## 7-2-67. H. Poincaré to Felix Klein, English translation

Paris, 12 May 1882

Sir,

I am late in responding and I hope you will forgive me, as
circumstance beyond my control led to my absence. I believe as you do
that our methods have much in common and differ less on principle than
in detail. Concerning the lemmas you mention, I established the first
via series development while you, as I gather, rely on the theorem you
wrote about in one of the letters you wrote last year. The second lemma presents no difficulties, and
it’s likely that we both prove it in the same way. Once these two
lemmas are established – and this is where I begin, just as you do
yourself – I employ *continuity*, as you do, although there are
many ways to do so, and it is possible that we differ on a few details.

You ask me how I establish the convergence of the series
$\frac{1}{(\gamma_{i}\eta+\delta_{i})^{2m}}$.^{1}^{1}endnote:
^{1}
Nörlund’s
transcription introduces the $\Sigma$ missing in the manuscript. I have two proofs, but they
are both too long to be included in a letter; I shall publish them
shortly. The first is based in principle on the fact that the surface of
the fundamental circle is finite. The second requires the same
assumption, but is based on non-Euclidean geometry. What is the lower limit
of the number $m$? It’s $m=2$.^{2}^{2}endnote:
^{2}
Poincaré responds here to
Klein’s query of 7 May (§ 4-47-21).
The limit is exact, assuming $m$ is an
integer. As far as series relating to *Zetafuchsian* functions
are concerned, I have, on the contrary, only an approximate limit. What most interested me in your
letter was what you had to say about functions admitting an infinity
of circles as their lacunary spaces. I also have encountered such
functions and provide an example in one or two of my notes. I
arrived at them, however, by a path absolutey different from yours.

It is probable that your functions and mine must be closely related; it is not at all obvious, nonetheless, that they are identical. I would be willing to believe that your method and mine can be very extensively generalized and that they would both lead to a large class of transcendental functions including as special cases those we have already met.

You aske me about offprints of my articles. Do you mean my notes in the
*Comptes Rendus*? I did not order offprints and it would now be
difficult to procure them, at least for the first ones.

I will send you in short order, and as soon as I receive them the
offprints of two more recent articles; the first “on curves defined
by differential equations”. This concerns the geometric form of
curves defined by first-order differential equations. Unfortunately,
only the first part of this memoir has yet been printed and contains
only the preliminaries.^{3}^{3}endnote:
^{3}
Poincaré (1881a). The second is on
the topic of cubic ternary forms, of which I wish to make an
arithmetic study.^{4}^{4}endnote:
^{4}
Poincaré (1881b). I wanted first
to recall certain algebraic results involved in the first part of the
memoir. This first part alone has appeared in volume 50 of the
*Journal de l’École Polytechnique*, and the remainder is to
appear in volume 51.^{5}^{5}endnote:
^{5}
Poincaré (1882). Thus the first part
will not be of much interest to you. It does contain, however, a study
of linear transformations and of certain *continuous* groups
contained in the linear group in 3 or 4 variables.

By the way, I don’t remember if I sent you my dissertation, or earlier
articles on differential equations and a work on functions related to
lacunary spaces.^{6}^{6}endnote:
^{6}
Poincaré (1879), reed.
Appell and Drach (1928, IL–CXXII);
Poincaré (1878).

Please be assured, sir, of my highest regard.

Poincaré

PTrL. Translated by S.A. Walter from the original French (§ 4-47-22). Previously translated by R. Burns in Saint-Gervais (2016).

Time-stamp: " 7.05.2021 00:39"

## References

- Œuvres d’Henri Poincaré, Volume 1. Gauthier-Villars, Paris. link1 Cited by: endnote 6.
- Note sur les propriétés des fonctions définies par les équations différentielles. Journal de l’École polytechnique 45, pp. 13–26. link1 Cited by: endnote 6.
- Sur les propriétés des fonctions définies par les équations aux différences partielles. Ph.D. Thesis, Faculté des sciences de Paris, Paris. link1 Cited by: endnote 6.
- Mémoire sur les courbes définies par une équation différentielle (I). Journal de mathématiques pures et appliquées 7, pp. 375–422. link1 Cited by: endnote 3.
- Sur les formes cubiques ternaires et quaternaires I. Journal de l’École polytechnique 50, pp. 190–253. link1 Cited by: endnote 4.
- Sur les formes cubiques ternaires et quaternaires II. Journal de l’École polytechnique 51, pp. 45–91. link1 Cited by: endnote 5.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-67. H. Poincaré to Felix Klein, English translation.