2\beta+1)\chi_0^2 \biggr]
</math>
  </td>
</tr>
</table>
</div>

In order for this to be a solution, we must have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2-3\chi_0^2 + \chi_0^4)x^{' '}
+ \biggl[ 8-22\chi_0^2 + 10\chi_0^4\biggr]\frac{x^'}{\chi_0} 
+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr]x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2a\beta \biggl[1 - a(2\beta+1)\chi_0^2 \biggr](2-3\chi_0^2 + \chi_0^4)
- 2a\beta (1 + a\chi_0^2) \biggl[ 8-22\chi_0^2 + 10\chi_0^4\biggr] 
+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr](1 + 2a\chi_0^2 + a^2\chi_0^4) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2a\beta (-2+3\chi_0^2 - \chi_0^4)
+ 2a^2\beta (2\beta+1) (2\chi_0^2-3\chi_0^4 + \chi_0^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2a\beta \biggl[ -8+22\chi_0^2 - 10\chi_0^4\biggr] 
+ 2a^2\beta  \biggl[ -8\chi_0^2+22\chi_0^4 - 10\chi_0^6\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \mathfrak{K} (1 + 2a\chi_0^2 + a^2\chi_0^4) 
+ 6\alpha (\chi_0^2 + 2a\chi_0^4 + a^2\chi_0^6) 
</math>
  </td>
</tr>
</table>
</div>



So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~\mathfrak{K} - 20a\beta </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~50a\beta + 4a^2\beta (2\beta-3) + 6\alpha + 2a\mathfrak{K}</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~- 22a\beta + 2a^2\beta (19-6\beta)  + 12a\alpha + a^2\mathfrak{K}</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~2a^2\beta (2\beta-9) + 6a^2\alpha</math>
  </td>
</tr>
</table>
</div>



'''Mode 3P'''

Try an eigenfunction of the form,
<div align="center">
<math>x = a + b\chi_0^2 + c\chi_0^4 \, ,</math>
</div>
in which case,
<div align="center">
<math>~\frac{x^'}{\chi_0} = \frac{1}{\chi_0}(2 b\chi_0 +4c\chi_0^3) = 2b+4c\chi_0^2</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~x^{' '} = 2 b + 12c\chi_0^2 \, . </math>
</div>
In order for this to be a solution, we must have,

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

<tr>
  <td align="right">
<math>~0</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{2}\biggl\{
(2-3\chi_0^2 + \chi_0^4)(2 b + 12c\chi_0^2 ) 
+ (8-22\chi_0^2 + 10\chi_0^4 )(2b+4c\chi_0^2 ) 
+ (\mathfrak{K}+6\alpha\chi_0^2 )( a + b\chi_0^2 + c\chi_0^4 ) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{2}\biggl\{
(4b+24c\chi_0^2 - 6b\chi_0^2 -36c\chi_0^4 + 2b\chi_0^4 + 12c\chi_0^6) 
+ (16b + 32c\chi_0^2 - 44b\chi_0^2 - 88c\chi_0^4 + 20b\chi_0^4 + 40c\chi_0^6 ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ (a\mathfrak{K} + b\mathfrak{K}\chi_0^2 + c\mathfrak{K}\chi_0^4 + 6a\alpha\chi_0^2 + 6b\alpha\chi_0^4 + 6c\alpha\chi_0^6 )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{2}\biggl[ (20b + a\mathfrak{K}) \chi_0^0 +(24c - 6b + 32c - 44b + b\mathfrak{K} + 6a\alpha)\chi_0^2
+ (-36c+2b -88c+20b +c\mathfrak{K} + 6b\alpha) \chi_0^4 + (12c + 40c + 6c\alpha )\chi_0^6 \biggr]
</math>
  </td>
</tr>
</table>
</div>


So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~20b + a\mathfrak{K}</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~56c + (\mathfrak{K}- 50)b + 6a\alpha</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~b(22+6\alpha) + c(\mathfrak{K}-124)</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~c(52 + 6\alpha)</math>
  </td>
</tr>
</table>
</div>


This is disappointing, as it does not result in nonzero coefficient values.

