MECHANICS. 
l»e the bodies in tlie fyftem, and the points A, B, C, their 
refpedlive centres of gravity ; join B C, and take BT : 
T C :: C : B; join A T, and take T E : E A :: A : 
B+C, or T E : T A :: A : A+B+C ; then will E be 
the centre of gravity of the fyftem, as before proved. 
Suppofe the momentum communicated to A would 
C-aufe it to move from A to x in T", and at x let the body 
be ftopped ; join T x , and take TF : T x :: A : A-J-B 
-j-C ; then F is the centre of gravity of the bodies when 
they are at x, B, C ; join E F ; and, fince T E : T A :: 
A : A-j-B-f-C :: T F : T*, E F is parallel to A*; and 
confequently the triangles TEF, TAr, are fimilar; 
therefore, EF : A* :: A : A+B-f-C. Hence, if one 
body A in the fyftem be moved from A to x, the centre 
of gravity is moved from E to F ; which point may be 
thus determined; draw EF parallel to Ax, and take 
E F : A * :: A : A+B+C. 
Next let a momentum be communicated to B, which 
would caufe it to move from B to y in T"; at y let the 
body be (lopped ; then, according to the rule above laid 
down, draw F G parallel to By, and take F G : By :: 
B : A-j-B-f-C ; and G will be the centre of gravity of 
the bodies when they are at x,y, C. In the fame manner, 
let a momentum be communicated to C, which would 
caufe it to move from C to z, in T", and at z let the 
body be ftopped ; draw G H parallel to C z, and take G H 
: Cz :: C : A+B+C; then H is the centre of gravity 
of the bodies when they are at x, y, z. If now the mo¬ 
menta, inftead of being communicated feparately, be com¬ 
municated at the fame injiant to the bodies, at the end of 
T" they will be found in x,y, z, refpeflively; therefore, 
at the end of T", their common centre of gravity will be 
in H. 
Now let E be a body equal to A+B + C, and let the 
fame momentum be communicated to it that was before 
communicated to A, and in the fame direction ; then, 
llnce E F is parallel to A*, EF is in the direblion in which 
the body E will move ; alfo, dince the quantities of mo¬ 
tion communicated to A and E are equal, their velocities 
are reciprocally proportional to their quantities of matter; 
or, E’s velocity : A’s velocity :: A : A+B-j-C :: EF : 
Ax -, therefore, EF and Ax are fpaces defcribed by E 
and A in equal times, or E will delcribe the fpace EF in 
T". In the fame manner F G is the fpace which the body 
E will delcribe in T", if the momentum, before commu¬ 
nicated to B, be communicated to it; and G H the fpace 
it will defcribe in T", if the momentum before commu¬ 
nicated to C be communicated to it; join EH; and, 
when the motions are communicated at the fame infant to 
E, it will defcribe E II in T". Hence it follows, that 
when the fame momenta are communicated to the parts 
of a fyftem, and to a body, equal to the fum of the bo¬ 
dies, placed in the common centre of gravity, this body 
and the centre of gravity are in the fame point at the end 
ofT"; and T may reprefent any time; therefore, they 
are always in the lame point. 
The fame demon ((ration may be applied, whatever be 
the number of bodies in the fyftem. 
Prop. XLII. If a body be fufpcnded by ary point, it will 
not remain at rejl till the centre of gravity is in the line which 
is drawn through that point, perpendicular to the horizon.— 
Let S be the point of fuipenfiori of the body ABC; fig. 70. 
G its centre of gravity ; join SG> and produce it; through 
S and G draw R ST and G H perpendiculars to the ho¬ 
rizon ; then the effort of gravity, to put the body in mo¬ 
tion, is the lame that it would be, were all the particles 
collected at G ; take G H to reprefent the force in that 
direction, and draw H I perpendicular to G I ; then the 
force G H is equivalent to the two G I, I H, of which 
G I is fultained by the re-a£lion at the point of fufpen- 
fion S, and I H is employed in moving the centre of gra¬ 
vity in a direftion perpendicular to SG; therefore the 
centre of gravity cannot remain at reft till I H vanifties ; 
that is, till the angle I G H, or G S T, or GSR, vanifhes, 
or SG coincides with R T. 
