3-15. George Howard Darwin
George Howard Darwin (1845–1912) was the fifth child of the British naturalist Charles Darwin. After graduating from Trinity College, Cambridge in 1868, Darwin studied law for six years, before returning to Cambridge, and publishing work in geophysics. In 1879, Darwin was elected fellow of the Royal Society, and became its president in 1900. In 1883, he was appointed Professor of Astronomy and Experimental Physics at Cambridge, succeeding James Challis. He was elected corresponding member of the Paris Academy of Sciences on 23 December, 1907, in the geography and navigation section (Académie des sciences de Paris, 1979).
Darwin’s contribution to cosmogony is original in that he put various hypotheses to the test of numerical calculation, in contrast to the qualitative arguments employed by Poincaré and others in the field.^{1}^{1}endnote: ^{1} Early appreciations of Darwin’s work include Love (1913) and Hough (1914). Later evaluations include Kopal (1971) and Barrow-Green (1997, 193).
The surviving Poincaré-Darwin correspondence is limited in scope, but substantial in one of its chosen subjects. The earliest letters concern the preparation of a volume in honor of George G. Stokes, and honors awarded Poincaré by the Royal Astronomical Society and the Royal Society, along with the classification of periodic orbits. The core topic of the Poincaré-Darwin correspondence, beginning with a letter from Darwin on 28 May, 1901 (§ 3-15-11) is the stability of the pear-shaped figure of equilibrium of a rotating fluid mass. The exchange generated much correspondence over the next twelve months, of which thirty-four letters have been preserved and are published here, including eighteen letters from Poincaré and sixteen from Darwin.
In what follows we provide some background information on the mathematical problem and on the notation introduced by Poincaré, Darwin and others.
Equilibrium figures of rotating liquid masses before 1885
The historical development of the theory of equilibrium figures of rotation is described in detail by Todhunter (1873a; 1873b), Oppenheim (1922), Chandrasekhar (1969), Lützen (1984), and Lichtenstein (1987). The present account concerns only the case of a homogeneous fluid and ellipsoidal figures of equilibrium, neglecting heterogeneous fluids, rigid kernels and annular figures.
The study of equilibrium figures began with Newton’s investigation of the figure of the earth (Newton, 1687, III, props. XVIII–XX). Generalizing Newton’s results to the case when the ellipticity caused by the rotation is large, Colin Maclaurin (1742) obtained a series of new equilibrium figures having the shape of oblate spheroids. For a long time, it was believed that these were the only possible equilibrium figures, and Lagrange attempted to prove this assertion in his Mécanique analytique (1811). However, Jacobi (1834) showed through an analysis of Lagrange’s argument that there is an additional series of equilibrium figures consisting of ellipsoids with three unequal axes (the shortest axis being the axis of rotation). Jacobi’s student C. O. Meyer (1842) showed that the Jacobi series bifurcates (as Poincaré later would say) from the Maclaurin series for an eccentricity of $0.81267$. Partly motivated by Jacobi’s result which he and other French mathematicians interpreted as a challenge, Liouville began to work on the topic around the same time; in particular, he developed a certain kind of function, introduced by Lamé in a different context, into a powerful tool for the investigation of equilibrium figures. Liouville also stressed (as Laplace had pointed out in the Mécanique Céleste livre 3, § 21) that the angular momentum and not the angular velocity is the relevant parameter for series of equilibrium figures. However, since angular velocity decreases monotonously in the case of the Jacobi series, it is reasonably taken as a parameter in this case.
The main tool used by Poincaré and Darwin in their investigations of figures of equilibrium of rotating fluids are ellipsoidal harmonic functions (or ellipsoidal harmonics, for short), also known as Lamé functions. The expressions for all relevant physical magnitudes are developed in series of these functions. This approach was developed by Lamé and Liouville, in analogy to the theory of spherical harmonics introduced around the end of the eighteenth century by Legendre, Laplace and others. We will follow the systematic presentation of Hobson (1931), although Darwin most likely relied on the presentation of spherical harmonics in Thomson and Tait (1879, 171–218).
A harmonic function of three variables is a solution $V(x,y,z)$ of Laplace’s equation
$\frac{d^{2}V}{dx^{2}}+\frac{d^{2}V}{dy^{2}}+\frac{d^{2}V}{dz^{2}}=0,$ |
which is the basic equation of classical potential theory. Solutions of this equation appropriate for spaces with boundaries of various forms (eg., spheres, spheroids, and ellipsoids) are often sought after. This is done by transforming the equation into a form in which the independent variables are the parameters $h_{1}$, $h_{2}$, $h_{3}$ of three orthogonal sets of surfaces. The values of $h_{1}$, $h_{2}$, $h_{3}$ at any point $(x,y,z)$ may be regarded as curvilinear coordinates at that point. Normal solutions to Laplace’s equation are then sought, i.e., solutions in the form
$V=\phi(h_{1})\psi(h_{2})\chi(h_{3})$ |
where $\phi$ denotes a function of $h_{1}$ only, $\psi$ of $h_{2}$ only, and $\chi$ of $h_{3}$ only.
In Cartesian coordinates, the triply orthogonal set of surfaces consists of the planes parallel to the coordinate planes, where $h_{1}$, $h_{2}$, $h_{3}$ are $x$, $y$, $z$, respectively. We will consider the case of spherical harmonics, a case which in itself is only of minor importance for the correspondence but gives the non-expert reader a good general idea of how the conceptually more complicated ellipsoidal harmonic functions are used.
In this case, $h_{1}$, $h_{2}$, $h_{3}$ are $r$, $\theta$, $\phi$ respectively, the polar coordinates of a point in 3-space, and the triply orthogonal set of surfaces consists of concentric spheres ($r=\text{const.}$), coaxial cones ($\theta=\text{const.}$), and planes through the axis ($\phi=\text{const.}$). It is easily shown (eg., by Hobson 1931, 10) that the normal solutions of Laplace’s equation, transformed in these coordinates, are of the form
$\begin{matrix}r^{n}\cdot u\cdot\begin{matrix}\cos\\ \sin\end{matrix}m\phi,\quad&r^{-n-1}\cdot u\cdot\begin{matrix}\cos\\ \sin\end{matrix}m\phi.\end{matrix}$ |
Here, $u=u_{n}^{m}$ is a solution to the following equation
$\frac{d}{d\mu}\left((1-\mu^{2})\frac{du}{d\mu}\right)+\left(n(n+1)-\frac{m^{2}% }{1-\mu^{2}}\right)u=0$ | (1) |
where $\mu=\cos\theta$, and $n,m$ are arbitrary constants (usually positive integers).^{2}^{2}endnote: ^{2} In the expression $u_{n}^{m}$, $m$ is an upper index, and not an exponent. This is standard notation for theories of spherical and ellipsoidal harmonics.
