on theory of 785. A short digression here on the theory of the potential, Digression and particularly on equipotential surfaces differing little from potential. concentric spheres, will simplify the hydrostatic examples which follow. First we shall take a few cases of purely synthetical investigation, in which, distributions of matter being given, resulting forces and level surfaces (§ 487) are found; and then certain problems of Green's and Gauss's analysis, in which, from data regarding amounts of force or values of potential over individual surfaces, or shapes of individual level surfaces, the distribution of force through continuous void space is to be determined. As it is chiefly for their application to physical sea level. geography that we admit these questions at present, we shall occasionally avoid circumlocutions by referring at once to the Earth, when any attracting mass with external equipotential surfaces approximately spherical would answer as well. We shall also sometimes speak of "the sea level" (§§ 750, 754) merely as a "level surface," or "surface of equilibrium" (§ 487) just enclosing the solid, or enclosing it with the exception of comparatively small projections, as our dry land. Such a surface will of course be an equipotential surface for mere gravitation, when there is neither rotation nor disturbance due to attractions of other bodies, such as the moon or sun, and due to change of motion produced by these forces on the Earth; but Level surit may be always called an equipotential surface, as we shall see tively to (§ 793) that both centrifugal force and the other disturbances centrifugal referred to may be represented by potentials. face rela gravity and force. of sea level than aver under 786. To estimate how the sea level is influenced, and how Disturbance much the force of gravity in the neighbourhood is increased or by denser diminished by the existence within a limited volume under- axe matter ground of rocks of density greater or less than the average, let us ground. imagine a mass equal to a very small fraction, 1/n, of the earth's whole mass to be concentrated in a point somewhere at a depth below the sea level which we shall presently suppose to be small in comparison with the radius, but great in comparison with 1/n of the radius. Immediately over the centre of disturbance, the sea level will be raised in virtue of the disturbing attraction, by a height equal to the same fraction of the radius of sea level than aver under ground. Intensity Disturbance that the distance of the disturbing point from the chief centre by a denser is of n times its depth below the sea level as thus disturbed. age matter The augmentation of gravity at this point of the sea level will be the same fraction of the whole force of gravity that n and direc- times the square of the depth of the attracting point is of the square of the radius. This fraction, as we desire to limit ourselves to natural circumstances, we must suppose to be very small. The disturbance of direction of gravity will, for the sea level, be a maximum at points of a circle described from A as centre, with D/2 as radius; D being the depth of the centre of disturbance. The amount of this maximum deflection will be √3a2/nD2 of the unit angle of 57°296 (§ 41), a denoting the earth's radius. tion of gravity altered by underground local ex cess above average density. B Let C be the centre of the chief attracting mass (1— n ̄1), and which is the equation of the equipotential surface in question. It gives This expresses rigorously the positive or negative elevation of the disturbed equipotential at any point above the undisturbed surface of the same potential. For the point 4, over the centre of disturbance, it gives which agrees exactly with the preceding statement: and it proves the approximate truth of that statement as applied to the sea level when we consider that when BP is many times BA, CP – a |