======Polytrope======

<font color="red">'''Case 3''' (n = 1 polytrope)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = \frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = \frac{\cos(\pi\chi_0)}{\chi_0} -  \frac{\sin(\pi\chi_0)}{\pi\chi_0^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p = \biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2 = \varpi^2</math>
  </td>
  <td align="center">
&nbsp;  
  </td>
  <td align="left">
&nbsp;  
  </td>
</tr>
<tr>
  <td align="right">
<math>~p^' = \frac{2\varpi}{(\pi \chi_0^2)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{\alpha p^'}{\chi_0 } = \frac{2\alpha \varpi}{(\pi \chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>



For the n = 1 polytropic case, therefore, the the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
px^{' '} + \biggl[ 4p + \chi_0 p^'\biggr]\frac{x^'}{\chi_0} + \biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr] \varpi x^{' '} 
+ \biggl\{ 4\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr] + \frac{2}{(\pi \chi_0)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggr\}\frac{\varpi x^'}{\chi_0} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\sigma^2  + \frac{2\alpha }{(\pi \chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] \biggr\}\varpi x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\varpi}{\pi \chi_0^2} \biggl\{
[\sin(\pi\chi_0)] \chi_0 x^{' '} + [ 2\sin(\pi\chi_0) + 2\pi\chi_0 \cos(\pi\chi_0) ] x^' 
+ \{(\pi \chi_0^2)\sigma^2  + 2\alpha \chi_0^{-1} [ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) ] \} x
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

==Exploration2==

Let's begin with the LAWE written in the following form (see, for example, the [[MathProjects/EigenvalueProblemN1#Context|related context discussion]]):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma^2  \mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{P}{\rho }\biggr)\frac{1}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \biggl(\frac{P^'}{\rho }\biggr) \frac{1}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

One advantage of beginning with this construction is that &#8212; as the following table shows &#8212; it might be reasonable to expect in general that both pre-factors &#8212; <math>~(P/\rho )</math>  and <math>~(P^'/\rho )</math> &#8212; will have a relatively simple mathematical form.  In addition, however, it appears as though both terms on the RHS ''want'' to be logarithmic derivatives.

<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="4"><font size="+1">Properties of Analytically Defined Astrophysical Structures</font></th>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~\rho(x)</math>
  <td align="center"><math>~\biggl[\frac{P(x)}{\rho(x)}\biggr]</math>
  <td align="center"><math>~\biggl[ \frac{P^'(x)}{\rho(x)} \biggr]</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~1</math>
  <td align="center"><math>~1 - x^2</math>
  <td align="center"><math>~-2x</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1-x</math>
  <td align="center"><math>~(1-x)(1 + 2x - \tfrac{9}{5}x^2)</math>
  <td align="center"><math>~-\tfrac{12}{5}x (4-3x)</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~1-x^2</math>
  <td align="center"><math>~(1-x^2)(1  - \tfrac{1}{2} x^2)</math>
  <td align="center"><math>~-x (5-3x^2)</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
  <td align="center"><math>~\frac{\sin }{ x}</math>
  <td align="center"><math>~\frac{\sin x}{x}</math>
  <td align="center"><math>~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr] </math>
</tr>
</table>

