FORCE. 
ing force R, given by the lane R fin.y = T; Re 
=Q;3y being the angle formed by R and 0. Babetuling 
Roh fin. y for T in the two firft oo. we have P = R 
cos. 9. fing 3 S = R fin. @. fin. 5 Q = R cosy. 
The fum of the fquares of thefe equations gives R® 
as The values . . and y are eafily found, ba 
compleat the problem. ore convenient to employ t 
angles z, 8, and y, which ne dire€ion of the refalting force 
R forms sblgen a re{pective forces F, , and Q. 
known that c in. y == cos. 3 and fin. @. fin. y = cos. 8. 
Therefore, inifead of “the preceding pans we have 
COS. %3 R ; O= Kc 3 from 
_ ay By yy are arn 
eneral cafe in shi the forces ats uppofed to have 
any ae whatev ry ee fins A. F, 
. 6. 
through any arbitrary point A: let A P be the axis 
The plane PAS will 
diveAtions form wit 
the 
a ay an 
By r were each of thefe er oe eee Cn. whofe 
) - 
direétions are parallel to the axes, we have for the producing AC x EH 
forces ee to ~ —_ 
Pp’, , P.cos. a!!, Pic '" 
y P’ cos. a P" cos. Br P" cos. we 
P’ , P” cos. y", PB” cosy!" 
Each of ee colleion of ae are equivalent to a fingle 
— equal 3 m. Since they are in the direc- 
tion of the fam ame int line, let X, Y, Z, be the three 
re parallel to ug aes axes, ad we fae 
X= Pe 
P” cos. «” + P" cos, al 
Let a, 8B, y, be the unknown angles which the relulting 
force R forms with the three axes; then Se cos. 8, 
R cos. y, will be the fe eae producing ores in the 
ieee of Xy Yy % e Rcos.«@ = X,Reos. 8 = Y, 
To. ena in the value of R and its dire€tion, add together ; 
f 
the {quares of thefe equations, and R? (cus 
+ cos.?y) = X? + ¥? + Z’, but cos.’ a + cos 
cos.” y = 1: gal 
J(X* + ¥? + Z’) and 
e2 
which equations test te that re refulting force is the 
diagonal of a parallelopiped, whofe three edges are X, 
and Z. 
B+ 
_ 
cos. a@ = BR 0% R08 7 = Re 
x',y', z', be the oo of the point to 
which thefe forces are applied, t 
line will therefore be 
y ae = 2 (x — x’) ‘upon the axis of J 
— 
Z— x % 
b(y—y) =a & — 2’) ye 
a and & being re tangents or the angles which the axis of 
sx forms with thefe projeCtions on the planes of x y andx x. 
If, therefore, through the point A, the lines AP, AS, 
drawn parallel to the axes x, y, 2, the projeétion 
for the refulting force R, on the plane PAS, will be 
) be fee re@angular pad ete pafling -ri : 
A T; therefore, the tangent a of the angle TAP = 8, 
or tang. 6 = 
<> = a, 
xX 
In the fame manner 4 = b. 
The equations of the Projections are therefore 
Coa x 
¥@—=ZG~s) 
Z(y— sh = YC 
If the fyftera ne in equibviom, 
Y=o,Z2= 
On parallel forces. Peal to the couifideration of pa 
rallel forces, it will be neceffary to netice a very fimp'e 
geometrical propofition, on which. forms, in fa, the bafis 
rha nd other French wiiters call the « Theo- 
etus of a power 1 
{fyftem of bodies; but modern writers, particularly the 
mathematicians of the continent, underftand by moment the 
produét of a power or force multiplied by the diltance of 
its dire&ion froma point, line, or plane 
If a point E, .) be taken any how fituated with 
refpeé to the Se A eee B , and the ‘gis endicu- 
culars E F, , be drawn from E to the two fides, 
AD x EG will be equal to 
and to the diayonal, 
+ABx Pro- 
EF. Join E A, EB,E D. 
duce D Btol; L will be the height of the 
triengle A C D, AC bene the bafe. 
Mie 
AEABE= —— 
A ABD=ACD= st SL 
A EBD = De xt t_ACxEt 
FAD—-APDXEG_ABx EF ACI 
ACx EI . ° : 
,ACx EI 
a ae ABxEF+ACx EH. 
x EG=PxEF+Qx EH. 
If ie on E be taken within the forall pian: then 
Rx EG= 
The force P, eclipled by ae diftance E F, is called the 
moment of the force P relatively to | the pon E ; hence by 
the preceding ording to thisde- 
finition, the moments of the producing. (tae are equal to 
the 
moment of the refulting forces ; and in cafe of equilibrium, 
the fum of the moments of all the forces are equal to ero 
If the point E be taken in the direGtion of one of the 
_ producing forces, as . then 
EG=Q=EH 
Ce G. 
If E be taken in the direétion of the jc forces 
EG =o,and P x EF=Q x Sere 
This propefition, which is quite independent of any idea 
of motion, rotation, or force, may be deduced generally, 
by means of the analytic equations already demonttrated for 
finding the refulting force, from any number of producing . 
forces foliciting "a point ; but the property being entirely 
geometrical, that form for the demonitration has been pre- 
