## 7-2-54. Felix Klein to H. Poincaré, English translation

Leipzig, 9 July 1881

Dear Sir!

By way of a quick reply to your letter, I have something like the following to say.

1. It is fine with me that you to have quoted that passage of my
letter. Up to now I have only your first Note of 27
June.^{1}^{1}endnote:
^{1}
Poincaré (1881). As for the name you gave that
class of functions, I was quite surprised; but then I myself had
no more than noticed the existence of these groups.^{2}^{2}endnote:
^{2}
Before sending this
letter to Poincaré, Klein showed it to Georges Brunel, who then
cited this remark in his own letter to Poincaré of 14 July, 1881
(§ 4-15-3). For my part, I
will use neither “Fuchsian” nor “Kleinian” but remain with my
“functions invariant under linear transformations”.

2. What I said about the value of Riemann’s principles was not precise
enough. There can be no doubt that Dirichlet’s principle must be abandoned
as not at all conclusive. However, it can be completely replaced by more
rigorous methods of proof. You will find this expounded in more detail in
a work by Schwarz that I have just recently seen (in connection with my
course) and in which you will find information on the determination of
the constants, which was only indicated in *Borchardt’s Journal* (you must in any case examine the memoirs published in Volumes
70, 74, and 75 of *Borchardt’s Journal*); the work of Schwarz in
question is in the *Berliner Monatsberichten* 1870,
pp. 767–795.

3. The general existence proof I mentioned last time remains valid,
naturally, for groups made up of arbitrary analytic (not necessarily
linear) substitutions. It is remarkable that in this sense every group of
operations defines functions remaining unchanged by them. “Discontinuous
groups” have the advantage that they have associated *single-valued*
functions, a very fundamental property, moreover. Might one be able to
master higher cases by means of *single-valued* functions of *
several* variables as was the custom in connection with the case treated
by Riemann in § 12 relating to the Jacobi inversion problem?

Enough for today. In the meantime, with Mr. Brunel I have looked over my works, notably the lecture notes from 1877–78 and 78–79 (which I had reworked back then), and he will shortly write to you about these.

With the greatest respect, your devoted

Prof. Dr. F. Klein.

PTrL. Translated by R. Burns from the original German (§ 4-47-9), in Saint-Gervais (2016), with slight modifications by S.A. Walter. See also the French translation (§ 7-2-28).

Time-stamp: "28.04.2021 16:52"

## References

- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92, pp. 1484–1487. link1 Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-54. Felix Klein to H. Poincaré, English translation.