Editing SSC/Stability/BiPolytropes/HeadScratching (section)

==LAWE==

Let's perform the LAWE integration in two parts: (1) Integrate from the center (where the derivative of the displacement function must be zero), through the core, up to the core-envelope interface; and (2) integrate from the surface (where the logarithmic derivative of the displacement function is negative one), through the envelope, down to the core-envelope interface.  Examine the discontinuity that results and see whether it makes sense in terms of the required "matching conditions" at the interface.

===Throughout the Configuration===

From the last couple of lines of an [[SSC/Stability/BiPolytropes#Foundation|accompanying ''Foundation'' presentation]], the relevant LAWE may be written as,

<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{d^2x}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{ M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \frac{\mathcal{H}}{r^*} \frac{dx}{dr*} 
+ 
\biggl[\biggl(\frac{\sigma_c^2}{\gamma_g}\biggr) \mathcal{K}_1 - \alpha_g \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\sigma_c^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\omega^2}{2\pi G\rho_c}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl(3 - \frac{4}{\gamma_g}\biggr) \, .
</math>
  </td>
<td align="center" colspan="4">&nbsp; </td>
</tr>
</table>
From a related discussion of [[SSC/Stability/BiPolytropes#Profile|interior structural profiles]], we appreciate that throughout the core we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{3}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]\biggl( \frac{2\pi}{3}\biggr)^{1 / 2} \frac{1}{\xi} 
= 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2\pi}{3}\biggr) \xi^{-2}
\, ;</math>
  </td>
</tr>
</table>

and, throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \eta^{-2}
\, .
</math>
  </td>
</tr>
</table>

===Surface Boundary Condition===
In an effort to [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|ensure finite-amplitude fluctuations]] at the surface, we will enforce the condition,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_0 \frac{d\ln x}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ,</math>
  </td>
</tr>
</table>
that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^* \frac{d\ln x}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl(\frac{2\pi}{3}\biggr)\frac{\sigma_c^2 (R^*)^3}{\gamma_g M^*_\mathrm{tot}} -\alpha_g \biggr]</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r^* = R^* \, ,</math>
  </td>
</tr>
</table>
where the asterisks <math>(*)</math> signal that we have employed the same variable normalizations as have been adopted in our [[SSC/Stability/BiPolytropes#Foundation|accompanying ''Foundations'' discussion]].  Since our analysis, here, is focused on the marginally unstable (minimum-mass) configuration in which we expect <math>\sigma_c^2 = 0</math>, the surface (envelope) constraint becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\alpha_g = -~1</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~\eta = \eta_s \, .</math>
  </td>
</tr>
</table>

===Interface===

Drawing from an [[SSC/Stability/BiPolytropes#Interface_Conditions|accompanying discussion]], the matching condition at the interface is given by the expression,
<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[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i \, .</math>
  </td>
</tr>
</table>
Given that <math>\gamma_c = 6/5</math> and <math>\gamma_e = 2</math>, this becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr) \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{5}{3}\biggl[ x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i \, .</math>
  </td>
</tr>
</table>

===Central Boundary Condition===

The central boundary condition is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dx_\mathrm{core}}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>

In order to kick-start the integration outward from the center of the configuration, we will following the procedure that has been detailed in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|accompanying discussion]].  At the center of the configuration <math>(\xi_1 = 0)</math>, we label the fractional displacement function as <math>x_1</math> &#8212; value to be set later, perhaps in an effort to help secure the proper matching conditions at the interface &#8212; then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center.  Specifically, given that <math>n = 5, \gamma_g = 6/5</math>, and <math>\alpha_g = -1/3</math> in the core, we will set,

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

<tr>
  <td align="right">
<math>
x_2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  \Delta_\xi^2}{60} \biggr] 
=
x_1 \biggl[ 1 - \frac{\mathfrak{F}  \Delta_\xi^2}{10} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\mathfrak{F}
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\alpha_g \biggr]
=
\frac{1}{6}\biggl[ 5 \sigma_c^2  + 4 \biggr]
\, .</math>
  </td>
</tr>
</table>

===Numerical Integration===
====Through the Core====
Throughout the core, the governing LAWE is,

<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{d^2x}{d\xi^2} + \frac{\mathcal{H}}{\xi} \frac{dx}{d\xi} 
+ 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi }{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 - 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, .
</math>
  </td>
</tr>
</table>

