George Howard Darwin à Henri Poincaré, 15 mai 1902 · Lettre · La correspondance d'Henri Poincaré

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 max^m for all deformations except the second zonal, and a max^m 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)¹¹Poincaré 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”.²²Darwin 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.³³Karl 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.

References

G. H. Darwin (1902) Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: 3-15-43. George Howard Darwin to H. Poincaré.
J. R. Lévy (Ed.) (1952) Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. External Links: Link Cited by: 3-15-43. George Howard Darwin to H. Poincaré.
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: 3-15-43. George Howard Darwin to H. Poincaré.

La correspondance d'Henri Poincaré

LettreGeorge Howard Darwin à Henri Poincaré, 15 mai 1902

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

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