Editing Appendix/Ramblings/ConcentricEllipsoidalT8Coordinates (section)

==Background==
Building on our [[Appendix/Ramblings/DirectionCosines|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[Appendix/Ramblings/T3Integrals|T3]] (concentric oblate-spheroidal) and [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system.  This is motivated by our [[ThreeDimensionalConfigurations/Challenges#Trial_.232|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].

Note that, in a [[Appendix/Ramblings/ConcentricEllipsoidalCoordinates|separate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7."  In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric  ellipsoidal coordinate system.  But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector.  In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system.  Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.