FLUXIONS. 
_ Therefore, feeing no impulfe, however great, can affeft 
the quantity of the area defcribed above the center S, in a 
given time, and becaufe the areas ASB, BSC, defcribed 
about that point, when no force a£ts, are as the bafes AB, 
BC, or the time's of their defcription, the propolition is 
manifeft. 
Cor. Hence the velocity of a revolving 
body, at any point Qjir R, is inverfely as 
the perpendicular SP or ST, falling-from 
the center of force upon the tangent at 
that point. 
For, let two other bodies m and n be 
fuppofed to move uniformly from Q^and 
R, along the tangents QP and RT, with 
velocities refpeftively equal to thofe of the revolving 
body at Q^and R ; then the diftances Qw and Rn, gone 
over in the fame time, will be to each other as thofe velo¬ 
cities ; and the areas QSw and RSrc will be equal, being 
equal to thofe defcribed by the revolving body in the 
fame time : whence QwX SP being — Rnx ST, it follows 
that Qm: Rn :: ST : SP :: JL : _L- 
Prop. LXXV 1 .— To determine the law of the centripetal 
force , tending to a given point C, whereby a body may deferioe a 
given curve AQH. 
152. Let QP be a tangent to the 
ctirveat any point Q^_; upon which, 
from the center C, let fall the per¬ 
pendicular CP ; put CQr:i, CP=?r ; 
and let the velocity of the projectile 
at Ch_be denoted by v. 
Therefore, fince v 2 is always as 
-L (by the Corol. to Lemma ) it is evi¬ 
dent, by taking the fluxions of both quantities, that vv. 
- II 
will alfo be as-: but the centripetal force, whether 
u 3 
the body moves in a right-line or a curve, is always as 
D 27 %i 
• - r . Therefore the centripetal force is likewife as —- 
^ u 3 s . 
The fame otherwife.—Let the ray of curvature QO be 
denoted by R : then, becaufe the centripetal forces in cir¬ 
cles are known to be as the fquares of the velocities di¬ 
rectly and the radii inverfely, it follows that the force,tend¬ 
ing to the point O, whereby the body might be retained 
in its orbit at Q__, or in tire circle whole radius is QO, will 
will be defined by 
— X -pr: whence, (by the refolution 
u 2 K 
of forces) it will be CP ( u) : CQ^(s) 
t< 2 R 
(the force in 
the direction QO) : —— 
^ t< 3 R 
the force in the direction QC : 
which, becaufe R= — will alfo be expreffed by—r- 1 
<f_E. I. 
Another way.—Let nq be the indefinitely fmall part of 
the right-line C q, intercepted by the curve and the tan¬ 
gent Q ^q, exprefling the effeCt of the centripetal force in 
the time of deferibing the area QC n. Now thele efteCts, 
or the dittances defcended by means of forces uniformly 
continued, are known to be in the duplicate ratio of the 
times, or of the areas denoting thofe times : therefore, 
the centripetal force at , or the diftance defeenued by 
means thereof in a given time, will be as nq applied to 
^ n( I 
the fecond power of the area Q Cq } eras Cw/s’ 
This expreflion. is the fame with that given by lir Ilaac 
Newton, in his Priticipia, Booki. Prop, 6, But, toadapt 
Vol. VII. No. 446. 
521 
it to a fluxional calculus; let QF be an. ordinate to the 
principal axis AG; and let (as iifual) AE—.v, EQ^v, 
he (or ) — x, (fjpzzz ; fuppofing eq (pa¬ 
rallel to EQ^) to interfeCt the curve and the tangent in 
m and q. 
Since Q__i q is conceived indefinitely fmall (or in its 
nafeent date),the triangle nmq may be taken as rectilineal 5 
alfo the angle » = CQJ? and the. angle m~Q^qt : whence, 
it will be (by trigonometry) as S. CQP- (//) : S. QoC 
(«0 :: mq : nq ; that is, as m q : nq — 
: which fubflituted above gives- 0 
C p X^ b cpsxq:^ 
for the meafure of the centripetal force at Q_: but mtf 
(fuppofing x to flow uniformly) is known to be as — -y : 
CO X O/x—v 
therefore the force at Q^, is as •—^— --,oritsequat 
cP*xCl „? 3 
- j y 
-r— ; where the divifor (u 3 £ 3 ) is as the cube of (QC n\ 
u 3 z 3 
the fluxion of the area AQC. 
The very fame theorem may likewife be deduced from 
that given by our fecond method : for, fince (R) the ray 
of curvature at Q^Js univerfally zz: the value of 
x y 
s 
——, (there found) will here, by fubftitution, become 
u s R 
__ —sxy ' 
u 3 z 3 
This expreflion, though in appearance lefs fimple thar\ 
u 
-firfl found, is, for the general part, more common 
u 3 s 
dious in practice. 
Cor. 1. If the point C be fo remote that all right-lines 
drawn from thence to the curve may be confidered as 
parallel to each other, the force will then (making 
——“SXV 
perpendicular to Cq) be as - -f -- ... or barely 
■ xy 
tA^XQr ) 3 
fince s (CQ^_) in this cafe may be rejefled. 
- 
From this expreflion, which is general, in all cafes 
where the force aids in the direction of parallel lines, it 
appears that the force, which always acting in the direc¬ 
tion of the ordinate QE, would retain the body in its 
— v 
orbit, is every where as —— ; becaufe Q£ Here coincides 
x 2 
with QJE, and becomes = x. 
Cor. 2. Becaufe the force, tending to the point C, is 
CQ_ 
univerfally as—~~ 
point c, will, by the fame argument, be as ±— 
cp l x '4.0° 
Hence the forces, to different centers C and c (about 
which equal areas are defcribed in the fame time) are t<? 
cp 3 c 7 i\ ■ 
each other in the ratio of —— to — inverfely. 
Cor. 3. Moreover, the ratio of -the velocity at Q^t® 
the velocity whereby the body might revolve in a circle 
about the center C, at the diftance CQ^., ,js egfily deduced 
from hence : for, fince the celerity at Qjs that whereby 
the body might revolve in a circle about the center O, 
and the forces tending to the centers O and C are to each 
other as u (CP) and s (CQ^) ; it therefore follows, if the 
(or J the force to any other 
ratio fought be aft'umed as v to w, that 
i (by Art. 148.) Y/hence alfo v 2 : u> 2 ; 
6 R 
QO ' QC 
: ax Oil (nR) : 
SXQ^ 
