## 3-15-21. George Howard Darwin to H. Poincaré

Aug. 8.01

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I am glad you have found my letters worthy of attention.^{1}^{1}See
Poincaré to Darwin (§ 3-15-20), where
Poincaré acknowledges Darwin’s two letters to Poincaré, of
31.07.1901 (§ 3-15-17) and
03.08.1901 (§ 3-15-18). I feel more & more
confident of my correctness. There is [a] portion of your
letter which I
could not follow & therefore I am not clear of the physical meaning
of your ${h}^{\prime}$.^{2}^{2}See Poincaré to Darwin
(§ 3-15-19). I feel no doubt that I have included all the sources of
energy & moment of inertia and therefore I believe your ${h}^{\prime}$ is
included. The ${h}^{\prime}$ has its counterpart in the problem I have solved &
the result is correct in
that case. How can it be that I have
omitted [it] in the pear problem?

I find it very hard to look at the problem through your eyes, as I feel as urgent a necessity to continually keep the physical meaning before me, as you (I fancy) feel it a relief to dismiss it from your mind & betake yourself to analysis.

Nevertheless I have no doubt that you will arrive at the truth, & I feel pretty confident you will agree with me.

I find that it is unnecessary to regard the shift in the centre of inertia of the pear because the terms involved are of the fourth order.

I have gone a long way towards the determination of the mass of the layer – which corresponds to the determination of $a$ in terms in ${a}_{0}$ – and I have no words to describe the immensity of the task.

I have not made an exact estimate but I think there will be 20 or 30 integrals to evaluate by quadratures. The integrals are all of this type

$${\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{sin}}^{2n}\theta d\theta}{{(1-{k}^{2}{\mathrm{sin}}^{2}y{\mathrm{sin}}^{2}\theta )}^{p}\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}\theta}}$$ |

and there is another group of the type

$${\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{cos}}^{2n}\varphi d\varphi}{{(1+{k}^{\mathrm{\prime}2}{\mathrm{tan}}^{2}y{\mathrm{cos}}^{2}\varphi )}^{p}\sqrt{1-{k}^{\mathrm{\prime}2}{\mathrm{cos}}^{2}\varphi}}$$ |

These will have to be found for $n=0$, $1$, $2$, $3$, $4$ combined with $p=0$, $1$, $2$, $3$. It would seem that there are then 40 integrals of this kind.

There are others which I suspect will fall into the same type but I have not got so far yet. Have I patience for the task? I suppose I shall not feel contented until I have done it – and that is the only reason I have for thinking I shall persevere.

I shall be much interested to hear what you think, altho’ as I have said we look at this thing from different points of view. Shall I return your original letter.

Yours sincerely,

G. H. Darwin

ALS 4p. Collection particulière, Paris.

Time-stamp: "21.01.2016 01:47"