Editing ThreeDimensionalConfigurations/RiemannTypeI (section)

===Regarding thought "D"===
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_x\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_x\biggr|_\mathrm{body}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_y\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_y|_\mathrm{body}\cdot \cos\theta + v_z|_\mathrm{body}\cdot \sin\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_z\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-v_y|_\mathrm{body}\cdot \sin\theta + v_z|_\mathrm{body}\cdot \cos\theta</math>
  </td>
</tr>
</table>

Setting <math>~v_z\biggr|_\mathrm{tip} = 0</math> implies,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~v_y|_\mathrm{body}\cdot \sin\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_z|_\mathrm{body}\cdot \cos\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \tan\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{v_z}{v_y}\biggr)_\mathrm{body}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \biggl[ \frac{b^2}{a^2 + b^2} \biggr]^{-1} \zeta_3^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{\zeta_2}{\zeta_3}\biggl[ \frac{(c/a)^2}{1 + (c/a)^2} \biggr] \biggl[ \frac{(b/a)^2}{1 + (b/a)^2} \biggr]^{-1} \, .</math>
  </td>
</tr>
</table>
For our particular example, then,


<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2.2794}{1.9637}\biggl[ \frac{(0.4703)^2}{1 + (0.4703)^2} \biggr] \biggl[ \frac{(1.25)^2}{1 + (1.25)^2} \biggr]^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2.2794}{1.9637}\biggl[ \frac{(0.4703)^2}{1 + (0.4703)^2} \biggr] \biggl[ \frac{(1.25)^2}{1 + (1.25)^2} \biggr]^{-1} = - 0.3448</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}(-0.3448) = - 19.0^{\circ} \, . </math>
  </td>
</tr>
</table>

Now, given that,
<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>~x_t \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_t \cos\theta - z_t \sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z_t \cos\theta + y_t \sin\theta \, ,</math>
  </td>
</tr>
</table>
we can generate the following expressions for the fluid velocity components ''in'' the tipped plane:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_z\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_x\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_x\biggr|_\mathrm{body}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl[ y_t \cos\theta - z_t \sin\theta \biggr] + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ z_t \cos\theta + y_t \sin\theta \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_t \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
+ z_t\biggl\{ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \sin\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \cos\theta  \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_y\biggr|_\mathrm{tip}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_y|_\mathrm{body}\cdot \cos\theta + v_z|_\mathrm{body}\cdot \sin\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \cdot \cos\theta - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \cdot \sin\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_t \biggl\{ \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta \biggr\} \, .</math>
  </td>
</tr>
</table>

Now, following our [[#Initial_Thoughts|earlier analysis for ''S-Type'' ellipsoids]], let's assume that,
<div align="center">
<math>~x_t = x_\mathrm{max}\cos(\varphi t)</math> &nbsp; &nbsp; &nbsp; and,  &nbsp; &nbsp; &nbsp; <math>~y_t = y_0 - y_\mathrm{max}\sin(\varphi t) \, .</math>
</div>

(Insertion of the extra term, y<sub>0</sub>, acknowledges that the center of the elliptical orbit in the "tipped" plane will be shifted off of the x-axis in the y<sub>t</sub>-direction; the size of this shift should correlate with z<sub>t</sub>, that is, with the vertical location of the tipped plane.)  We have, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_y\biggr|_\mathrm{tip} = \frac{dy_t}{dt} = -y_\mathrm{max} \varphi \cos(\varphi t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max} \cos(\varphi t) \biggl\{ \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta \biggr\} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_x\biggr|_\mathrm{tip} = \frac{dx_t}{dt} = -x_\mathrm{max}\varphi \sin(\varphi t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ y_0 - y_\mathrm{max} \sin(\varphi t) \biggr]\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
+ z_t\biggl\{ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \sin\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \cos\theta  \biggr\} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- y_\mathrm{max} \sin(\varphi t) \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ y_0  \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
+ z_t\biggl\{ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \sin\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \cos\theta  \biggr\} \, .
</math>
  </td>
</tr>
</table>
So, in terms of the value of z<sub>t</sub>, the offset in y<sub>t</sub> must be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ - y_0  \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ z_t\biggl\{ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \sin\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \cos\theta  \biggr\} \, ;
</math>
  </td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ -y_\mathrm{max} \varphi \cos(\varphi t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max} \cos(\varphi t) \biggl\{ \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta \biggr\} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \varphi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl\{ \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta \biggr\} \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-x_\mathrm{max}\varphi \sin(\varphi t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- y_\mathrm{max} \sin(\varphi t) \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \varphi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\} \, .
</math>
  </td>
</tr>
</table>
In order for both of these expressions to be simultaneously true, we need,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl\{ \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta - \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta  
-
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \sin\theta  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \cdot \cos\theta 
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2 
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 \biggl\} \frac{\zeta_2 \sin\theta }{(a^2 + c^2)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 \biggr\} \frac{\zeta_3 \cos\theta }{(a^2 + b^2)} \, .
</math>
  </td>
</tr>
</table>


