Editing PGE/FirstLawOfThermodynamics (section)

===Substantiation===
To further substantiate this claim, we note that,

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  <td align="right">
<math>\frac{\tau}{\rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\epsilon^{1/\gamma_g} \cdot \rho^{1/\gamma_g - 1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ln\biggl(\frac{\tau}{\rho}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[ \ln\epsilon - ( \gamma_g-1)\ln\rho \biggr] \, .
</math>
  </td>
</tr>
</table>

Now, from the first law, we can write,
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  <td align="right">
<math>ds</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{T} \biggl[
d\epsilon - \frac{P}{\rho} {d\ln\rho}
\biggr]
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_V~ d\ln\epsilon - \frac{\Re}{\mu} ~{d\ln\rho}
</math>
  </td>
</tr>

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  <td align="right">
<math>
\Rightarrow ~~~ \frac{ds}{c_P}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{c_V}{c_P}~ d\ln\epsilon - \frac{\Re/\mu}{c_P} ~{d\ln\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[  d\ln\epsilon - (\gamma_g-1){d\ln\rho} \biggr] \, ,
</math>
  </td>
</tr>
</table>
which, upon integration, gives,

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  <td align="right">
<math>\frac{s}{c_P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[  \ln\epsilon - (\gamma_g-1)\ln\rho \biggr] + \mathrm{constant} \, .
</math>
  </td>
</tr>
</table>

To within an additive constant, the right-hand side of this relation is precisely the expression for the logarithm of the entropy tracer, as provided immediately above.  Hence, we see that,
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<math>s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, ,</math>
</div>
that is, we see that the variable, <span title="Entropy tracer"><math>\tau</math></span>, traces the fluid entropy just as {{ Template:Math/VAR_Density01 }} traces the fluid mass.

<span id="EntropyLL75">We have found</span> one other instance in the literature &#8212; although there are undoubtedly others &#8212; where the role of this ''entropy tracer'' previously has been identified.  In chapter IX of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>] we find that, "apart from an unimportant additive constant," the specific entropy is,
<div align="center" id="LL75">
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  <td align="right">
<math>s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr) \, .</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;80, Eq. (80.12)
</div>

Given that <math>\tau \propto P^{1/\gamma_g}</math>, this is clearly the same expression as we have derived for the specific entropy of the fluid.