## 7-2-56. Felix Klein to H. Poincaré, English translation

Leipzig, 4 December 1881

Sophienstraße 10/II

Dear Sir!

After having long reflected only in passing on the questions of
mutual interest to us, this morning I took the opportunity of reading
together the different communications that you have published successively
in the *Comptes rendus*. I see that you have now definitely proved
(as of 8 August): “que toutes les équations différentielles
linéaires à coefficients algébriques s’intègre par les fonctions
zétafuchsiennes” and “que les coordonnées des points d’une
courbe algébrique quelconque s’expriment par des fonctions fuchsiennes
d’une variable auxiliaire”.^{1}^{1}endnote:
^{1}
Poincaré (1881), reed.
Nörlund and Lebon (1916, 29–31). “[T]hat every linear differential equation with
algebraic coefficients is integrable by means of zetafuchsian functions
and that the coordinates of any algebraic curve whatever can be expressed
via fuchsian functions of an auxiliary variable.” While congratulating you on the results you have
obtained, I would also like to put a proposition to you respecting
both your interests and mine equally. I would ask you to to send me, for
*Mathematische Annalen*, a more or less long article, or, if you don’t
have the time to write up such a work, then a *letter*, expounding,
in broad strokes, your points of view and results. I would then write an
accompanying note in which I would describe how I view the whole matter
and how at this juncture the research program that you are now pursuing
has served as a fundamental guiding principle for my work on modular
functions. Of course, I would submit this note to you for your approval
prior to publication. Such a publication would have a double effect:
first, it would definitely draw the attention of the readers of *Math. Annalen* to your work, undoubtedly a desirable outcome for you;
and second, your work would be presented to a large
general readership, at the same time demonstrating the connections
with my work as they actually are. As I know from what you wrote to me,
you intend to analyse these equations in a detailed memoir; but writing it
will take time, and I would like an announcement to be made in the *Annalen* also.

For my part, I have meanwhile written up a little treatise on
“Riemann’s theory” which may be of interest to you since it
presents a version of the concept of a Riemann surface that I believe R.
himself actually worked on.^{2}^{2}endnote:
^{2}
Klein (1882), reed.
Fricke et al. (1923, 499–573). Perhaps Mr. Brunel has told you of this. I
have also been busy lately with different existence proofs designed to
replace Dirichlet’s principle, and I am convinced that the methods
expounded by Schwarz in the *Berliner Monatsberichten*, 1870, p. 767
*et seqq.* in any case suffice completely to yield, for
example, the general theorem I wrote to you about once or twice this past
summer.^{3}^{3}endnote:
^{3}
Schwarz (1870).

With the greatest respect,

F. Klein

PTrL. Slightly emended from the translation by R. Burns of the original German (§ 4-47-10) in Saint-Gervais (2016). See also the French translation (§ 7-2-36).

Time-stamp: "20.09.2023 18:44"

### Notes

- 1 Poincaré (1881), reed. Nörlund and Lebon (1916, 29–31). “[T]hat every linear differential equation with algebraic coefficients is integrable by means of zetafuchsian functions and that the coordinates of any algebraic curve whatever can be expressed via fuchsian functions of an auxiliary variable.”
- 2 Klein (1882), reed. Fricke et al. (1923, 499–573).
- 3 Schwarz (1870).

## References

- Felix Klein Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. link1 Cited by: endnote 2.
- Ueber Riemann’s Theorie der algebraischen Functionen und ihrer Integrale. Teubner, Leipzig. link1 Cited by: endnote 2.
- Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 1.
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 93, pp. 301–303. link1 Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-56. Felix Klein to H. Poincaré, English translation.
- Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen. Monatsberichte der königliche Akademie der Wissenschaften zu Berlin, pp. 767–795. link1 Cited by: endnote 3.