My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to the interior of the unit circle. These fragments date from 1839, much later then his better-known article from 1822 on conformal mappings. The relevant pages from Gauss's nachlass are in volume 10-1, p. 311-320. Schlesinger comments on these fragments of Gauss on p.192 of his essay, where he mentions that his results match the formula found much later by Hermann Schwarz.

These fragments are very noteworthy, first of all because they seem to anticipate the Riemann's mapping theorem (why did he attempt to find such a solution at all?), and secondly because they introduce powerful mathematical tools for solving the problem of explicit construction of conformal mapppings to the unit circle (this problem seems to be even more complex than the so-called Schwarz-Christoffel maps for mappings of polygonal interiors by elliptic integrals) .

I've already added a very partial answer at HSM stack exchange, of which i'm not very satisfied. According to several articles i found, this is a very dificult problem and surprisingly i didn't find any comment on Gauss's solution to it in the literature (except Schlesinger's comment). So i'll be glad if anyone will explain what is going on there in Gauss's writings from a modern viewpoint.