## 3-15-43. George Howard Darwin to H. Poincaré

May 15. 1902

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I am drawing very near to the end of the arithmetic of the ‘Pear’, and in the course of it a point has turned up on which I should be glad of confirmation.

If we refer to the critical Jacobian I find

 $\displaystyle\frac{1}{5}R_{3}S_{3}$ $\displaystyle=.4933=\;\text{my}\;\mathfrak{P}_{2}\mathfrak{Q}_{2}\quad\text{(% second zonal)}$ $\displaystyle\frac{1}{3}R_{2}S_{2}=\frac{1}{7}R_{5}S_{5}$ $\displaystyle=.3517$ Also $\displaystyle\frac{1}{5}R_{4}S_{4}$ $\displaystyle=.2153=\;\text{my}\;\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}\quad% \text{(second sectorial)}$

(I use the $R$, $S$ in the senses defined in foot note to Roy. Soc. paper p. 336.)

Thus for the second zonal

 $\frac{1}{3}R_{2}S_{2}-\frac{1}{5}R_{3}S_{3}$

is negative. It follows that my function $E$ (see Pear-shaped Figure) is a minimax being a maxm for all deformations except the second zonal, and a maxm for the second zonal.

I have however verified that the function

 $\overline{U}=-\frac{1}{2}\int\frac{dm_{1}dm_{2}}{D_{12}}+\frac{1}{2}A\omega^{2}$

is an absolute minimum, for it is certainly a minimum for all deformations except the second zonal – moment of momentum being kept constant—and for the second zonal the increment of $\overline{U}$ due to the moment of inertia is such as to outweigh the diminution due to the negative value of

 $\frac{1}{3}R_{2}S_{2}-\frac{1}{5}R_{3}S_{3}.$

In other words

 $\frac{1}{3}R_{2}S_{2}-\frac{1}{2n+1}R_{n}S_{n}$

is not the complete coefficient of stability for deformations of the second order.

I do not see this point referred to explicitly in your papers, but in the Royal Society paper (p. 362)11 1 Poincaré 1902, 362; Lévy 1952, 191. the signs in the expression

 $y_{0}-\frac{Q_{3}y_{3}}{2G_{3}}-\frac{Q_{4}y_{4}}{2G_{4}}$

seem to me to show that I am correct, since I agree with them when I use these values of $R_{2}S_{2}$, $R_{3}S_{3}$.

I am sure that I am right in my values of $\mathfrak{P}_{2}\mathfrak{Q}_{2}$, $\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}$, since I have computed from the rigorous formulæ and entirely independently from the approximate formulæ of my paper on “Harmonics”.22 2 Darwin 1902, 488. The two values of $\mathfrak{P}_{2}\mathfrak{Q}_{2}$ agree within about 1 percent, and of $\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}$ within about 3 percent.

The great trouble I have had is that my formulæ for the integrals tend to give the results as the differences between two very large numbers. I suspect that the same difficulty would occur in your more elegant treatment – for I think that I have arrived at nearly the same way of splitting the integrals into elementary integrals. I do not understand Weierstrasse’s method enough to trust myself in using it.33 3 Karl Weierstrass (1815–1897).

I hope this letter will not give you much trouble.

I remain, Yours Sincerely,

G. H. Darwin

P. S. On looking back I am not sure whether I have used the suffixes to your $R$, $S$ in the same sense as you do, but I think you will understand my point. I use notation of Roy. Soc. paper and not of the Acta.

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "14.01.2016 01:49"

## References

• G. H. Darwin (1902) Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: footnote 2.
• J. R. Lévy (Ed.) (1952) Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. External Links: Link Cited by: footnote 1.
• 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 Cited by: footnote 1.