22

Of the Precession of the Equinox ,

Fig. 12 . Let OAP ap be the earth, under the form of an oblate sphe-roid ; let AIæL be the plane of its equator, and HICL a planepassing thro' S the center of the fun, making right-angles withthe plane of the meridian HAP Cp and with the plane GG. Itis found, in Prob. I. that the force whereby a particle, at anypoint E in the equator AELæI, tends from the plane GG, is inproportion to that respecting the highest point A, as the di-stance EF to the distance AK, or as ED to AO (supposing EFparallel to AK, and ED to AO); whence it is evident that theforce on the particle at E, in a direction perpendicular to theplane of the equator, must be to the force on a particle at A,in the like direction, in the very fame ratio of ED to AO, thatis, in the ratio of the cosine of the arch AE to the radius.But this, by Corol. I. Lem. II. appears to be the law of theforces under which a ring of particles AELæI, detached fromthe earth, may continue in equilibria , in the fame plane, undera twofold motion about the center O, and about the line A a asan axis.

Imagine now this ring to be exceeding dense, so that itsmomentum of rotation about its center O, may be equal to thatof the earth itself, or so that the two bodies may equally en-deavour to persevere in the fame state and direction of motion,in opposition to any new force impressed. Then it is evident,that, were the forces whereby the two bodies tend to turnabout the line LI, through the fun’s attraction, to be also equal,the same effect, or alteration of motion, would be produced inboth ; and consequently, that the effects produced, when theforces applied are unequal, will be in proportion directly as theforces. Now the force whereby a particle at A is urged from

the plane GG, is found to be fi x x ^ (by Prop. I. Co-rol. I.) ; which, in a direction perpendicular to the plane of theequator or ring, will be (6 x x XO X AO = * X TT X mn '

Therefore, the force acting on a particle at E, in a like di-rection, being expressed by fi x -1^ X mn x the effect

thereof to turn the ring about the line IL will be expressed

by