Editing 3Dconfigurations/RiemannEllipsoids (section)

====''v'' and ''w'' Ratios====

From the first modification (mod.01) of equation <math>(\alpha.4)</math>, we find that,

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

<tr>
  <td align="right">
<math>(b+c+2a)v' w'</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
-(b+c-2a) vw
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{ v'}{v}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a - b - c)}{(2a + b + c)}\cdot \frac{ w}{w'} \, .
</math>
  </td>
</tr>
</table>
And from the first modification (mod.01) of equation <math>(\alpha.5)</math>, we find,

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

<tr>
  <td align="right">
<math>(2a - b + c)v' w </math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>(2a + b - c) v w' </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{ v' w}{vw'}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a + b - c)}{(2a - b + c)} \, .
</math>
  </td>
</tr>
</table>

<span id="vw">Combining these two expressions gives</span>,

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

<tr>
  <td align="right">
<math>\frac{(2a - b - c)}{(2a + b + c)} \cdot \frac{w}{w'} \biggl[\frac{ w}{w'} \biggr] </math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\biggl[\frac{(2a + b - c)}{(2a - b + c)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{ w'}{w} \biggr]^2 </math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a - b - c)}{(2a + b + c)} \cdot \frac{(2a - b + c)}{(2a + b - c)} \, ;
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{ v'}{v} \biggr]^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a - b - c)}{(2a + b + c)} \cdot \frac{(2a + b - c)}{(2a - b + c)} \, .
</math>
  </td>
</tr>
</table>

From the first modifications (mod.01) of <math>(\alpha.6)</math> through <math>(\alpha.9)</math>, we also see that,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{ v'}{v} \biggr]^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(c+a)^4}{(c-a)^4} \times \mathrm{constant} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>\biggl[ \frac{ w'}{w} \biggr]^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(a-b)^4}{(a+b)^4} \times \mathrm{constant} \, .
</math>
  </td>
</tr>
</table>

Hence, following Riemann (1861) &#8212; see the bottom of p. 183 of [http://www.kendrickpress.com/Riemann.htm BCO2004] &#8212; we can write,
<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">

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

<tr>
  <td align="right">
<math>\biggl[ \frac{ v'}{v} \biggr]^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a - b - c)}{(2a + b + c)} \cdot \frac{(2a + b - c)}{(2a - b + c)} 
=
\biggl( \frac{c+a}{c-a}\biggr)^4 \times \mathrm{constant} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \frac{ w'}{w} \biggr]^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{(2a - b - c)}{(2a + b + c)} \cdot \frac{(2a - b + c)}{(2a + b - c)} 
=
\biggl( \frac{a-b}{a+b} \biggr)^4 \times \mathrm{constant} \, .
</math>
  </td>
</tr>
</table>

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

As Riemann (1961) points out &#8212; see the top of p. 184 of [http://www.kendrickpress.com/Riemann.htm BCO2004] &#8212; <font color="orange">"Taking this together with</font> the constant volume constraint <math>(\alpha.10)</math> <font color="orange">we see that <math>a, b,  c</math> are constant, and consequently <math>v, v', w, w'</math> are constant."</font>