Now, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\xi}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\xi^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math>
  </td>
</tr>
</table>
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>.  Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 2 \rightarrow N</math>,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x_+ - 2x_i + x_-}{\Delta_\xi^2}  \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{x_+ }{\Delta_\xi^2}  \biggr] + \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ }{2 \Delta_\xi} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{2x_i - x_-}{\Delta_\xi^2}  \biggr]
+ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi }{2 } \biggr] \biggr\}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2
- 
\biggl(\frac{\Delta_\xi^2}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr] \biggr\} 
-  x_-\biggl\{1 - \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi}{2 } \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
<div align="center">'''Check Against Independent Derivation'''</div>
We have dealt with this identical LAWE in connection with our [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|analysis of the stability of pressure-truncated n = 5 Polytropic configurations]].  Let's see whether that derivation matches our current one.  In that case, we found,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_+ \biggl\{ 2\theta +\frac{4\Delta_\xi \theta}{\xi} - \Delta_\xi (n+1)(- \theta^')\biggr\} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{4\theta - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}
- x_- \biggl[2\theta - \frac{4\Delta_\xi \theta}{\xi} + \Delta_\xi (n+1)(- \theta^') \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{\theta}\biggl[ \frac{5\sigma_c^2}{6}  + 
\frac{2}{3} \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta}\biggr] \biggr\} 2\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ \mathcal{H} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{4\pi}\biggl[ 5\sigma_c^2 \cdot \frac{4\pi}{6\theta}  + 
\frac{8\pi}{3} \biggl(- \frac{3\theta^'}{\xi \theta}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ \mathcal{H}\biggr] \biggr\} 2\theta \, .
</math>
  </td>
</tr>
</table>
Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi}{6\theta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}
~~\rightarrow ~~ \mathcal{K}_1
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp;&nbsp; &nbsp; </td>
  <td align="right">
<math>\frac{4\pi}{3}\biggl(-\frac{3\theta'}{\xi\theta}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{-1}
~~\rightarrow ~~ \mathcal{K}_2 \, ,
</math>
  </td>
</tr>
</table>
we can confirm that the two expressions are identical.
</td>
</tr>
</table>

====Through the Envelope====
Throughout the envelope &#8212; that is, for <math>\eta_i \le \eta \le \eta_s</math> &#8212; the governing LAWE is,

<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{d^2x}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} 
+ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \biggl[\biggl(\frac{\sigma_c^2}{2}\biggr) \mathcal{K}_1 -  \mathcal{K}_2\biggr]  x\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Keep in mind that, once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( 1 + \frac{1}{3}\xi^2_i \biggr)^{-1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3}~\theta_i^2 \xi_i \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} + \biggl( \frac{d\phi}{d\eta}\biggr)_i 
= 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{3} \xi_i \theta_i^2} - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
B + \pi \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

From a related discussion of [[SSC/Stability/BiPolytropes#Profile|interior structural profiles]], we appreciate that throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \eta^{-2}
\, .
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1} 
=
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\rho^*}{P^*}\biggr)\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2\cdot \frac{d\ln \phi}{d\ln \eta}
=
2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}
=
4 -2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]
=
2\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2} 
\, .
</math>
  </td>
</tr>
</table>

Finally, restructuring the radially dependent coefficient of the linear term in the LAWE, we have,

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

<tr>
  <td align="right">
<math>\frac{d^2x}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \mathcal{K}_2
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}   \mathcal{K}_1 
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\cdot
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \cdot
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x \, .
</math>
  </td>
</tr>
</table>

Again, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\eta}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\eta}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\eta^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} \, .</math>
  </td>
</tr>
</table>
For the envelope, we will integrate from the surface, into the core-envelope interface.  So, this time these "finite-difference" expressions will provide an approximate expression for <math>x_- \equiv x_{i-1}</math>, given the values of <math>x_+ \equiv x_{i+1}</math> and <math>x_i</math>.  If the surface of the configuration is denoted by the grid index, <math>i=N</math>, then for zones, <math>i = (N-1) \rightarrow ??</math>,

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

<tr>
  <td align="right">
<math>
\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
\biggl[ \frac{x_-}{\Delta_\eta^2} \biggr]
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_- }{2 \Delta_\eta}  \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-
\biggl[ \frac{x_+ - 2x_i }{\Delta_\eta^2} \biggr] 
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~x_- \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_+  \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''[[#Surface_Boundary_Condition|Surface Boundary Condition]]'''</div>

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-1
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; at, &nbsp; &nbsp; &nbsp; <math>\eta = \eta_s</math>.</td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>
\biggl[\frac{x_+ - x_-}{2 \Delta_\eta}\biggr]_s
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\biggl[\frac{dx}{d\eta} \biggr]_s = -\frac{x_s}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{x_{N+1} - x_{N-1}}{2 \Delta_\eta}\biggr]
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>-\frac{x_N}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
x_{N+1}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \, .</math>
  </td>
</tr>
</table>
Inserting this expression for "<math>x_+</math>" in the finite-difference representation of the envelope's LAWE allows us to determine the value for <math>x_- = x_{N-1}</math>.  Specifically,