A special case of particular interest is where $m=0$; the resulting equation is called Legendre’s equation. The complete solution of this equation is
$u=AP_{n}(\mu)+BQ_{n}(\mu)$ |
where $A$ and $B$ are arbitrary constants, $P_{n}$ a polynomial of degree $n$ (called Legendre’s polynomial), and $Q_{n}$ another function of a certain type.
$P_{n}$ can be expressed as
$\frac{1\cdot 3\cdot 5\ldots(2n-1)}{n!}(\mu^{2}-\alpha_{1}^{2})(\mu^{2}-\alpha_% {2}^{2})\ldots(\mu^{2}-\alpha_{\frac{n}{2}}^{2})$ |
for $n$ even and in a similar fashion (involving a factor $\mu$) for $n$ odd. Now we construct the surface of revolution with polar equation
$r=a+bP_{n}(\cos\theta);$ |
this surface will cut the sphere $r=a$ in the points for which $P_{n}(\cos\theta)$ vanishes. These points lie, according to the above expression for $P_{n}(\cos\theta)$, upon $n$ circles whose planes are perpendicular to the axis. This system of $n$ circles on the sphere is called the system of nodal lines of the function $P_{n}(\cos\theta)$. Since these lines divide the spherical surface into zones, $P_{n}(\mu)$ is called a zonal harmonic. The function $r^{n}P_{n}(\mu)$ is called a solid zonal harmonic of degree $n$; the function $P_{n}(\mu)$ is called a surface zonal harmonic (Hobson 1931, 20).
Corresponding to $P_{n}$ and $Q_{n}$ are solutions $P_{n}^{m}$ and $Q_{n}^{m}$ of (1) in its general form (i.e., for $m\neq 0$). These are given as follows (Hobson 1931, 89):
$P_{n}^{m}(\mu)=(-1)^{m}(1-\mu^{2})^{\frac{1}{2}m}\frac{d^{m}P_{n}(\mu)}{d\mu^{% m}}=\frac{(-1)^{m}}{2^{n}n!}(1-\mu^{2})^{\frac{1}{2}m}\frac{d^{n+m}}{d\mu^{n+m% }}(\mu^{2}-1)^{n}$ | (2) |
(and similarly for $Q_{n}^{m}$). The functions $\begin{matrix}\cos\\ \sin\end{matrix}m\phi\cdot P_{n}^{m}(\mu)$ are tesseral surface harmonics when $m\neq n$ and sectorial surface harmonics when $m=n$. The geometric motivation for this terminology is similar to that of the zonal harmonic: the surface of a sphere is divided into tesseræ (i.e., spherical squares) or sectors (Hobson 1931, 94). Every continuous function can be developed into a series with respect to these functions.
The case of ellipsoidal harmonics is in many respects analogous to that of spherical harmonics. Here, the triply orthogonal set of surfaces consists of confocal ellipsoids, one-sheeted hyperboloids, and two-sheeted hyperboloids. The parameters (usually called ellipsoidal or more simply elliptic coordinates) are the three roots $\rho$, $\mu$, $\nu$ of the equation^{3}^{3}endnote: ^{3} While Poincaré uses this form of the equation in the Acta paper (Poincaré 1885a), he switched to the following form in his paper on the stability of the pear-shaped figure (Poincaré 1902b): $\frac{x^{2}}{\rho^{2}-a^{2}}+\frac{y^{2}}{\rho^{2}-b^{2}}+\frac{z^{2}}{\rho^{2% }-c^{2}}=1$ with $a^{2}>b^{2}>c^{2}$.
$\frac{x^{2}}{\rho^{2}}+\frac{y^{2}}{\rho^{2}-b^{2}}+\frac{z^{2}}{\rho^{2}-c^{2% }}=1.$ |
By choosing a value of $\rho$, we select a particular ellipsoid from a series of confocal ellipsoids, and a subsequent choice of $\mu$, $\nu$ determines a particular point on the surface of the selected ellipsoid. In the theory of ellipsoidal harmonics, the role of Legendre’s equation is played by Lamé’s equation (Poincaré 1885a, 302):
$(\rho^{2}-b^{2})(\rho^{2}-c^{2})\frac{d^{2}R}{d\rho^{2}}+(2\rho^{2}-b^{2}-c^{2% })\rho\frac{dR}{d\rho}=(n(n+1)\rho^{2}-B)R$ | (3) |
For $B$ an appropriate number, there exist particular solutions $R(\rho)$ which are polynomials in $\rho^{2}$ (one of them playing the role of the $P_{n}$ in the spherical case above), or such polynomials multiplied by one, two or three of the factors $\rho$, $\sqrt{\rho^{2}-b^{2}}$, $\sqrt{\rho^{2}-c^{2}}$, where $n$ is the total degree of $R(\rho)$. Replacing $\rho$ by $\mu$ or $\nu$, one obtains functions $M(\mu)$ or $N(\nu)$, respectively. The functions $RMN$ are then normal solutions to Laplace’s equation in elliptic coordinates.
There is another type of solutions to Lamé’s equation, namely functions $S(\rho)$, corresponding to a given $R(\rho)$ and defined as
$S=(2n+1)R\int_{\rho}^{\infty}\frac{d\rho}{R^{2}\sqrt{(\rho^{2}-b^{2})(\rho^{2}% -c^{2})}}$ | (4) |
(these functions correspond roughly to the spherical $Q$ functions discussed above). Lamé introduced his functions in the context of a problem of heat conduction, but it was Liouville who applied them first in the context of equilibrium figures and found several important properties of these functions. First of all, let
$l=\frac{1}{\sqrt{\rho^{2}-\mu^{2}}\sqrt{\rho^{2}-\nu^{2}}};$ | (5) |
further, let $(\rho,\mu,\nu)$, $(\rho,\mu^{\prime},\nu^{\prime})$ be two points of the ellipsoid, $\Delta$ their distance, and $d\omega^{\prime}$ a surface element having $(\rho,\mu^{\prime},\nu^{\prime})$ as its center. Write $M^{\prime}$, $N^{\prime}$, $l^{\prime}$ for the functions of $\mu^{\prime}$, $\nu^{\prime}$ corresponding to the functions $M$, $N$, $l$ of $\mu$, $\nu$. Then
$\iint\frac{l^{\prime}M^{\prime}N^{\prime}d\omega^{\prime}}{\Delta}=\frac{4\pi RSMN% }{2n+1},$ |
where the integral extends to all surface elements of the ellipsoid. Another important property is the following orthogonality relation: given four Lamé functions $M$, $M_{1}\neq M$, $N$, $N_{1}\neq N$, we have
$\iint lMNM_{1}N_{1}d\omega=0,$ | (6) |
where $d\omega$ is defined as before. Perhaps the most important property, however, is that an arbitrary function of $\mu$, $\nu$ can be developed into a series of the form
$\sum A_{i}M_{i}N_{i}$ |
where the $A_{i}$ are constant coefficients.
