# Explanation of Gauss's late fragments dealing with “the conformal image of the ellipse”

My question refers to some not very well known 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. I guess the mathematics involved here is connected somehow with the later Riemann's mapping theorem, which states that every simply-connected subset of the complex plane is "conformally equivalent" to the interior of unit circle.

The relevant pages from Gauss's nachlass are in volume 10-1, p. 311-320. Gauss's formulas in this fragment are interesting since they seem to be very complex - this stands in contrast to the reltively simple solution to this problem using the "Joukowsky transform" (as can be seen from this post: https://math.stackexchange.com/questions/1582608/conformally-mapping-an-ellipse-into-the-unit-circle). I even found (in the last pages) some formulas that look like theta functions. It's noteworthy that in his essay, Schlesinger comments on this fragment of Gauss at p. 192, so i think it might be of importance (i didn't find comments about it the literature).

So just to be clear, i ask this question simply because i don't understand the mathematical framework of this problem. I think that unlike other questions i asked before (like the question on Gauss's "conception of braids"), this question can be answered by a mathematician who is not an historian, but has enough knowledge in the theory of conformal mappings. So i hope for a good answer!