George Darwin's lectures on Hill's lunar theory

Abstract

In the late 1870s two papers were published in which G W Hill presented a novel method for calculating the orbital motion of the Moon. These much-acclaimed papers were a turning point in the history of the three-body problem and proved inspirational to mathematicians such as Poincaré and G H Darwin in their work on periodic orbits.

 Two decades later Darwin gave a series of lectures on Hill's lunar theory at the University of Cambridge. His lectures were an attempt to make Hill's theory more accessible, particularly to students of astronomy, and also mark the beginning of his own research in the field. This article, which expands upon a talk first given at the BSHM's Research in Progress meeting in March 2008, describes both the content and context of Darwin's lectures.

Notes

¹ George Darwin's interest in this is indicated by the paper he read before the Statistical Society of London on the effects of marriages between first cousins (Darwin 1875).

² Charles Pritchard was appointed Savilian Professor of Astronomy at Oxford in 1870. He was noted for his teaching abilities and designed and supervised the building of the new University Observatory (Fauvel et al. 2000, 183).

³ The Smith's prize is awarded each year to two junior research students in theoretical physics and mathematics at the University of Cambridge. In 1868, when Darwin won the 2nd Smith's prize, it was still based upon an examination held shortly after the announcement of the Tripos results. For an account of the history of the Smith's Prize competition see Barrow-Green (1999).

⁴ Most of Darwin's papers are contained in the first four volumes of his Scientific papers (Darwin 1907–16).

⁵ Nathaniel Bowditch was a friend of Strong's and had given him a presentation copy of his much acclaimed translation of the Mécanique céleste (Bowditch 1829–39).

⁶ Charles-Eugène Delaunay published his monumental work La théorie du mouvement de la lune in two volumes, each over 900 pages long, in 1860 and 1867.

⁷ The convergence of the series derived by the various astronomers was a subject later studied in detail by Poincaré (1892–99, vol 2). Hill (1878, 8) acknowledges that he was unable to prove the convergence of the series he derived and many years later Moulton (1914, 397) comments: ‘Since Hill's work there have been few attempts at proving the convergence of the series used in the lunar theory, and no real progress in the matter has been made.’

⁸ In 1772, Lagrange found solutions in which the three bodies lie on a line or at the vertices of an equilateral triangle and move such that their orbits are similar conic sections of arbitrary eccentricity.

⁹ About 5′ of the variation arises from terms involving the lunar and the solar eccentricity, which are both set equal to zero in the determination of the variational orbit. The variational orbit also includes the contribution from all the higher-order harmonics of the variation, to which Brown (1896, 125) gives the general name ‘variational inequalities’.

¹⁰ James Whitbread Lee Glaisher was a tutor and lecturer at the University of Cambridge whose interests included the history of mathematics. He was President of the Royal Astronomical Society from 1886 to 1888 and again from 1901 to 1903.

¹¹ Hill did not attend this presentation, which took place at the anniversary meeting of the Royal Astronomical Society in February 1887. Having received notification by letter of the award only several weeks earlier, Hill excused himself (Hill to Huggins, 31 January 1887, RAS letters) and was delivered the Gold Medal shortly afterwards via the Foreign Secretary of the Society, William Huggins, and the US Department of State (Hill to Huggins, 15 March 1887, RAS letters).

¹² John Couch Adams is best remembered in connection with the controversy over the discovery of Neptune. He held the chair of Lowndean Professor of Astronomy and Geometry between 1858 and his death in 1892.

¹³ Shortly after Hill's paper on the motion of the lunar perigee was published, Adams wrote a short paper (Adams 1877) in which he described how several years earlier he had, whilst investigating the motion of the lunar node, arrived at a differential equation of the same form as that found by Hill and consequently at the same infinite determinant.

¹⁴ Volume 5 of Darwin's Scientific papers was edited by F J M Stratton and J Jackson. Stratton was a former student of Darwin's, who later (in 1928) became Professor of Astrophysics at Cambridge.

¹⁵ Sydney Hough is best remembered for his revision of Laplace's dynamical theory of the tides and his work as a practical astronomer. However, his 1901 Acta mathematica paper On certain discontinuities connected with periodic orbits (Hough 1901), which was reprinted in volume 4 of Darwin's Scientific papers, is evidence of his continued interest in celestial mechanics. Philip Cowell wrote his first paper as a result of Darwin's lectures and followed it with a series of papers in which the various constants and coefficients for the lunar orbit were obtained to high accuracy. He later became Superintendent of the Nautical Almanac Office and won the Gold Medal of the Royal Astronomical Society in 1911.

¹⁶ Hill (1886, 3) describes his intermediate orbit as containing the lunar inequalities which are both independent of the eccentricity and have the argument of the variation. Therefore it does not include the parallactic inequalities, which have a period of one mean synodic month and would, if included, slightly distort the variational curve in the direction of the Sun, see Brown (1896, 125–127).

¹⁷ The force function is simply another name for the potential function when its definition does not include a minus sign.

¹⁸ This has the effect of making τ equivalent to the mean elongation, i.e. the difference in the mean longitude of the Moon and the Sun.

¹⁹ The equation , where a and q are constants, is known as Mathieu's equation. In physical problems, the periodicity of the boundary conditions leads to the choice of a as a certain function of q. The even and odd periodic solutions are called Mathieu functions. In the lunar theory the constants Θ₀, Θ₁, Θ₂, … are already known, so that the problem of searching for periodic solutions does not arise. The solution of Hill's equation is not periodic.

²⁰ Hill was not the first to make use of an infinite determinant, but prior to his paper on the motion of the lunar perigee the idea was not widely known. Its convergence (in the form of an infinite series) remained unproved for a further decade until a proof was provided by Poincaré. See Barrow-Green (1997, 27) for further details.

²¹ This simple expression is correct to terms in m⁴ and gives a value for 1 − c which is only about 1/60 in excess of the value 0.008452 given by observation, which, as Hill (1886, 23) notes, is mainly due to the neglect of the inclination of the lunar orbit. Hill also derives another formula for c which, whilst again only involving the coefficients Θ₀ and Θ₁, is two orders more exact and gives a value for 1 − c which is correct to the sixth decimal.

²² The two constants are related by c = (1 + m)c = c/(1 − m).