Editing VE (section)

===Setting the Stage===

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations.  The ''virial method'' &hellip; is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account.  The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font>  In this context, Chandrasekhar's focus is on two of the four [[PGE#Principal_Governing_Equations|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
('''Momentum Conservation''')

{{ Template:Math/EQ_Euler01 }}
</div>
and the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Template:Math/EQ_Poisson01}}
[[File:OriginButton.jpg|125px|link=PGE/PoissonOrigin]]
</div>

In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the Euler equation first appears in &sect;11 (p. 20) as equation (38) and is written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho \frac{du_i}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} \, ,</math>
  </td>
</tr>
</table>
</div>
and the Poisson equation appears in &sect;10 (p. 20) &#8212; specifically, the left-most component of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] equation (37) &#8212; as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\nabla^2 \mathfrak{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4\pi G \rho \, .</math>
  </td>
</tr>
</table>
</div>
It is clear, therefore, that Chandrasekhar uses the variable <math>\vec{u}</math> instead of <math>\vec{v}</math> to represent the inertial velocity field.  More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>- \Phi = \mathfrak{B} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
</div>
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.