===New Form of LAWE===

Multiplying the LAWE through by <math>~[\rho/(P\mathcal{G}_\sigma)]</math> gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma^2  \biggl(\frac{\rho}{P}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^'}{\mathcal{G}_\sigma }\biggr)\frac{1}{(x^4 \mathcal{G}_\sigma^')} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \biggl(\frac{P^'}{P}\biggr) \frac{1}{(x^\alpha \mathcal{G}_\sigma)}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \cdot \frac{d\ln( x^4 \mathcal{G}_\sigma^' )}{dx}
+ \frac{d\ln P}{dx } \cdot \frac{d\ln( x^\alpha \mathcal{G}_\sigma )}{dx}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma^' )}{dx} \biggr]
+ \frac{d\ln P}{dx } \biggl[ \frac{d\ln( x^\alpha  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma )}{dx} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma^' )}{dx} + \frac{d\ln P}{dx } \biggr]
+ \frac{d\ln P}{dx } \biggl[ \frac{d\ln( x^\alpha  )}{dx}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ - \biggl[ \sigma^2  \biggl(\frac{\rho}{P}\biggr) + \frac{d\ln P}{dx } \cdot \frac{d\ln( x^\alpha  )}{dx}  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4 P  \mathcal{G}_\sigma^' )}{dx} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ - \frac{\mathcal{G}_\sigma}{P}\biggl[ \sigma^2  \rho + \frac{\alpha P^'}{x}  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{x^4 P } \biggl[ \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~  \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathcal{G}_\sigma \biggl[ \sigma^2  x^4 \rho + \alpha x^3 P^'  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x^4 \rho \mathcal{G}_\sigma \biggl[ \sigma^2   +  \frac{\alpha P^'}{x\rho}  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

===Trial Logarithmic Eigenfunction===
Defining,
<div align="center">
<math>\mathcal{F}(x) \equiv \biggl[ \sigma^2   +  \frac{\alpha P^'}{x\rho}  \biggr] \, ,</math>
</div>

the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x^4 \rho \mathcal{F} \mathcal{G}_\sigma \, .</math>
  </td>
</tr>
</table>
</div>

====First Try====
Let's try an eigenvector of the form,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}_\sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A(x) \ln P + B(x) \, ,</math>
  </td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}_\sigma^' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A^' \ln P + \frac{A\cdot P^'}{P}+ B^' </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ x^4 P \mathcal{G}_\sigma^' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^4\biggl[ PA^' \ln P + A\cdot P^'+ PB^' \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\mathrm{LHS} ~~\equiv \frac{d}{dx}\biggl[x^4 P \mathcal{G}_\sigma^' \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^4\biggl[ P'A^' \ln P + PA^{' '} \ln P + PA^' \biggl(\frac{P^'}{P}\biggr)         
+ A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 4x^3\biggl[ PA^' \ln P + A\cdot P^'+ PB^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^4 \biggl\{\ln P \biggl[P'A^' + PA^{' '} + \frac{4}{x} \cdot PA^'
\biggr] + 
\biggl[  A^' P^' + A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' + \frac{4}{x} \biggl( A\cdot P^'+ PB^'  \biggr) \biggr]\biggr\} \, .
</math>
  </td>
</tr>
</table>

</div>
Now, in order for this expression to match the RHS of the LAWE, we must have, first of all,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{F} A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{1}{\rho}\biggl[ P'A^' + PA^{' '} + \frac{4}{x} \cdot PA^'\biggr] \, ;</math>
  </td>
</tr>
</table>
and, second,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{F} B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{1}{\rho}\biggl[A^' P^' + A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' + \frac{4}{x} \biggl( A\cdot P^'+ PB^'  \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>

</div>

<font color="red">'''Case 1''' (Parabolic)</font>:
<div align="center">
<math>\mathcal{F}(x) \equiv \biggl[ \sigma^2   - \alpha(5 - 3x^2) \biggr] = f_0 + f_2 x^2 \, ,</math>
</div>
where,
<div align="center">
<math>f_0 \equiv \sigma^2 - 5\alpha</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>f_2 \equiv 3\alpha \, .</math>
</div>
Also, the first condition is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- 2( f_0 + f_2 x^2 )A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2( -5 + 3x^2 ) xA^' + (2-3x^2 + x^4) \biggl[A^{' '} + \frac{4A^'}{x}  \biggr] \, .</math>
  </td>
</tr>
</table>

