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

28th 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.11endnote: 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.22endnote: 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"

### Notes

• 1 Kurt Hensel (1861–1941) was a Privatdozent at the University of Berlin.
• 2 See Poincaré (1891), the inspiration for which Poincaré credited Sylvester’s paper in Messenger of Mathematics (Sylvester, 1891a, b).

## References

• H. Poincaré (1891) Extension aux nombres premiers complexes des théorèmes de M. Tchebicheff. Journal de mathématiques pures et appliquées 8, pp. 25–68. Cited by: endnote 2.
• J. J. Sylvester (1891a) On arithmetical series (I). Messenger of Mathematics 21, pp. 1–19. Cited by: endnote 2.
• J. J. Sylvester (1891b) On arithmetical series (II). Messenger of Mathematics 21, pp. 87–120. Cited by: endnote 2.