Note 118, [p. 44]. The force of gravity, &c. Gravity at the equator acts in the direction Q C, fig. 30. Whereas the direction of the centrifugal force is exactly contrary, being in the direction C Q; hence the difference of the two is the force called gravitation, which makes bodies fall to the surface of the earth. At any point, m, not at the equator, the direction of gravity is m b, perpendicular to the surface, but the centrifugal force acts perpendicularly to N S, the axis of rotation. Now the effect of the centrifugal force is the same as if it were two forces, one of which acting in the direction b m, diminishes the force of gravity, and another which, acting in the direction m t, tangent to the surface at m, urges the particles towards Q, and tends to swell out the earth at the equator.

Fig. 30.

Note 119, [p. 45]. Homogeneous mass. A quantity of matter, everywhere of the same density.

Note 120, [p. 45]. Ellipsoid of revolution. A solid formed by the revolution of an ellipse about its axis. If the ellipse revolve about its minor axis Q D, fig. 6, the ellipsoid will be oblate, or flattened at the poles like an orange. If the revolution be about the greater axis A P, the ellipsoid will be prolate, like an egg.

Note 121, [p. 45]. Concentric elliptical strata. Strata, or layers, having an elliptical form and the same centre.

Note 122, [p. 46]. On the whole, &c. The line N Q S q, fig. 1, represents the ellipse in question, its major axis being Q q, its minor axis N S.

Note 123, [p. 46]. Increase in the length of the radii, &c. The radii gradually increase from the polar radius C N, fig. 30, which is least, to the equatorial radius C Q, which is greatest. There is also an increase in the lengths of the arcs corresponding to the same number of degrees from the equator to the poles; for, the angle N C r being equal to q C d, the elliptical arc N r is less than q d.

Note 124, [p. 46]. Cosine of latitude. The angles m C a, m C b, fig. 4, being the latitudes of the points a, b, &c., the cosines are C q, C r, &c.

Note 125, [p. 47]. An arc of the meridian. Let N Q S q, fig. 30, be the meridian, and m n the arc to be measured. Then, if Zʹ m, Z n, be verticals, or lines perpendicular to the surface of the earth, at the extremities of the arc m n they will meet in p. Q a n, Q b m, are the latitudes of the points m and n, and their difference is the angle m p n. Since the latitudes are equal to the height of the pole of the equinoctial above the horizon of the places m and n, the angle m p n may be found by observation. When the distance m n is measured in feet or fathoms, and divided by the number of degrees and parts of a degree contained in the angle m p n, the length of an arc of one degree is obtained.