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

15R3S3\displaystyle\frac{1}{5}R_{3}S_{3} =.4933=my𝔓2𝔔2(second zonal)\displaystyle=.4933=\;\text{my}\;\mathfrak{P}_{2}\mathfrak{Q}_{2}\quad\text{(% second zonal)}
13R2S2=17R5S5\displaystyle\frac{1}{3}R_{2}S_{2}=\frac{1}{7}R_{5}S_{5} =.3517\displaystyle=.3517
[or𝐏11𝐐11=𝔓3𝔔3\displaystyle[\text{or}\;\mathbf{P}^{1}_{1}\mathbf{Q}^{1}_{1}=\mathfrak{P}_{3}% \mathfrak{Q}_{3} =.3517]\displaystyle=.3517]
15R4S4\displaystyle\frac{1}{5}R_{4}S_{4} =.2153=my𝔓22𝔔22(second sectorial)\displaystyle=.2153=\;\text{my}\;\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}\quad% \text{(second sectorial)}

(I use the RR, SS in the senses defined in foot note to Roy. Soc. paper p. 336.)

Thus for the second zonal


is negative. It follows that my function EE (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


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 U¯\overline{U} due to the moment of inertia is such as to outweigh the diminution due to the negative value of


In other words


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)11endnote: 1 Poincaré 1902, 362; Lévy 1952, 191. the signs in the expression


seem to me to show that I am correct, since I agree with them when I use these values of R2S2R_{2}S_{2}, R3S3R_{3}S_{3}.

I am sure that I am right in my values of 𝔓2𝔔2\mathfrak{P}_{2}\mathfrak{Q}_{2}, 𝔓22𝔔22\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”.22endnote: 2 Darwin 1902, 488. The two values of 𝔓2𝔔2\mathfrak{P}_{2}\mathfrak{Q}_{2} agree within about 1 percent, and of 𝔓22𝔔22\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.33endnote: 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 RR, SS 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: " 4.05.2019 00:12"


  • 1 Poincaré 1902, 362; Lévy 1952, 191.
  • 2 Darwin 1902, 488.
  • 3 Karl Weierstrass (1815–1897).


  • G. H. Darwin (1902) Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: endnote 2.
  • J. R. Lévy (Ed.) (1952) Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. link1 Cited by: endnote 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. link1 Cited by: endnote 1.