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 =.4933=my𝔓2𝔔2(second zonal)
13R2S2=17R5S5 =.3517
Also
15R4S4 =.2153=my𝔓22𝔔22(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

13R2S2-15R3S3

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

U¯=-12dm1dm2D12+12Aω2

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¯ due to the moment of inertia is such as to outweigh the diminution due to the negative value of

13R2S2-15R3S3.

In other words

13R2S2-12n+1RnSn

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

y0-Q3y32G3-Q4y42G4

seem to me to show that I am correct, since I agree with them when I use these values of R2S2, R3S3.

I am sure that I am right in my values of 𝔓2𝔔2, 𝔓22𝔔22, 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 𝔓2𝔔2 agree within about 1 percent, and of 𝔓22𝔔22 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.

Time-stamp: "14.01.2016 01:49"

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