FORCE, 
ferred. The above prety is a very important one in the 
general equations of equilibr 
be extended to paralle Nees | by fuppofing the ae A to 
eae = an infinite diftance by the revolution 
A any two points es N; then the lente will take 
the ‘gies as a ee in ig? . And fince 
A 
a x E 
RL EGEO x EH 
P:QuEH: EE 
R:Q:EH:EG 
_ Coro). fince P:GH: Q:EF::R: FH 
P :GH+EF: :R:FH 
An : Be +Q=R. 
and P — O = R, when they a& in oppofite direCtions. 
Or the ine preponte on may be demenitrated thus, 
_ without the nee of confidering the point as removed to 
an peu difta 
wo parallel cae f and g, ( fig. 8.) acting in the fame 
diredtion be applied t e two extremities E, f the 
line E F, to which the are perpendicular ; then Foppofe 
shed at the extremities FE, 
+ 4 
forces p and f', the tangent of the angle PE gs! = £ 
~ EO! 
g- aa 3 
known principle of the lev 
wo parallel forces P P and Q are oblique to the = 
and if see refulting force be R, th 
e force, and fuppole B, C, 
=Qx 
EO 
;hence p'= p. KO} in the fame manner ‘g! = 
therefore p x EO =q x OF, which is the 
thefe forces. 
I 
EG=a, EF=,,FG= 
a=p+qgR=P+Q,Pp=Q7. 
ef Miwa i quantities, three being given, the remainder 
may be 
Let us 5 a confider the effe& of any fyftem of parallel 
forces P', P’, pm ; acting on different points of any bo 
whatever : let x’, y! 2%, xl, y!", 2, be the refpeCtive co-or- 
ey of thefe points to the three rectangular planes ; let 
K, (fig. g.) be the points of application of the sae es Hi ‘4: 
a G that of their refulting ea R, and let a’, 8, 
the three co-ordinates o 5 
A 
ad if D K ‘« ead ‘ it meets 
LG=P'xLD+P" 
AE= 
R! = P+ P', 
the plane of x yin L; ee 
. The fame propofition may . 
x LK; and fincee LG, LD, L K, are proportional te 
chein projections Rial = Plx! + Px", for the fame rea- 
fon 
RQipl— Fly! + Pi yt, and Ri cl = Plat 4 Pigtt, 
In which ¢ equations great attention muft b Paid to the 
figns which are ey pofitive ; they are fufficient 
eve th the magnitude and dire@tion or “the eegae 
This heels force R! muft be taken with the next 
srce Pit ain a new force, and this procels conti- 
ee till ail = forces are combined ; R! being fubftitut- 
ed for P! and P’, and combined with PM, we have 
+ RY, Ra" = Ral + Pu 
Ri pt— ‘aes = }! + pu i, Ri'¢ a = R! c! + pu. sa 
peered for R’, R’a', R’ 3', R’ a their values, the v 
of R", a’, Rb", Rc" are obtained, and finally 
a e lene are R; and its dire€tion is obtained in the 
following equation 
= Pi+ Pl + &e. 
The firft equation gives the magnitude of the force 3 the 
co-ordinates x, yy 2%, 0 he point of application, are 
P' « a P's " + 
forces. 
It now se remains to inveftigate the general cafe, where 
any nu of me are fuppoied to a& on a body in any 
diveetions ae 
t the a be fuppofed a oe and the forces 
likewife nee in the plane of the figu 
et P’, P’ &c., veprefent t the ee x', 9', 2", 9", the 
co- eines of the points on which they act ; a’, a", the 
angles which thefe co-ordinates make with the axis of x 
Let P’ be refolved into two forces X’, Y’, parallel to dive 
axes x5 J, be ie me X", Y", &e. 
Pic X', P' cos. a! = 
Pp! fn. a ~ Yi, P" fin. oi = = YH, &e. 
eaice parallel 2 the axis ae ; ia 
+X" 4 &es YoY! + Y"4 &e, 
= Vio + Y" cl + &e; 
w be combined into a fingle 
one, which will be- ‘pplied in their point of interfection, 
whofe co-ordinates are b 
As the point of appliation of R may be taken in any 
part of its dire€tion, it is neceffary to determine the equa- 
tion of the ftraight line eanieatne this direétion, fince it 
paffes 
