Editing Appendix/Ramblings/ForCohlHoward (section)

==Understanding the Dimensionality of EFE Index Symbols==
Howard put together a Mathematica script intended to provide &#8212; for any specification of the semi-axis length triplet <math>(a, b, c)</math> &#8212; very high-precision, numerical evaluations of any of the index symbols, <math>A_{ijk\ldots}</math> and <math>B_{ijk\ldots}</math> as defined by Eqs. (103 - 104) in &sect;21 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  Originally I suggested that, without loss of generality, he should only need to specify the ''pair'' of length ratios, <math>(1, b/a, c/a)</math>.  In response, Howard pointed out that evaluation of all but a few of the lowest-numbered index symbols &#8212; as defined by [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; ''does'' explicitly depend on specification of (various powers of) the semi-axis length, <math>a</math>.

<font color="red">Joel's response:</font>&nbsp; Howard is correct!  He should leave the explicit dependence of <math>a</math> &#8212; to various powers &#8212; in his Mathematica notebook's determination of all the EFE index symbols.

Instead, what we should expect is that the evaluation of various ''physically relevant'' parameters will produce results that are independent of the semi-axis length, <math>a</math>; these evaluations should involve combining various index symbols in such a way that the dependence on <math>a</math> disappears.  Consider, for example, our [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Virial_Equilibrium|accompanying discussion]] (click to see relevant expressions) of the virial-equilibrium-based determination of the frequency ratio, <math>f \equiv \zeta/\Omega_f</math>, in equilibrium S-Type Riemann Ellipsoids.  Although most of the required index symbols, <math>A_1, A_2, A_3</math> and <math>B_{12}</math>, are dimensionless parameters, the index symbol <math>A_{12}</math> has the unit of inverse-length-squared.  Notice, however, that when <math>A_{12}</math> appears along with any of these other ''dimensionless'' parameters in the definition of <math>f</math>, it is accompanied by an extra "length-squared" factor, such as <math>a^2</math>.  Hence, although I strongly agree that Howard should continue to include various powers of <math>a</math> (etc.) in his Mathematica notebook expressions, I suspect that, without loss of generality, in the end we will always be able to set <math>a=1</math> and only need to specify the ''pair'' of length ratios, <math>(1, b/a, c/a)</math>.