What were Gauss's results on the attraction of homogenous ellipsoids in his paper from 1813?

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?

• So why don't you look into Gauss' paper? Apr 20 '18 at 12:25
• I looked into Gauss's paper ("Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum"), but as you know, Gauss's papers are very difficult to read and understand. They are written in a masterly fashion that only very skilled mathematicians can follow their chain of reasoning. I'm not a mathematician, and i'm looking for more accesible articles that summarize his work or for experts that can explain to me in Gauss's results in more approachable way. Apr 20 '18 at 12:39

This is a very partial answer, but i had to write down the information i collected up to now.

One of the best sources on Gauss's article on ellipsoids attraction is Harald Geppert's treatise on Gauss's activities in fields related to classical mechanics. According to this treatise, the structure of Gauss's paper can be described in this way:

• Historical background - this takes 2 pages of Gauss's article and lists the contributions made to this problem from the time of Newton up to more recent work made primarily by the french school of analysis ( Lagrange, Legendre, Laplace).
• The first part of the article - several integral theorems - this apparently includes the known Gauss's divergence theorem. It includes six integral propositions, that fall into two parallel groups - the first three are based on the idea of spatial division of space, while the following three are based on a division of space into elementary cones.
• The second part of the article contains the special application of the preceeding theorems to the attraction of ellipsoid.

For the calculation of the attraction of ellipsoids, Gauss first employs the parametric representation of locations on the ellipsoids, and then applies Maclauren's theorem on attraction of confocal ellipsoids in order to carry out several integrations.

Then Gauss breaks the problem of ellipsoid attraction into two sub-problems, the first of which is done by solving cubic equation with one real root, while the second requires integration of a certain expression that leads to elliptic integrals of the second kind, which in the case of spheroid ("ellipsoid of revolution") reduce to elementary functions.

Geppert remarks that Gauss's solution is of great elegence and avoids convergence issues that arise in previous solutions to this problem.