The precise relation between Lamé functions and the spherical harmonics discussed before will become clearer below in connection with Darwin’s work on ellipsoidal harmonics. Note that the terminology of zonal, tesseral and sectorial harmonic was taken over to the ellipsoidal case in the English literature.
In the second edition of their Treatise on Natural Philosophy, Thomson and Tait inserted a paragraph absent from the first edition, containing a stimulating list of open problems and intuitive conjectures about equilibrium figures of rotating masses of liquid and their stability. Of particular interest to us is their assertion that Jacobi ellipsoids cease to be stable beyond a certain angular momentum. This led them to remark:
We have a most interesting gap between the unstable Jacobian ellipsoid when too slender for stability, and the case of smallest moment of momentum consistent with stability in two equal detached portions. The consideration of how to fill up this gap with intermediate figures, is a most attractive question, towards answering which we at present offer no contribution. (Thomson and Tait, 1883, § 778”(i))
This passage suggested to Poincaré to consider the possibility of pear-shaped figures of equilibrium. He published a long paper in Acta mathematica which resolved several of the open questions mentioned by Thomson and Tait, and supplied complete proofs of some of their claims.^{4}^{4}endnote: ^{4} Poincaré 1885a. Poincaré announced some of the results contained in this paper, in particular the existence of new figures of equilibrium which are non-ellipsoidal and pear-shaped, in two short notes in the Comptes rendus (Poincaré 1885b, 1885c).
Poincaré’s paper made good use of Lamé functions, and outlined new concepts and methods: stability coefficients, linear series of equilibrium figures, points of bifurcation of such series, and the principle of exchange of stabilities at such a point of bifurcation. These conceptual ideas were motivated in part by his parallel work on the three body problem; in both cases, he used a result from Analysis situs: the Kronecker index (Poincaré 1885a, 268).
The basic idea is the following (Poincaré 1885a, 261ff): suppose that the position of the system is defined by $n$ quantities $x_{1},x_{2},\ldots,x_{n}$ and that there is what Poincaré calls a force function (a potential) $F(x_{1},x_{2},\ldots,x_{n},y)$ (where $y$ is a parameter), such that we have equilibrium in case
$\frac{dF}{dx_{1}}=\frac{dF}{dx_{2}}=\ldots=\frac{dF}{dx_{n}}=0.$ |
This system of equations has a certain number of roots varying continuously with $y$. The necessary and sufficient condition that a given root belongs to more than one linear series is that the Hessian of $F$, i.e., the discriminant $\Delta$ of the quadratic form
$\Phi=\sum\frac{d^{2}F}{dx_{i}dx_{k}}X_{i}X_{k}\qquad(i=1,2,\ldots,n;\;k=1,2,% \ldots,n)$ | (7) |
changes its sign. In his proof, he uses a result by Kronecker (Kronecker 1869a, 1869b; Poincaré 1885a, 268). He calls such a form of equilibrium a bifurcation form, or limit form if the equilibrium equations become imaginary (and thus one of the series “disappears”) after the point of intersection. For instance, the MacLaurin ellipsoids form a linear series which admits of a point of bifurcation (where the series of Jacobi ellipsoids starts; see below). In his correspondence with Poincaré, Darwin (1886) calculated the numerical values corresponding to this point of bifurcation; see (§ 3-15-11).
From his study of the function $F$, Poincaré drew information not only about bifurcations, but about stability as well. The first step in this direction was to bring $\Phi$ into the following form (which is always possible with quadratic forms):
$\Phi=\sum\alpha_{i}Y_{i}^{2}\qquad(i=1,2,\ldots,n)$ | (8) |
where $Y_{i}$ is a linear function of the $X$. $\Delta=0$ is then equivalent to $\alpha_{i}=0$ for some $i$. Now, Poincaré wrote that the equilibrium is stable if and only if $F$ is maximum, i.e., if and only if $\Phi$ is negative definite. This is the case if and only if all the $\alpha_{i}$ are negative. Consequently, he called the $\alpha_{i}$ coefficients of stability. His study of the stability of various figures of equilibrium, especially of his newly discovered pear-shaped figures (see below), relied on the study of the sign of the coefficients of stability.
A central tool of Poincaré’s analysis of stability is what he calls the principle of exchange of stabilities. Let the coordinate system be such that a bifurcation occurs in the origin, so that the expressions $\frac{d^{2}F}{dx_{i}^{2}}$ are the stability coefficients, and $y=0$ is the critical parameter. Now, Poincaré keeps $x_{1}$ fixed in $F$ and eliminates the other variables with the aid of the equilibrium equations (assuming the Hessian of $F$ as a function of $x_{2},\ldots,x_{n}$ is not zero, or equivalently, that $\frac{d^{2}F}{dx_{1}^{2}}$ is the only stability coefficient that vanishes). According to Schwarzschild (1898, 41), this restriction poses no difficulty in practice.
Poincaré wrote $F$ as a function of the two variables $x_{1}$ and $y$. The equilibrium equation $\frac{dF}{dx_{1}}=0$ defines a curve in the $(x_{1},y)$-plane with a double point (in agreement with the assumption that $\frac{d^{2}F}{dx_{1}^{2}}$ is zero for a value of $x_{1}$, hence that there occurs a bifurcation). Accordingly, Poincaré was able to express $x_{1}$ as a function of $y$ in two different ways, say
$x_{1}=\psi_{1}(y),\qquad x_{1}=\psi_{2}(y).$ |
Next he expressed $\Delta$ as a function of the two variables $x_{1}$ and $y$ as well and then, using $\psi_{1}$, $\psi_{2}$, as a two-branch function of $y$. Thus he was able to study what happens to the sign of the coefficients of stability on each branch when the parameter $y$ changes the sign. In particular, he found that if there is stability on one linear series $B$ and instability on another $B^{\prime}$ before a point of bifurcation, $B$ will be unstable and $B^{\prime}$ stable afterwards. This he calls the principle of exchange of stabilities.
