490 
FLUXIONS. 
perfevcre in its motion, or which is the fame, to prevent 
any change in its motion, is called its inertia. 
Pp„op . XXVI .—To find the center of gyration of a body. 
60. Let a body be conceived to be made up of the par¬ 
ticles A, B, C, Sec. whofe diftances from the ax!is are 
a , b, c, Sec. and let x be the diflance of the center of gy¬ 
ration from the axis ; then by Art. 59. the inertia of A, 
B, C, See. will be as AX«% Bx^S Cxc 1 , &c. and the 
inertia of all the matter at the dilfance x will be as 
A^-B-f-C + . X^ 2 ; now as the moving force is the fame 
in both cafes, the inertia muftbe the fame when the fame 
angular velocity is generated; hence, + B +C-j- Sec. 
X x* — A X a 2 -h B X b* + C X o'- &c. therefore x zz. 
V 
\X« ! +Bx4 , -(-Cxc !! +, &c. 
that is, if s be the 
A + B-f-C-f, Sec. 
fluxion of the body at the diflance z from the axis, * = 
jflu. z*s. 
V - 
s 
Ex. r. Let the ftraight line CA revolve about C ; to 
^nd O the center of gyration. Put zzzCp, then szzz, 
and s—z, .•. z*s—z*z, 
y-_O_^ whofe fluent is -j z 3 = 
(when z=CA) A 3 ; 
hence, COzzfjCA* 
R 
/ 
5c 
r 
V 
-CA J 3 . 
Ex. 2. Let a circle AB re¬ 
volve in its own plane about 
its center C ; to find its cen¬ 
ter of gyration. Put fzz 
6,28318, Sec. the circumfe¬ 
rence of a circle whofe radius 
Jp> =z 1, 2 — C p •, then the 
circumference pq—pz, and 
pzzzzs ; hence the fluent of 
z-s, or of pz 3 z , is J pz^zz 
(whenz=CA=:?')|pr 4 . Alfo, 
the area of the circle = \pr*\ 
hence, CO = y a r*—rf%. 
Cor. The fame mud be true for a cylinder revolving 
about its axis, it being true for every fettion parallel to 
the end. 
Ex. 3. Let RADB be a fpliere revolving about the 
diameter RD ; to find its center of gyration. Draw CA 
I and spr parallel to RD ; put Crzzr, Cp—z, then pr 
~ \J r* —z 2 ; and if p—6, 28318, 8ec. the furface of the 
cylinder generated by sr revolving about RD, is pzx 
if r* — z -; hence, s zz ipzif r- — z 2 , and z*s—.2pz 3 z 
3/r 2 —z 2 . Now to find this fluent, put r 2 — z z zzy-, then 
x*zzr z —_y 2 > and z 4 =r 4 - 
_ _______ 
y 3 j ; hence, 2 pz 3 z\/ r 2 —z 2 = 2 px — r y a j'fj 4 j, whofe 
fluent is 2 px —iO'M -\y 5 > and when z—o this fluent 
ought to vanifh, but y is then —r, and the fluent be- 
comes 2 px — r? ? ' 5 > hence the correct fluent is 2 px 
tV’ 6 —i r2 J 3 +i> 5 5 an,d the whole fluent when z—r (in 
which cafej—o) will be fijpr*. Now the content of the 
fpherc = | pr 3 ; hence, CO zz\/^r 2 —rf^. 
On the CENTER of PERCUSSION. 
61. The center of percuflion is that point in the axis 
of a vibrating or revolving body, which flriking againfl: 
an immoveable obflacle, tIre whole motion is deftroyed, 
or the body inclines neither way. 
Prop. XXVII.— To find the center of percufion of a body. 
62. Let ARD be a plane paffing through the center 
of gravity G of the body, and perpendicular to the axis 
of fufpenlion which paffes through C ; and conceive the 
whole body to be projected upon this plane in lines per¬ 
pendicular to it, or parallel to the axis; then as each par¬ 
ticle is thus kept at the fame diflance from the axis, the 
efleit, from the rotatory motion about the axis, will not 
fee altered, nor will the center of gravity be changed. 
