MECHANIC S. 
t>4S 
To find the centre of gravity mechanically, it is only 
requifite to difpofe the body fucceffively in two poiitions 
of equilibrium, by the aid of two forces in vertical direc¬ 
tions, applied in fucceffion to two different points of the 
body ; the point of interfeftion of thefe two directions 
will (bow the centre required. This may he exemplified 
by p3rticularifing a few methods. If the body have plane 
fules, as a piece of board, hang it up by any point, then a 
a plumb-line fufpended from the fame point will pafs 
through the centre of gravity ; therefore mark that line 
upon it; and after fufpending the body by another point, 
apply the plummet to find another fuch line, then will 
their interfeclion fhow the centre of gravity. Or thus: 
Hang the body by two firings from the fame point fixed 
to different parts of the body ; then a plummet hung from 
the* fame tack will fall on the centre of gravity. Another 
method : Lay the body on the edge of a triangular priim, 
or fuch like, moving it to and fro till the parts on both 
fidesare in equilibrio, and mark a line upon it clofe by the 
edge of the prifm : balance it again in another pofition, 
and mark the frefh line by the edge of the prifm : the ver¬ 
tical line palfmg through the interfeclion of thefe lines, 
will like wife pais through the centre of gravity. The 
fame thing may he effected by laying the body on a table ; 
till it is jtill ready to fall off, and then marking a line upon 
it by the edge of the table : this done in two poiitions of 
the body, will in like manner point out the centre of 
gravity. 
When a plane or a line can be fo drawn as to divide a 
folid ora plane into two parts equal and fimilar, or fo that 
its moleculae fhall be difpofed two by two, in the fame 
manner, with rel'peCt to fuch plane or fuch line, we may 
call the body fymmetrical with regard to that plane or 
axis. And in all fuch bodies, it is obvious that the fum 
of the momenta-of its feveral moleculae, with relation to 
fuch plane or axis, will be nothing ; for, if we take two 
particles difpofed in the fame manner but on different fides, 
their momenta will be equal, but with contrary figns ; and, 
confequently, their fum will be equal to o; and the fame 
may be fhown of every other pair of moleculae, fimilarly 
fituated ; whence, as there are none but what are fimilarly 
fituated, the refultant of the lyftem will be in fuch plane, 
or line ; and, of' confequence, its centre of gravity will 
be there alfo. The fame reafoning may be extended to 
the centre of figure or of magnitude, that is, the point 
with refpefl to which a whole body fliall be fymmetrical. 
Hence, we conclude that the centre of gravity of a right 
line, or of a parallelogram prifm or cylinder, is in its mid¬ 
dle point; as is alfo that of a circle, or of its circumfer¬ 
ence, or of a fphere, or of a regular polygon ; that the cen¬ 
tre of gravity of a triangle is fomewhere in a line drawn 
from any angle to the middle of the oppofite fide; that of 
an elliple, a parabola, a cone, a conoid, a fpheroid, &c. 
fomewhere in its axis. And the fame of all fymmetrical 
figures whatever. 
Prop. XXXIII. To deduce fome general theorems which may 
he ufieful in finding the centre of gravity of any prop fed body .— 
If pip'i P"i &c. (fig- 6a.) be equal material particles, and 
g the point through which the refultant R of the gra¬ 
vitating forces upon thefe particles always pafles; and 
A B C D be a vertical plane, on which perpendiculars from 
pi p'i p"i and g, are let fall, then will the fum of the pro¬ 
ducts. of the forces upon p,p', p", into their refpeClive 
dillar.ces from A BCD, be equal to the produCt of the 
refultant R into its diftance, where the force R would be 
equal to thofe upon p+p'+p"- The fame would like- 
wife obvioufly hold with refpeCl to perpendiculars upon 
the other plane AECG; and, fince the fame will alfo 
obtain with relation to any vertical plane, although the 
polition of p, p', and p", be changed, provided they retain 
their relative fituations, it will of courfe obtain when the 
pofition of the fyllem is fo varied that AE B F becomes a 
vertical plane; confequently the equality of the products 
may be affirmed with regard to all the three planes at the 
fame time ; and, if the difiances from the feveral planes be 
referred to the re&angular co-ordinates AX, AY, A Z, 
we may readily appropriate equations to our prefent pur- 
pofe. Denote the force of gravity by g, the diftances re¬ 
ferred to AX by d,d\ cl", See. the diftances referred to 
AY, by D, D', D", See. and thofe referred to Z by a, S', S", 
Sec. the diftances from the centre of parallel forces to the 
lame axis being denoted by X, Y, and Z: then we (hall have 
RX —gp d gp'd'-{-gp"d"- (- Sec. 
