## 4-83-6. James Joseph Sylvester to H. Poincaré

28^{th} Dec 1891

New College

Dear Monsieur Poincaré,

For the last 4 months and more I have been incapable of working. The part I
sent you was *composed* before that period.

The “asymptotic limits” referred to are *limits* to the asymptotic values; *not* the
asymptotic values themselves.

I carried on this part of the inquiry to a point *much* beyond where it stood when I
had the pleasure of seing you in Oxford, but have been
unable to fix my thoughts
on the subject since composing the part on Stigmatic Series. A *portion* of my
results, I wrote to Mr. Hensel in Germany a pupil of Kronecker to whom he gave
a copy of what I wrote to him, but I went *much further* afterwards than what is
contained in that note; trusting to my *memory* to reproduce what I had thought
out, but at present my memory plays me false.^{1}^{1}endnote:
^{1}
Kurt Hensel (1861–1941)
was a Privatdozent at the University of Berlin.

I paid £ $5_{12}/()$ about 140 francs to a calculator to work out a certain table connected with this subject (a mere particular case).

I congratulate you on your remarkable discovery relating to primes $4n+1$, $4n-1$;
I never proved or hoped to prove this by my method.^{2}^{2}endnote:
^{2}
See
Poincaré (1891), the inspiration for which Poincaré credited
Sylvester’s paper in *Messenger of Mathematics* (Sylvester, 1891a, b).

My opinion is worth little, but as you ask me the question I may say that I do think
the Asymptotic Coeffcient in the Stigmatic series may be brought indefinitely near
to unity.
*Every* harmonic scheme (I think it may be shown) has its own proper asymptotic
coeffcient.

I am to have a years leave of absence from Oxford on medical grounds.

Pray forgive any officialness in my answer and believe me

Yours with the highest respect and sincere regard,

J. J. Sylvester

You did not mention to me when you were in Oxford that you had been considering “Stigmatic Series”.

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "30.10.2021 20:17"

## References

- Extension aux nombres premiers complexes des théorèmes de M. Tchebicheff. Journal de mathématiques pures et appliquées 8, pp. 25–68. Link Cited by: endnote 2.
- On arithmetical series (I). Messenger of Mathematics 21, pp. 1–19. Link Cited by: endnote 2.
- On arithmetical series (II). Messenger of Mathematics 21, pp. 87–120. Link Cited by: endnote 2.