We can express this in a more intuitive, geometrical manner, following Poincaré’s own presentation (Poincaré 1902a, 167ff).^{5}^{5}endnote: ^{5} For a similar, but more detailed approach, see Schwarzschild (1898). Let $F$ be expressed as $\psi(x_{1},y)$, and recall that there will be stability if $\frac{d^{2}\psi}{dx_{1}^{2}}<0$. Poincaré wrote that the curve $\frac{d\psi}{dx_{1}}=0$ divides the $(x_{1},y)$-plane in two regions, one in which $\frac{d\psi}{dx_{1}}$ is positive, and one in which it is negative. If the positive region is situated below the curve, $\frac{d^{2}\psi}{dx_{1}^{2}}$ will be negative. If two curves arrive at a point of bifurcation, one stable (having the positive region above), the other unstable (having the positive region below), the situation will be exchanged after the passage through the point of bifurcation.
There is another way of expressing the stability condition, by studying the energy of the system. In this approach, there are two types of stability: that destroyed by friction, or ordinary stability, and that unaffected by friction, or secular stability. Following Poincaré (1902b, 333), let $U$ denote the potential energy of the fluid mass, $\omega$ the angular velocity, $J$ the moment of inertia, and $\mu=\omega J$ the angular momentum. For stability, either the expression
$U+\frac{1}{2}\omega^{2}J$ | (9) |
is a minimum, $\omega$ being given, or the expression
$U-\frac{\mu^{2}}{2J}$ | (10) |
is a minimum, $\mu$ being given. For Poincaré, the first condition is necessary and sufficient for secular stability (with an assumption about friction). For an isolated mass, the first condition is sufficient, while the second condition is necessary and sufficient. In a similar fashion, Darwin (1902b, 315) distinguished between the first condition, applied in the static case, and the second condition, for the non-static system. In the correspondence with Poincaré, condition (9) is used (§§ 3-15-18, 3-15-22), while Liapunov employed condition (10); see (§ 3-32-5).
Poincaré assumed $F$ to be a holomorphic function, and noted that the position of the systems under consideration is typically not defined by a finite number of quantities, although generalization to an infinite number of variables is at hand 1885a, 276). According to Schwarzschild (1898, 18), this transition is possible using Lamé functions.
Let us look closer at how Poincaré employs Lamé functions in the study of the Jacobi series. Note first that Poincaré uses two different notations for the $R$ functions, a systematic one and an ad hoc one. The systematic notation (Poincaré 1885a, 308) is $R_{n,i}^{(k)}$.^{6}^{6}endnote: ^{6} However, as Darwin points out (§ 3-15-40), on occasion Poincaré writes $R_{i,n}^{(k)}$ for $R_{n,i}^{(k)}$. Here, $k$ is 1, 2, 3, or 4, according to whether $R$ is a polynomial, involves the factor $\sqrt{\rho^{2}-b^{2}}$, the factor $\sqrt{\rho^{2}-c^{2}}$, or both; $n$ is the total degree (hence corresponds to $n$ in our presentation of spherical harmonics above), and is consequently called the degree of the function; the order of the function, $i$, corresponds to our $m$ above in the sense that $R_{n,i}^{(k)}$ reduces to
$A(\rho^{2}-c^{2})^{\frac{i}{2}}D^{i+n}(\rho^{2}-c^{2})^{n}$ |
for $b^{2}=c^{2}$, where $A$ is a constant and $D^{i+n}$ is the operator $\frac{d^{i+n}}{d\rho^{i+n}}$.
The alternative notation involves indexing the various $R$ as they arrive. Thus Poincaré refers to the Lamé function $\sqrt{\rho^{2}-c^{2}}$ first as $R_{1,1}^{(3)}$, then more simply as $R_{1}$ (Poincaré, 1885a, 317).
Let $g$ denote the force resulting from gravitation and centrifugal force in a point of the surface of an ellipsoid in equilibrium. Then, since the fluid is supposed to be in equilibrium, the direction of $g$ is always perpendicular to the surface. Poincaré gave the expression of $g$ at the pole (where the centrifugal force is zero) in elliptic coordinates and, using the quantity $l$ (see equation 5), obtained
$gl=\frac{4}{3}\pi R_{1}S_{1},$ | (11) |
this expression being valid for every point of the surface of the ellipsoid. He then developed the normal displacement $\zeta$ into a series of the form
$\zeta=\sum A_{i}lM_{i}N_{i},$ | (12) |
which is possible since Poincaré showed that $l$ is a function of $\rho$ only, hence can be considered as constant on the surface. By using this and Liouville’s theorems about Lamé functions discussed above in an evaluation of the energy of the system, he arrived at the following formula for the total potential energy $W$:
$W=W_{0}-\frac{4\pi}{2}\sum A_{i}^{2}\int\left(\frac{R_{1}S_{1}}{3}-\frac{R_{i}% S_{i}}{2n+1}\right)lM_{i}^{2}N_{i}^{2}d\omega,$ | (13) |
where $W_{0}$ is the potential energy of the ellipsoid, $d\omega$ a surface element, and $n$ the degree of the Lamé function $R_{i}$. The form of the deformed surface is then determined by the $A_{i}$, which play the role of the $Y_{i}$ in (8). Thus, the coefficients of the $A_{i}$ in the above sum are the stability coefficients, and their vanishing is equivalent to the following fundamental equation:
$\frac{R_{1}S_{1}}{3}-\frac{R_{i}S_{i}}{2n+1}=0.$ | (14) |
Why is the latter equation responsible for bifurcations? By developing into a series of Lamé functions the part of the potential of the deformed body due to the perturbing forces, and denoting by $B_{i}$ the coefficients, Poincaré showed that the $A_{i}$ (determining the figure) emerge from the equations
$4\pi A_{i}\left(\frac{R_{1}S_{1}}{3}-\frac{R_{i}S_{i}}{2n+1}\right)=B_{i},$ |
unless the final factor on the left hand side is zero. Thus for a solution of equation (14), the form of the deformed surface is no longer uniquely determined, and there is bifurcation (Poincaré 1885a, 321).
