The book attempts to draw together the various strands of evidence that have led to present knowledge of the distribution of density throughout the interior of the Earth. Details are also given of other properties with which the density is closely linked, including pressure, compressibility and compression, rigidity, seismic velocities, Poisson's ratio and gravitational intensity. Questions of thermodynamics and chemical composition and phase enter discussions where they bear more or less sharply on the density determination; but the book does not purport to be a comprehensive text on the Earth's internal tem- perature distribution and composition. The density distributions of other planets are discussed. The quest for clues on the Earth's internal densities has long been an exciting one, and an aim of the book is to present a developing story which has fascinated the author over much of his worl(ing life. The early chapters refer to key developments from ancient times to around 1930. The later chapters recount, in greater detail, develop- ments since 1930. The stage has now been reached where numerous published papers bearing on the Earth's density seem to do little more than fidget around the resolving power of long accumulated observational data. So the present seems an appropriate time to try to put some perspec- tive into the story. Of course, the story will never be quite finished: in describing the interior of the Earth, there will always be extra decimal places to add as further significant evidence arises.

Table of Contents:
1. Noted ancient investigations.- 1.1 Size of the Earth.- 1.2 Contributions of Newton and contemporaries.- 1.3 The period after Newton.- References.- 2. Determination of the mean density of the Earth.- 2.1 Bouguer’s experiments in Peru.- 2.2 The Schiehallion experiment.- 2.3 Other pendulum and plumb-line experiments.- 2.4 The Michell-Cavendish experiment.- 2.5 Other small-scale experiments.- References.- 3. Spherical harmonics.- 3.1 Solutions of Laplace’s equation.- 3.2 Legendre polynomials.- 3.3 Integrals of products of spherical harmonics.- 3.4 Tesseral harmonics.- 3.5 Expansions iQ spherical harmonics.- 3.6 Spherical harmOnICs and Earth oscillation theory.- References.- Theory of the Earth’s gravitational attraction.- 4.1 General theorems on gravitational attraction.- 4.2 Attractions due to some particular mass distributions.- 4.3 MacCullagh’s formula.- 4.4 Representation of the Earth’s surface.- 4.5 Attraction due to spheroidal Earth model.- References.- 5. The figure and moment of inertia of the Earth.- 5.1 The geopotential function.- 5.2 Forms of surfaces of equal density inside rotating Earth model.- 5.3 Relations involving ?a and h.- 5.4 Clairaut’s equation on the internal variation of ?.- 5.5 Estimation of the moment of inertia of the Earth.- 5.6 Numerical results on the hydrostatic theory.- 5.7 Use of artificial satellites.- 5.8 International reference systems.- 5.9 Ellipticities of internal surfaces of constant densitys.- References.- 6. Early models of the Earth’s density variation.- 6.1 Earth models.- 6.2 Clairaut’s equation and the density problem.- 6.3 The Legendre-Laplace density law.- 6.4 Background theory in density determination.- 6.5 Other early model density laws.- 6.6 Numerical results for early models.- 6.7 Some furthernineteenth century results.- 6.8 Early evidence on the Earth’s rigidity.- 6.9 Early twentieth century models.- 6.9 Early twentieth century models.- 7. Representation of elasticity in the Ear.- 7.1 Stress.- 7.2 Strain and rotation.- 7.3 Model stress-strain relations.- 7.4 Stress-strain relations for the Earth.- 7.5 Interpretation of coefficients in perfect elasticity.- 7.6 Strength of a material.- 7.7 The terms ‘fluid’ and ‘solid’.- 7.8 Pressure and finite strain.- 7.9 Thermodynamic considerations.- References.- 8. Seismic wave transmission.- 8.1 Earthquakes and other sources of seismic waves.- 8.2 Equations of motion of seismic disturbances.- 8.3 Bodily seismic waves.- 8.4 Scalar and vector potentials.- 8.5 Surface seismic waves.- 8.6 Refraction and reflexion of bodily seismic waves.- References.- First approximation to seismic P and S distributions in the Earth.- 9.1 Seismic rays.- 9.2 Effect of the Earth's ellipticity on seismic travel times.- 9.3 Normal and abnormal seismic velocity variation.- 9.4 Bodily seismic phases.- 9.5 Evolution of travel-time tables.- 9.6 Derivation of P and S velocity distributions in the Earth.- References.- 10. Earth models of type A.- 10.1 Introductory theory of density variation in the Earth.- 10.2 Historical background on the Earth’s internal layering.- 10.3 The regions A, B, C, D, E, F and G.- 10.4 Density near the Earth’s surface.- 10.5 Early evidence on inhomogeneity insitle the upper mantle.- 10.6 Minimum central density.- 10.7 Earth models of type A.- 10.8 Corrections for temperature and inhomogeneity.- Critique of A-type models.- References.- 11. Evidence on compressibility in the Earth.- 11.1 Compression in the Earth.- 11.2 Variation of incompressibility in homogeneous regions.- 11. 3 Some further implications offinite-strain theory.- 11.4 Compressibility-pressure hypothesis.- 11.5 Theory for inhomogeneous regions.- 11.6 Degrees of inhomogeneity in particular regions of the Earth.- 11.7 Solidity of the inner core.- 11.8 Earth models of type B.- References.- 12. Some second approximations.- 12.1 P and S velocities in the mantle.- 12.2 Structure of the outer core.- 12.3 Structure of the inner core.- 12.4 Radius of the Earth’s core.- 12.5 The Earth’s central density.- 12.6 Further evidence bearing on rigidity in lower core.- 12.7 Improved B-type models.- References.- 13. Evidence from seismic surface waves.- 13.1 Underlying principles in applying surface-wave data.- 13.2 More complex model structures.- 13.3 Direct observation of phase velocities.- 13.4 Allowance for Earth’s curvature and gravity.- 13.5 Evidence on crustal structure.- 13.6 Evidence on mantle structure.- 13.7 Surface waves and density variation.- 13.8 Further remarks.- References.- 14. Evidence from free Earth oscillations.- 14.1 Free oscillations of a dynamical system.- 14.2 Approach to the theory of Earch oscillations.- 14.3 Equations of motion of an oscillating Earth model.- 14.4 Solving the equations of motion.- 14.5 Observational data.- 14.6 Early inferences from free Earth oscillation data.- 14.7 The model HB1.- 14.8 Other models using free Earth osciIlation data.- 14.9 Oscillation evidence on solidity of inner core.- References.- 15. Miscellaneous developments.- 15.1 Equations of state, and related equations, for the Earth’s interior.- 15.2 Some miscellaneous Earth models.- 15.3 Monte Carlo techniques.- 15.4 The general problem of ‘inverting’ observation data.- 15.5 Density and seismic wave amplitudes.- 15.6 Implications of wave-scattering investigations.- 15.7 Deviations from sphericalsymmetry.- 15.8 Changes in gravitational constant.- References.- 16. Optimum and standard Earth models.- 16.1 General requirements of Earth models.- 16.2 Consequence of non-uniqueness.- 16.3 Approaches to the optimum model problem.- 16.4 Progress towards an optimum Earth model.- 16.5 The problem of a standard Earth model.- References.- 17. Application to other planets and the Moon.- 17.1 Planetary observational data.- 17.2 Assumptions on the Earth’s internal composition.- 17.3 Earth, Venus, Mars.- 17.4 Mercury.- 17.5 Moon.- 17.6 Jupiter and Saturn.- 17.7 Uranus and Neptune.- 17.8 Pluto.- 17.9 Further remarks.- References.