Introduction

The Earth, as will appear, is not typical of the planets. It is the largest of the inner planets, it is the only one on which active tectonic development of the surface appears to be going on at present and, so far as we know, it has the most complex structure. Yet it is the only one which can be studied in detail; from it we may derive empirically equations of state of the materials of the inner planets; and the methods that have been used to study the structure of the Earth are those we should like to use, but are inhibited from using by the difficulties of observing, in the investigation of the other planets. For these reasons it is helpful to preface an account of the methods used to study the planets and of the results that have been obtained with a review of the way in which the Earth is examined and what has been discovered.

Our knowledge of the internal structure of the Earth comes by two routes. In the first place the mass, size and density of the Earth provide a rough idea of the overall composition and of the central pressure, while the value of the moment of inertia shows that the density increases strongly towards the centre. Naturally a wide range of models could be constructed to fit just three facts and so it is necessary to turn, in the second place, to seismology to provide more detailed information.

The planets, which have always been objects of wonder and curiosity to those with the opportunity or need to lift their eyes to the heavens, now in our times shine with new and strange lights revealed to us by the far seeing instruments carried upon space craft. The Moon, Mars, Venus and Mercury all bear on their surfaces the crater scars of innumerable meteorites that have fallen upon them from the beginning of the solar system. The Earth alone has an active surface that has obliterated those scars. The fluid surface of Jupiter is in constant and vigorous motion, driven by heat flowing out from the interior or, it may be, brought to it by the ultra-violet radiation from the Sun or by the solar wind. The Medicean satellites of Jupiter now' present to us strange and individual faces: would Galileo who first saw the mountains on the Moon or the spots on the Sun have been surprised by the eruption of sodium and sulphur from Io and the cloud of gas within which it moves, or by the strange stress patterns upon other of the satellites? Seeing these strange and varied faces of the planets, each apparently different from any other, who can forbear to ask, what bodies are these, how are they made up, that their appearances are so distinctive? Why are some active, and others apparently dead, some dry, and others thickly covered with atmosphere or ocean?

Introduction

Leaving aside the special case of the Moon, the properties of planets that can at present be determined are certain gross quantities descriptive of a planet as a whole; these are the size, the spin angular velocity, the mass and mean density, the moments of inertia and the coefficients in a spherical harmonic expansion of the gravitational potential, together with some features of the magnetic field and possibly electromagnetic induction in the planet. The Moon alone is open to the study of the variation of properties with depth by seismology. The investigation of the internal state of a planet depends on what can be inferred from the measured gross properties, and fails unless those properties can be measured with precision. Given only integral properties, a wide range of internal distributions of density is consistent with the data, but the more precisely the integral properties are known the more restricted the range of possible distributions.

The various dynamical properties of a planet are not independent, for all are determined by three factors: the spin, the chemical composition and the temperature. Suppose the spin acceleration at the surface at the equator (where it has its greatest value) to be small compared with the acceleration of the self-gravitational attraction. Then composition and temperature together determine the equation of state, the latter mainly by its control of the occurrence of any polymorphic phase changes.

Introduction

After the Earth, the Moon is much the best known body of the solar system. Almost all physical measurements that have been made on the Earth have also to some extent been made on the Moon. Artificial satellites have been placed in orbit about the Moon and have enabled the components of the gravitational potential to be estimated. The physical librations, the equivalent of the luni-solar precession of the Earth, have been observed, especially by laser ranging to the retroreflectors left on the Moon by Apollo astronauts. The Apollo astronauts took with them seismometers that have recorded impacts of meteorites and rockets on the surface and moonquakes within the Moon. The flow of heat through the surface of the Moon was measured.

The magnetic field of the Moon has been studied intensively, globally by satellites at a distance from the surface and in detail by others close to it, while the magnetization of rock samples brought back by the Apollo astronauts has been studied in the laboratory. In addition, electro-magnetic induction in the Moon has been studied. Thus, there is some prospect of being able to construct models of the interior of the Moon using much the same methods as are followed for the Earth, whereas there is at present no such prospect for any of the planets. However, there are major gaps in our knowledge of the Moon as compared with the Earth, and the principal one is that seismic data are comparatively very sparse because there are only four seismic stations on the Moon and all of them are on the same hemisphere and, furthermore, because free oscillations of the Moon have never been observed.

The field of dynamical astronomy is a wide one and it is obvious that it will be impossible to consider even in the most elementary manner all branches of it; for it embraces all those effects in the heavens which may be attributed to the effects of gravitation. In the most extended sense of the term it may be held to include theories of gravitation itself. Whether or not gravitation is an ultimate fact beyond which we shall never penetrate is as yet unknown, but Newton, whose insight into physical causation was almost preternatural, regarded it as certain that some further explanation was ultimately attainable. At any rate from the time of Newton down to to-day men have always been striving towards such explanation—it must be admitted without much success. The earliest theory of the kind was that of Lesage, promulgated some 170 years ago. He conceived all space to be filled with what he called ultramundane corpuscles, moving with very great velocities in all directions. They were so minute and so sparsely distributed that their mutual collisions were of extreme rarity, whilst they bombarded the grosser molecules of ordinary matter. Each molecule formed a partial shield to its neighbours, and this shielding action was held to furnish an explanation of the mutual attraction according to the law of the inverse square of the distance, and the product of the areas of the sections of the two molecules.

