of the rotation, at the end of the aforesaid time (_m’_). But, this angle being exceeding small, the tangent may be taken to represent the measure of the angle itself; and, if Z be assumed to represent the arch described by A, in the same time (_m’_) about the center O, we shall also have (_m’_/_m_) = (Z/AR) = (Z/AO), and consequently OA × (_F_/_ncf_) x (_m_/_m’_) = Z × (_F_/_ncf_). From whence it appears, that the angle expressing the change of the direction of the rotation, during any small particle of time, will be in proportion to the angle described about the axe of rotation in the same time, as _F_/_ncf_ is to _unity_. _Q.E.I._ Altho’, in the preceding proposition, the body is supposed to be a perfect sphere, the solution, nevertheless, holds equally true in every other species of figures, as is manifest from the investigation. It is true, indeed, that the value of _n_ will not be the same in these cases, even supposing those of _c_, _f_ and _F_ to remain unchanged; except in the spheroid only, where, as well as in the sphere, _n_ will be = ⅖; the momentum of any spheroid about its axis being 2-5ths of the momentum of an equal quantity of matter placed in the circumference of the equator, as is very easy to demonstrate. But to shew now the use and application of the general proportion here derived, in determining the regress of the equinoctial points of the terrestrial spheroid, let AE_a_F (_Fig. 2._) be the equator, and P_p_ the axis of the spheroid: also let HECF represent the plane of the ecliptic, S the place of the sun, and HAPNH the plane of the sun’s declination, making right-angles with the plane of the equator AE_a_F: then, if AK be supposed parallel, and OKM perpendicular, to OS, and there be assumed _T_ and _t_ to express the respective times of the annual and diurnal revolutions of the earth, it will appear (from the _Principia_, B. III. prop. xxv.) that the force, with which a particle of matter at A tends to recede from the line OM in consequence of the sun’s attraction, will be expressed by (_3tt_/_TT_) × (AK/OA) × _f_; _f_ denoting the centrifugal force of the same particle, arising from the diurnal rotation. Hence, by the resolution of forces, (_3tt_/_TT_) × (AK/OA) × (OK/OA) × _f_ will be the effect of that particle, in a direction perpendicular to OA, to turn the earth about its center O. [Illustration: FIG. 2.]