## 3-15-11. George Howard Darwin to H. Poincaré

May 28.01

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I think it will interest you to hear that I have determined the
limiting stability of the Jacobian ellipsoid.^{1}^{1}
1
One of
Poincaré’s major results concerning the stability of equilibrium figures
is the proof of the fact, asserted but not rigorously proved by
Lord Kelvin (1879, § 778" g), that
beyond an upper limit of the angular momentum, Jacobian ellipsoids
cease to be stable
(Poincaré 1885, 373).
Poincaré proved that
there is an upper limit but gave no figures for the
critical Jacobian ellipsoid (which is actually the first point of
bifurcation of the series).
Darwin gave numerical results concerning the bifurcation of the Jacobi
series from the Maclaurin series as early as 1886
(Darwin 1886);
these earlier numerical results are taken up in his later paper
(Darwin 1902b, 326).
Darwin cited Kelvin along with Poincaré’s
Acta paper
(Poincaré 1885), but addressed stability
only briefly, without determining the limiting
stability
(Darwin 1886, 328).
It occurs
when $\frac{{\omega}^{2}}{2\pi \rho}=.14205$.
The axes of the ellipsoid
are^{2}^{2}
2
In the formula, $\omega $ denotes angular velocity
(rotation around $c$) and $\rho $ the density of the body.
In the final publication of these results, Darwin gives slightly
different values (translated in the present notation):
$\frac{{\omega}^{2}}{2\pi \rho}=0.14200$, $a=1.885827$, $b=0.814975$,
$c=0.650659$
(Darwin 1902b, 325).
For comparison, Chandrasekhar found
$\frac{{\omega}^{2}}{2\pi \rho}=0.142015$, $a=1.885641$,
$b=0.815034$, $c=0.650676$
(Chandrasekhar 1969, 110).
Darwin was apparently unaware that Liapunov had shown
$$ in 1884; see Liapunov
(1904, 101) and
Liapunov to Poincaré, 12.11.1886
(§ 3-32-3).

$a$ | $=1.8853$ | ||

$b$ | $=\mathrm{\hspace{0.33em}\hspace{0.17em}.8152}$ | ||

$c$ | $=\mathrm{\hspace{0.33em}\hspace{0.17em}.6507}$ |

— the scale being chosen so that $abc=1$. I enclose a rough drawing
of the section through $a$ and $c$, together with your figure — the
ellipsoid being dotted & the pear being in firm line.^{3}^{3}
3
The
original drawings have not been located, but were likely those
reproduced in Poincaré’s and
Darwin’s papers (Figs. 1, 2).
Fig. 1, Poincaré (1885, 347)
Fig. 2, Darwin (1902b, 329)
Darwin (1902b, 328) stressed that the figure is
considerably longer than Poincaré had supposed. Schwarzschild was
struck by this result, as he wrote to Darwin “Ihr numerisches Resultat
verändert die Anschauung, die man bisher nach Poincaré’s Skizze hatte,
doch sehr wesentlich” (Schwarzschild to Darwin, 22.04.1902,
§ 3-48-6). Schwarzschild
inserted a figure resembling that of Poincaré in his thesis
(Schwarzschild 1896, 46).

