3. When two or more bodies perform their revolutions in equal times, but at different distances from the center they revolve about, the forces requisite to re tain them in their orbs will Ge to as the distance they revolve at from the center : for instance, if one revolves at twice the distance the other does, it will require a double force to retain it, &c.
4. When two or more bodies revolving at different distances from the centre are retained by equal centripetal forces, their velocities will be such, that their periodi cal times will be to each other as the square roots of their distances. That is, if one revolves at four times the distance another does, it will perform a revolution in twice the time that the other does ; if at nine times the distance, it will revolve in thrice the time.
5. And, in general, whatever be the dis tances, the velocities, or the periodical times of the revolving bodies, the retain ing forces will be to each other in a ratio compounded of their distances di rectly, and the squares of their periodical times inversely. Thus, for instance, if one revolves at twice the distance another does, and is three times as long in moving round, it will require two-ninths, that is, two-ninths of the retaining power the other does.
6. If several bodies revolve at differ ent distances from one common center, and the retaining power lodged in that center decrease as the squares of the distances increase, the squares of the periodical times of these bodies will be to each other as the cubes of their dis tances from the common center. That is, if there be two bodies, whose distances, when cubed, are double or treble, &c. of each other, then the periodical times will be such, as that when squared only they shall also be double, or treble, &c.
7. If a body be turned out of its rec tilineal course by virtue of a central force, which decreases as you go from the seat thereofas the squares of the distances increase ; that is, which is inversely as the square of the distance, the figure that body shall describe, if not a circle, will be a parabola, an ellipsis, or an hyperbola ; and one of the foci of the figure will be at the seat of the retaining power. That is,
if there be not that exact adjustment be tween the projectile force of the body and the central power necessary to cause it to describe a circle, it will then describe one of those other figures, one of whose foci will be where the seat of the retaining power is.
8. If the force of the central power de creases as the square of the distance in creases, and several bodies revolving about the same describe orbits that are elliptical, the squares of the periodical times of these bodies will be to each other as the cubes of their mean distances from the seat of that power.
9. If the retaining power decrease something faster as you go from the seat thereof (or, which is the same thing, in crease something faster as you come to wards it) than in the proportion mention ed in the last proposition, and the orbit the revolving body describes be not a circle, the axis of that figure will turn the same way the body revolves : but if the said power decrease (or increase) some what slower than in that proportion, the axis of the figure will turn the contrary way. Thus, if a revolving body, as ll, (fig. 11) passing from A towards B, de scribe the figure A D B, whose axis A B at first points, as in the figure, and the power whereby it is retained decrease faster than the square of the distance in creases, after a number of revolutions, the axis of the figure will point towards P, and after that towards. It, &c. revolving round the same way with the body ; and if the retaining power decrease slower than in that proportion, the axis will turn the other way.
Thus it is the heavenly bodies, viz. the planets, both primary and secondary, and also the comets, perform their respec tive revolutions. The figures in which the primary planets and the comets re volve are ellipses, one of whose foci is at the sun ; the areas they describe, by lines drawn to the center of the sun, are in each proportional to the times in which they are described. The squares of their periodical times are as the cubes of their mean distances from the sun. The secondary planets describe also circles or ellipses, one of whose foci is in the cen ter of their primary ones, &c.