## 2-56-3. William Thomson to H. Poincaré

Nov. 7/91

The University, Glasgow

Dear Mr. Poincaré,

I am sorry you had the trouble of returning the little proof. I ought
to have written at the top of it “need not be returned”, as I have
now done! But I only send it to you again
in writing to thank you for kindly returning it to me when you
naturally thought it had been enclosed by mistake with the other
papers. It appeared in the Philosophical Magazine at the
beginning
of this month.^{1}^{1}endnote:
^{1}
According to Poincaré’s letter
(§ 2-56-2),
the proof in question is a printer’s page proof, corresponding to
Thomson’s demonstration of
a theorem in point mechanics, or what Thomson calls “kinetic
trigonometry”
(W. Thomson 1891a).

The December number will contain a continuation of my paper on
Periodic Motion, to which will be appended a short (too short)
reference to your great work on
the subject, which only came to my knowledge last week in Cambridge
where Mr. Cayley lent me the volume of the Acta formed by your
paper. I have been much interested in it so far as I have yet been
able to look through it.^{2}^{2}endnote:
^{2}
Thomson published a postscript
(Thomson, 1891b)
to his earlier paper on periodic motion of a
finite conservative system
(Thomson, 1891c). As Barrow-Green
(1997, 148) observes, Thomson finds
Poincaré’s conjecture on the denseness of periodic solutions to be less general than a similar
statement made by Maxwell, but notes that his own investigation of
the instability of periodic motion concurs with that of Poincaré
(1890), and comments: “M. Poincaré’s investigation and mine are as different as two
investigations of the same subject could well be, and it is very
satisfactory to find perfect agreement in conclusions.” On
Poincaré’s paper and its several consequences see the correspondence
with Mittag-Leffler (§§ 1-1-89 et
1-1-90),
the overview by Nabonnand (1999), and detailed
studies by Barrow-Green (1994) and
Andersson (1994).

Believe me, yours very truly,

William Thomson

ALS 1p. Private collection, Paris 75017.

Time-stamp: "30.06.2023 11:02"

### Notes

- 1 According to Poincaré’s letter (§ 2-56-2), the proof in question is a printer’s page proof, corresponding to Thomson’s demonstration of a theorem in point mechanics, or what Thomson calls “kinetic trigonometry” (W. Thomson 1891a).
- 2 Thomson published a postscript (Thomson, 1891b) to his earlier paper on periodic motion of a finite conservative system (Thomson, 1891c). As Barrow-Green (1997, 148) observes, Thomson finds Poincaré’s conjecture on the denseness of periodic solutions to be less general than a similar statement made by Maxwell, but notes that his own investigation of the instability of periodic motion concurs with that of Poincaré (1890), and comments: “M. Poincaré’s investigation and mine are as different as two investigations of the same subject could well be, and it is very satisfactory to find perfect agreement in conclusions.” On Poincaré’s paper and its several consequences see the correspondence with Mittag-Leffler (§§ 1-1-89 et 1-1-90), the overview by Nabonnand (1999), and detailed studies by Barrow-Green (1994) and Andersson (1994).

## References

- Poincaré’s discovery of homoclinic points. Archive for History of Exact Sciences 48 (2), pp. 133–147. Cited by: endnote 2.
- Oscar II’s prize competition and the error in Poincaré’s memoir on the three body problem. Archive for History of Exact Sciences 48 (2), pp. 107–131. Cited by: endnote 2.
- Poincaré and the Three Body Problem. AMS/LMS, Providence. Cited by: endnote 2.
- The Poincaré-Mittag-Leffler relationship. Mathematical Intelligencer 21, pp. 58–64. Cited by: endnote 2.
- Sur le problème des trois corps et les équations de la dynamique. Acta mathematica 13, pp. 1–270. link1 Cited by: endnote 2.
- On a theorem in plane kinetic trigonometry suggested by Gauss’s theorem of Curvatura Integra. Philosophical Magazine 32, pp. 471–473. link1 Cited by: endnote 1.
- On instability of periodic motion, being a continuation of article on periodic motion of a finite conservative system. Philosophical Magazine 32, pp. 555–560. link1 Cited by: endnote 2.
- On periodic motion of a finite conservative system. Philosophical Magazine 32, pp. 375–383. link1 Cited by: endnote 2.