516 
FLUXIONS. 
metrical focus for all rays falling on all points of the 
fphere ; and d will be tof :: as m to n, and the rays after 
refraftion will diverge from O. And on the contrary 
rays converging to O, and falling on the lphere, will all 
foe accurately refracted to their focus L. 
uniformly over any diftance b in one fecond ; 
and let AB (x) denote the diftance defcended 
in any propofed time t ; which time let it be 
denoted by PQ^j making Bfcrxand C \jl~t 
£ 
-C 
d 
j - ~-e 
Ex. 6. Let parallel rays fall upon the fpheroid AM, thenitwi11 be, as r;*::*:(&) thediftancethat Q 
would be uniformly defcribed in i, with the „j_ 
i 
velocity at B: alfo i ; t:: the faid diftance 
let tranfverfe ~b, parameter —a, then zz—ax —-xx, and 
/ 
z%—\ax — —xx ; and fince d 
b 
is infinite, therefore the 
() equation for AO becomes n— 
r ma . ma 
vij—mx - -]—-—x 
2 0 —, whence f 
J 
ax — -xxflf—x 
will be found ; and in the vertex where x=o, mf—\ma 
„ btna 
~>fl, or f =-— 
J ’ m—n 
Cor. And to find if the fpheroid has a geometrical 
focus, we have from the above equation, « 2 X : ff 
—• - - 2 2 
max ma 
- ma X —7- mx +- v \ 
o b l 
— 2 fxflxx —ma | 
, a -mf- 
flax— jxx 
then equalling the coefficients of x and x 2 , n-ya —2/ 
ma —yincL ma 
zmf—ma X — — m , alfo nn —-— =2 — 
former equation 2m 
:m z a- 
2—2 
b 
277Z 2 <2 
■yf, or 2 m 2 — zn 2 
X- 
m- 
xma 
2/i2 m~- 
n 2 a, hence we find a—- 
m-a 
n b m- 
And from the fecond equation, we get likevvife a— 
m 2 — n -, 
711 •= 
-b\ tlierefore we may conclude that the fpheroid 
lias a geometrical focus, when a— 
by, m 2 — n 2 
p \ma mfln^ _,; 
bn 
but — =: diftance of the 
zm 
focus from the center ; therefore f— diftance of the re¬ 
moter focus from the vertex. And therefore in this 
cafe all parallel rays falling on all points of the fpheroid, 
will be accurately refra£ted to the further focus of the 
figure. And on the contrary, rays i(Tiling from the fur¬ 
ther focus of this fperoid and refratted at the furface, 
will all emerge parallel to the axis. 
OF DETERMINING THE MOTION OF BODIES AFFECTED 
BY CENTRIPETAL FORCES. 
Prop. LXVIII. — The motion, or velocity, acquired by a 
body freely defc endingfrom rejl , by theforce of an uniform gravity, 
is proportional to the time of its defeent ; and the Jpacc gone over , 
as thefquare of that time. 
143. The firft part of the propofttion is almoft felf- 
evident: for, iince any motion is proportional to tlie force 
by which it is generated, that generated by the force of an 
uniform gravity muft be as the time of defeent; becaufe 
the whole effeft of fucli a force, aiding equally every in- 
(lant, is as that time. 
Let, now, the velocity acquired during a defeent of 
one fecond of time, be fuch as would carry the body 
B 
(It) to bttizzx. By taking the fluent whereof 
we get b bt 2 —x. Therefore the diftance de- fS 
feended (|<f« 2 ) is as the fquare of the time. Q^_E. D. 
Otherwife, without fluxions. —Conceive the time ( PQ^) 
of falling through AB to be divided into an indefinite 
number of very fmall equal particles, reprefented, each, 
by m ; and let the diftance defcended in the firft of them 
be Ac, in the fecond cd, in the third de, 8 c c. See. Then, 
the velocity being always as the time from the begin¬ 
ning of the defeent, it will in the middle of the firft of 
the faid particles be defined by ; in the middle of 
the fecond by j in the middle of the third by z\ m * 
&c. &c. But, fince the velocity at the middle of any 
particle of time, is a mean between thofe at the two ex¬ 
tremes, or betwixt any other two equally remote from it, 
the correfponding particle of the diftance AB may, there¬ 
fore, be confidered as defcribed by that mean velocity. 
And fo, the fpaces Ac, cd, de, efl, &c. defcribed in equal 
times, being refpeftively as the faid mean celerities \m, 
1 \m, 2fin, 3i?72, Sec. it follows, by addition, that the dif- 
tances, Ac, Ad, Ac, Afl, 8 c c. gone over from the begin_ 
. m 4 m am j 6 m „ 
From the nin S are t0 one another as —j —, - > &c. or 1,. 
222 2 
4, 1 6, 25, &c. that is, as the fquares of the times, 
Q^E. D. 
Cor. i_ Since the diftance that might be uniformly 
run over in one fecond, with the velocity at B, is ex- 
prefled by bt, the diftance that might be defcribed with 
the fame velocity in the time t will therefore be expreffed 
by btxt, or bt 2 ; whence it appears, that the fpace AB- 
(\bt 2 ) through which the body falls in any given time t, 
is juft the half of. that which would be uniformly defcrib¬ 
ed with the celerity at B, in the fame time. 
Therefore, fince it is found from experiment, that a 
body near the earth’s furface (where the gravity may be 
taken as uniform) defeends about i 6 7 V feet in the firft 
fecond, it follows that the value of b (is in this cafe) — 
2X 32-1: and Qonfequentlv the number of feet de¬ 
fcended* in t leconds, equal to . 
Cor. 2. It is evident, whatever force the body de¬ 
feends by, the value of b will always be as that force; 
fince a double force, in the fame time, generates a double 
velocity ; a treble force, a treble velocity, &c. There- 
277 A a 
-72- 
b. 
and then 
fore, feeing our equation \bl 2 =zx, alfo gives —, and 
X 
b — it follows, 
2 1 
1. That the diftance defcended is, univerfally, as the 
force and the fquare of the time conjunftly. 
2. That the time is always as the fquare-root of the 
diftance applied to the force. 
3. And that the force is as the diftance applied to the 
fquare of the time. — And it may be further obferved, 
that, whatever is here faid with regard to the time, alfo 
holds in the velocity, being proportional to the time. 
Prop. LXIX .—To determine the velocity and tune of de¬ 
feent, of a body along an inclined plane AC. 
144. From any pointT, in AC, draw FE perpendicu¬ 
lar to the vertical line AD, and make Fi 3 and CD per¬ 
pendicular 
