CENTRAL FORCES. 
distaniees. That is, by how many times 
greater the distance a body revolves at is 
from the center, so many times less force 
will retain it. 
3. When two or more bodies perform 
their revolutions in equal times, but at dif- 
ferent distances from the centre they re- 
volve about, the forces requisite to retain 
them in their orbs will be to each other 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. Wiien two or more bodies revolving 
at dilferent distances from the center are 
retained by equal centripetal forces, their 
velocities will be such, that their periodical 
times will be to each other as the square 
roots of their distances. That is, if one re- 
volves 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 tlirice 
the time. 
5. And, in general, wliatever be. the dis- 
tances, the velocities, or the periodica! 
times of the revolving bodies, the retaining 
forces will be to each other in a ratio com- 
pounded of their distances directly, and the 
squares of their pefilSdical 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 dilferent 
distances from one common center, and 
the retaining power lodged in that center 
decrease as the squares of the distances in- 
crease, the squares of the periodical times 
of these bodies will be to each other as the 
cubes of their distances 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 periodi- 
cal times wilt be such, as that when squared 
tsnly they shajl also be double, or treble, 
&c. 
7. If a body be turned out of its rectili- 
neal course by virtue of a central force, 
which decreases as you go from the seat 
thereof, as the squares of the distances in- 
crease; that is,' which is inversely as the 
square of the distance, the figure that body 
shall describe, if not a circle, will be a para- 
bola, 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 between 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 tlie 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 some- 
thing faster as you go from the seat there- 
of (or which is the same thing, increase 
something faster as you come towards it) 
than in the proportion mentioned 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) somewhat 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 describe the figure ADB, 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 increases, 
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 retain- 
ing power decrease slower than in that pro- 
portion, the axis will turn the other way. 
Thus it is tiie heavenly bodies, viz. the 
planets, both primary and secondary, and 
also the comets, perform their respective 
revolutions. The figures in which the pri- 
mary planets and the comets revolve are 
ellipses, one of whose foci is at the sun : the 
areas they describe, by lines di awn 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 ellip.ses, one of whose foci is in 
the center of their primary ones, &c. 
From what has been said may be deduced 
tlie velocity and periodic time of a body re- 
volving in a circle, at any given distance from 
the earth’s center, by means of its own gravi- 
ty. Put g = 16Jj feet, the space described by 
gravity, at the surface, in the first second 
of time, viz. =: A M ; tlien, putting r = the 
radius A Cj it is AE = y/AB x AM = 
\/agr the velocity in a circle at its surface 
in one second of time ; and hence, putting c = 
