FLUXIONS. 
517 
A pendicular to AC, meeting AD in B and 
D. Becaufe (by the principles of mecha¬ 
nics) the force of gravity in the direction 
.n CF, whereby the body is made to defeend 
along the plane, is to the abfolme force 
thereof, as AF to AB, or as AC to AD; 
and fince (by Cafe i. Cor. 2.) the dii- 
tances defeended in equal times are as the 
sJD forces, it follows that the time of defeent 
through AF will be equal to the time of the perpendi¬ 
cular defeent through AB: and confequently the time 
of defeent through AC equal to that through AD; 
which is given by Prop, lxviii. Moreover, becaufe the 
velocities at F and B, acquired in equal times, are as the 
forces, or as A F to AB; and it appears from P . 1 . : i. 
that the velocity at E is to that f B, as -j/aE : y/ Ait, 
or as f A E x A B (:=AF) : p 7 a BX a B (=AB) it fol¬ 
lows, by equality, that the celerity at F is equal to that 
at E'; which is therefore given, by the preceding propo- 
fition. Q^_E. I. 
Cor. Hence the time of defeent along the chord AC 
of a femicircle ACD is equal to the time of defeent along 
the vertical diameter AD : and, if the chord DG be of 
the lame length with AC (its inclination to tire horizon 
being alfo the (ante) the time of defeent along it will alio 
be equal to that along the vertical diameter. 
Prop. LXX .—If, from two points A and D, equally re¬ 
mote from the center of attraction C, two bodies move, with 
equal celerities, the one along the right-line AC, the other in a 
curve-line D B their celerities, at all other equal di/lances 
from the center, will be equal. 
145. For, let CB and CF be any two fuch diftances ; 
let the arch BF be deferibed from the ,center C, and alfo 
cb, indefinitely near to it, tutting CB in 
-A. n : let the centripetal force at the dif- 
.F tance of CB, or CF, be reprefented by f 
and the velocity at B, by v. By the re- 
folntion of forces, the efficacy of the force 
(f) in the direction B b, whereby the ve¬ 
locity of the body is accelerated, will be 
-— xf. Draw the line CP perpendicular 
Bb 
to B b, and the triangle Bbn will be fimi- 
the force f, in the direction BP, being re- 
folved into two, viz. one in the direction CP, and the 
other in the cireCtion Bb, thefe forces will be propor¬ 
tional to thofe lines, when it will be as BP : Bb -.:/ ; 
X f— the force in the direction Bb-, but, by fimilar 
Bn 
triangles, BP : BC:: Bb : Bs ::/: ~gl'Xf Alfo the time of 
moving over B b (being as the diftance applied to the ve- 
locity) is reprefented by —: therefore the increafe of 
velocity, in moving through Bb, being as the force and 
#me conjunCtly, will be defined by — X f X — > or its 
D b v 
B 71 
equal -—X/• I n tfie fame manner, the velocity at F 
v 
being denoted by w, the time of falling through Fc will 
Fc 
be reprefented by —, and the velocity generated in that 
Fc 
time by —X/ ; which is to that 
w 
lar to BCP: 
BP 
£ 
falling through the arch B b % 
1 1 
as - to 
w v 
X/J acquired in 
Therefore, fee¬ 
ing the correfponding increments of velocity are always, 
reciprocally, as the velocities themfelves, it is rnanifeft, 
Vor,. VII. No 446, 
if thofe velocities are equal, in any two correfponding po- 
ficions of the bodies^, they will be fo in all others, being 
always increaled alike. But they are equal at A and D 
by fupftofition : therefore, &c. E. D. 
Prop. LXXI— To find the ratio of the velocities, and limes 
of defeent, of bodies, in curves-, the force of gravity being con- 
fidered as uniform. 
146. Let ARD reprefent a curve of any kind, along 
which a body defeends, by the force ^ 
of its own gravity from A; let AC, -— - - 
RB, &c. be parallel, and CD per¬ 
pendicular, to the horizon ; more¬ 
over, let Rtz touch the curve at R ; 
and let CB~tt, AR^at, and Rn~w. 
Since the points B and R (as well as C and A) may be 
looked on as equally remote from the earth’s center, (to 
which the gravitation tends,) the velocity acquired in 
defeendiug through the arch AR will ( y the lart propo¬ 
rtion) be equal to that acquired by falling freely through 
the right-li ne C B ; which luft velocity (by Prop, lxviii.) is 
always as y/CB (ora|). Therefore the celerity, whether 
the body moves in a right-line, or a curve, is always in 
the fubduplicate ratio of the perpendicular defeent ; and 
fo, the time in which R« (a>) would be uniformly de- 
zo 
feribed, with that celerity, will be univerfally as—; 
whofe fluent is as the time of falling through AR. 
Q^E.I. 
Ex. Let the curve ARD be any portion of the com. 
m<>n cycloid ; whereof the vertex is D and axis DC ; and 
whofe nature (putting DCinr, and the ray of curvature 
at D=2tf) is defined by the equation ua X DB = DR 2 . 
Here, we have DR (= y/fay y D B ) 2= f~Ia X c— u 2 i 
. „ _ . 1 — ■ 
whofe fluxion — 4/2 a X—--with a contrary fign, is 
c ^ 2 
zu 
■ku 
therefore —— Wz a y~7~ 
if V cu- 
the value of R?z or w ; 
17"' uu 
the loweft point D, where u be. 
be equal to 4/2 a multiplied by 
whofe fluent, at 
comes 222 c, will 
^3.14159, &c. ^ 
j half the meafure of the periphery of 
the circle whofe diameter is unity. Which fluent (and 
confequently the time of defeent) will therefore continue 
the fame, lee the arch DA be what it will. 
Prop. LXXII. —To determine the paths of projebltlcs near 
the earth's fur face ; ( negleEling the refijlance of the atmofphere.) 
J 47 - R et a body be projected from the point A, in the 
direction of the line AC, with a v locity {Efficient to carry 
it uniformly over the diftance d in the 
time/; and let the fpace through which 
it would freely defeend, by its own gra¬ 
vity, in that time, be denoted by b ; alfo 
let the fine of the angle of elevation B AC 
(to the radius r) be put —s, its co-fine 
222 c, and the diftance of the point A from_ 
the ordinate H m (conlidered as moving A H 
parallel to it'felf along with the body) ~x-, then, bv 
Trig. HG (perpendicular to AB) will be 2=—, and AG 
Becaufe the projeCtile is turned afide, Continually 
from a rectilinear path, by the earth’s attraction -t 
mufl: deferibe a curve-line AmEmB to which AC is a 
tangent at the point A : hut that attraction, aCting 
always in a direction (H m) perpendicular to the horizon 
can have no efteCt upon that part of the velocity with 
which the body approaches the line BC, parallel to H m - 
therefore the right-line HG (in which the body is always 
6 found) 
