### 5.2 Super-potential-in-series method

As all what we need to do is to evaluate the surface integrals in the general form of equations of
motion (111), we need an expression for the gravitational field only around the star. In fact, we have
developed such a method in [91] for the source term of the following form
where and are integers and , . Note that . Then, we take spatial
derivatives out of the Poisson integral,
Note that the integration region is and therefore is nonsingular in . For this kind of
source term, we have given a method in [91] to find a field in the neighborhood of star in the
following sense:
We have checked at 3 PN order that the resulting field from this method is equal to the field obtained
from the usual (super-potential) method whenever the super-potentials are available. Unfortunately,
however, this method is not perfect and we need another method to derive the equations of motion which
we explain now.