Page:Popular Astronomy - Airy - 1881.djvu/300

286 of gravity at the bottom of a mine would be less than the force at the top. To show this, suppose that the mine reached half-way to the centre of the earth. Then (since the volumes of spheres vary as the cubes of their diameters) the quantity of matter nearer to the earth's centre than the bottom of the mine would be only one-eighth of the whole quantity of matter in the earth. But the attraction of a quantity of matter at the earth's centre would be more powerful on a body at the bottom of a mine than on one at the top, in the inverse ratio of the squares of the distances of the bodies from the earth's centre: that is in the present case in the ratio of four to one. Hence the attraction on a body at the bottom of the mine would be, on the whole, less than the attraction on a body on the top in the. ratio of one to two.

If, however, the earth be not of uniform density, but its density increase towards the centre, then though the attracting mass which acts on a body at the bottom of a mine be smaller, yet the diminution in the force of gravity so occasioned may be more than compensated by the comparative nearness of the attracted body to the denser parts of the earth. From the two laws of the attraction of spheres, which have been stated above, it is possible to calculate the ratio which the force of gravity at the bottom of the mine would bear to that at the top, on any supposition we choose to make as to the ratio which subsists between the mean density of the earth and the density of the surface; so that if we know one of these ratios we can immediately infer the other. Now, pendulum observations afford us the means of determining the force of gravity at any place, (page 248,) and therefore, if the times of