## 3-15-27. George Howard Darwin to H. Poincaré

Aug. 27. 1901

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I have your letter and although I have not yet studied it as much as it deserves I hasten to write to you.11Poincaré to Darwin, ca. 13.08.1901 (§ 3-15-25). I quite understand the double layer now.22Darwin refers here to Poincaré’s “double couche”. I see it involves the use of what we call a “doublet” layer of surface density. I have used the same kind of idea in working out my result without using the word.

I see now where our disagreement comes in and it makes me no less confident in my correctness. In fact the very key of my method consists in the neglect of those terms and I believe it to be perfectly justifiable.33Darwin came to agree with Poincaré’s view; see Darwin to Poincaré, 01.09.1901 (§ 3-15-31. In the simple problem of the sphere and ellipsoid which I sent you in the first solution I wrote out the equation to the ellipsoid explicitly in harmonics and had a certain function of $e$ as the coefficient of the fourth harmonic.44Darwin did not send these detailed formulæ; see Darwin to Poincaré, 31.07.1901 (§ 3-15-17). That coefficient corresponds to your $\xi^{\prime}$.55By $\xi^{\prime}$ Poincaré denotes an infinity of different coefficients, including the coefficient of the fourth harmonic $\mathfrak{P}_{4}(\mu)$. In a second letter I applied the same method on the supposition that this result was unknown and I think I wrote the coefficient as $f$.66Darwin to Poincaré, 03.08.1901 (§ 3-15-18. All I was supposed to know was that $f$ is of order $e^{2}$ at least. I worked the whole through retaining $f$ (and $g$ for the sixth harmonic) and the analysis was pretty long. (I did not send this). I was surprised when I got to the end to find that $f$ and $g$ disappeared. I then looked round to find why they disappeared and found, as I believe, the reason. $f$ being of order $e^{2}$ the energy with itself & with the sphere is proportional to $f^{2}$ and therefore to $e^{4}$, and negligible ex hypothesi (so also for $g$).

It only remains to consider the energy corresponding to $f$ in combination with the layer. Now $f$ being of order $e^{2}$ and the layer of order $e$, it follows that their mutual energy is of order $e^{3}$. Hence we may regard the layer as concentrated. Since the layer & the $f$ term are of different orders of harmonics their mutual energy vanishes.

Hence $f$ does not come in at all in the $E$ terms. Also it does not enter in the moment of inertia since it is a harmonic of the fourth order.77See Darwin to Poincaré, 06.06.1901 (§ 3-15-13). The same considerations mutatis mutandis apply to the pear.

I had written thus far when I returned to your letters again and I think I see that you are developing to one term higher than I do. At no stage do I attempt to retain fourth powers of $e$. In my final expression I have a single term dependent on $e^{3}$ in $E$, the term in $e^{2}$ vanishing because of the bifurcation, and a single term dependent on $e^{2}$ in the moment of inertia say.88Cf. the annotation of Poincaré to Darwin (§ 3-15-19). $Ce^{3}+\frac{\varpi^{2}}{4\pi\rho}De^{2}$ for the whole, where the actual angular (velocity)${}^{2}$ of the pear is $\omega^{2}+\varpi^{2}$. Making this f${}^{\text{n}}$ stationary I have:

 $3Ce^{2}+2\frac{\varpi^{2}}{4\pi\rho}De=0$

one root is $e=0$, leading back to Jacobian.99Here $e$ is proportional to the thickness of the layer distinguishing the pear from the Jacobian ellipsoid. The other is

 $\frac{\varpi^{2}}{4\pi\rho}=-\frac{3}{2}\frac{C}{D}e$

This is what I started out to get.

I do not obtain any higher approx. to the shape of the pear. I think you will do so — if the work is carried out.

Does this clear up our difficulties at all?

If you write your paper should you like to let it appear in the Phil. Trans. along with mine?1010Poincaré agreed; see Darwin (1902), and Poincaré (1902). I think it would be convenient in that way and I should like it to be so very much. I hope to get my work finished in the autumn.

AL 4p. Collection particulière, Paris 75017.

Time-stamp: "21.01.2016 02:00"

## References

• 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.
• H. Poincaré (1902) Sur la stabilité de l’équilibre des figures piriformes affectées par une masse fluide en rotation. Philosophical Transactions of the Royal Society A 198, pp. 333–373. External Links: Link