relative to plane rectangular and circular plates. Ex- amples, with Tables and Diagrams, of Polar Har- monics of the sixth and seventh orders, Digression on the theory of the Potential. Disturbance by matter underground, of greater than average density. Harmonic spheroidal levels. Parallel mountain-ridges. Practical conclusions, Potential through space determinate when its value is given over every point of a surface. Determina- tion from this datum. Determination from form of approximately spherical equipotential surface sur- rounding the mass. Resultant force; for gravity alone; for gravity and centrifugal force. Clairault's Theorems. Figure of sea level determinable from measures of gravity. Results of geodesy, . Hydrostatic examples. Liquid with a centrobaric nucleus. Mutual attraction of portions of the liquid omitted -Example (1.)-Centrifugal force. Example (2.) -Attraction of distant body. Example (3.)-For Tides. Results of ordinary equilibrium theory. Equilibrium theory of the Tides corrected for extent of sea, attraction of water neglected. Lunar and Solar Augmentation of results by mutual gravitation of water. Stability of the ocean. Local influence of high water on gravity. Centrifugal force incompetent to ex- plain the actual ellipticity of sea-level, Equilibrium of rotating spheroid of heterogeneous liquid. Laplace's hypothetical law of the earth's interior density. Dynamical origin of Precession and Nuta- tion. Precession gives information as to the distri- bution of the earth's mass, while surface-gravity does Constant of Precession derived from Laplace's Abrupt changes of interior density-Case of two non- Rigidity of the earth. Shows that the geological hypo- earth. Influence on the tides in the surface-liquid. SECTIONS 831 832-846 Effects of elastic yielding on Precession: not yet dis- APPENDIX D.-On the Secular Cooling of the Earth. 847-849 DIVISION I. PRELIMINARY NOTIONS. CHAPTER I.- KINEMATICS. 1. THERE are many properties of motion, displacement, and deformation, which may be considered altogether independently of such physical ideas as force, mass, elasticity, temperature, magnetism, electricity. The preliminary consideration of such properties in the abstract is of very great use for Natural Philosophy, and we devote to it, accordingly, the whole of this our first chapter; which will form, as it were, the Geometry of our subject, embracing what can be observed or concluded with regard to actual motions, as long as the cause is not sought. 2. In this category we shall first take up the free motion of a point, then the motion of a point attached to an inextensible cord, then the motions and displacements of rigid systems-and finally, the deformations of surfaces and of solid or fluid masses. Incidentally, we shall be led to introduce a good deal of elementary geometrical matter connected with the curvature of 1.nes and surfaces. point. 3. When a point moves from one position to another it must Motion of a evidently describe a continuous line, which may be curved or straight, or even made up of portions of curved and straight lines meeting each other at any angles. If the motion be that of a material particle, however, there can be no such abrupt changes of direction, since (as we shall afterwards see) this would imply the action of an infinite force. It is useful to consider at the outset various theorems connected with the geometrical notion of the path, described by a moving point, A |