So, if we adopt a polynomial expression for the function, <math>~A(x)</math>, of the form,
<div align="center">
<math>~A(x) = a_0 + a_2 x^2 + a_4 x^4 \, ,</math>
</div>


<div align="center">
<math>~\Rightarrow ~~~~ A^' = 2a_2 x + 4a_4 x^3</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>A^{' '} = 2a_2 + 12a_4 x^2 \, ,</math>
</div>
the condition becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- 2( f_0 + f_2 x^2 )(a_0 + a_2 x^2 + a_4 x^4)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( -10 + 6x^2 ) (2a_2 x^2 + 4a_4 x^4) + (2-3x^2 + x^4) \biggl[(2a_2 + 12a_4 x^2) + 4 (2a_2  + 4a_4 x^2) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ ( f_0 + f_2 x^2 )(a_0 + a_2 x^2 + a_4 x^4)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( 10 - 6x^2 ) (a_2 x^2 + 2 a_4 x^4) - (2-3x^2 + x^4) (5a_2 + 14a_4 x^2)  </math>
  </td>
</tr>
</table>

So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0 a_0 + 10 a_2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0  a_2 + f_2 a_0 -10a_2 + 28a^4 - 15a_2</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0 a_4 + f_2 a_2 -20a_4 +6a_2 -42a_4 + 5a_2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_2 a_4 +12a_4 +14a_4</math>
  </td>
</tr>
</table>
</div>
This does not seem to work.

===Another Trial===

Start with a form of the LAWE found midway through the [[SSC/Structure/OtherAnalyticRamblings#New_Form_of_LAWE|above derivation]], namely,

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

<tr>
  <td align="right">
<math>~- \biggl[ \sigma^2  \biggl(\frac{\rho}{P}\biggr) + \frac{d\ln P}{dx } \cdot \frac{d\ln( x^\alpha  )}{dx}  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4 P  \mathcal{G}_\sigma^' )}{dx} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and multiplying through by <math>~x^2</math> gives,

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

<tr>
  <td align="right">
<math>~- \biggl[ \sigma^2  \biggl(\frac{x^2 \rho}{P}\biggr) + \alpha \cdot \frac{d\ln P}{d\ln x } \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggl[ \frac{d\ln( x^4 P  \mathcal{G}_\sigma^' )}{d\ln x} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\frac{d\ln P}{d\ln x }  \biggl[ \sigma^2  \biggl(\frac{P^'}{x \rho}\biggr)^{-1} + \alpha \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggl[ \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x} + \frac{d\ln( P )}{d\ln x} 
+ \frac{d\ln( x^4  )}{d\ln x} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\frac{d\ln P}{d\ln x }  \biggl[\alpha +  \sigma^2  \biggl(\frac{P^'}{x \rho}\biggr)^{-1} + \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggl[ \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x} +4 \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{d\ln \mathcal{G}_\sigma}{d\ln x } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\ln \mathcal{G}_\sigma}{d\ln x } \cdot \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma} \, .</math>
  </td>
</tr>
</table>
</div>

====Consider Parabolic Case====
In the case of a parabolic density distribution, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~\frac{2x^2(5-3x^2)}{(1-x^2)(2-x^2)}  \biggl[\alpha -  \sigma^2  \biggl(5-3x^2\biggr)^{-1} + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +4 \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  
</math>
  </td>
</tr>
</table>
</div>