Poincaré then sought functions $R_{i}$ for which this is actually the case. One obvious solution is $i=1$, corresponding to an infinitesimal movement of translation of the ellipsoid which does not alter the state of equilibrium. He found other functions $R_{i}$, and called $R_{2}$ the function (of degree 2) for which the Jacobi series bifurcates from the MacLaurin series (Poincaré 1885a, 335) such that:
$\frac{R_{1}S_{1}}{3}-\frac{R_{2}S_{2}}{5}=0.$ | (15) |
In order for a Jacobi ellipsoid to be an ellipsoid of bifurcation, a second stability coefficient has to cancel out, which leads to the two equations (Poincaré 1885a, 341):
$\frac{R_{1}S_{1}}{3}=\frac{R_{2}S_{2}}{5}=\frac{R_{i}S_{i}}{2n+1}.$ | (16) |
Poincaré shows that the only $R_{i}$ satisfying these equations are the $R_{n,0}^{1}$, the zonal harmonics of degree $n$, for $n\geq 3$ (Poincaré 1885a, 343).^{7}^{7}endnote: ^{7} The exact expression of $R_{3,0}^{1}$ is $R_{3,0}^{1}(\rho)=\frac{1}{5}\rho(5\rho^{2}-2(b^{2}+c^{2})+\sqrt{4b^{4}-7b^{2}% c^{2}+4c^{4}}).$ Note that the notation for Lamé functions is slightly inconsistent throughout the Acta paper: Poincaré writes this function $R_{0,3}^{1}$ on p. 343, but it is clear from the fact that the function is a polynomial in $\rho$ of degree $3$ that he should have written it $R_{3,0}^{1}$ in agreement with the earlier definition. Compare also Darwin to Poincaré (§ 3-15-40). The coefficient of stability corresponding to this case is called $C_{n}$. By considering the equation $\zeta=0$ (where $\zeta$ is the normal displacement for $R_{3,0}^{1}$ and thus a function involving $M_{3,0}^{1}$ and $N_{3,0}^{1}$), Poincaré obtained the lines of intersection of the ellipsoid with the deformed figure corresponding to $C_{3}$, and provided a rough sketch (reproduced in § 3-15-11) showing a pear-shaped figure. Similarly, the coefficients $C_{n}$ for $n\geq 4$ will give rise to other bifurcating series composed of figures with $n-2$ constrictions. In a popular presentation, Poincaré (1892) provided sketches of some of these figures, and an intuitive interpretation of the harmonic functions involved as oscillations giving rise to normal displacements.
With the discovery of the pear-shaped figure of equilibrium, Poincaré thought he had closed the gap discussed by Thomson and Tait (op. cit.). But what about the stability of the new figures? Poincaré thought it admissible to apply the principle of exchange of stabilities in this case. This principle applies, for instance, at the point of bifurcation of the MacLaurin series: the MacLaurin ellipsoids cease to be stable at this point, and the Jacobi ellipsoids are stable from this point on. Poincaré held the situation to be similar at the first point of bifurcation of the Jacobi series, where the series of pear-shaped figures begins (he showed in his paper that the Jacobi ellipsoids actually cease to be stable at the point of bifurcation leading to the series of pear-shaped figures, thus proving Thomson and Tait’s assertion, mentioned above).
Over the next decade, Poincaré added no further insight but affirmed the stability of the pear-shaped figures, while Karl Schwarzschild’s doctoral thesis (1898) held the argument of exchange of stability to be fallacious in the case of the series of pear-shaped figures. Schwarzschild thus reopened the question of the stability of this series.
Darwin’s work on equilibrium figures prior to 1901
Darwin became interested in figures of equilibrium very early in his career. His interest was motivated by problems of cosmogony and geophysics.^{8}^{8}endnote: ^{8} The roots of G. H. Darwin’s geophysical research in his father Charles Darwin’s evolutionary theory, and his uncomfortable role as an intermediary between his father and W. Thomson (Lord Kelvin) are discussed by Kushner (1993). In July 1881, while reflecting on the rapid motion of a Martian satellite, Darwin wondered if the earliest form of a satellite might not be annular (Darwin 1881, 534). Some years later he presented two papers, one on the Jacobi ellipsoid (Darwin 1886), and another in which he investigated pear-shaped figures of equilibrium (Darwin 1887). These papers were written independently of Poincaré’s Acta paper (Poincaré 1885a). Like Poincaré, Darwin was motivated by Thomson’s discussion of the “most interesting gap” (Darwin 1887, 420). However, unlike Poincaré, who had found pear-shaped figures by investigating the series of Jacobian ellipsoids for decreasing angular velocity, Darwin took the configuration of two separated spherical bodies as his point of departure, and studied the further development of the equilibrium figure “backwards in time”, so to say, in agreement with his motivation from cosmogony. In this paper, he confined himself to the use of spherical harmonic analysis.
Also important for understanding the Poincaré-Darwin exchange is Darwin’s approach to the theory of ellipsoidal harmonic functions. He sought to develop this theory in a form more appropriate for numerical calculations (Darwin 1901, 462). His major innovations in this respect were a new approach to elliptic coordinates, and the introduction of a particular notation for the functions, analogous to the notation used in the spherical case, and deviating from Poincaré’s notation. Darwin’s elliptic coordinates are defined with respect to the three roots $u_{1}$, $u_{2}$, $u_{3}$ of the cubic in $u$
$\frac{x^{2}}{a^{2}+u^{2}}+\frac{y^{2}}{b^{2}+u^{2}}+\frac{z^{2}}{c^{2}+u^{2}}=1.$ |
Darwin then writes
$u_{1}^{2}=k^{2}\nu^{2},\qquad u_{2}^{2}=k^{2}\mu^{2},\qquad u_{3}^{2}=k^{2}% \frac{1-\beta\cos 2\phi}{1-\beta},$ |
where $\beta$ and $k$ are given by the three axes of the fundamental ellipsoid of reference^{9}^{9}endnote: ^{9} In his paper on the stability of the pear-shaped figure, Darwin (1903a) considers as an ellipsoid of reference for a certain value of $\nu_{0}$ the critical Jacobian ellipsoid enclosing the whole pear; compare, for instance, Darwin to Poincaré, 21.06.1901 (§ 3-15-16). In a loose sense, Darwin’s fundamental ellipsoid of reference figures in the construction of the coordinates, as explained by Darwin in the case of spheroidal coordinates. In this case $\nu$, $\mu$ and $\phi$ can reasonably be thought of as the radial, latitudinal, and longitudinal coordinates, respectively. In the general case, the geometry involved is more difficult, but Darwin affirms that the above interpretation of the coordinates is still acceptable (Darwin 1901, 549).
