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 $\frac{1}{(\gamma_{i}\eta+\delta_{i})^{2m}}$. 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 $m=2$. 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. The second is on the topic of cubic ternary forms, of which I wish to make an arithmetic study. 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. 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.

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"

Notes

• 1 Nörlund’s transcription introduces the $\Sigma$ missing in the manuscript.
• 2 Poincaré responds here to Klein’s query of 7 May (§ 4-47-21).
• 3 Poincaré (1881a).
• 4 Poincaré (1881b).
• 5 Poincaré (1882).
• 6 Poincaré (1879), reed. Appell and Drach (1928, IL–CXXII); Poincaré (1878).