BEFORE his death Sir George Darwin expressed the view that his lectures on Hill's Lunar Theory should be published. He made no claim to any originality in them, but he believed that a simple presentation of Hill's method, in which the analysis was cut short while the fundamental principles of the method were shewn, might be acceptable to students of astronomy. In this belief we heartily agree. The lectures might also with advantage engage the attention of other students of mathematics who have not the time to enter into a completely elaborated lunar theory. They explain the essential peculiarities of Hill's work and the method of approximation used by him in the discussion of an actual problem of nature of great interest. It is hoped that sufficient detail has been given to reveal completely the underlying principles, and at the same time not be too tedious for verification by the reader.

During the later years of his life Sir George Darwin collected his principal works into four volumes. It has been considered desirable to publish these lectures together with a few miscellaneous articles in a fifth volume of his works. Only one series of lectures is here given, although he lectured on a great variety of subjects connected with Dynamics, Cosmogony, Geodesy, Tides, Theories of Gravitation, etc. The substance of many of these is to be found in his scientific papers published in the four earlier volumes.

Introduction.

An account of Hill's Lunar Theory can best be prefaced by a few quotations from Hill's original papers. These will indicate the peculiarities which mark off his treatment from that of earlier writers and also, to some extent, the reasons for the changes he introduced. Referring to the well-known expressions which give, for undisturbed elliptic motion, the rectangular coordinates as explicit functions of the time—expressions involving nothing more complicated than Bessel's functions of integral order—Hill writes:

I propose to take advantage of the circumstance that this is the first of the lectures which I am to give, to say a few words on the Mathematical School of this University, and especially of the position of a professor in regard to teaching at the present time.

There are here a number of branches of scientific study to which there are attached laboratories, directed by professors, or by men who occupy the position and do the duties of professors, but do not receive their pay from, nor full recognition by, the University. Of these branches of science I have comparatively little to say.

You are of course aware of the enormous impulse which has been given to experimental science in Cambridge during the last ten years. It would indeed have been strange if the presence of such men as now stand at the head of those departments had not created important Schools of Science. And yet when we consider the strange constitution of our University, it may be wondered that they have been able to accomplish this. I suspect that there may be a considerable number of men who go through their University course, whose acquaintance with the scientific activity of the place is limited by the knowledge that there is a large building erected for some obscure purpose in the neighbourhood of the Corn Exchange.

George Howard, the fifth child of Charles and Emma Darwin, was born at Down July 9th, 1845. Why he was christened George, I cannot say. It was one of the facts on which we founded a theory that our parents lost their presence of mind at the font and gave us names for which there was neither the excuse of tradition nor of preference on their own part. His second name, however, commemorates his great-grandmother, Mary Howard, the first wife of Erasmus Darwin. It seems possible that George's ill-health and that of his father were inherited from the Howards. This at any rate was Francis Galton's view, who held that his own excellent health was a heritage from Erasmus Darwin's second wife. George's second name, Howard, has a certain appropriateness in his case for he was the genealogist and herald of our family, and it is through Mary Howard that the Darwins can, by an excessively devious route, claim descent from certain eminent people, e.g. John of Gaunt. This is shown in the pedigrees which George wrote out, and in the elaborate genealogical tree published in Professor Pearson's Life of Francis Galton. George's parents had moved to Down in September 1842, and he was born to those quiet surroundings of which Charles Darwin wrote “My life goes on like clock-work and I am fixed on the spot where I shall end it.”

The scientific work of Darwin possesses two characteristics which cannot fail to strike the reader who glances over the titles of the eighty odd papers which are gathered together in the four volumes which contain most of his publications. The first of these characteristics is the homogeneous nature of his investigations. After some early brief notes, on a variety of subjects, he seems to have set himself definitely to the task of applying the tests of mathematics to theories of cosmogony, and to have only departed from it when pressed to undertake the solution of practical problems for which there was an immediate need. His various papers on viscous spheroids concluding with the effects of tidal friction, the series on rotating masses of fluids, even those on periodic orbits, all have the idea, generally in the foreground, of developing the consequences of old and new assumptions concerning the past history of planetary and satellite systems. That he achieved so much, in spite of indifferent health which did not permit long hours of work at his desk, must have been largely due to this single aim.

The second characteristic is the absence of investigations undertaken for their mathematical interest alone; he was an applied mathematician in the strict and older sense of the word. In the last few decades another school of applied mathematicians, founded mainly by Poincaré, has arisen, but it differs essentially from the older school. Its votaries have less interest in the phenomena than in the mathematical processes which are used by the student of the phenomena.