Let O be the center of percuflion, and draw pnw perpen¬ 
dicular to pC, andOa; perpendicular to pw ; alfo pv per¬ 
pendicular to Cn. As 
the velocity of any par¬ 
ticle pixpC, the mo¬ 
mentum of p in the di¬ 
rection pzooipXpC, it 
being as the velocity 
and quantity of matter 
conjointly ; and by the 
property of the lever, 
the efficacy of this force 
to turn the body about O is as pxpCxOwzz (becaufe 
On : Ow :: pC : <vG) pXvCxOn—pXvCxGo — Cti—pX 
vCx CO —p x vC x Cn— (as Cn : Cp :: Cp : vC) pXvCx 
CO-pxCp°. Now that the efficacy of all the par¬ 
ticles to turn the body about O may be =0, we 
mult make the fum of all the quantities pxvCx CO_. 
fum of all the quantities p x C p* — o ; hence, CO — 
fum of all the pxCp 2 fum of all the pxCp 2 
fum of all the pXv C bodyX CG ’ t..Te tu o 
denominators being equal from the property of the 
center of gravity, (Art. 56.) 
On thf. CENTER of OSCILLATION. 
63. The center of olcillation is that point in the axis 
of a vibrating body, at which if a particle were fnfpended 
from the axis of motion, it would vibrate in the fame 
time the body does. 
Prop. XXVIII .—To find the center of of dilation of a body. 
64. Let ABD be a body projected upon a plane per¬ 
pendicular to the axis ot rotation, as in Art. 62. the axis 
palling through C ; and let G be the center of gravity, 
O the center of ofcilla- 
tion ; draw Cv parallel to ^ 
the horizon, Om, G g, 
pr, perpendicular to it. 
Then by the property of 
the lever, the force of 
gravity to turn the par¬ 
ticle p about C oc/’X Cr; 
hence, the force of gra¬ 
vity to turn the whole 
body about Coc the fum 
of all the pxCr. Alfo, the force of gravity to turn a 
Angle particle O at O about CccOxwC. Now by 
Art. 59. the inertia of patpXpC -; and therefore the 
inertia of the whole body cc the fum of all the pXpC-. 
Alfo, the inertia ofOccOxOC 2 . Now that the acce¬ 
leration of the body about C may be equal to that 
of the particle O, the moving forces muft be in pro¬ 
portion to the inertiae ; becaufe, if the powers to pro¬ 
duce motion be as the powers to oppofe it, the accele¬ 
ration muft be the fame. Hence, fum of all pxCr : 
OxmC :: Jum of all pxCp’ 1 : OxOO, therefore 0 C=: 
fum of all pxCp^xCm fum of all pxCp 1 
fum of all pxCrxOC ~ body X CG b^anfe 
(by fim. triangles) Cm : CO :: Cg : CG, and therefore 
Cm Cg 
—-zz-——, and by the property of the center of gra- 
CO CG 
vity, Jum of all pxCr—lody x Cg. Hence, the cen¬ 
ter of ofcillation is the fame as the center of percuflion. 
Or if s be the body, * the diflance of s from the axis of 
_flu. x-s flu. x 2 s 
fufpenfion, then CO —-r =-. 
fl u. xs sx CG 
65. Join/>G; and draw Po perpendicular to CG ; then 
C/> 2 — CG 2 + G/> 2 — 2CG x Go, therefore pxQp-—px 
CG*-\-pxGp* —2CGXAXGo, and th e fum of all px 
C/y 2 — Jum of all pxCG-fi- Jum of all pXGp'—iCOx 
fum of all pxGo; but the jum of all pxGozz o, from 
the property of the center of gravity ; and the J'um of all 
pXCG* — body XCG 2 ; hence, fum of all fx^-P"" — 
body 
