CENTRAL FORCES. 



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 peiform 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, 8cc. 

 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 thereof as 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 D, 

 (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. R, &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. 



From what has been said may be de- 

 duced the velocity and periodic time of 

 a body revolving in a circle, at any given 

 distance from the earth's center, by means 

 of its own gravity. Put ^-=16^ feet, the 

 space described by gravity, at the surface, 

 in the first second of time, viz. = A M j 

 then putting- r = the radius A C ; it is 

 AE =V A B X A M = ^/ TJ~r the 

 velocity in a circle at its surface in one 

 second of time ; and hence, putting c = 

 3.14159 &c.the circumference of the earth 

 being 2 cr = 25,000 miles, or 132,000,000 



/2r 



feet, it will be ^/ 2 g r : 2 cr :: I" : c<J 



= 5078 seconds nearly, or l h 24 m 38 s , the 

 periodic time at the circumference : also 

 the velocity there, or x /2j7"is =26,000 

 feet per second nearly. Then, since the 



