Let F denote the given force, whereby the motion about the axis Pp is disturbed, supposing f to represent the centrifugal force of a small particle of matter in the circumference of the equator, arising from the sphere's rotation; and let the whole number of such particles, or the content of the sphere, be denoted by c: let also the momentum of rotation of the whole sphere, or of all the particles, be supposed, in proportion to the momentum of an equal number of particles, revolving at the distance OA of the remotest point A, as n is to unity.
It is well known, that the centripetal force, whereby any body is made to revolve in the circumference of a circle, is such, as is sufficient to generate all the motion in the body, in a time equal to that, wherein the body describes an arch of the circumference, equal in length to the radius. Therefore, if we here take the arch AR = OA, and assume m to express the time, in which that arch would be uniformly described by the point A, the motion of a particle of matter at A (whose central force is represented by f) will be equal to that, which might be uniformly generated by the force f, in the time m; and the motion of as many particles (revolving, all, at the same distance) as are expressed by cn (which, by hypothesis, is equal to the momentum of the whole body), will, consequently, be equal to the momentum, that might be generated by the force f × cn, in the same time m. Whence it appears, that the momentum of the whole body about its axe Pp is in proportion to the momentum generated in a given particle of time m', by the given force F in the direction AL, as ncf × m is to F × m', or, as unity to F/ncf × m'/m (because the quantities of motion produced by unequal forces, in unequal times, are in the ratio of the forces and of the times, conjunctly). Let, therefore, AL be taken in proportion to AM, as F/ncf × m'/m is to unity (supposing AM to be a tangent to the circle ABCD in A), and let the parallelogram AMNL be compleated; drawing also the diagonal AN; then, by the composition of forces, the angle NAM (whose tangent, to the radius OA, is expressed by OA × F/ncf × m'/m) will be the change of the direction 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 × 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 AEaF ([Fig. 2.]) be the equator, and Pp 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 AEaF: 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.
But it is demonstrated by Sir Isaac Newton, and by other authors, that the force of all the particles, or of all the matter in the whole spheroid AP ap, to turn it about its center, is equal to ⅕th of the force of a quantity of matter, placed at A, equal to the excess of the matter in the whole spheroid above that in the inscribed sphere, whose axis is Pp. Now this excess (assuming the ratio of π to 1, to express that of the area of a circle to the square of the radius) will be truly represented by 4π/3 × OP × (OA² - OP²); and, consequently, the force of all the matter in the whole earth, by 3tt/TT × AK/OA × OK/OA × 4π/15 × OP × (OA²- OP²). Let, therefore, this quantity be now substituted for F, in the general formula F/ncf, writing, at the same time, 4π/3 × OA² × OP, and ⅖, in the place of their equals c and n; by which means we have (here) F/ncf = 3tt/2TT × OA² - OP²/OA² × AK × OK/OA². Put the given quantity 3tt/2TT × OA² - OP²/OA² = k; and let the angle EAe represent the horary alteration of the position of the terrestrial equator, arising from the force F (here determined), and let the arch Ee be the regress of the equinoctial point E, corresponding thereto: then, in the triangle EAe (considered as spherical) it will be sin. e∶ sin. AE (∷ sin. EAe∶ sin. Ee) ∷ EAe ∶ Ee (= sin. AE x EAe/sin. E) = k × sin. AE/sin. E × AK × OK/OA² = k × sin. AE × cos. AH × sin. AH/sin. E. But in the triangle EHA, right-angled at A (where HA is supposed to represent the sun's declination, AE his right ascension, and HE his distance from the equinoctial point E[207]) we have (per spherics)
| sin. AE | ∶ | 1 (rad.) | ∷ | co-t. E | ∶ | co-t. AH, |
| (sin. AH)² | ∶ | (sin. EH)² | ∷ | (sin. E)² | ∶ | 1² (rad.²) |