FORCE, 
C63. a a tos. @ = = 608. y = a 
fo - nothing remains but to employ the three pene 
N, to determine the valié of the c6-6 Haars x, Js % and 
ica to determine the point of applicatio of R.: ie 
L = (xX ‘x') + he 
y+ & 
x, Y, Z, L, M, N, will be inoat ananien and the 
equations i N) will gi ve, 
L=Xy— sM=Zx—X2z;N=Y¥2-—-Zy 
by which X, Y, ‘Z are oe and by muliiplying ae 
equations by Z, Y, and X, and addi ing, 
NX = 
without which equation the three cede equations can- 
not fubfift together: it therefore expreffes the conditions of 
the roblem, neceffary for all the forces to be 
reducible into a fingle one, 
ce it appears aa ta fyftem of forces in fpace cannot 
be generally maintained in equilibrium by a fingle force 
But when the equation LZ 4+ MY + NX fubhfts, then 
there will be a fingle refulting force, Shee direGtion and 
intenfity may be found; and its point of application may be 
anywhere in the ftraight line whofe equations are X 
Be other — there will be two refulting 
not be combined into one. 
'y is ° fix ed either ona gal or an axis, a force 
mutt be introduced he laste? fyftem, 
The preflure which a fixed point or axis futtains from the 
action of a fyftem is always equal, and in a contrary direc- 
tion to gd sae st mu 
librium yftem, in e nese 
therefor, ‘for the  inveligation of the preffure ona point o 
e the fame as thofe aus = finding aie refulting force a 
a ce of. erty: fore 
“The. fixed axis of a item of bodies-is gee oe on 
fewhere is fom 
it is oe RM, the ndig force. 
mine t e preffure on A ani the force R mutt be refolved 
into two ethers applied to the points A, B, according to 
the rules given a 
In the cafe of e equili ibrium about a fixed axis, every 
force parallel to the axis may be neglected ; 3 but to deter- 
mine the preffure on certain points of this axis, thefe forces 
be-:taken into na lara For inftance, fuppofe it 
were required to determine the preffure which a force R, 
fig. U1.) parallel to ae axis A B, produced on two fixed 
A B =a; and fuppofe at the 
M, Q, app plied in the direCtions 
int B, the force S dire&ted to- 
roduce equilibrium in the fyftem 
C ABD, then the efforts exercifed on A and B will be 
cetermined. If the point A be taken as the origin of the 
co-ordinates.x, y 3 “AB for the axis «; A C that of y; the 
oe of eqiitibrinas' ‘become 
Rr — Sa 
From the firkt it appear dl rat ap the body or Gian is foli- 
eited in the direction e fame as if the force R acted 
direftly m that line:; and ae other equations give = 
r 
‘o 
Hence, the force R tends to turn the axis on the 
Ae A and B, with an equal aGion in oppofite direce 
1 the Motion of Bodies direGed. aS Centres Of Force 
To. ra thefe problems, we muft refer to the general dife 
ferencial equations of varied motions, Thefe are os = 
d d 
= F; a= OQ; 0e S; F.dx= vdv;3 where 
x =z fpace, and ¢ = time. 
aod a bo dy — = a ftate of reft towards a — sd 
en diftan 
x ai hen the velo. 
a is obtained, it may be f{ubftituted in this expreffion 
x . 8 . ° 
= for v; and the integral of it will be equal to the time 
dx 
- employed, fince d¢ = — 
Fore oy become an ebjeét of calculation, as the caufe 
of motion, when it is either ea conitant, or 
meg according to fome given law 
If a force a€ts on any particle of matter inftantaneoufly, 
or, if after acting on it a certain time, its adtion ceafes, the 
motion produced is uniform ; and 
ee uniform motion alone that 
Let any per een. unacquainted with me fu xional or dif. 
ie calculus, rae 4 to folve the eafieit of the follow- 
ee S, an 
on is He will fir 
ten fae parts, next into one hun 
ead o 
f fitite aaa pee it is fuppofed divided i ‘ate a number of 
infinitely {mall ones. 
Let C (fig. 12) bea centre, toward which bodies are ate 
trated with forces whic 
{quares of the diftancefrom 
from A, b t 
is required to oe the velocity acquired at any given 
point O of its defcen 
Let f reprefent sm ftandard force, as that of gravity, 
re G be the are at which C = a force equal to f. 
Then F (the force at O) : Force at G) iC Gi: CO% 
*3 (a—x)%, 
for 
————.3 but fince Fd = vdy, 
Thetefore, F = Pia x 
d x 
