Analytic solution found for Holchronous galaxy rotation curves

Up to now I had been calculating the gravitational potential arising from a collapsing galactic gas cloud, under the Holochronous model, using a numerical simulation in Excel. Obviously this is not ideal when it comes to presenting the results in a scientific publication, and I was confident that an analytical solution must exist, if only I could get me head around the concept of integrating a 4-dimensional density function. I am pleased to say that I did eventually managed to come up with the solution with nothing more than basic A-level calculus. I am still not quite sure what the underlying meaning is of the process that I have been using. It seems akin to the integration of a Lagrangian 4-density, in which case the end result driving the gravitational potential is a scalar action. Go figure! Anyway, the results agree with the simulation, which is good, and the 3 parameters of the resulting formula allow fitting of the calculated rotation curves to a wide sample of elliptical galaxy observed rotation curves, which is even better.