## 3-15-32. George Howard Darwin to H. Poincaré

Sep. 8. 01

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I am interested to hear of the progress you have made.11 1 Poincaré reported his progress by letter to Darwin in early September, 1901 (§ 3-15-31).

The difficulty you refer to as to the difference between the potentials of the ellipsoid outside & inside is the reason why I adopted the artifice of enveloping the whole pear in an external Jacobian ellipsoid.22 2 See Darwin to Poincaré, 12.08.1901 (§ 3-15-23, and Darwin (1903, 309). I attempted in the case of the sphere to use different formulæ for the external & internal parts and met with failure. By making the ellipsoid of reference a Jacobian, the potential is of the form

 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}.$

I suspect that you will be driven to the same course.

I have determined to send in my paper to the R.S. as I find I have finished it as far as it goes. My reason for doing so is that I fear that as soon as our term begins I shall not have time to go on with the approximation (to be suggested by you), and I think if it ever comes it will be quite enough to stand by itself.33 3 Darwin (1902, received 21.10.1901), determined the axes of the critical Jacobian, on which see Darwin to Poincaré, 28.05.1901 (§ 3-15-11). Darwin went on to publish a putative demonstration of the stability of the pear-shaped figure on 19.06.1902 (Darwin 1903), which Jeans (1915) and Cartan (1928) both found to be flawed, as noted by Lévy (1952, 625–626).

I do not remember whether I told you that I have got the next six coefficients of stability, and find that they all vanish for Jacobians which would not perceptibly differ from the one already drawn.

Three of the figures are remarkable (as you will easily see) from the fact that the northern & southern quasi-hemispheres are different.44 4 See Poincaré’s reply (§ 3-15-33), where the symmetries of Darwin’s figures are shown to conflict with the idea of a bifurcation figure. Darwin refers not to the six coefficients corresponding to zonal harmonics for degrees 4 to 9, shown to vanish by Poincaré (1885, 321), but to the six coefficients corresponding to harmonics $\mathfrak{P}_{3}^{s}$, $\mathsf{P}_{3}^{s}$ of degree 3 and of orders $s=1,2,3$. See also Darwin to Poincaré, 12.10.1901 (§ 3-15-35).

The next one to vanish is one in which the mean axis of the Jacobian is sharpened at one end & bluntened at the other. I am reminded of Roche’s result of the two figures of equilibrium under the action of a distant force.55 5 Roche (1849) showed that for a small satellite there are two possible figures of equilibrium that approximate each other when approaching a certain distance from a central body; see Darwin (1887, 414; 1906, 161), Schwarzschild (1896, 46); Chandrasekhar (1969, 12).

The presentation of my paper now will make no difference as to the “reading” of yours along with mine at the end of November, but it hastens the formalities of reference to reporters & order for printing.66 6 Both papers were presented on 21.11.1901. I am glad to get it off my mind as I am going away for three weeks — letters will however be forwarded.

Will you some time give me a reference to the investigations about double layers by Neumann (I think you said).77 7 Neumann 1877; see Poincaré to Darwin §§ (3-15-25, 3-15-33). I have not read about it.

Yours very sincerely,

G. H. Darwin

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "22.01.2016 17:16"

## References

• É. Cartan (1928) Sur la stabilité ordinaire des ellipsoïdes de Jacobi. pp. 9–17. Cited by: footnote 3.
• S. Chandrasekhar (1969) Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven. External Links: Link Cited by: footnote 5.
• G. H. Darwin (1887) On figures of equilibrium of rotating masses of fluid. Philosophical Transactions of the Royal Society of London 178, pp. 379–428. Cited by: footnote 5.
• G. H. Darwin (1902) On the pear-shaped figure of equilibrium of a rotating mass of liquid. Philosophical Transactions of the Royal Society A 198, pp. 301–331. Cited by: footnote 3.
• G. H. Darwin (1903) The approximate determination of the form of Maclaurin’s spheroid. Transactions of the American Mathematical Society 4 (2), pp. 113–133. Cited by: footnote 2, footnote 3.
• G. H. Darwin (1906) On the figure and stability of a liquid satellite. Philosophical Transactions of the Royal Society A 198, pp. 161–248. Cited by: footnote 5.
• J.C. Fields (Ed.) (1928) Proceedings of the International Mathematical Congress Toronto 1924. University of Toronto, Toronto. Cited by: É. Cartan (1928).
• J. H. Jeans (1915) On the potential of ellipsoidal bodies, and the figures of equilibrium of rotating liquid masses. Philosophical Transactions of the Royal Society A 215, pp. 27–78. Cited by: footnote 3.
• J. R. Lévy (Ed.) (1952) Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. External Links: Link Cited by: footnote 3.
• C. G. Neumann (1877) Untersuchungen über das logarithmische und Newton’sche Potential. Teubner, Leipzig. External Links: Link Cited by: footnote 7.
• H. Poincaré (1885) Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta mathematica 7 (1), pp. 259–380. External Links: Link Cited by: footnote 4.
• É. Roche (1849) Mémoire sur la figure d’une masse fluide soumise à l’attraction d’un point éloigné. Mémoires de l’Académie des sciences et lettres de Montpellier 1, pp. 243–262. External Links: Link Cited by: footnote 5.
• K. Schwarzschild (1896) Die Poincarésche Theorie des Gleichgewichts einer homogenen rotierenden Flüssigkeitsmasse. Ph.D. Thesis, Ludwig-Maximilians-Universität zu München, Munich. Cited by: footnote 5.