before, but all of the same density, viz. 1-100th part of that of superficial rock. The numbers then are

Subtract each line from the line below (except the 3rd line)

and we obtain the following

The horizontal dimensions of the spaces will be somewhat contracted in passing downwards owing to the convergence of the sides towards the centre of the earth: but the densities from the distribution downwards in slender prisms of uniform mass will increase in a corresponding degree: and the masses of the spaces will be all the same.

The last change we shall make is this. We shall increase the density of the semi-cubic space as its depth increases, so as to make it 1-100th part, not of the superficial density as at present, but of the density of the earth's mass at the centre of the space.

If D be the density of the surface, a the earth's radius, the usually received law of density of the interior is

2aD

density at depth d=24) sin (5a-d),

a

6

EFFECT OF EXCESS OR DEFECT IN MASS BELOW. 65

when d = 50, 150, 250, 950 miles, this gives the ratio of the density at these depths to the superficial density = 1.17, 1.21, 1.35, 2.39. Multiply the deflections last found by these numbers, and we have finally

The defect or excess in density which we have taken, viz. 1-100th, might have been chosen larger, and the deflections proportionably increased. For there are many kinds of rock, as granite, which differ so in density in the different specimens that the difference between the extremes is greater even than 1-10th of the mean. And if this difference exists at the surface, it does not seem to be improper to suppose that great variations may exist also below, from the effect of the cooling down and solidifying of the crust, even much greater than 1-100th.

66. We have taken a semi-cubic space as our example: but the same result is true of a space of the same volume and of any form so long as its dimensions in one direction are not much larger than in another. This follows from Art. 64.

P. A.

5

FIGURE OF THE EARTH.

INTRODUCTION.

67. It is easy to show in a general way, that the earth is a more or less spherical mass.

The globular form is seen in the shadow which the earth casts on the moon in eclipses in a variety of positions. The comparison of the distance at which ships at sea lose sight of each other's decks, with the height of the decks from the water, shows all over the world that the sea is of a globular form; and an approximation to the diameter of the globe is thus obtained by simple geometry. The distance of the horizon at sea as seen from cliffs and hills, the height of which is known, leads to the same result. The distance north and south between two places, measured, for instance, by a perambulator, is always found to be nearly in proportion to the difference of latitude; this could not be the case, if the curve of the meridian were not nearly circular.

After it was known that the earth is of a globular form, Newton was the first who demonstrated that it is not a perfect sphere. From theoretical considerations and also from the discovery that a pendulum moves slower at the equator than in higher latitudes, he arrived at the conclusion that its form is that of an oblate spheroid-the form being derived from rotation in a fluid state. This subject we propose now to consider. We shall in the first Chapter treat it on the hypothesis that the Earth was a fluid mass when it assumed its present general form. The calculation is one of great difficulty, and would indeed be impracticable did we not know that the figure differs but little from a sphere. In the second Chapter we shall show how the actual form is found by geodesy.

CHAPTER I.

THE FIGURE OF THE EARTH CONSIDERED AS A FLUID MASS.

§ 1. The Earth considered to be a fluid homogeneous mass.

As a first approximation we shall inquire whether a homogeneous fluid mass revolving about a fixed axis can be made to maintain a spheroidal form according to the laws of fluid pressure.

PROP. A homogeneous mass of fluid in the form of a spheroid revolves with a uniform velocity about an axis: required to determine whether the equilibrium of the surface left free is possible.

=

68. Let a and b be the semi-axes of the spheroid referred to three axes of rectangular co-ordinates, b being that about which it revolves: also let ba2 (1-e). The forces which act upon the particle (xyz) are the centrifugal force and the attraction of the spheroid parallel to the axes: these latter are given in Art. 12, and are

Let these be represented by Ax, By, Cz. Let w be the angular velocity of the rotation, then w√x2+ y2 is the cen

trifugal force of the particle (xyz), and the resolved parts of it parallel to the axes of x, y, z are wx, w3y, 0. Hence X, Y, Z, the forces acting on (xyz) parallel to the axes, are

X = − (A — w2) x, Y=-(B—w2) y, z=- Cz.

These make Xdx + Ydy + Zdz a perfect differential, and therefore so far the equilibrium is possible.

The equation of fluid equilibrium gives

is the equation to the surface; and this is a spheroid, and therefore the equilibrium is possible, the form of the spheroid being properly assumed. The eccentricity is given by the

condition

fugal force at the equator to gravity at the equator. Hence

By expanding in powers of e and neglecting powers higher