I'm very curius to learn of the solution to the problem of the gravitational attraction of triaxial ellipsoid, both in his internal and external parts. From what i read, i understood that the solution for the attraction of spheroid (biaxial ellipsoid) in his internal part was given implicitly by Newton in his Principia, and the mathematical formulas involved were made explicit by Maclaurin (see Chandrasekhar: Newton's Principia for the common reader).
However, the solution for the external part of the spheroid is complicated even for a homogenous ellipsoid, as is being said here https://www.sciencedirect.com/science/article/pii/S003206331730257X, and according with this article, it was made for the first time by Gauss and later by Dirichlet. I also read that Gauss gave different expression for the internal part of ellipsoids, involving quadratic transfomation of the coordinates with coefficient values that are elliptic integrals (see https://hal.archives-ouvertes.fr/hal-01592829/document).
So what was the method of potential theory used by Gauss? how did he tackle this unapproachable problem? in paricular, can anyone give an account of the results and the structure of Gauss's 1813 article?