FLUXIONS. 
497 
body XCG 2 -}- fum of all /xG^ ! ; confequently CO 
body X CG'-f fum of dll p X Gp 2 
body x CG 
:CG- 
fum of all p X G/j- 
body X CG 
i fum of all pyGp 2 
hence, GO= — -————. Now as the numerator 
body x CG 
is conftant, GO varies inverfely as CG ; hence, if we 
find GO for any one value of CG, we fiiall know every 
other value of GO from that of CG. 
Cor. If x be the diftance from C to the center of 
gyration; then by Art. 6o. x-s— fum of all pyCp--• 
and by Art. 64. COxixCG= fum of all p X Cp "; 
hence, x 2 — CO x CG, and CG : x :: x : CO. 
Ex. 1. Let CD be a ftraight line fufpended at C ; to 
find the center of of- 
—j-— cillation. Put \-Cp-, 
then the fluent of x 2 s 
(when CD) 4CD 3 . Alfo, body 
X CG—CDxiCD—J-CD® ; hence, CO=§CD. 
Ex. 2. Let the line AB vibrate lengthways in a ver¬ 
tical plane about C, which is equidiftant from A and B; 
to find its center of ofcillation. Draw CG perpendicular 
to AB ; and put CG =a, Gp—x •, 
then pC 2 —a 2 -\-x 2 ; and the fluent 
U 
= flu. 
Ex. 5. Let CG be perpendicular to the plane of the 
circle ABV, and let the circle vibrate about an axis paf- 
fing through C and parallel 
to AB ; to find the center 
of ofcillation. Draw GPV 
perpendicular to AB, and 
EF parallel to AB. Put 
AG—r, CG—a, G P—x, 
then CP ! =a ! +.v', PE = A' 
3/ r- —x 2 , and EF = 2 
\/r" —x 2 ; hence, EFxCP 2 
=ra 2 -|-x 2 X 2 \Jr 2 — x 2 , which 
multiplied by x gives a 2 x-\-x 2 x x *V r ‘—* 2 for f ' ie 
fluxion of the fum of the products of each particle of the 
area ABFE multiplied into the fquare of its diftance 
from the axis of vibration. Now to find the fluent, we 
have the fluent of a"- x 2 \Jr 2 —x 2 , x x—a 2 X area ABFE 
by Art. 49. and when x—r, the fluent —a'xAVB ; and 
as the fame is true for the other femicircle, the whole 
fluent is a 2 X circle. The fluent of the lecond part, 
2x 2 x f r*— x 2 , may be found thus. Let x f r 2 — x 2 — A, 
x 9 -x 3/ r 2 — x 2 , — B, and x X r 2 —x^I = P ; then by tak- 
of Cp-ys —fluent of a 2 x-\-xfx — ing the fl llx ; on of the 1 aft, we have P,— xyr 2 
■0 
2a 2 y AG-ff AG 3 
a 2 x-f^x 3 — (when x—AG) a 
X AG-j-jAG 3 ; hence, for the 
B ‘whole line AB, it becomes 2a 2 y 
AG-f §AG 3 . Alfo, body xCG 
=«X A B—«x 2 AG ; hence, CO 
AG 2 
' d 
3 x *x 3/ r 2 —x 2 — xy r' —x 2 X f r 2 —x 2 —f 2 x f r 2 — x 2 — 
r2X{/r 2 — x 2 —4 x 2 x\/r 2 — x 2 , that is, P — r- A—4 B, 
hence, (by taking the fluents) P=r 2 A—4 B, and B 2=2 
r 2 A—P . - . .-. r 2 A—P 
-therefore the fluent of 2 x 2 x\Jr 2 — x‘ is 
ay 2A G 3 a 
Ex. 3. Let DAE be any parabola vibrating flatways, 
or about an axis parallel to PMN ; to find the center of 
ofcillation. Put AC—d, AM— x, 
PM =y t then ax"-=iy ; hence, 2yx= 
2ax r x=zs; and the fluent of CM 2 
ys, or 2.d-\-x* X ax”x, or 2.d 2 ax"xf- 
2 d 2 ax”+ i 
a r dax" J r l x-\-2ax’ ,J o 2 x, is —yyy -b 
4 
r 2 A 
4 dax '’+ 2 
+ 
72-^— I 
which vaniflies 
n-\-2 ' n-f 3 
when x—o, and therefore it wants no 
correction. Alfo, the fluent of 
. _ 2dax” J r i . 2 ax "+ 2 . . c 
CMxr, or d-\-xy 2ax"x is —7—--, — » l iei 'ce, it 
the former be divided by the latter, we ge t (by red uction) 
CO— n-\-2.n-ft.d 2 f-n -\-1 ,n-\-2- 2dx-\-n + 1 .n-\-2.x- . 