Let's try,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^m +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2x)</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{-1}(2a_2x^2)  +m  (b_0 + b_2x^2)^{-1}(2b_2x^2) </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ n a_2 (b_0 + b_2x^2)  +mb_2  (a_0 + a_2x^2) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^{' '}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n m (a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) + n(a_0 + a_2x^2)^{n-1}(2a_2) \cdot (b_0 + b_2x^2)^m  + n(n-1)(a_0 + a_2x^2)^{n-2}(2a_2x)^2 \cdot (b_0 + b_2x^2)^m </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+m n(a_0 + a_2x^2)^{n-1}(2a_2x)  \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2) 
+m(m-1) (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-2}(2b_2x)^2</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~8n m a_2b_2 x^2 (a_0 + a_2x^2)^{-1}\cdot (b_0 + b_2x^2)^{-1} + n2a_2 (a_0 + a_2x^2)^{-1}   + n(n-1)4a_2^2 x^2 (a_0 + a_2x^2)^{-2}  
+m2b_2 (b_0 + b_2x^2)^{-1} +m(m-1)4 b_2^2 x^2 (b_0 + b_2x^2)^{-2}</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>
</table>
</div>

So, we have for the LAWE:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)(a_0 + a_2x^2) (b_0 + b_2x^2)}  
\biggl\{ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{8}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(a_0 + a_2x^2)(b_0 + b_2x^2)} \biggl\{ 2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)  
+ 8(n a_2 b_0 + mb_2 a_0)  + 8(na_2 b_2+ mb_2 a_2)x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[8n m a_2b_2+ \frac{4n(n-1) a_2^2(b_0 + b_2x^2) }{ (a_0 + a_2x^2)}  
+ \frac{4m(m-1) b_2^2(a_0 + a_2x^2) }{(b_0 + b_2x^2)} \biggr]x^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Putting these together gives,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(b_0 + b_2x^2) + m b_2  (a_0 + a_2x^2)  + 4(n a_2 b_0 + mb_2 a_0)  + 4(na_2 b_2+ mb_2 a_2)x^2+ 4n m a_2b_2x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{(1-x^2)(2-x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}\biggl[2n(n-1) a_2^2(b_0 + b_2x^2)^2   + 2m(m-1) b_2^2(a_0 + a_2x^2)^2 \biggr]x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

----
=====First Guess=====
Now, if we are very lucky, we will find that,
<div align="center">
<math>~(a_0 + a_2x^2) = (1-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~a_0 = 1</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~a_2 = -1</math>;
</div>
and, simultaneously,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>.
</div>
In this case, the fractional coefficient in the last term of the LAWE will become unity and the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(1-x^2) (2-x^2) - 2x^2(2n  + m)  + 2x^4 (n+ m) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[\sigma^2 - n (2-x^2) - m  (1-x^2)  - 4(2n  + m)  + 4(n + m )x^2+ 4n m x^2 \biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[2n(n-1) (2-x^2)^2   + 2m(m-1) (1-x^2)^2 \biggr]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2\alpha - x^2(3\alpha +4n  + 2m)  + x^4 (\alpha + 2n+ 2m) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[(-\sigma^2 + 10n + 5m  )  - x^2( 5n+5m+4nm  )\biggr] (2-3x^2+x^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [8n(n-1) +2m(m-1) ] - [8n(n-1) + 4m(m-1)  ] x^2 + [ 2n(n-1)+ 2m(m-1) ] x^4 \biggr\} x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~10\alpha + 2(-\sigma^2 + 10n + 5m  )</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-6\alpha - 5(3\alpha +4n  + 2m) -2( 5n+5m+4nm  ) - 3(-\sigma^2 + 10n + 5m  ) - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~5(\alpha + 2n+ 2m)+3(3\alpha +4n  + 2m) + (-\sigma^2 + 10n + 5m  )  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-3(\alpha + 2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ] </math>
  </td>
</tr>
</table>
</div>


Now let's begin simplification.  The <math>~x^0</math> coefficient implies,
<div align="center">
<math>~(-\sigma^2 + 10n + 5m  )=-5\alpha \, .</math>
</div>

Hence,
<div align="center">
<table border="1" cellpadding="8" align="center">

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-6\alpha - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~9\alpha + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-3\alpha -3(2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ] </math>
  </td>
</tr>
</table>
</div>

