Of the Precession of the Equinox ,

forces proportional to the dijlances from the said line , such that theforce aSling at a given difiance a , may be expressed by'a givenquantity y ; it is required to find the whole efficacy of all theseforces , to turn the ellipse about its center O.

If BC be supposed parallel to 66, intersecting OT, perpen-dicular thereto, in D ; then the force with which a particle, atany place V in that line, is urged in the direction wV parallel

to OD, will be expressed by ^ x Vw, or ^ x 00; and con-sequently it’s efficacy to turn the ellipse about its center by

— x OD X Ow, or — x OD x DV. Let there be taken Cv —

a a

DV; and the efficacy of a particle at v will, in like manner,be had equal to ^ x OD x Dv; which, added to that of the

5. former particle at V, gives -J x OD x DC. Therefore, seeing

the joint action of any two particles in DC, equally distantfrom the middle one I, is expressed by the fame quantity

— x OD x DC, the efficacy of all the particles must conse-quently be equal to that quantity drawn into half the numberof the particles; and so is truly expounded by — xiODxDC 1 .

By the fame argument, the force*of all the particles in the lineBD to turn the ellipse about its center, the contrary way, will

be — x yOD x BD 1 . Therefore the difference of these two

a

values, — x ~0D x BD 1 — CD% is the whole force of all the

a

particles in the line BC, to turn the ellipse about its center(downwards) ; which expression, if Ff the conjugate dia-meter to MN be drawn, bisecting BC in E, will become

— x qOD x BD4-CD x BD —CD = 1 x OD x BC x DE.

Put, now, OF=r, OM = d, FH (perpendicular to MN) —f,OH —and let OE and OD, considered as variable, be de-noted by x and y, respectively. Then, by the property of

the