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

20.3.99

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I think you will be interested to learn that I have at last discovered (as I believe with confidence) the natural history, so to speak, of the $A$ orbits & the figure-of-8 which I will call $A^{\prime}$.11 1 Following up on a previous letter to Poincaré of 28.02.1899 (§ 3-15-1), Darwin returns here to his classification of periodic orbits (1897), which Poincaré considered in the third volume of the Méthodes nouvelles de la mécanique céleste (Poincaré 1899, 352). Somewhere between $C=40$ & $39.6$ an orbit of ejection which is periodic arises of this form:

As $C$ falls in value the loop diminishes becomes a cusp & then becomes rounded & coalesces with $A$ when both vanish.

For a very slightly smaller value of $C$ another orbit of ejection arises of this form:

In this case also the loop diminishes, becomes a cusp, then rounded & we pass on to the figures-of-8 $A^{\prime}$ which I have traced.

I missed these families because they have so transitory an existence. I have my idea of how they will behave as to stability.

Now pass to the side of $J$ remote from $S$ & it is obvious that the $C$ orbit ends in an orbit of ejection thus:

For a very slightly smaller value of $C$ a new orbit of ejection arises thus:

(I should have drawn the other half of these.)

This orbit is obviously the parent of a new series of figures-of-8 thus

The quasi-coalescence & disappearance of the figure-of-eight $A$ (or say $A^{\prime}$) with $B$ may take place through the figure-of-8 terminating in an orbit of ejection, & $B$ with another. But I have not yet the materials for deciding this point. These ideas throw a flood of light on the march of the orbits, but I will not trouble you with more.

I think I shall ask Mittag Leffler to let me have another paper in the Acta but I fear it will be some months before I shall be ready.22 2 Darwin published his thoughts only a decade later (Darwin 1909). A few years after Darwin sent this letter to Poincaré, Hough (1901) found a family of figure-of-eight orbits that Darwin had missed (Barrow-Green 1997, 196).

I remain, Yours sincerely

G. H. Darwin

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "14.01.2016 03:44"

## References

• J. E. Barrow-Green (1997) Poincaré and the Three Body Problem. AMS/LMS, Providence. Cited by: footnote 2.
• G. H. Darwin (1897) Periodic orbits. Acta mathematica 21, pp. 101–242. External Links: Link Cited by: footnote 1.
• G. H. Darwin (1909) On certain families of periodic orbits. Monthly Notices of the Royal Astronomical Society 70, pp. 108–143. External Links: Link Cited by: footnote 2.
• S. S. Hough (1901) On certain discontinuities connected with periodic orbits. Acta mathematica 24, pp. 257–288. External Links: Link Cited by: footnote 2.
• H. Poincaré (1899) Les méthodes nouvelles de la mécanique céleste, Volume 3. Gauthier-Villars, Paris. External Links: Link Cited by: footnote 1.