Editing Appendix/Ramblings/ToroidalCoordinates (section)

==Preamble==
As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal &#8212; or at least a toroidal-like &#8212; coordinate system.  Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?

I should begin by clarifying my terminology.  In volume II (p. 666) of their treatise on ''Methods of Theoretical Physics'', Morse &amp; Feshbach (1953; hereafter MF53) define an orthogonal toroidal coordinate system in which the Laplacian is separable.<sup>1</sup>  (See details, below.)  It is only this system that I will refer to as ''the'' toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse &amp; Feshbach's coordinate system will be referred to as ''toroidal-like.'' 

I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student).  Howie's dissertation research uncovered a ''Compact Cylindrical Greens Function'' technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.<sup>2,3</sup>  The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind &#8212; see [http://dlmf.nist.gov/14.19 NIST digital library discussion] &#8212; where, if I recall correctly, the argument of this special function (or its inverse) seemed to resemble the ''radial'' coordinate of Morse &amp; Feshbach's orthogonal toroidal coordinate system &#8212; see more on this, [[#Relating_CCGF_Expansion_to_Toroidal_Coordinates|below]].