FLUXIONS. 
519 
in a circle at its furface, will be f 2 rd ■, and the time of 
3.14159. &c. X. J r _, iUKQi 0 c. x 
revolution =2 
: 3 ' 59 > 
V 2)'/ 
which two expreffions, becaufe r is — 
(feconds) 
21000000 feetand </=i6 t V, " ill therefore be nearly equal 
26000 feet and 5075 feconds, refpeftively. Let R be now 
put for the radius of any other circle defcribed by a pro¬ 
jectile about the earth’s center : then, becaufe the force 
of gravitation above the furtac.e is known to vary accord¬ 
ing to the fquare of the diftance inverfly, vve have (by 
_i _L F )' 
Cafe 4, Cor. 5.) r 2 : R 2 :: (26000/ the velocity (per 
I r 
fecond) at the furface, to 26000 X "Vp } the velocity 
in the circle whofe radius is R : and 
R 2 ' 
the periodic time at the furface : to 5075X 
■J 
■J_ 5 ° 7 
R 3 
the 
time of revolution in the circle R. Which, if R be af- 
-fumed equal to.(6or) the diftance of the moon from the 
s. D 
earth, will give 2360000, or 27.3 nearly, for the periodic 
time at that diftance. 
In like fort the ratio of the forces of gravitation of the 
moon, towards the fun and earth, may be computed. 
For, the centrifugal forces in circles, being univerfally 
as the radii applied to the fquares of the times of revo¬ 
lution, it will be as ^' 8 ‘ 00o0Q °^ the femi-diameter of 
the magmis orbis divided by the fquare of one year (the 
periodic time of the earth and moon about the fun) is to 
(240000 x 178) the diftance of the moon from the earth 
divided by —the fquare of the periodic time of the 
17S 
tnoon about the earth, fo is 1.9 to 1 nearly ; and fo is the 
gravitation of the moon towards the fun to her gravita¬ 
tion towards the earth. 
Alfo, after the fame manner, the centrifugal force of a 
body at the equator, atifing from the earth’s rotation, is 
derived. For iince it is found above, that 5075 feconds is 
the time of revolution, when the centrifugal force would 
become equal to the gravity,, and it appears (by Cafe 2. 
Cor. 2.) that the forces, in circles having the fame radii, 
are inverfly as the fquares of the periodic times, we there¬ 
fore have, as 86i6cl 2 (the fquare of the number of feconds 
H M _ 
in; (23 56) one whole rotation of the earth) to 5075) 2 (the 
fquare of the number of feconds above given) io is the 
289 
force of gravity (which we will denote by unity to 
the centrifugal force of a body at the equator arifing from 
the earth’s rotation. 
But, to determine, in a more general manner, the 
ratio of the force of a body revolving in any given 
circle, to its gravity., we have already given 3.14, &c. x 
j— 
*!—-for the time of revolution at the furface of the 
V d 
earth, when the gravity and centrifugal force are equal : 
therefore, if the time of revolution in any circle w hofe 
radius is a, be denoted by t, it follows, from Cor. 2. laft 
r a 
Prop, that, 
T^Al Z &c. X-r 12 
a 
— :: the gravity of the 
body: to its centrifugal force in that circle; which, 
r . 3U41 2 ^c. X 
therefore, is as unity to —■ - ; or as 1 to 
’ J dt 2 
1.228 x -p very nearly ; where a denotes the number of 
feet Lathe radius of the propofed circle, and 't the number 
of feconds in one entire revolution. So that, if the 
length of 9. fling, by which a ftone is whirled about, be 
two feet, tjnd the time of revolution -*• of a fecond, the 
force by which the (tone endeavours to fly off, will be to 
its weight as 9.824 to unity. 
From this general proportion, the centrifugal force and 
periodic time of a pendulum deferibing a conical furface 
may likewife be deduced.—For, let SR, S 
the length of the pendulum, be denoted by 
g; the altitude CS of the cone, by c ; the 
femi-diameter CR of the bafe by a ; and the 
time of revolution by t : then the force of 
gravity being reprefented by unity, the force with which 
the revolving body at R, the end of the pendulum, 
tends to recede from the center C, will be defined by 
3.14, i/c.] 2 X 2a 
--——- ’ as has been already (hewn. Therefore, 
becaufe the body is retained in the circle RR by the adfion 
of three different'powers, i. e. the centrifugal force 
3.14, &c .| 2 x 2(2 \ 
- : - - -Jin the direflion CR, the force of 
dt 2 J 
gravity (1) in a direction parallel to SC, and the force of 
the thread or wire RS, compounded of the former two ; 
it follows, from the principles of mechanics, that as SC 
(c) to CR (»), fo is the weight of the body at R, to the 
force with which it adds upon the thread or w ire RS ; and 
3.14, &c. I 2 X la 
dt 2 
CS (c) : CR (a) : whence 
dt 2 —3. 14, Cde ,\ 2 x 2c, and 7=3.14,^. ~ 1 > l0S 
nearly. Becaufe dt 2 , or its equal 3.14, (3c. | 3 X 2c, 
expreftes the fpace a heavy bony will defeend, by its 
Own gravity, in the time t ; and fince 1 2 : 3.14, GV.) 2 :: 
2c : 3.14, <S?c.] 2 X 2 c (—dt 2 ) ir therefore appears that, as 
the fquare of the diameter iff any circle, is to the fquare of 
its periphery, fo is twice the perpendicular altitude of the 
cone, to the diftance a heavy body will freely defeend in 
the time of ore whole gyration of the pendulum, let the 
bafe of the cone and the length of the pendulum be what 
they will. 
Prop. LXXI V. —To determine the ratio of the velocities of 
bodies defending, or afeending , in right-lines, when accelerated, 
or retarded, by forces, varying according to a given law. 
149. Suppofe a body to move in the right-line CH, and 
let the force whereby it is urged towards C, or H, be as 
any variable quantity F : moreover, let the velo- -tjj 
city of the body be reprefented by v ; putting its \ 
D 
diftance CD, from the point C—x, and D d=x. 
Then, fince the time wherein the fpace Da? ( x ) 
would be uniformly defcribed, with the velocity at 
D, is known to be as -, the velocity that would be — ^ 
v 
uniformly generated, or deftroyed, in that time by the 
force F (being as the time and force conjundlly ) will con- 
F jc 
fequently be .as—— : which therefore muft be equal to, 
the uniform increafe or decreafe of celerity in that 
time ; and confequently ± vv—Tx. From whence, 
when tire value of F is given in terms of x, or », the 
value of v will likewife be known. Q^E. I. 
Cor. x. Hence, the law of the velocity being given, that 
of the force is deduced : -for, fince F.v= ± vv, it is evb. 
vv 
dent that F = ± — 
x ’ 
Cor. 2. Hence, alfo, the ratio of the velocity at D to 
th*t 
