On the equilibrium between radiation and matter in Einstein's closed universe

In two preceding articles,(1) the principles of thermodynamics have been expressed in a form suitable for applications in general relativity, and then applied to Einstein's closed universe, employing a set of coordinates in which the line element assumes the form ds^2 = -R^2dX^2 - R^2 sin^2 X(dθ^2 + sin^2 θdp^2) + dt2.