518 
FLUXIONS. 
found) will continue to move uniformly towards BC, the 
fame as if gravity were not to ait ; and the diftance Gm 
defcended from the tangent'AC, by means of ihe attrac¬ 
tion, will be the very fame as if the body was to defcend 
from reft: along the line GH. This being premifed, it is 
R 
have, as F X — :/X 
V 
whence (multiplying the 
antecedents by —and the confequents by ® } it will be 
R 
”) ■■ 
evident, that as d : AG^-^ 
defcribing Am ; and, as t- 
■ (*») 
the time of 
F 
V 2 
R 
therefore the forces are as the 
d 2 
fpace (Gm) through which a body would freely defcend 
s x bv^ 
in that time (by Prop. Ixviii.) Hence 
csd 2 x — br-x 2 . 
is a general value for the ordinate Hm: 
csd 2 
~ c 2 d 2 
putting which =:o, we get x. 
.. 2 2\ fqtiaresof the velocities direilly, and as the radii invcrfely. 
. b . f j the Otherwife .—Let the indefinitely little arch AB be the 
v j diftance that the body moves over in a given or con- 
ftant particle of time ; and let the centripetal force at B 
be meafured by twice the fubtenfe or fpace AE through 
which the body is drawn from the tangent A Dun that 
time. Then, by the nature of.thecircle, AB 2 :s:.\Hx 
AB 2 
AE =3 AC X zAE, and confcquently zAE — 
c 2 d 2 
by 
tude of the projeftion. 
br 2 
to nothing, we have x— 
■ — AB ~ the anxpli- 
But, by putting its fluxion equal 
csd 2 
AC 
therefore, the force is as the fquare of the velocity applied 
to the radius of the circle, (as before. ) 
2 br 2 
/Inch fubftituted for x in 
s 2 d 2 
the value of Hat, gives —- for the altitude DE of the 
4 br 2 
projection. CL_E. I. 
Cor. If another body be projected, with the fame ce¬ 
lerity, in the vertical diredfion AS ; then, s becoming 
( S y 2 
Cor. i. Becaufe, F : f 
v :: RF : \/ rf, and R : t 
L., it follows that V : 
r 
Cor. 2. 
by that of ] 
being as 
the altitude of that projeftion 
d 2 
come —7= AS 
4 b 
0 
will be- 
^ 4 br 2 ^ 
which call k, and let this value be fub¬ 
ftituted in thofe of AB and DE, and they will become 
4/ics fhs 2 
-p r and —— relpectively. 
Hence, if from the point Q^yvhere the line of direction 
V r f : 
V 2 
R~ 
V 2 v 2 . 
' "f ' 7 
If the ratio of the periodic times be denoted 
P to p ; then the ratio of the velocities V, v, 
to I, we (dialI have, by equality, 5/RF : 
P 
R 
pi 
whence alfo F 
P 
and Iv 
R 
T 
FP 2 ,fp 2 . 
Cor. 3. If the meafure of the force, or the velocity that 
might be uniformly generated in a given time (r), be ex¬ 
pounded by any power a" of the radius AC (a) ; then 
the diftance through which a body would freely defcend 
in the fame time, by that force uniformly continued, 
will be exprefled by \a*. Therefore, the diftances de- 
AC cuts a femicircle defcribed upon AS, the lines SQ^ fcended, by means of the fame force, uniformly con 
’ ' ' ■ - - tinned, being as the fquares of the times, it is evident, if 
the time of moving through AB be denoted by t, that the 
diftance AE defcended in that,time, will be denoted by 
ft ■ —.___• 1 1 
— Xls" : and fo we Audi have AB (p2AExAC) inl¬ 
and QP be drawn, the latter perpendicular to AB, the 
the triangles ASCfjmd AQP being finrilar, we fhall have 
r:-s:vk (AS) : i^ = ACL, 
r 
r:s::tt (AQJ : t !A — PQ= DE 
* r r 2 
*JL (aqj 
sc/i 
— AP— 4 -AB. 
h 
Prop. LXXIII.— To determine the ratio of the forces, 
•whereby bodies, tending to the centers of given circles , are made 
to revolve in the peripheries thereof. 
148. Let ABH and abh be any two propofed circles, 
- whereof let AB and ab be fimilar 
arcs; in which, let the velocities 
Jf of the revolving bodies be refpec- 
/j\ tively as V to v ; make DBK and 
'y dbk parallel to the radii AC and 
ac, putting AC— R, ac=.r , and the 
ratio of the centripetal force in 
ABH to that in abh, as F to f 
It is plain, becaufe the angles ABD and abd are equal, 
that the velocities at B and b, in the directions BK and 
bk, with which the bodies recede from the tangents AD 
and ad, are to each other as the abfolute celerities V and 
v. But thofe velocities, being the effetfts of the centri¬ 
petal forces acting in correfponding, fimilar, directions 
during the times of defcribing AB and-u 5 , will therefore 
be as the forces themfelves when the times are equal ; 
but when unequal, as the forces and times conjunCtly. 
Therefore, the times being univerfally as '^-5 to—, or 
X a 2 ; which being the diftance defcribed by the re¬ 
volving body in the time t , it follows that the fpace gone 
over in the given-time (1) will be a 2 : which, there¬ 
fore, is the true meafure of the celerity in this cafe. The 
fame conclufion might have been derived in much fewer 
words from Cor. 1. but, as a thorough underftanding 
hereof is abfolutely neceflary in what follows hereafter, 
I have endeavoured to make it as plain as poflible. 
Cor. 4. Hence the time of revolution is alfo derived; 
» +1 
for it will be as a 2 : 3.14x59, &c. x *a (the whole 
3.14, &c. X za, 
periphery) :: 1 : --— or 3.14591 
»+1 
1-» n 2 
(Sc. X 
as— to 
V „ 
(becaufe the arcs AB and ab are fimilar) we 
2 a 2 , the true meafure of the periodic time. 
Cor. 5. Therefore, if n be expounded by r, o, —*r, 
—2, and —3, fuccefiively, then the velocitycorrefponding 
will be as a, ai, 1, a i> and a ‘ ; and the time of 
revolution, as 1, ai, ai, and a 2 , refpeClively. 
Scholium.— From the preceding propofition, and its 
fubfequent corollaries, the velocity and periodic time of 
a body revolving in a circle, at any given diftance from 
the earth’s center, by means of its own gravity, may be 
deduced : for let d be put for the fpace through which a 
heavy body, at the furface of the earth, defeends in the 
firft fecond of time, then 2 d will be the meafure of the 
force of gravity at the furface : and therefore, the radius 
of the earth being denoted by r, the velocity, per fecond, 