$k\sqrt{\nu_{0}^{2}-\frac{1+\beta}{1-\beta}},k\sqrt{\nu_{0}^{2}-1},k\nu_{0}.$ |
Then $\infty>\nu\geq 0$, $1\geq\mu\geq-1$ and $2\pi\geq\phi\geq 0$ are the coordinates of a point. As Darwin observed, this presentation is less symmetric than Lamé’s, and instead of Lamé’s equation (3) and corresponding equations in the coordinates $\mu$, $\nu$ (in the sense of Lamé), Darwin had to consider three differential equations the third of which differs from the two other in the same manner as the equation for $u_{3}$ differs from the equations for $u_{1}$, $u_{2}$ (Darwin 1901, 467). Of course, a product of three functions, each of which is a solution of one of the three equations will be a normal solution of Laplace’s equation in Darwin’s coordinates. The functions in $\nu$ and in $\mu$ are related to the spherical $P$ and $Q$ functions discussed above, while the functions in $\phi$ involve a cosine or sine of a multiple of $\phi$.
As announced, Darwin’s notation for these functions is analogous to the notations of the spherical harmonics. Note, however, that he writes $P_{i}^{s}$ where we wrote $P_{n}^{m}$ above, and correspondingly for the $Q$ functions; thus, $i$ is the complete degree of Legendre’s polynomial, and $s$ the order of derivation. Darwin observes that contrary to the spherical case, there are two types of $P$ functions, divisible or not by the following factor:
$\Omega(\nu)=\sqrt{\frac{\nu^{2}-\frac{1+\beta}{1-\beta}}{\nu^{2}-1}}$ |
($\Omega(\mu)$ being defined similarly). The indivisible functions are written $\mathfrak{P}_{i}^{s}$, and the divisible ones $\mathsf{P}_{i}^{s}$. There are also two types of functions of $\phi$, expressed as $\mathfrak{C}$ and $\mathsf{C}$ (or $\mathfrak{S}$ and $\mathsf{S}$, according to whether they involve a cosine or a sine of $\phi$).^{10}^{10}endnote: ^{10} These functions are related to the spherical harmonics in that the functions of the $\mathfrak{P}$ type can be developed in a certain manner into a sum of ordinary $P_{i}^{s}$ functions, while the functions of the $\mathsf{P}$ type can be so developed only up to the factor $\Omega$. The precise form of the $P$ function to be taken in the normal solution depends on the evenness or oddness of $i$ and of $s$ and on whether a cosine or a sine of $\phi$ is involved; thus, Darwin has to distinguish eight cases (four of which involve a $\mathfrak{P}$ function and the other four a $\mathsf{P}$ function).
The $Q$ functions relate to the $P$ functions much like the $S$ functions relate to the $R$ functions in Poincaré’s notation of Lamé functions (see equation 4 above).^{11}^{11}endnote: ^{11} The precise relation of Darwin’s and Poincaré’s notation was explained by Darwin in his letter to Poincaré of 22.10.1901 (§ 3-15-40) in the case of $P$ functions and in a note to Poincaré’s paper (Poincaré 1902b) in the case of $Q$ functions. Since $P$ functions fall into two classes, the same is true for $Q$ functions (which are written $\mathfrak{Q}$ or $\mathsf{Q}$), and we have
$\mathfrak{Q}_{i}^{s}(\nu_{0})=\mathfrak{P}_{i}^{s}(\nu_{0})\int_{\nu_{0}}^{% \infty}\frac{d\nu}{(\mathfrak{P}_{i}^{s}(\nu))^{2}\sqrt{\nu^{2}-1}\sqrt{\nu^{2% }-\frac{1+\beta}{1-\beta}}}.$ | (17) |
Darwin determined these functions rigorously for $i=2$, $s=0$, 1, 2, and gave approximate formulæ for the higher functions (Darwin 1902a, 488); in his subsequent paper concerned with the determination of the limiting stability of the Jacobi series, he added rigorous determinations of the functions for $i=3$, $s=0$, 1, 2 which are crucial for this purpose, $\mathfrak{P}_{3}^{0}=\mathfrak{P}_{3}$ being the third zonal harmonic (Darwin 1902b, 307ff).
Darwin provided the formulæ of the normal displacement, the surface density, and the internal and external potentials for a given surface harmonic (Darwin 1901, 505).^{12}^{12}endnote: ^{12} The expression for a normal displacement of an element of the ellipsoid for the third zonal harmonic was given in Darwin’s letter to Poincaré of 28.05.1901 (§ 3-15-11). The stability coefficients are (Darwin 1902b, 320)
$\mathfrak{K}^{s}_{i}=1-\frac{\mathfrak{P}^{s}_{i}(\nu_{0})\mathfrak{Q}^{s}_{i}% (\nu_{0})}{\mathsf{P}^{1}_{1}(\nu_{0})\mathsf{Q}^{1}_{1}(\nu_{0})},\qquad% \mathsf{K}^{s}_{i}=1-\frac{\mathsf{P}^{s}_{i}(\nu_{0})\mathsf{Q}^{s}_{i}(\nu_{% 0})}{\mathsf{P}^{1}_{1}(\nu_{0})\mathsf{Q}^{1}_{1}(\nu_{0})},$ | (18) |
and Poincaré’s equation (14) for the vanishing of the stability coefficients in Darwin’s notation reads (Darwin 1902b, 321)
$\mathsf{P}^{1}_{1}\mathsf{Q}^{1}_{1}-\mathfrak{P}^{s}_{i}\mathfrak{Q}^{s}_{i}=0.$ |
Darwin explained (p. 514) that in order to develop an arbitrary function of $\mu$, $\phi$ in a series of ellipsoidal harmonics it is necessary to know the integrals over the surface of the ellipsoid of the squares of surface harmonics multiplied by the perpendicular on the tangent plane. He provided a table (p. 548) collecting these integrals showing, once again, his interest in numerical results.
Poincaré’s and Darwin’s strategies for proving stability
The correspondence shows that it was Darwin and not Poincaré who started their joint attack of the question of the stability of the pear-shaped figure. From the outset, the goal of providing a new setting for the theory of ellipsoidal harmonic functions was geared towards obtaining exact numerical results with respect to this figure (Darwin 1901, 462). As a early achievement of this program, he obtained good numerical data about the critical Jacobi ellipsoid — a victory he shared with Poincaré on 28.05.1901 (§ 3-15-11).
In the same letter, Darwin expressed doubts about Schwarzschild’s objection against Poincaré’s stability argument for the pear-shaped figure. Poincaré did not harbor such doubts, and in reply, sketched a program designed to decide the stability question, and which formed the basis of their exchange and the related papers (§ 3-15-13). The pursuit of the project required more precise numerical data concerning the physical magnitudes, and the shape of the figures. Poincaré hoped that Darwin’s new tools might afford the means to acquire this data, and Darwin presented a strategy for their calculation.
