Volume 5 of Gauss's works contains a section which includes "Essays on various objects in mathematical physics". Just to emphasize the importance of this section, i'll mention that it includes, for example, his theory of falling objects on the rotating earth (precursor to the theory of Coriolis force), as well as his treatise about the achromatic double lenses, which was the subject of a previous unanswered post (What was Gauss's theoretical work related to his invention of the "Double-Gauss lens"?). The same section includes a 2-pages "letter to Brandes on the same subject" (the subject of achromatic double lenses). This letter is of great interest to me, because it includes uncommon and interesting formulas in optical theory - it involves, for example, certain integrals (so this letter doesn't seem to deal with the traditional aspects of optics).
I searched in Clemens Schaefer's treatise for information about this, and in article 84, it states that his letter to Brandes is a "result of a new calculation, which takes the intensity of light into account". Another useful piece of information which i inffered from Schaefer's treatise is that the calculation Gauss made to arrive at his results is reproduced in J.C.E Schmidt's "textbook on analytical optics" (published in 1834).
- What was the specific problem which Gauss solved in this letter (Gauss only gives final results)? unfortunately, i was not able to understand much from the section on optics in Schaefer's treatise, as it immediately throws the reader into the details of Gauss's occupation with optics and doesn't give much background and supporting motivation.
While my posted answer clarified a little bit the context of writing of Gauss's letter, it still doesn't give a direct answer to the title question. Therefore, in order to futher eludicate the answer, i'll focus now more on the details of Gauss's letter.
Gauss writes in his letter:
I now find through a more in-depth investigation that the indistinctness, which depends on the fourth power of the distance of the striking rays in the expression for the length deviation, has the smallest possible total influence, if one constructs the objective in such a way...
There, the main riddle to dechiper now is:
- what is the meaning of the term "indistinctness" which Gauss uses? why does it depend on the fourth power of the distance of the striking rays in the expression for the length deviation? (and what is "the expression for the length deviation"?)