## 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.^{1}^{1}endnote:
^{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.^{2}^{2}endnote:
^{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.^{3}^{3}endnote:
^{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.^{4}^{4}endnote:
^{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.^{1}^{1}
1
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.^{5}^{5}endnote:
^{5}
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).^{6}^{6}endnote:
^{6}
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: " 4.05.2019 00:12"

### Notes

- 1 Poincaré reported his progress by letter to Darwin in early September, 1901 (§ 3-15-31).
- 2 See Darwin to Poincaré, 12.08.1901 (§ 3-15-23, and Darwin (1903, 309).
- 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).
- 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).
- 5 Both papers were presented on 21.11.1901.
- 6 Neumann 1877; see Poincaré to Darwin §§ (3-15-25, 3-15-33).

## References

- Sur la stabilité ordinaire des ellipsoïdes de Jacobi. See Proceedings of the International Mathematical Congress Toronto 1924, Fields, pp. 9–17. Cited by: endnote 3.
- Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven CT. Cited by: footnote 1.
- On figures of equilibrium of rotating masses of fluid. Philosophical Transactions of the Royal Society of London 178, pp. 379–428. Cited by: footnote 1.
- 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: endnote 3.
- The approximate determination of the form of Maclaurin’s spheroid. Transactions of the American Mathematical Society 4 (2), pp. 113–133. Cited by: endnote 2, endnote 3.
- On the figure and stability of a liquid satellite. Philosophical Transactions of the Royal Society A 206, pp. 161–248. link1, link2 Cited by: footnote 1.
- Proceedings of the International Mathematical Congress Toronto 1924. University of Toronto, Toronto. Cited by: É. Cartan (1928).
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- Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. link1 Cited by: endnote 3.
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- Die Poincaré’sche Theorie des Gleichgewichts einer homogenen rotierenden Flüssigkeitsmasse. Ph.D. Thesis, Ludwig-Maximilians-Universität zu München, Munich. link1 Cited by: footnote 1.