Poincaré promised Darwin that he would work on the problem (§ 3-15-15); he communicated his result in the form of a long letter (later supplemented by an equally long commentary) in which he presented his strategy (§§ 3-15-19, 3-15-20).
The determination of deformations of an equilibrium figure yielding new equilibrium figures involves studying the behavior under deformation of terms entering the equilibrium condition (the energy $W=U+\frac{1}{2}\omega^{2}J$ and the moment of inertia $J$). The normal displacement $\zeta$ is developed in a harmonic series (see equation 12 above), and the coefficients of the development determine the shape of the new figure. Poincaré expressed these coefficients as $A_{i}$ and later $\theta_{i}$ in the Acta paper, but wrote them as $\xi_{i}$ in the new paper; we follow the latter convention. The terms $W$ and $J$ are developed in a series with respect to powers of the $\xi_{i}$, which reveals the influence that the variables $\xi_{i}$ determining the shape of the new figure have on $W$ and $J$. This is not only important for equilibrium but for stability as well, since $W$ and $J$ enter into the stability condition.
The determination of the figure that Poincaré provided in the Acta paper is correct to the first order (Poincaré 1885a, 347); $\xi_{i}^{2}$ and higher powers were neglected. Obtaining a higher-order approximation has an effect in terms of harmonic functions, as Darwin observed:
The pear-shaped figure is a deformation of the critical Jacobian ellipsoid, and to the first order of small quantities it is expressed by the third zonal harmonic with respect to the longest axis of the ellipsoid. In the higher approximation a number of other harmonic terms will arise, and the coefficients of these new terms will be of the second order of small quantities. (Darwin 1903a, 253)
What this means is that while the harmonic series for $\zeta$ in the first order reduces to a single term (the coefficient corresponding to $R^{(1)}_{3,0}$ alone are retained), other terms must be considered for a higher-order approximation. Poincaré tried to determine the coefficients of the $\xi_{i}$ in the development of $W$ and $J$ up to the second order, and found he had to calculate infinitely many elliptic integrals (Poincaré to Darwin, § 3-15-19).
The stability question
Despite his efforts, Poincaré was unable to either prove or disprove the stability of the pear-shaped figure (Poincaré 1902b). He provided the inequality that was satisfied in case of stability but did not pursue a numerical evaluation. After evoking the possibility if evaluating the integrals involved with Weierstrass-Schwarz theory of elliptic functions, Poincaré concluded:
La détermination de chacune des intégrales ne présente […] aucune difficulté, et le calcul serait en somme facile si ces intégrales n’étaient en nombre infini. (Poincaré 1902b, 368)
He further identified two terms that depend on infinitely many integrals, denoted $\sum\frac{Q_{i}^{2}}{2G_{i}}$ and $H^{\prime}$, both of which he had mentioned in his exchange with Darwin.^{13}^{13}endnote: ^{13} See Poincaré to Darwin (§ 3-15-19). The first term is expressed as $\sum\frac{{h^{\prime}}^{2}}{a^{\prime}}$ in the letter, while $a_{1}$ denotes the problematic portion $H_{0}$ of $H^{\prime}$. Poincaré then took up the difference of the two problematic terms, and observed:
Heureusement il ne s’agit pas de calculer la valeur exacte de cette quantité, mais de reconnaître si elle satisfait à une certaine inégalité. (Poincaré 1902b, 369)
The problem remains quite difficult. After some pages of preliminary calculations, Poincaré ends his memoir with the following remark:
Si donc il y a instabilité, c’est-à-dire si l’inégalité précédente n’a pas lieu, il suffira pour le constater de calculer un nombre fini de termes du premier membre. Si au contraire il y a stabilité, on ne pourra s’en assurer qu’en calculant la somme des termes positifs du premier membre qui sont en nombre infini, ou en évaluant une limite supérieure de cette somme.
Did Darwin attempt to carry out this program? In his next paper on the stability of the pear-shaped figure, Darwin explained his motivation as follows:
At the end of [my previous paper] it was stated that the stability of the figure could not be proved definitely without approximation of a higher order of accuracy. After some correspondence with M. Poincaré during the course of my work, I made an attempt to carry out this further approximation, but found that the expression for a certain portion of the energy entirely foiled me. Meanwhile he had turned his attention to the subject, and he has shown […] by a method of the greatest ingenuity and skill how the problem may be solved. He has not, however, pursued the arduous task of converting his analytical results into numbers, so that he left the question of the stability of the pear still unanswered.
M. Poincaré was so kind as to allow me to detain his manuscript on its way to the Royal Society for two or three days, and I devoted that time almost entirely to understanding the method of his attack on the key of the position – namely, the method of double layers […]. (Darwin 1903a, 252)
Darwin was certainly aware of the necessity of calculating infinitely many terms:
Let $S_{i}^{s}$ denote any surface harmonic, so that $S_{i}^{s}$ is the same thing as $[\mathfrak{P}_{i}^{s}(\mu)$ or $\mathsf{P}_{i}^{s}(\mu)]\times[\mathfrak{C}_{i}^{s}(\phi)\;\text{or}\;\mathsf{% C}_{i}^{s}(\phi)]$. The third zonal harmonic deformation will then be $eS_{3}$ or $e\mathfrak{P}_{3}(\mu)\mathsf{C}_{3}(\phi)$, where $e$ is of the first order of small quantities. On account of the symmetry of the figure, the new terms cannot involve the sine functions $\mathfrak{S}$ or $\mathsf{S}$, and moreover, the rank $s$ must necessarily be even.
Suppose that the new terms are expressed by $\sum f_{i}^{s}S_{i}^{s}$ for all values of $i$ from 1 to infinity, and with $s$ equal to $0$, $2$, $4$, …, $i$ or $i-1$. Then all the $f$’s are of order $e^{2}$, excepting $f_{3}$ which is zero. (Darwin 1903a, 253)
Darwin commented on his calculations in a letter to Schwarzschild of June 1, 1902, where he announced his intention to give a talk on the question at the astronomers’ meeting of August 1902 in Göttingen:
I have just completed the proof of the stability in question — not, I regret to say, a rigorous algebraic proof but one which renders the result practically certain. I have determined the coefficients of 8 new harmonic terms of the second order of small quantities in the expression for the pear. In order that there may be stability the sum of the squares of those coefficients each multiplied by a certain quantity must be less than a certain other quantity. Taking 8 harmonics it is decisively less, and the coefficients of the higher harmonics contribute very little to the sum in question. It seems practically impossible that the 8^{th}, 10^{th}, 12^{th} etc. harmonics should present large coefficients in the expression for the pear — on the contrary the evidence shows the coefficients to diminish rapidly as far as the 6th harmonic. Hence I consider the stability assured.
