FLUXIONS, 
405 
hence, S—pyz—paf a 2 -\-z 2 fi-C~pyz — pa 2 —paxf-C; but 
when x—o, 7=0, and S—o, therefore C— -pa 2 — o, and 
C —pa 2 ; lienee, S—pyz—pax the furface generated by 
the curve CF revolving about the axis CE. 
ON THE CENTER OF GRAVITY. 
56. If there be any number of bodies A, B, C, and G 
be their center of gravity ; and to any plane xy, perpendi- 
T/ 
PvT 
f 
G- \ 
dinate TF in N, 
the parts on each 
fide LR will al¬ 
ways balance 
each other, and 
therefore the 'co- 
_ dv will balance 
R. V itfelf upon LR; 
confequently the center of gravity mu ft be fomewhere in 
that line. Put LN— x, TN—7, TL— z, and draw xy pa¬ 
rallel to TF ; then if we conceive this body to be made 
up of an indefinite number of corpufcles, and multiply 
each corpufcle by its diftance from xy, the fum of all the 
products divided by the fum of all the corpufcles, or by 
the whole body, will give LG by Art. 56. Now to get 
the fum of all thefe products, we muft firft get the fluxion 
of the fum, and the fluent will be the fum itfelf. Put 
5 for the fluxion of the body at the diftance x from xy, 
then will ,vs be the fluxion of the’fum of all the products ; 
alfo, s is the fluxion of the fum of all the corpufcles; 
flu. xs 
therefore by Art. 56. LG -—. 
flu. s 
1 ft. If the body be an area , then .1—27^ by Art. 49 ; 
flu. 2 yxx flu .yxx , . , , , 
hence, LG=-- ld - If the bod y be » 
flu ,py*xx 
flu. py'x 
flu. 27* flu .yx 
fetid, then py 2 x—s by Art. 52 ; hence, LG= 
fl u .pyxx 
flu. y*x 
s — pyz. by Art. 55 
4>h. If th 
3d. If the body be the furface of a folid, then 
fl u .pyxz, fl tj .yxz. 
hence, .LG = n .— „ 
nu. pyz flu .yz 
body be a curve line FT, then s=zzz ; hence, 
Ex. 2. Let y—ax” 3 to find the center of gravity of the 
folid generated by the revolution of this curve about its 
flu.2X:i flu.YZ flu.-rz; 
LG=z . - 
fl 11.2a flu.z 2; 
Ex. 1. Let y—ax" be the equation to any parabola ; to 
find its center of gravity. Asy=zax", .‘.yxx—ax n + i x, 
whofe fluent is —-; alfo, yx=zax’‘x, whofe fluent is 
n-\- 2 
2 n-Y 1 k 4 -i 
x—grrH—;—X-v, 
"+1 ~ 
cx’ , + 1 , ax 
- -5 hence, (Art. 57.) LG=- 
71 - j-i 
ax' :+1 ' n-\-i ' 
If n —\, then y—ax 2 , .•. y 2 -a-x, which is the common 
parabola ; hence, LG—-|x. If n—i, tiien7=:ax, and the 
figure is a triangle ; hence, LG=^v. 
axis. As y 2 zz.a 2 x 2 " 
a 2 x 2’>.\-2 
- - alfo, y 2 x\ 
. y 2 xxzza 2 x 2 ’ 1 *’ 1 x, whofe fluent is 
zn-\-2 
hence, by Article 57. LG — 
21? 4-1 
a 2 x 2 ”x, whofe fluent is 
a*x*’-\-* 
X 
a 2 x 2 ' : +* 
2n-\-i 
21?—1 
2)1 4 - 2 
2iif-2 a”x 2 "A i 
X x. If n—\, the folid becomes a paraboloid, and 
If n— 1, the folid becomes a cone, and LG 
LG=|k 
=|v. „ 
Ex. 3. Let ALV be a hemifpheroid ; to find its center 
of gravity. Put LR=«, AR=;i; then a 2 : b 2 :: 2ax — x 2 
b 2 - . b 2 - 
: 7 2 =i—X 2 ax — x 2 ; hence, y 2 xx——-se 2 ax 2 x — x 3 x ,whofe 
a 2 n2 
fluent 
b 2 
culars AP, BQ, CR, GL, be let fall, then (by Vince’s Me- 
, . „ . , T . AxAP+BxBQ + CxCR 
chamcs, Art. 173.) LG=- A-'p flT -f-C' -- 
Prop. XXV .—To find the center of gravity of a body , con- 
fidcred as an area, folid, furface of a [olid, or curve line. 
57. Let ALV be any curve, RL the axis, in which 
the center of gravity muft lie ; for as it bifefts every or- 
tx- 
1S o X NX 
a 2 
b 2 - 
whofe fluent is — X ax% 
a 2 
\ax 3 — £x 4 
alfo, 
b 2 
and when x—a, LG 
y 2 x —: — X 2 axx — x 2 x, 
a 2 
hence, by Art. 37. LG 
for 
As this is independent of b, if bz=a, 
fame, and the folid becomes an hemi- 
ax 2 — j;x 3 
the whole folid. 
LG remains the 
fphere. 
Ex. 4. Let ARV be a femicircle ; to find its center of 
gravity. Put LN=x, TN=7, !L=r ; then x”-\-y”— r* ; 
hence, xx-\-yyz=zo, .-. 
yxx— —7*7,whofe fluent 
> is — %y 3 fi-C ,which muft 
vanifli when TF coin¬ 
cides with AV, or y—r ; 
therefore put r for y, 
and — |r 3 +C=:o, .’. 
C—^r 3 ; hence, the cor¬ 
rect fluent of yxx is 
alfo, the fluent of yx is (Art. 49.) the area 
■y 3 
• and when 
—iy 
ATNL ; hence, by Art. 57. LG=f X 
r 3 
ATNL’ 
7=0, LG; 
for the femicircle. 
3ARL 
Ex. 5. To find the center of gravity of the arc ARV* 
Put LN— x, NT=7, RTr:z; then, (Art. 46) 21:7:: r% 
x, therefore xzz=rj, whofe fluent is ry ; lienee, by Art. 
y y r 2 
57. LG—— ; and when y~r, LG^;——. The fame is 
z -K. A 
true for RV ; therefore the center of gravity of ARV lies 
at the fame diftance. 
Ex. 6. To find the center of gravity of the furface ARV 
of an hemifphere. Put x=RN,yz:TN, z=RT, and a— 
TL ; then (Art. 46.) we have z : x :: a : y, therefore yk 
—ax ; hence, yxz— axx, whofe fluent is ^ax” ; alfo, the 
fluent of yz, or ax, is ax; hence, by Art. 57. RG— 
X.QX 2 
- - x ; ar.d when x — RL=r, then RG=tr for the 
ax 
hemifphere. 
On the CENTER of GYRATION. 
58. The center of gyration is that point of a body re¬ 
volving about an axis, into which if the whole quantity 
of matter were collected, the fame moving force would 
generate the fame angular velocity in the body. 
59. Let a body / revolve about C, and let a force aft 
at D to oppofe its motion. Then the momentum of p 
varies as p X its velocity, or as / x pC, which we may 
confider as a power 
afting at p in oppo- _;_ q_ 
fition to the force j? 1 
at D ; but this power 
afting at the diftance pC from the center of motion, its 
effeft to oppofe a force at D muft (by the property of the 
lever) be as py.pCXpC—py.pC\ This effeft of p to 
perfeverc 