Vql. XIV. No. 1002, 
Cor. Hence it follows, that, if a body he nifpended 
fucceftively by different points, and perpendiculars to the 
horizon be drawn through the points of fufpenfion, and pals 
through the body, th’e centre of gravity will lie in each 
of thele perpendiculars, and, confequently, in the point 
of their interfedlion. 
Of MOTION, UNIFORM and VARIED. 
Motion of Bodies in Right Lines, a6ted upon by 
conftant or variable Forces. 
A body which lias received only a fingle impulfron 
will, according to the firft law of motion, perfevere in its 
motion with the fame velocity and in the fame direction 
it had at the firft inftant; but, if it receives a new im- 
pulfion, either in the fame direction or in a direction con¬ 
trary to the firft, it will then move with a velocity equal 
to either the fum or the difference of the two velocities 
which it received fucceftively. If, therefore, we conceive 
that at fucceftive intervals of time the body receives new 
impreftions, either in the fame or contrary directions, it 
w'ill be transferred to different parts of fpace with a varied, 
or unequable, motion ; its velocity will be different at the 
commencement of each interval of time. In variable 
motions, the velocity undergoing repeated changes, it 
is ufual to eftimate it at any time whatever by the fpace 
it is capable of palling over during a unit of time, if its 
motion for that interval continued the fame as at the in¬ 
ftant where we would confider the velocity. Or, in va¬ 
riable motions, the velocity of a body at any determinate 
inftant is the fpace which it would run over iri every unit 
of time, if at that inftant the aCtion of the power ceafed, 
and the motion became uniform. 
We call in general any force which afts on a body fo 
as to make it vary its motion an accelerating force ; when, 
in equal intervals of time, it afts equably, or the velocity 
undergoes equal mutations, we call it a confant or uniform 
accelerating force, or a confant retarding force, according as 
it lends to augment or diminifli the actual velocity of the 
moving body. 
When a fingle body is aCted upon by a conftant force, 
there are four quantities which become the objects of 
mechanical confideration, viz. the fpace defcribed, the 
time of defcription, the velocity acquired, and the force 
which produces it ; any three of which being given, the 
Other may be afcertained. But, when different forces aCl 
upon bodies of different maffes, thefe are two additional 
quantities for confideration, making in the whole fix 
kinds of magnitudes which sffeCt the difcufiion. 
Prop. XLIII. The velocities generated in equal bodies by 
the action of confant forces are in the compound ratio of the 
forces and times of ailing. —For, when the times are the 
fame, the velocities generated each inftant are as the 
forces of acceleration, and confequently the velocities 
generated at the end of equal times are as thofe forces; 
and, if the forces are the lame, the velocities generated 
are as the times wherein the forces ail; becaufe, when 
the force is given, equal velocities are generated in equal- 
times, and confequently the whole velocities acquired are 
as the times wherein the given force ails ; wherefore, both 
times and accelerating forces being different, the velo¬ 
cities generated will be as the forces and times of ailion, 
jointly. 
Cor. 1. The momenta generated in unequal bodies are 
alfo conjointly as the forces and their times of action. 
This is evident, becaufe momenta in unequal bodies may 
be fubftituted for proportional velocities in equal bodies, 
throughout the whole re a foiling. 
Cor. z. The momenta loft or deftroyed in any times 
are likewife conjointly as the retarding forces and their 
times of action. For, whatever momenta any force ge¬ 
nerates in a given time would an equal force deltroy in 
an equal time, by aiding in a contrary direction. And 
the fame is true of the increafe or dscreafe of motion, by 
forces that either confpire with, oroppofe, the motions of 
bodies. 
8 B 
Cor, 
