Consistently Implementing the Fields Self-Energy in Newtonian Gravity

We consider in a pedagogical fashion alterations to Newtonian gravity due to the postulate that all energy corresponds to active gravitational mass when applied to the self-energy of the gravitational field. We show why a simple addition of $\frac{1}{c^2}$ times the gravitational field energy to the matter density in Newton’s field equation is inconsistent. A consistent prescription is shown and discussed. The connection to general relativity is pointed out.