3-15-30. George Howard Darwin to H. Poincaré

Sep. 1. 01

Newnham Grange – Cambridge

Dear Monsieur Poincaré,

In order to save you trouble I hasten to make confession.¹¹endnote: ¹ Voir (§ 3-15-29). My work is right but insufficient. It is necessary to include 4^th powers of $e$ so that you are right. I see this in the following way. If I change the sign of $e$ , I change the stalk into the blunt end of the pear & vice versa. This cannot change the value of $E$ and hence $e^{3}$ must be absent. I found this to be the case with the transition from the Maclaurin to the Jacobian Ellipsoid, but I was delayed by (my usual) small algebraic mistake which prevented the $e^{3}$ term from vanishing. In this case the change of sign of $e$ changes the azimuth of the Jacobian and the same result follows. But in the case of the sphere and Maclaurin the change of sign of $e$ changes oblate into prolate ellipsoid and therefore $E$ is not the same in the two cases.

I must admit that at present I do not see how to calculate the term in $e^{4}$ in my way. Your method will probably indicate it to me. But the calculation of the $e^{3}$ term was so troublesome that I suppose the $e^{4}$ term will be enormously worse. I do not think I can attempt it now.

I had written thus far when I received your letter in which you point out the same thing. I am sorry to have given you so much trouble.

I shall be much interested to see what you write.

I remain, Yours very sincerely,

G. H. Darwin

ALS 3p. Collection particulière, Paris 75017.

Time-stamp: " 4.05.2019 00:12"

Notes

1 Voir (§ 3-15-29).