RY g p D g p' D' + g p"i>" + &c. 
R Z=zgpSf-gp''S’+g'p" See. 
But R=zgp-{-gp'-\-gp"-\-Sec. and M—p-\-p'~]-p"-\-Scc, whence 
X= pd+P'd'+ P "d'<+ See. 
M 
*>D-!-//D'-f/>"D"+ See. 
~ M 
pZ-fip'S'+ffJr See. 
Z ~ M 
Here, if we adopt the language of fluxions, and put 
x, y, and z, for the variable diftances from A upon AX, 
AY, and A Z, refpeCtively, we may convert thefe equa¬ 
tions into the following form, which will render it more 
ufeful in many inveftigations : 
x = 
fluent of x M 
fluent x M 
fluent of M 
M 
V — 
fluent of y M 
fluent^/ M 
fluent of M 
M 
T — 
fluent of z M 
fluent of z M 
fluent of M 
— M 
As thefe values together determine only one point, we 
fee that a body has but one centre of gravity ; of which 
the three equations determine the three co-ordinates, and 
of confequence the diftances of the centre from three 
planes refpeCtively perpendicular to each other. 
Thefe refults being entirely independent of g, that is, 
of the force of gravity, fome philosophers have preferred, 
the term centre of inertia to that of centre of gravity: others 
have, on account of other properties, preferred the terms 
centre of pofition, and centre of mean difiances. 
Drop. XXXIV. To find the centre of gravity of any num¬ 
ber of bodies, whatever be their pofition .—Let A,B, C, D, 
(fig. 63.) beany numberof bodies influenced by the force 
of gravity. Suppofe the bodies A, B, connected by the 
inflexible line AB confidered as devoid of weight; then 
find a point F, fo that The weight of A : the weight of 
B—BF : FA. The bodies A, B, will therefore be in 
equilibrio about the point F in every pofition (as proved 
in the lever) and the prefiure upon F will be equal to 
A+B. Join FC, and find the point/, fo that A-J-B : 
C~Cf : /F; the bodies A, B, C, will confequently be 
in equilibrio upon the point f which will fultain a pref- 
fure equal to A-f-B-^-C. Join T>f and take the point (p x 
fo that A-j-B-{-G : D=<pD : (pf\ the bodies A,B, C, D, 
will therefore be in equilibrio about the point tp, which 
will be their common centre of gravity, and which fup- 
ports a weight equal to A-f-B+C-J-D. In the fame man¬ 
ner we may find the centre of gravity of any lyftem of 
bodies, by merely connecting the lafl fulcrum with the 
next body by an inflexible right line, and finding a new 
fulcrum from the magnitude of the oppofite weights which 
it is to fuftain. 
Cor. 1, If the weights of the bodies A, B, C, D, be In- 
creafed or diminifhed in a given ratio, the centre of gra¬ 
vity of the fyllem will not be changed ; for the poiitions 
of the points F,f, <p, are determined by the relative and 
not by the abfolute weights of the bodies. 
Cor. 2. A motion of rotation cannot be communicated 
to a body by means of a force afting upon its centre of 
gravity ; for the refinances which the gravity of each par¬ 
ticle oppofes to th§ communication of motion act in paral¬ 
lel' 
