Page:Proceedings of the Royal Society of London Vol 69.djvu/446

428 and in whatever new direct/ion the singular surface is propagated, the motion of the medium outside is such as represents equal and opposite momentum, as when a bubble is rising in a glass of water.

In exactly the same way, for inequalities resulting from an excess of grains, the momentum resulting from the displacement of the medium would be positive.

The principal stresses in the medium outside the singular surface of a negative inequality are to a first approximation two equal tangential pressures equal in all directions :

and a normal pressure :

the mean of these pressures being everywhere the mean pressure of the medium equal in all directions.

Efforts, proportional to the inverse square of the distance, to cause two negative inequalities to approach from a finite distance, are the result of those components of the dilatation (taken to a first approxi- mation) which are caused by the variation of those components of the inward strains which cause curvature in the normal piling of the medium.

The other components of the strain, being parallel distortions, which satisfy the condition of geometrical similarity, do not affect the efforts. If the grains were indefinitely small there would be no effort ; thus the diameter of a grain is the parameter of the effort, and, multiplying this diameter by the curvature of the medium and again by the mean pressure of the medium, the product measures the intensity of the effort.

The dilatation diminishes as the centres of the negative inequalities approach, and work is done by the pressure in the medium outside the singular surfaces to bring the negative inequalities together.

The efforts to cause the negative inequalities to approach correspond exactly to the gravitation, if matter represents negative mass.

Taking the mean density of the earth as - 5*67 as compared with water (-1),

The reciprocal of the density of the medium being 0'10~ 4 ,

The mean pressure of the medium .................. 1 -172 x 10 14 ,

<r the diameter of a grain .............................. 5 '534 x 10~ 18 ,

The mean radius of the earth .......................... 6-3709 x 10 s ,

the effort to cause approach between the earth and a unit of matter on the earth's surface ( - 1) is the product of these quantities multi- plied by 4:r/3, or :

- pa- x 10~ 4 x f TT x 5-67 x 6-3709 x 10 s = 9-81 x 10 2