CEOTER. 
icoMe. As the cone consists of an infinite 
number of circular areas, which may be 
considered as so many weights, the center 
ef gravity may be found as before, by put- 
ting B E = r (fig. 2) B G = a, the circu- 
lar area D 1' E = y, and A G C = 6 ; and 
from the nature of the cone, a? :x^::b:y = 
— • but X v= — ’ = fluxion of the weights ; 
■ a‘ 
and vxx = ^ = fluxion of the momen- 
ta, whence the fluent of the latter, viz, 
, ’ 4 f) ^3 
— divided by the fluent of the former 
4 “ 
will give for the center of gravity of the 
part D B E F, consequently the center of 
gravity of the cone A B C G is distant from 
the vertex B | of the side B G, in a circle 
parallel to the base. 
To find the center of gravity in a paralle- 
logram and parallelepiped, draw the diago- 
nal AD and E G (fig. 3,) likewise C B and 
H F ; since each diagonal A D and C B 
divides the pallelogram A C D B into two 
equal parts, each passes through the center 
of gravity ; consequently the point of inter- 
section, I, must be the center of gravity of 
the parallelogram. In like manner, since 
both the plane CBFHandADGE divide 
the parallelopiped into two equal parts, 
each passes through its center of gravity, so 
that the common intersection IK is the 
diameter of gravity, the middle whereof is 
the center. After the same manner may 
the centre of gravity be found in prisms and 
cylinders, it being the middle point of the 
right line that joins the center of gravity of 
their opposite bases. 
The center of a gravity of a parabola is 
found as in the triangle and cone. Thus, 
let B F in the parabola ABC (fig. 4) be 
equal to x, D E = y, then will y x be the 
fluxionary weight, and y x x the fluxion of 
tlie momenta ; but from the nature of the 
curve we have y = X 2 ; whence y x = x^ x, 
1 . 2 ~ . 
and yxx — x^xx, whose fluent r divid- 
ed - the fluent of x^ x will give ^ x = - 
S o o 
B F for the distance of the center of gravity 
from the vertex B in the part of D B E ; 
and so I of B G is that center in the axis 
of the whole parabola ABC from the ver- 
tex B. 
The center of gravity in the human body 
is situated in that part which is called the 
pelvis, or in the middle between the hips. 
For the center of gravity of segments. 
parabolic, conoids, spheroids, &c. we refer 
to Wolfius. 
Center of gravity of two or more bodies, 
a point so situated in a right line joining the 
centers of these bodies, tliat if this point be 
suspended the bodies will equiponderate 
and rest in any situation. In two equal 
bodies it is at equal distances from both : • 
when the bodies are unequal it is nearer to 
the greater body, in proportion as it is 
greater than the other ; or the distances 
from the centers are iriVersely as the bodies. 
Let A (fig. 5,) be greater than B, join A B, 
upon which take the point C, so that C A : 
CB;:B ; A, orthat A X CA = B X CB ; 
then is C the center of gravity of the bodies 
A and B. If the center of gravity of three 
bodies be required, first find C the center of 
gravity of A and B ; and supposing a body 
to be placed there equal to the sum of A 
and B, find G the center of gravity of it and 
D ; then shall G be the center of gravity of 
the three bodies A, B, and D. In like 
manner the center of gravity of any number 
of bodies is determined. 
The sum of the products that arise by 
multiplying tlie bodies by their respective 
distances, from a right line or plane given 
in position, is equal to the product of the 
sum of the bodies multiplied by the dis- 
tance of the center of gravity from the same 
right line or plane,when all tlie bodies are on 
the same side of it : but when some of them 
are on the opposite side, their products, 
when multiplied by their respective distances 
from it, are to be considered as negative, or 
to be subducted. Let I L (fig. 6,) be the 
right line given in position, C the center of 
gravity of the bodies A and B ; A a, B 6, 
C c, perpendiculars to I L in the points a, h, 
and c ; tlien if the bodies A and B be on the 
same side of I L we shall find A x A a -j- b 
X B6 = A-f B X Cc. For drawing through 
C, the right line M N parallel to I L meet- 
ing A a in M, and B 6 in N, we have A : 
B :: B C ; A C by the property of the center 
of gravity, and consequently A : B :: B N : 
AM, or A X AM = B x BN ; but A X A 
a-4-BxB6 = AxCc-fAxAM-f- 
B X Cc — Bx BN=:AxCc-l-Bx 
Cc = A -j- B X Cc. Wlien B is on the 
other side of the right line I L (fig. 7,) and 
C on the same side with A, then A x A a 
— BxB6 = AxC c-|- a X A M — B 
X B N -1- B X Cc = A -f. B X C c; and 
when the sum of the products of the bodies 
on one side of I L multiplied by their dis- 
tances from it, is equal to the sum of the 
products of the bodies multiplied by their 
