3-15-15. George Howard Darwin to H. Poincaré
June 21.01
Newnham Grange–Cambridge
Dear Monsieur Poincaré,
I do not see how it is possible to solve the question without carrying the investigation through completely^{1}^{1}endnote: ^{1} Variant: “almost completely”. to the second order of small quantities. I had come to this conclusion before hearing from you,^{2}^{2}endnote: ^{2} See Poincaré to Darwin, § 3-15-14. but when you said that my figure afforded the means of deciding the question of increase of momentum I thought you must be right, and besides forgot my previous reflections on the subject.^{3}^{3}endnote: ^{3} See Poincaré to Darwin, § 3-15-12.
As it is possible to carry the theory of the figure of the earth to the second order of small quantities by means of spherical harmonics, so I think it is possible to proceed to the second order here with ellipsoidal harmonics. I believe I see how it is to be done numerically, proceeding from a specified Jacobian ellipsoid; but I do not see much prospect of doing it analytically. But the arithmetic would be very heavy, since it would involve the determination of a considerable number of integrals by quadratures – integrals theoretically perhaps expressible in Elliptic integrals, but with coefficients of stupendous complication. The fundamental difficulty is thus. Suppose we have an inequality on the ellipsoid represented by (curvilinear normal)
$\delta n=p.\varepsilon\mathfrak{P}_{3}(\mu)\mathsf{C}_{3}(\phi)\quad\left[3^{% \text{rd}}\text{ zonal}\right]$ |
($p\perp$ on tangent plane); then we have to find the mass standing on unit area of the ellipsoid as far as squares of $\varepsilon$.
I have obtained the expression in question. The term in $\varepsilon$ is obvious by ellipsoidal analysis – that in $\varepsilon^{2}$ depends on $p[\mathfrak{P}_{3}(\mu)\mathsf{C}_{3}(\phi)]^{2}$ multiplied by a complicated function of the two coordinates $\mu$, $\phi$.
In order to proceed further, this function must be expressed in ellipsoidal harmonics and it will involve harmonics of orders 0, 2, 4, 6…and of even orders of tesserality.^{4}^{4}endnote: ^{4} Darwin (1903, 253) linked the even-order harmonics to the figure’s symmetry.
I believe that by quadratures & a sufficient degree of labour I can get out those harmonics – but whether or not I have the patience to do so, is another question.
This result shows that we must start by supposing that the figure of equilibrium consists not only of a 3^{rd} harmonic of amplitude $\varepsilon$ (whose square is retained), but also of harmonics of orders 0, 2, 4, 6…& of amplitudes unknown save that they are of the order $\varepsilon^{2}$.^{5}^{5}endnote: ^{5} Variant: “save that it is of the second order $\varepsilon^{2}$.” These new harmonics will affect the moment of inertia.
The harmonic 0 is what Kelvin calls a focaloid shell & indicates that the elevation of the 3^{rd} zonal must be computed from a surface confocal to the primitive ellipsoid – otherwise the mass of the pear will not be equal to that of the ellipsoid.^{6}^{6}endnote: ^{6} The focaloid was defined by Thomson & Tait (1883, §494 g) as an infinitely-thin shell bounded by two concentric similar ellipsoidal surfaces; see also the generalization proposed by Routh (1902, 97). Thomson and Tait used the focaloid to prove MacLaurin’s theorem, whereby any two confocal homogeneous solid ellipsoids of equal mass give rise to equal attraction. Clairaut’s theory of heterogeneous masses also employed such shells; see the notes to Darwin to Poincaré, 31.07.1901 (§ 3-15-17). This harmonic then enters in the moment of inertia, as also does that of order 2; the remaining harmonics do not do so.^{7}^{7}endnote: ^{7} Variant: “do not do so. Hence if I worked out these two only I could decide the question of the moment of momentum with”.^{8}^{8}endnote: ^{8} Cf. Poincaré to Darwin (§ 3-15-19), where Poincaré shows that $\xi_{3}$ (the coefficient of $R_{2,0}^{1}=\mathfrak{P}_{2}$ in the notation of Poincaré (1902), here $\eta$) and $\xi_{4}$ (the coefficient of $R_{2,2}^{1}=\mathfrak{P}_{2}^{2}$ in Poincaré’s notation, here $\zeta$) enter the computation with the second order while the third zonal enters with the first order.
If I assume these difficulties overcome, then I think (also with sufficient labour) I can overcome the remainder.
In order to find the gravitational terms I should proceed thus:
Draw a concentric and similar ellipsoid touching the stalk of the pear. It is a Jacobian ellipsoid, but its mass depends on the determination of the exact figure of the pear. Nevertheless I can find the expression for its internal gravity although that expression involves $\varepsilon$.^{9}^{9}endnote: ^{9} On Darwin’s solution see Darwin to Poincaré, 12.08.1901 (§ 3-15-23).