It remains for me to compute the deformation represented by these terms & so to obtain a better figure. This is no small task but I expect to have done it long before the meeting. (Darwin to Schwarzschild, 01.06.1902, Cod. Ms. K. Schwarzschild Briefe 154, Niedersächsische Staats- und Universitätsbibliothek Göttingen)
Darwin believed he could determine the stability question without calculating the infinite number of terms relevant to the problem according to Poincaré.^{14}^{14}endnote: ^{14} Darwin (1903b, 312) advances a similar argument. Even so, he was quite aware that his considerations fell short of a rigorous proof, and that Liapunov had found the pear-shaped figure to be unstable in 1905.
Part of the motivation for studying pear-shaped figures of equilibrium stemmed from their potential application in the cosmological realm. Poincaré observed that the theory of equilibrium figures of a homogeneous rotating mass of fluid does not directly apply to problems of cosmogony, but both he and Darwin saw a future for their work in this domain. Darwin expressed as much in the conclusion to his second paper:
Notwithstanding the caveat which M. Poincaré enters as to the dangers of applying these results to heterogeneous masses and to cosmogony, I cannot restrain myself from joining him in seeing in this almost life-like process a counterpart to at least one form of the birth of double stars, planets, and satellites. (Darwin 1903b, 314)
The discovery of spectroscopic binary stars in 1889 led to a rapid rise in studies of variable stars in the 1890s. The Poincaré-Darwin fission theory of homogeneous fluid masses was a plausible model for the origin of these celestial objects, and young astrophysicists like Schwarzschild and G. W. Myers were quick to latch on to the possibility that pear-shaped figures of equilibrium were a key to understanding stellar evolution.^{15}^{15}endnote: ^{15} On the Poincaré-Darwin fission theory, see Poincaré to Darwin (§ 3-15-11), Myers to Poincaré (§ 3-36-1), and Walter (2024, forthcoming).
Time-stamp: " 1.08.2023 22:07"
Notes
- ^{1} Early appreciations of Darwin’s work include Love (1913) and Hough (1914). Later evaluations include Kopal (1971) and Barrow-Green (1997, 193).
- ^{2} In the expression $u_{n}^{m}$, $m$ is an upper index, and not an exponent. This is standard notation for theories of spherical and ellipsoidal harmonics.
- ^{3} While Poincaré uses this form of the equation in the Acta paper (Poincaré 1885a), he switched to the following form in his paper on the stability of the pear-shaped figure (Poincaré 1902b): $\frac{x^{2}}{\rho^{2}-a^{2}}+\frac{y^{2}}{\rho^{2}-b^{2}}+\frac{z^{2}}{\rho^{2% }-c^{2}}=1$ with $a^{2}>b^{2}>c^{2}$.
- ^{4} Poincaré 1885a. Poincaré announced some of the results contained in this paper, in particular the existence of new figures of equilibrium which are non-ellipsoidal and pear-shaped, in two short notes in the Comptes rendus (Poincaré 1885b, 1885c).
- ^{5} For a similar, but more detailed approach, see Schwarzschild (1898).
- ^{6} However, as Darwin points out (§ 3-15-40), on occasion Poincaré writes $R_{i,n}^{(k)}$ for $R_{n,i}^{(k)}$.
- ^{7} The exact expression of $R_{3,0}^{1}$ is $R_{3,0}^{1}(\rho)=\frac{1}{5}\rho(5\rho^{2}-2(b^{2}+c^{2})+\sqrt{4b^{4}-7b^{2}% c^{2}+4c^{4}}).$ Note that the notation for Lamé functions is slightly inconsistent throughout the Acta paper: Poincaré writes this function $R_{0,3}^{1}$ on p. 343, but it is clear from the fact that the function is a polynomial in $\rho$ of degree $3$ that he should have written it $R_{3,0}^{1}$ in agreement with the earlier definition. Compare also Darwin to Poincaré (§ 3-15-40).
- ^{8} The roots of G. H. Darwin’s geophysical research in his father Charles Darwin’s evolutionary theory, and his uncomfortable role as an intermediary between his father and W. Thomson (Lord Kelvin) are discussed by Kushner (1993).
- ^{9} In his paper on the stability of the pear-shaped figure, Darwin (1903a) considers as an ellipsoid of reference for a certain value of $\nu_{0}$ the critical Jacobian ellipsoid enclosing the whole pear; compare, for instance, Darwin to Poincaré, 21.06.1901 (§ 3-15-16). In a loose sense, Darwin’s fundamental ellipsoid of reference figures in the construction of the coordinates, as explained by Darwin in the case of spheroidal coordinates. In this case $\nu$, $\mu$ and $\phi$ can reasonably be thought of as the radial, latitudinal, and longitudinal coordinates, respectively. In the general case, the geometry involved is more difficult, but Darwin affirms that the above interpretation of the coordinates is still acceptable (Darwin 1901, 549).
- ^{10} These functions are related to the spherical harmonics in that the functions of the $\mathfrak{P}$ type can be developed in a certain manner into a sum of ordinary $P_{i}^{s}$ functions, while the functions of the $\mathsf{P}$ type can be so developed only up to the factor $\Omega$. The precise form of the $P$ function to be taken in the normal solution depends on the evenness or oddness of $i$ and of $s$ and on whether a cosine or a sine of $\phi$ is involved; thus, Darwin has to distinguish eight cases (four of which involve a $\mathfrak{P}$ function and the other four a $\mathsf{P}$ function).
- ^{11} The precise relation of Darwin’s and Poincaré’s notation was explained by Darwin in his letter to Poincaré of 22.10.1901 (§ 3-15-40) in the case of $P$ functions and in a note to Poincaré’s paper (Poincaré 1902b) in the case of $Q$ functions.
- ^{12} The expression for a normal displacement of an element of the ellipsoid for the third zonal harmonic was given in Darwin’s letter to Poincaré of 28.05.1901 (§ 3-15-11).
- ^{13} See Poincaré to Darwin (§ 3-15-19). The first term is expressed as $\sum\frac{{h^{\prime}}^{2}}{a^{\prime}}$ in the letter, while $a_{1}$ denotes the problematic portion $H_{0}$ of $H^{\prime}$.
- ^{14} Darwin (1903b, 312) advances a similar argument.
- ^{15} On the Poincaré-Darwin fission theory, see Poincaré to Darwin (§ 3-15-11), Myers to Poincaré (§ 3-36-1), and Walter (2024, forthcoming).
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