A Scientific Analysis of Galaxy Tangential Speed of Revolution Curves III

I last reported on my preliminary analysis of 350$+$ spiral, lenticular, irregular, polar ring, ring, and dwarf elliptical galaxies' tangential speed of revolution curves [TSRCs; and not rotation (\textit{sic}) curves]. I now know that the consensus opinion in the literature---for which I can find no geometrical, numerical, statistical, nor scientific testing in 2,500$+$ publications---that the TSRC, v$_{\mathrm{B}}$(r), in the central bulges of these galaxies, is a linear function of the radial distance from the minor axis of symmetry r---is false. For the majority (\textgreater 98{\%}) v$_{\mathrm{B}}$(r) is rarely well represented by v$_{\mathrm{B}}$(r) $=$ $\omega_{\mathrm{B}}$r (for which the unique material model is an homogeneous, oblate, spheroid). Discovered via a scientific analysis of the gravitational potential energy computed directly from the observational data, v$_{\mathrm{B}}$(r) is almost exactly given by v$_{\mathrm{B}}^{\mathrm{2}}$(r) $=$ ($\omega _{\mathrm{B}}$r)$^{\mathrm{2}}$(1 $+ \quad \eta $r$^{\mathrm{2}})$ with $|\eta|$ \textless 10$^{\mathrm{-2}}$ and frequently orders of magnitude less. The corresponding mass model is the simplest generalization: a two component homoeoid. The set of possible periodic orbits, based on circular trigonometric functions, becomes a set of periodic orbits based on the Jacobian elliptic functions. Once again it is possible to prove that the mass-to-light ratio can neither be a constant nor follow the de Vaucouleurs R$^{1/4}$ rule.