Consider the shaded pear to consist of negative matter; then by quadratures I can find the energy due to the gravitation of the larger ellipsoid and this shaded matter.
The mutual energy of the layer as far as it depends on the harmonics 0, 2, 4, 6 …is determinable at once by harmonic analysis. That corresponding to the 3^{rd} zonal is to the first order the same as tho’ it were concentrated on the surface. To go to the second order we must first concentrate in surface density on ellipsoid and then deform that surface density – so as to bring it to stand half way between the ellipsoid & the true surface.^{10}^{10}endnote: ^{10} This is the double layer strategy used later by Poincaré; see Poincaré to Darwin (§ 3-15-25). This last may be done thus: If we have a layer of surface density with potential $v_{(+)}$ externally and potential $v_{(-)}$ internally, the force acting on an element $d\sigma$ is^{11}^{11}endnote: ^{11} The second factor involves Poisson’s equation (Darwin 1901, 507); the last fraction has been corrected from $dv_{(-1)}/dn$.
$-\frac{1}{2}\left(\frac{dv_{(+)}}{dn}+\frac{dv_{(-)}}{dn}\right)\cdot\frac{1}{% 4\pi}\left(\frac{dv_{(+)}}{dn}-\frac{dv_{(-)}}{dn}\right)\cdot d\sigma.$ |
If the element is carried outwards through a distance $\frac{1}{2}\delta n$, the work done is
$\frac{1}{16\pi}\left[\left(\frac{dv_{(+)}}{dn}\right)^{2}-\left(\frac{dv_{(-)}% }{dn}\right)^{2}\right]\delta n\cdot d\sigma$ |
Integrating all over the surface one may obtain the whole work done in deformation.
In this sketch of procedure I have left a good deal vague, but I think the processes involved are legitimate.
I do not see that any analytical skill could make the processes necessary other than very laborious – unless indeed it were possible to obtain expressions for harmonic functions applicable to the pear-shaped figure.
I am afraid this letter is very ill expressed, but I hope it is intelligible and that you will not detect faults of principle.
I remain, Yours sincerely
G.H. Darwin
ALS 8p. Collection particulière, Paris 75017.
Time-stamp: "20.07.2021 18:13"
Notes
- ^{1} Variant: “almost completely”.
- ^{2} See Poincaré to Darwin, § 3-15-14.
- ^{3} See Poincaré to Darwin, § 3-15-12.
- ^{4} Darwin (1903, 253) linked the even-order harmonics to the figure’s symmetry.
- ^{5} Variant: “save that it is of the second order $\varepsilon^{2}$.”
- ^{6} The focaloid was defined by Thomson & Tait (1883, §494 g) as an infinitely-thin shell bounded by two concentric similar ellipsoidal surfaces; see also the generalization proposed by Routh (1902, 97). Thomson and Tait used the focaloid to prove MacLaurin’s theorem, whereby any two confocal homogeneous solid ellipsoids of equal mass give rise to equal attraction. Clairaut’s theory of heterogeneous masses also employed such shells; see the notes to Darwin to Poincaré, 31.07.1901 (§ 3-15-17).
- ^{7} Variant: “do not do so. Hence if I worked out these two only I could decide the question of the moment of momentum with”.
- ^{8} Cf. Poincaré to Darwin (§ 3-15-19), where Poincaré shows that $\xi_{3}$ (the coefficient of $R_{2,0}^{1}=\mathfrak{P}_{2}$ in the notation of Poincaré (1902), here $\eta$) and $\xi_{4}$ (the coefficient of $R_{2,2}^{1}=\mathfrak{P}_{2}^{2}$ in Poincaré’s notation, here $\zeta$) enter the computation with the second order while the third zonal enters with the first order.
- ^{9} On Darwin’s solution see Darwin to Poincaré, 12.08.1901 (§ 3-15-23).
- ^{10} This is the double layer strategy used later by Poincaré; see Poincaré to Darwin (§ 3-15-25).
- ^{11} The second factor involves Poisson’s equation (Darwin 1901, 507); the last fraction has been corrected from $dv_{(-1)}/dn$.
References
- The pear-shaped figure of equilibrium of a rotating mass of liquid. Proceedings of the Royal Society of London 69, pp. 147–148. link1 Cited by: endnote 11.
- The approximate determination of the form of Maclaurin’s spheroid. Transactions of the American Mathematical Society 4 (2), pp. 113–133. Cited by: endnote 4.
- 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 8.
- A Treatise on Analytical Statics, Volume 2. Cambridge University Press, Cambridge. link1 Cited by: endnote 6.
- Treatise on Natural Philosophy, Volume 1, Part 2. Cambridge University Press, Cambridge. link1 Cited by: endnote 6.