You probably know that Schwarzschild (Ann. of Munich
Ob. vol. III) expressed a doubt as to your conclusion as to the
exchange of stabilities in this case.^{4}^{4}
4
Poincaré had thought it
possible to prove the stability of the pear-shaped figure by an
application of the principle of exchange of stabilities (Poincaré
1885, 377–378). Schwarzschild contested Poincaré’s
analysis (Schwarzschild 1898, 275), observing that if
Poincaré’s figure is rotated 180° about the $z$ axis
(the axis of rotation in Schwarzschild’s setting; take Poincaré’s
original illustration as a section along the $(x,z)$-plane), one
obtains a figure with the same angular momentum which must be
counted as a new figure of equilibrium, since the original figure is
not symmetrical with respect to the $(y,z)$-plane. The coordinate
system rotates with the body, such that a fixed point of the surface
of the critical Jacobian is subject to different displacements in
the two cases. Consequently, there are two branches of pear-shaped
figures starting at the point of bifurcation and extending to the
same angular momentum value. Furthermore, one of the two branches of
the series of Jacobi ellipsoids extends to the same angular momentum
value, such that the angular momentum of the pear-shaped figures
determines the stable branch of the Jacobi ellipsoids (i.e, the one
with angular momentum lower than that of the critical Jacobian).
However, Schwarzschild points out first that the principle of
exchange of stabilities applies systematically whenever there are
precisely two branches on each side of the point of bifurcation
extending to the same angular momentum value (p. 38); and secondly,
that the principle applies if and only if a particular coefficient
vanishes (p. 40f). The pear-shaped figures belong
to the second situation, where the supplementary coefficient is to
be studied. Schwarzschild suggested a method involving difficult
calculations (p. 45).
Poincaré adopted the viewpoint of Schwarzschild (for whom the
stability of the pear-shaped figure is a function of angular
momentum) in his response to Darwin
(§ 3-15-12) and in his article
(Poincaré 1902, 333). The reason he assigns (p. 45)
is that your figures appear in pairs & that if the figure is turned
round it does not reproduce itself. Now it seems to me that he is
quite wrong & that it does reproduce itself in the only sense which
is material. In my notation (wh[ich] will I think be intelligible to you)
the normal displacement which gives the figure is
$\epsilon {\U0001d513}_{3}(\mu ){\U0001d5a2}_{3}(\phi )$ (a zonal surface harmonic of
the third order). The investigation of the coefficient of stability
shows that it is equally justifiable to take $\epsilon $ $+$ or $-$,
and the change of sign of $\epsilon $ produces the same result as
the turning of the figure about the axis of rotation thro’
180°.^{5}^{5}
5
Poincaré
(1885) observed that
the first point of bifurcation of the Jacobi series occurs when the
stability coefficient corresponding to the third zonal harmonic
vanishes. Darwin
gave the following intuitive explanation of the fact that harmonics
of third order are the
first harmonics relevant to the problem of the pear-shaped figure:
An harmonic of the first order merely denotes a shift of the centre
of inertia along one of the three axes; one of the second order
denotes a change of ellipticity of the ellipsoid. Since we must keep
the centre of inertia at the origin, and since the ellipticity is
determined by the consideration that the ellipsoid is a Jacobian,
these harmonics need not be considered, and we may begin with those
of the third order.
(Darwin 1902b, 320)
Here the symbol ${\U0001d5a2}_{3}$ differs
from the coefficient of stability ${C}_{3}$ encountered in Poincaré
(1885).
In Darwin’s papers, one finds analogous
expressions for the normal displacement $\delta n$ (which will also be
denoted $\zeta $) for an arbitrary harmonic
(Darwin 1902a, 508;
1902b, 319).
In the case of the third zonal harmonic, Darwin evaluates the
expression by using formulæ for the evaluation of the functions
involved, arriving at an expression involving only the cartesian
coordinates $x$, $y$, $z$ of the point and the long axis $c$ of the
critical Jacobian. The role of the quantity $\epsilon $ (which is
designated $e$ in the sequel of the correspondence and in Darwin’s paper)
is explained by Darwin as follows:
The expression has been arranged so that when $x=y=0$, $z=c$, we
have $\delta n=e$. Hence $+e$ and $-e$ are the normal displacements
at the stalk and blunt end of the pear respectively. (Darwin
1902b, 328)

Thus as I understand it Schwarzschild’s argument breaks
down.^{6}^{6}
6
A rotation of 180° corresponds to a sign change
in the term for the normal displacement, but a fixed point on the
surface of the critical Jacobian is subject to different
displacements in the two cases, such that the flipped figure is a
new figure of equilibrium. I should be very glad if you could let
me hear what you think of this. I have several other interesting
computations to make & may perhaps come across some other points of
interest.

I remain, Yours very sincerely

G. H. Darwin

ALS 2p. Collection particulière, Paris 75017.

Time-stamp: " 7.05.2016 18:29"

## References

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- Ellipsoidal harmonic analysis. Philosophical Transactions of the Royal Society A 197, pp. 461–557. Cited by: footnote 5.
- On the pear-shaped figure of equilibrium of a rotating mass of liquid. Philosophical Transactions of the Royal Society A 198, pp. 301–331. Cited by: footnote 1, footnote 2, footnote 3, footnote 5.
- Sur la stabilité des figures ellipsoïdales d’équilibre d’un liquide animé d’un mouvement de rotation. Annales de la faculté des sciences de Toulouse 6 (1), pp. 5–116. External Links: Link Cited by: footnote 2.
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