## 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

$\frac{1}{5}}{R}_{3}{S}_{3$ | $=.4933=\text{my}{\U0001d513}_{2}{\U0001d514}_{2}\mathit{\hspace{1em}}\text{(second zonal)}$ | |||

$\frac{1}{3}}{R}_{2}{S}_{2}={\displaystyle \frac{1}{7}}{R}_{5}{S}_{5$ | $=.3517$ | |||

Also | ||||

$\frac{1}{5}}{R}_{4}{S}_{4$ | $=.2153=\text{my}{\U0001d513}_{2}^{2}{\U0001d514}_{2}^{2}\mathit{\hspace{1em}}\text{(second sectorial)}$ |

(I use the $R$, $S$ in the senses defined in foot note to Roy. Soc. paper p. 336.)

Thus for the second zonal

$$\frac{1}{3}{R}_{2}{S}_{2}-\frac{1}{5}{R}_{3}{S}_{3}$$ |

is negative.
It follows that my function $E$ (see Pear-shaped Figure) is a minimax being a max^{m}
for all deformations except the second zonal, and a max^{m} for the
second zonal.

I have however verified that the function

$$\overline{U}=-\frac{1}{2}\int \frac{d{m}_{1}d{m}_{2}}{{D}_{12}}+\frac{1}{2}A{\omega}^{2}$$ |

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

$$\frac{1}{3}{R}_{2}{S}_{2}-\frac{1}{5}{R}_{3}{S}_{3}.$$ |

In other words

$$\frac{1}{3}{R}_{2}{S}_{2}-\frac{1}{2n+1}{R}_{n}{S}_{n}$$ |

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)^{1}^{1}
1
Poincaré 1902, 362;
Lévy 1952, 191.
the signs in the expression

$${y}_{0}-\frac{{Q}_{3}{y}_{3}}{2{G}_{3}}-\frac{{Q}_{4}{y}_{4}}{2{G}_{4}}$$ |

seem to me to show that I am correct, since I agree with them when I use these values of ${R}_{2}{S}_{2}$, ${R}_{3}{S}_{3}$.

I am sure that I am right in my values of ${\U0001d513}_{2}{\U0001d514}_{2}$,
${\U0001d513}_{2}^{2}{\U0001d514}_{2}^{2}$, since I have computed from the
rigorous formulæ and entirely independently from the approximate
formulæ of my paper on “Harmonics”.^{2}^{2}
2
Darwin
1902, 488. The two values
of ${\U0001d513}_{2}{\U0001d514}_{2}$ agree within about 1 percent, and of
${\U0001d513}_{2}^{2}{\U0001d514}_{2}^{2}$ 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.^{3}^{3}
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 $R$, $S$ 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"

## References

- Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: footnote 2.
- Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. External Links: Link Cited by: footnote 1.
- 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. External Links: Link Cited by: footnote 1.