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

<tr>
  <td align="right">
<math>x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] \biggl[ x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_{N-1} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_s \biggl[ \frac{2\Delta_\eta }{\eta_s} \biggr] x_N
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 1 
+
\biggl[ \frac{\Delta_\eta }{\eta_s} \biggr]
+
2\biggl[ \frac{\Delta_\eta^2 }{\eta^2_s}\biggr]  
+
\biggl(\frac{\sigma_c^2}{2^2\cdot 3}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta_s}{A}\biggr]  
\biggr\} x_N  \, .
</math>
  </td>
</tr>
</table>
Note that, in the last term of this last expression, we have acknowledged that, <math>(\eta_s - B) = \pi ~~\Rightarrow ~~ \sin(\eta_s - B) = -1</math>.

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


<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''Slope at the Interface'''</div>

We will need to determine the slope that is associated with the envelope's eigenfunction, <math>[dx/d\eta]_\mathrm{env}</math>, precisely at the interface. While the envelope's eigenfunction does not actually exist on the "core" side of the interface, we can ''project'' what its value at <math>x_-</math> ''would'' be if the envelope's eigenfunction were to continue smoothly just one small step beyond the interface, then use this ''projected'' value to determine the function's slope ''at'' the interface location.  Labeling the interface at <math>i = J</math>, first we have, 

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

<tr>
  <td align="right">
<math>\biggl[
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr]_J\biggl[x_-\biggr]_\mathrm{project} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_J x_J  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_J x_{J+1}  \, .
</math>
  </td>
</tr>
</table>
Then we conclude that,

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

<tr>
  <td align="right">
<math>[x_\mathrm{env}^']_J \equiv \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]_J</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{x_{J+1} }{2 \Delta_\eta}
- 
\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .
</math>
  </td>
</tr>
</table>

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

===Feeble Analytic Attempt===

Noice that if we assume <math>\sigma_c^2 = 0</math>, the governing envelope LAWE is,

<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{d^2x}{d\eta^2} 
+ 
\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta} \frac{dx}{d\eta}
-
\biggl\{ 
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta^2}  
\biggr\} x \, .
</math>
  </td>
</tr>
</table>
Let's try &hellip;

<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>
\eta^{m}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dx}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m\eta^{m-1}[\tan(\eta-B)]^{k}
+
k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d^2x}{d\eta^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
km\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-3}\sin(\eta-B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
\biggl[\frac{2}{\eta} + \frac{2}{\tan(\eta-B)}\biggr] 
\biggl\{ 
m\eta^{m-1}[\tan(\eta-B)]^{k}
+
k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-2
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]  
\eta^{m-2}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
2m\eta^{m-2}[\tan(\eta-B)]^{k}
+
2k\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
2m\eta^{m-1}[\tan(\eta-B)]^{k-1}
+
2k\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
2  \eta^{m-2}[\tan(\eta-B)]^{k}
+
2 \eta^{m-1}[\tan(\eta-B)]^{k-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
+ 
2m\eta^{m-2}[\tan(\eta-B)]^{k}
-
2  \eta^{m-2}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
+
2k\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
+ 
2m\eta^{m-1}[\tan(\eta-B)]^{k-1}
+
2 \eta^{m-1}[\tan(\eta-B)]^{k-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}\eta^{m-2}[\cos(\eta-B)]^{-2} \biggl\{
[m(m-1) +2m - 2][\cos(\eta-B)]^{2}
+
2k\eta^{2} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ [\tan(\eta-B)]^{k-1}2(m+1)\eta^{m-1}[\cos(\eta-B)]^{-2}\biggl\{
k + [\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ k[\tan(\eta-B)]^{k-2}[\cos(\eta-B)]^{-4}\eta^{m}
\biggl\{
(k-1) 
+
2[\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>
</table>
If <math>m = -1</math>, the second group of terms disappears and we have,

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

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}\biggl\{\biggl[
2k\eta^{2} 
- 2[\cos(\eta-B)]^{2}
\biggr]
+ k[\sin(\eta-B)]^{-2}\eta^{2}
\biggl[(k-1) +2[\cos(\eta-B)]^{2}\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}[\sin(\eta-B)]^{-2}\biggl\{
\biggl[
2k\eta^{2} 
\biggr][\sin(\eta-B)]^{2}
-
\biggl[
2[\cos(\eta-B)]^{2}
\biggr][\sin(\eta-B)]^{2}
+ k\eta^{2}(k-1) 
+ 2k\eta^{2}
[\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}[\sin(\eta-B)]^{-2}\biggl\{
k(k+1)\eta^{2} 
-
2[\cos(\eta-B)]^{2}[\sin(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>
</table>