When we initially determined that the change in the gravitational potential energy of an object as it moves up or down is equal to `\pm mg \Delta h `, we had assumed that the gravitational force acting on the object was constant.  This was true near the surface of the Earth.  If I have a set of point masses, however, the gravitational force that a point mass feels as it moves in the presence of other masses is not constant.  Therefore, we can’t use `\Delta U_{g}=mg \Delta h` to calculate the change in the gravitational potential energy of a point mass in the presence of other masses.  We need to go back to our definition for the change in a potential energy to determine an equation we can use for the gravitational potential energy due to the interactions between point masses.