## 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

$\displaystyle\frac{1}{5}R_{3}S_{3}$ | $\displaystyle=.4933=\;\text{my}\;\mathfrak{P}_{2}\mathfrak{Q}_{2}\quad\text{(% second zonal)}$ | |||

$\displaystyle\frac{1}{3}R_{2}S_{2}=\frac{1}{7}R_{5}S_{5}$ | $\displaystyle=.3517$ | |||

$\displaystyle[\text{or}\;\mathbf{P}^{1}_{1}\mathbf{Q}^{1}_{1}=\mathfrak{P}_{3}% \mathfrak{Q}_{3}$ | $\displaystyle=.3517]$ | |||

Also | ||||

$\displaystyle\frac{1}{5}R_{4}S_{4}$ | $\displaystyle=.2153=\;\text{my}\;\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}\quad% \text{(second sectorial)}$ |

(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

$\overline{U}=-\frac{1}{2}\int\frac{dm_{1}dm_{2}}{D_{12}}+\frac{1}{2}A\omega^{2}$ |

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

$y_{0}-\frac{Q_{3}y_{3}}{2G_{3}}-\frac{Q_{4}y_{4}}{2G_{4}}$ |

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 $\mathfrak{P}_{2}\mathfrak{Q}_{2}$,
$\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}$, since I have computed from the
rigorous formulæ and entirely independently from the approximate
formulæ of my paper on “Harmonics”.^{2}^{2}endnote:
^{2}
Darwin
1902, 488. The two values
of $\mathfrak{P}_{2}\mathfrak{Q}_{2}$ agree within about 1 percent, and of
$\mathfrak{P}^{2}_{2}\mathfrak{Q}^{2}_{2}$ 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}endnote:
^{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: " 4.05.2019 00:12"

## References

- Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: endnote 2.
- Œuvres d’Henri Poincaré, Volume 7. Gauthier-Villars, Paris. link1 Cited by: endnote 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. link1 Cited by: endnote 1.