n + 2. n + 3. d\n + 1 . n-\- 3. x 
If d— o, and n— 1, the figure becomes a triangle, and 
AOzz-jt. 
If n— i, it becomes the common parabola, and 
AO— 4 x. 
Ex. 4. Let the parabola vibrate edgeways, and let it 
be fufpended at A ; to fine! the center of ofcillation. By 
Ex. 2. the fum of the produCtsof each particle of the line 
PN into the fquare of its diftance from A, is 2x 2 yy+f )) 3 
~2x 2 yax”+^a 3 x 3 ^ hence, 2ax“+ 2 x-\-%a 3 x 3 ”x is the 
fluxion of the fum of the products for the whole body ; 
but when x=r 9 P—o : and tlie fluent becomes 
2 
r 2 
— X circle, becaufe A :=r \ of the circle when x=r; and 
o 
r 2 
for both femicircles it becomes — x circle; hence, the 
— 1 -— . 4 
whole fluent is a 2 -\-\r 2 y circle, which is the fum of the 
products of each particle of the circle X the fquare of its 
diftance from the axis of vibration. Alfo, a y circle =: 
the denominator for the value of CO ; hence, by divid- 
7 ~ 2 
ing the former by the latter, we get CO— a-\ -. 
43 
Ex. 6. Let the folid formed by the rotation of any 
curve DAE about its axis AB, vibrate about C ; to find 
the center of ofcillation. By Ex. 
5. the fum of the products of 
each panicle of the circle MN 
into the fquare of its diftance from 
the axis = CP 2 -)-aPN 2 X cir¬ 
cle MN = CF+lPN^X/'XPN 2 
(/> being —3.14159, &c.) —/> X 
CP 2 X PN 2 + i PN 4 — p x 
a+xl* yy 2 +iy 4 ; hence, px y 
d-\-x~yy 2J r\y^'^ the fluxion of the 
film of all Inch produfts for the D f 
whole body ; the fluent of which 
divided by CGxbody, gives CO. 
C 
G 
P- 
\ 
- 
23 X ’'+ 3 2 a 3 X 
3 v 3 » + 1 
whofe fluent is , -- 
r t 3 
AMxs is the fame as before, d being now —o ; hence, 
n-\-2.x a 2 .n-\-2.x 2 "— 1 
Ex. 7. Let the folid be a paraboloid ; to find the cen- 
Alfo, the fluent of terofofcillatio n. Hereg.v—y 2 ; he nce, pxydfx^yy^f-yy^ 
is equal to pxyd-fx 2 yax-\-%a 2 x z , whole fluent is ^pitWx* 
AO — 
n+3 
3-3 n + 1 
5 X 
If ?it is the common parabola; and AO_ — + —. 
a triangle. If a — 1, 
If n 
AO: 
3f 
4 
a a r 
j -for 
4 
-\-£padx 3 -\-ipax*-\-frpa 2 x 3 ; alfo (Art. 52. Ex. 1.), the 
body —±pax 2 ; and Art. 57. Ex. 2.) AG=|v; .'.CG— 
d -p fv ; lienee CG X body = ^padx 2 f^pax 3 ; dividing 
therefore the above fluent by this quantity, we have CO 
_ 6d 2 -\-%dxf-T,x 2 T ax ' 
(>d-\-o,x. 
If C coincide with A, d—o, and CO=:lilLf. 
AO— x. 
Vol.VII. No. 444. 
'6 L 
1 v. 
