7271. H. Poincaré to Felix Klein, English translation
Nancy, 22 September 1882
Sir,
Here are a few details concerning functions whose natural limit is formed by an infinity of circles, and about which I spoke in my note in the Annalen.^{1}^{1}endnote: ^{1} See Poincaré (1882c), and Klein’s letter to Poincaré of 19 September, where he claims to know “sehr wohl” such functions (§ 44725). To simplify the exposition, I shall take by way of example a very special case. Suppose we have four points $a$, $b$, $c$, $d$ on the fundamental circle and four circles meeting it orthogonally: the 1^{st} at $a$ and $b$, the 2^{nd} at $b$ and $c$, the 3^{rd} at $c$ and $d$, and the 4^{th} at $d$ and $a$. We have thereby a curvilinear quadrilateral. Let us consider two substitutions (hyperbolic or parabolic), the 1^{st} sending the circle $a\,b$ to the circle $a\,d$, the 2^{nd} sending the circle $c\,b$ to the circle $c\,d$. The Wiederholungen of our quadrilateral will cover the surface of the fundamental circle or only part of that surface; but in every case the group will obviously be discontinuous. It is easy to see that the fundamental circle will be entirely covered in one case only: when the four points $a\,b\,c\,d$ are harmonic and the two substitutions $(ab,ad)$ and $(cb,cd)$ are parabolic. We are dealing here with the modular function. In every other case, we find that the Wiederholungen in question cover only a region bounded by an infinity of circles. Now the whole plane can be abgebildet on our quadrilateral and in such a way that two corresponding points on the perimeter correspond to the same point of the plane. This Abbildung defines a function that exists only in the region covered by the Wiederholungne. Here, however, there is an important remark that must be made. The group generated by the two substitutions $(ab,ad)$ and $(cb,cd)$ may be considered as generated in another way. Let’s consider four circles $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$ meeting the fundamental circle orthogonally and not intersecting each other such that any is exterior. Consider two substitutions changing $C_{1}$ to $C_{2}$ and $C_{3}$ to $C_{4}$; the group they generate is obviously discontinuous and, for a suitable choice of the 4 circles, it will be the same as the group considered above. The part of the plane exterior to the 4 circles is a quadrilateral of sorts which can be abgebildet on a Riemann surface of genus 2 and which thereby determines a function defined on the whole plane. Here we have one and the same group giving rise to two essentially different functions. One can ask in this connection a great many delicate questions which I can not go into here.
In summary, you see that we can indeed have functions defined only in a region bounded by an infinity of circles and yet which are still “Fuchsian functions”, since all the group’s substitutions preserve the fundamental circle. Each of the circles of the boundary is preserved by one of the substitutions from the group, which also preserves the fundamental circle. You know, of course, that every hyperbolic substitution preserves all circles passing through the two double points.
I learn with pleasure that you are preparing a substantial piece of work of the topic of mutual interest. I shall read it with the greatest pleasure. As Mr. MittagLeffler told you, I am writing a paper on that theme myself; in light of its length I divided it in five memoirs:

The 1^{st} will appear this year, on groups of real substitutions (those I called Fuchsian groups)^{2}^{2}endnote: ^{2} Poincaré (1882e).

the 2^{nd} on Fuchsian functions; I will soon finish writing this up^{3}^{3}endnote: ^{3} Poincaré (1882b).

the 3^{rd} on the more general groups and functions that I call Kleinian.^{4}^{4}endnote: ^{4} Poincaré (1883).

In the 4^{th}, I will address a range of issues sidelined in the second memoir, i.e., the proof of the existence of functions satisfying certain conditions, for instance the proof of the fact that to every Riemann surface there corresponds an analogous function and the determination of the relevant constants.^{5}^{5}endnote: ^{5} Poincaré (1884b).

Finally, in the 5^{th}, I will discuss Zetafuchsian functions and the integration of linear equations.^{6}^{6}endnote: ^{6} Poincaré (1884a).
I have to return to Paris the day after tomorrow, so I will be there Mr. Lie’s visit. I would be sad to miss the opportunity to meet this famous geometer.
You should have received the first part of my paper on curves defined by differential equations. I will send you the second part shortly, along with my memoir on cubic forms.^{7}^{7}endnote: ^{7} Poincaré (1881a), (1882a), (1881b), (1882d).
Please be assured, sir, of my highest regard,
Poincaré
PTrL. Translated by R. Burns in SaintGervais (2016) from the original French (§ 44726), with minor revisions by S.A. Walter.
Timestamp: "15.05.2021 23:30"
References
 Mémoire sur les courbes définies par une équation différentielle (I). Journal de mathématiques pures et appliquées 7, pp. 375–422. link1 Cited by: endnote 7.
 Sur les formes cubiques ternaires et quaternaires I. Journal de l’École polytechnique 50, pp. 190–253. link1 Cited by: endnote 7.
 Mémoire sur les courbes définies par une équation différentielle (2nde partie). Journal de mathématiques pures et appliquées 8, pp. 251–296. link1 Cited by: endnote 7.
 Mémoire sur les fonctions fuchsiennes. Acta mathematica 1, pp. 193–294. link1 Cited by: endnote 3.
 Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires. Mathematische Annalen 19, pp. 553–564. link1 Cited by: endnote 1.
 Sur les formes cubiques ternaires et quaternaires II. Journal de l’École polytechnique 51, pp. 45–91. link1 Cited by: endnote 7.
 Théorie des groupes fuchsiens. Acta mathematica 1, pp. 1–62. link1 Cited by: endnote 2.
 Mémoire sur les groupes kleinéens. Acta mathematica 3, pp. 49–92. link1 Cited by: endnote 4.
 Mémoire sur les fonctions zétafuchsiennes. Acta mathematica 5, pp. 209–278. link1 Cited by: endnote 6.
 Sur les groupes des équations linéaires. Acta mathematica 4, pp. 201–311. link1 Cited by: endnote 5.
 Uniformization of Riemann Surfaces: Revisiting a HundredYearOld Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7271. H. Poincaré to Felix Klein, English translation.