====First Trial Value of &theta;====
Now, from above, we determined that the "tip" angle must obey the expression,

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

<tr>
  <td align="right">
<math>~\tan\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \biggl[ \frac{b^2}{a^2 + b^2} \biggr]^{-1} \zeta_3^{-1} \, .</math>
  </td>
</tr>
</table>
Hence, we require,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \cdot \biggl[ \frac{b^2}{a^2 + b^2} \biggr]^{-1} \zeta_3^{-1}
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2 
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 \biggl\} \frac{\zeta_2 }{(a^2 + c^2)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 \biggr\} \frac{\zeta_3 }{(a^2 + b^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~  
\biggl\{
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 
-  \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2 
\biggl\} \frac{c^2 \zeta_2^2 }{(a^2 + c^2)^2} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]
a^2 \biggr\} \frac{b^2\zeta_3^2 }{(a^2 + b^2)^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~  
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
\frac{a^2 c^2 \zeta_2^2 }{(a^2 + c^2)^2} 
- 
\frac{c^4 \zeta_2^2 }{(a^2 + c^2)^2} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b^4 \zeta_3^2 }{(a^2 + b^2)^2} 
- \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
\frac{a^2 b^2\zeta_3^2 }{(a^2 + b^2)^2} \, .
</math>
  </td>
</tr>
</table>
That is,

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

<tr>
  <td align="right">
<math>~  
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{b^4 \zeta_3^2 }{(a^2 + b^2)^2} 
+ 
\frac{c^4 \zeta_2^2 }{(a^2 + c^2)^2} \biggr]
\biggl[ \frac{a^2 c^2 \zeta_2^2 }{(a^2 + c^2)^2} 
+ 
\frac{a^2 b^2\zeta_3^2 }{(a^2 + b^2)^2} \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ b^4 \zeta_3^2 (a^2 + c^2)^2 
+ 
c^4 \zeta_2^2 (a^2 + b^2)^2 \biggr]
\biggl[ a^2 c^2 \zeta_2^2 (a^2 + b^2)^2 
+ 
a^2 b^2\zeta_3^2 (a^2 + c^2)^2 \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

So for the example parameters provided above, we have,

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

<tr>
  <td align="right">
<math>~  
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 14.04 
+ 
1.669 \biggr]
\biggl[ 7.546  
+ 
8.985 \biggr]^{-1} = 0.9502 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~  \Rightarrow~~~
\frac{y_\mathrm{max}}{x_\mathrm{max}}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.9748
</math>
  </td>
</tr>
</table>


====Second Trial Value of &theta;====

Instead, what if we obtain the tip angle straight from the tabulated expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\zeta_3}{\zeta_2} \, .</math>
  </td>
</tr>
</table>

Then we have,

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

<tr>
  <td align="right">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2 
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 \biggl\} 
\frac{1 }{(a^2 + c^2)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 \biggr\} 
\frac{1 }{(a^2 + b^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2  (a^2 + b^2)
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2  (a^2 + b^2)  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2 (a^2 + c^2)
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 (a^2 + c^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ c^2  (a^2 + b^2)- b^2 (a^2 + c^2) \biggr]
\biggl[ a^2  (a^2 + b^2)  - a^2 (a^2 + c^2) \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ c^2 - b^2 ]
[ b^2  - c^2 ]^{-1} = -1 \, .
</math>
  </td>
</tr>
</table>

And this is not physically acceptable.  What if, instead, we set,

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

<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\zeta_3}{\zeta_2} ~~~\Rightarrow ~~~ \theta = -40.74^\circ \, .</math>
  </td>
</tr>
</table>

Then we have,

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

<tr>
  <td align="right">
<math>~
- \biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2 
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 \biggl\} 
\frac{1 }{(a^2 + c^2)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2  
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 \biggr\} 
\frac{1 }{(a^2 + b^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
- \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] c^2  (a^2 + b^2)
+
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2  (a^2 + b^2)  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]  b^2 (a^2 + c^2)
-
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] a^2 (a^2 + c^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] [ a^2  (a^2 + b^2)  + a^2 (a^2 + c^2) ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] [ b^2 (a^2 + c^2) +  c^2  (a^2 + b^2) ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ b^2 (a^2 + c^2) +  c^2  (a^2 + b^2) ] [ a^2  (a^2 + b^2)  + a^2 (a^2 + c^2) ]^{-1} \, .
</math>
  </td>
</tr>
</table>

In our particular example, this means,

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

<tr>
  <td align="right">
<math>~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ 2.4749 ] [ 3.7837 ]^{-1} = 0.6541 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \bigg] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.8088 \, . 
</math>
  </td>
</tr>
</table>