Using the <math>~x^6</math> coefficient to define <math>~\alpha</math>, that is, setting,
<div align="center">
<math>~3\alpha = \{-3(2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ]\} \, ,</math>
</div>

means that the other two coefficient expressions are,
<div align="center">
<table border="1" cellpadding="8" align="center">

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~6(2n+ 2m) + 2 ( 5n+5m+4nm  ) + 4 [ n(n-1)+ m(m-1) ] - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-9(2n+ 2m) - 3( 5n+5m+4nm  ) - 6[ n(n-1)+ m(m-1) ] + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>
</table>
</div>

The only question remaining is, what pair of values for <math>~(n, m)</math> result in both of these expressions going to zero?  Simplifying the first expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6(2n+ 2m) + 2 ( 5n+5m+4nm  ) + 4 [ n(n-1)+ m(m-1) ] - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2m  -8n -4n(n-1) + 2m(m-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2[m^2 - 2n(n+1)] \, . </math>
  </td>
</tr>
</table>
</div>

Simplifying the second expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-9(2n+ 2m) - 3( 5n+5m+4nm  ) - 6[ n(n-1)+ m(m-1) ] + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4n - 2m   + 2n(n-1) - 2m(m-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2[n(n+1) -m^2] </math>
  </td>
</tr>
</table>
</div>

Okay.  Because it is not possible for both of these last two constraints to be simultaneously satisfied, I conclude that this last, specific eigenfunction guess is incorrect.  

----
=====Second Guess=====
Let's try again, keeping the same values of the <math>~b_0</math> and <math>~b_2</math> &#8212; that is,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>
</div>

&#8212; but leaving the values of <math>~a_0</math> and <math>~a_2</math> unspecified.  In this case, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (2 - x^2) + 2x^2(2n a_2 - m a_0)  - 2x^4 (na_2 + m a_2) \biggr](5-3x^2)(a_0 + a_2x^2) 
-\sigma^2 (a_0 + a_2x^2)^2 (2 - x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(2 - x^2) -m  (a_0 + a_2x^2)  + 4(2n a_2 - m a_0)  - 4(na_2 + m a_2)x^2 - 4n m a_2 x^2
\biggr](1-x^2)(2-x^2)(a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[2n(n-1) a_2^2(2 - x^2)^2   + 2m(m-1) (a_0 + a_2x^2)^2 \biggr]x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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] 
-\sigma^2 [ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ] (a_0+a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ ( 5m a_0 - 10n a_2)  +  (4n m a_2 + 5na_2 + 5m a_2)x^2 \biggr] (1-x^2)[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^4    \biggr\} x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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] 
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ [ ( 5m a_0 - 10n a_2)  ] + [(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ]x^2 - [ 4n m a_2 + 5na_2 + 5m a_2  ]x^4 \biggr\}
[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]x^2 + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^4 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^6    \biggr\} (1-x^2) \, .
</math>
  </td> 
</tr>

</table>
</div>


So, the coefficients of each even power of <math>~x^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~
10a_0^2 \alpha - 2a_0^2\sigma^2 + 2a_0[ ( 5m a_0 - 10n a_2)  ]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <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 + a_0(2a_2-a_0)]
</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></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <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><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></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">&nbsp; : &nbsp;</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">&nbsp; : &nbsp;</td>
  <td align="left">
<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>
  </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">&nbsp; : &nbsp;</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">&nbsp; : &nbsp;</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">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha (10a_2^2 - 22a_0a_2+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_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>~+ 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">&nbsp; : &nbsp;</td>
  <td align="left">
<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>~-10n(n-1) a_2^2  - 2m(m-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">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \{ 3\alpha + [ 4n m  + 11n + 11m   ] + [ 2n(n-1)   + 2m(m-1) ]\}a_2^2
</math>
  </td>
</tr>
</table>
</div>



----

=====Third Guess=====

Let's try again, keeping the same values of the <math>~b_0</math> and <math>~b_2</math> &#8212; that is,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>
</div>

