3-15-42. George Howard Darwin to H. Poincaré

Jan. 14.02

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I have made a preliminary correction of your proof sheets & have removed at least the worst of the misprints. A revise will be sent to you soon together with your M.S. I venture to draw your attention to one point in the treatment of the “double layer,” because it puzzled me. Your standard is where the layer is external to the ellipsoid and you take

$$\frac{d\sigma}{d\sigma^{\prime}}-1=k\ell.$$

That is to say that if $k$ were positive and $\ell$ positive and $d\sigma^{\prime}$ outside $d\sigma^{\prime}$ would be less than $d\sigma$. But it is clear that $d\sigma^{\prime}$ must be greater than $d\sigma$ and so $k$ is negative if $\ell$ is measured outwards. In order to be sure that I understood your formula I worked out independently the case of a sphere of radius $a$ charged with surface density $-\delta$, with a stratum superposed extending to radius $a(1+e)$ of density $\rho$. $\delta$ is connected with $\rho$ by the equation

$$\delta=\frac{1}{3}\rho a[(1+e)^{3}-1].$$

The formula in your paper gives the right result if $k$ is $-2$, which is just what I thought it should be.

Is it worth while to change the sign of $k$ all through to prevent a similar doubt in others.

I have gone someway in my own working of the question, which of course follows yours pretty nearly, but not quite. I have got out two of the integrals, but there are many more. I am afraid I must suspend work at it for the present.

I should much like some copies of your paper now in the press. The Society gives you 100 separate copies – is it possible that you will be so generous as to spare some of them for me.

I was very glad to hear of the award to you of a medal by the R.S. I was unfortunately unable to come to the meeting & dinner.

I remain, Yours sincerely,

G. H. Darwin

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: " 4.05.2019 00:12"

Notes

• 1 The proofs are those of Poincaré (1902).
• 2 Poincaré was awarded the first Sylvester Medal, on the history of which see G. Cantor (2004) and I. Grattan-Guinness (1993).