&#8212; but leaving the values of <math>~a_0</math> and <math>~a_2</math> unspecified.  In this case, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (2 - x^2) + 2x^2(2n a_2 - m a_0)  - 2a_2 x^4 (n + m ) \biggr](5-3x^2) 
-\sigma^2 (a_0 + a_2x^2) (2 - x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[-  n a_2(2 - x^2) + m  (a_0 + a_2x^2)  - 4(n a_2 2 - m a_0)  + 4(na_2 + m a_2)x^2 + 4n m a_2 x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{ (a_0 + a_2x^2)}\biggl[2n(n-1) a_2^2(2 - x^2)^2   + 2m(m-1) (a_0 + a_2x^2)^2 \biggr](1-x^2)x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{[2a_0\alpha   ] + [\alpha(- a_0 +2a_2) + (4n a_2 - 2m a_0)  ]x^2 + [-a_2\alpha  - 2a_2 (n + m )  ]x^4  \biggr\} (5-3x^2) 
-\sigma^2 [2a_0 + (-a_0 + 2a_2)x^2 -a_2x^4]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[( -10na_2 + 5ma_0 )  + (5ma_2 + 5na_2  + 4n m a_2 )x^2 \biggr](2-3x^2 +x^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{ (a_0 + a_2x^2)}\{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^4  \} (x^2-x^4) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ [10a_0\alpha - 2a_0\sigma^2  ] + [5\alpha(- a_0 +2a_2) + 5(4n a_2 - 2m a_0) -6a_0\alpha  + (a_0 - 2a_2)\sigma^2  ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ [-5a_2\alpha  - 10a_2 (n + m ) -3\alpha(- a_0 +2a_2) -3 (4n a_2 - 2m a_0) + a_2\sigma^2]x^4  
+ [3a_2\alpha  +6a_2 (n + m )  ]x^6  \biggr\}  (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\biggl\{ [2( -10na_2 + 5ma_0 )]  + [2 (5ma_2 + 5na_2  + 4n m a_2 ) -3( -10na_2 + 5ma_0 )]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ [ -3 (5ma_2 + 5na_2  + 4n m a_2 ) + ( -10na_2 + 5ma_0 )]x^4  + (5ma_2 + 5na_2  + 4n m a_2 )x^6 
\biggr\} (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^4  \} (x^2-x^4) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ [10a_0\alpha - 2a_0\sigma^2  ] + [\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0)   + (a_0 - 2a_2)\sigma^2  ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ [\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m)  + a_2\sigma^2]x^4  
+ [3a_2\alpha  +6a_2 (n + m )  ]x^6  \biggr\}  (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl\{ [ -20na_2 + 10ma_0 ]  + [10ma_2 + 40na_2  + 8n m a_2   -15ma_0 ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ [ -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)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]x^2  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^4 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^6  \} (1-x^2) 
</math>
  </td>
</tr>
</table>
</div>

So, the coefficients of each even power of <math>~x^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a_0[10a_0\alpha - 2a_0\sigma^2  ] + a_0 [ -20na_2 + 10ma_0 ]</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a_0  [\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0)   + (a_0 - 2a_2)\sigma^2  ] + a_2[10a_0\alpha - 2a_0\sigma^2  ] 
+ a_0[10ma_2 + 40na_2  + 8n m a_2   -15ma_0 ] 
</math><p><math>
+ a_2[ -20na_2 + 10ma_0 ] - [ 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">&nbsp; : &nbsp;</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 - 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>
~ - [ -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">&nbsp; : &nbsp;</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 - 25na_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">&nbsp; : &nbsp;</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>


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">&nbsp; : &nbsp;</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">&nbsp; : &nbsp;</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">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
\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>
~ + 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">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
\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>~
- 10 n(n-1)a_2^2 -2m(m-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">&nbsp; : &nbsp;</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>

=See Also=


{{ SGFfooter }}