FORCE, 
paffes through ive points, whofe co-ordinates are aide its 
equation is Y — 6 = tang. a(x — a); or fince tang. 
= X y—Yux=Xb-—Y a, and by fubftitution for 
the i Ky. You = Xl y!— ¥' x! + 
xX! Y's + & 
wil 
Here x and y are t the co-ordinates of the point of ap- 
plication of the Soe oS x! x', the difference 
of the moments ? - es X', ¥'; the ae of the other 
forces: therefore, if p’; é 97s reprefeut the ae let 
fall from the origin to the direétions of the for 
ya yu 
Rr=P'p'+ Pip" + &e. . (m) 
If x reprefent the fum of the moments, R cos.2 = X, R 
newzY, Rr=r=X 
The conditions of equilibrium of a fyftem are obtained 
by fuppofing it to exilt, and then ene of the forces P to 
be removed ; ‘ae refulting force R may then be confidered as 
an as Cae Xy — Y«=>a, fhould be that 
Py the fore e by fubftitution ce equations 
of afi aun are ned ; 
Xl 4 KM + eae 
Yr+ Y' + &e.= - l (n) 
ae + Pil pl + Re. { 
Ky — ve a Ki y weeds =o 
In re fame form.are the terms oe P, & 
If the fy tem be fixed by a point pet which it may 
take a motion i lelleovia the equilibrium may fubfitt 
without the forces deftroying each other, provided the re- 
fultiig force oe throngh ae fixed point. ‘Tf this point 
e the origin re) nates, = Oy = dr= 0; 
b 
and aes + Pip" a 
= 0, an 
P and 
which cersiee is indeterminate as one of th 
er; oe spndidod therefore 
ey the preflure on the 
fixed oe may es in quantity and direction fome given 
force, 
Let us imagine a folid body fituated in {pace, the re- 
fpeGive points of which are folicited by any fyftem of 
forces, whofe intenfity and dire€tions are exprefled by 
the fame chads or sara as before, but fince the forces 
are not ee me any. one point, the a of 
each muft be era oe that of one of the points of 
its dire€tion ; for inftance, on es of its application to 
the fyftem; let .2', y’, ", y", x", be the co-ordinates 
of bhele points of the ae P,P", & Refolvé each 
force at the point of its application es ‘three others pa- 
to the axes of the angular co-ordinates; let 
re 
» be the Si ene forces of P’; X", Y",Z", 
zy 
bial x; ‘pi cos, a! == X" 
3 P" cose" = Yl 
ole P' cos. y! = VAP 
Leta plane be imagined fixed in the folid body, and moveable 
with it; let this be the plane of x y, and fuppofe each force 
roduced till it meet this plane; the requanons of the ds ight 
line reprefenting the forée P’ are x (x I—y 
x 
= B(z—2x!); and, as above, A =F and B= —— 
To obtain the point, where the point in which i. line-in- 
Vou. XV. 
terfeCtsthe plane of « y, — eh —— gives for the 
co-ordinates a’, 4!, of this poi 
 Zhad KE! 
al. 
_ 2 Zy! — Xs! 
a = 77 am cart, ae ; 
And by changing the: te analogous values: are ob- 
tained for the other forc 
If every force be peed to be applied at the point 
where its direétion interfe@s the plane of xy, it may be re- 
folved into two, one in the diredtion of the plane; and the 
thefe 
ora once to z fhould ‘fatisfy the three equations 
PH 
Pix' + P's i" ] t= ‘tang. o (Pix! i Pi at + &c.) 
Ply! + Py" + &c. = tang. B (P's! + P! gl 4+. &e.) 
and as the force Z! is applied : toa point, of which a! and 
b! are we co- ordinates, the | moments of Z! are Z! a! and 
Z!' Bo x! — X's!, and Z’ y! — 
In like manner are Rianne thofe of on, ia a 
Zi + Z" + &e. 
Zz! x! — X'al + Zi xl — ae + &e. = 0 
Zl y! — Vial + Zilyt — Vil ve 
The forces fituated in the plane Hast alfo Crake des 
equations (7) of’ the laft problem. ythem, every 
force muft be se ~ two others, par allel tox and ¥; 
P' into X’, Y', and P’ i 0X", Y", and a’, b', fubftituted 
for ieee mad a" b" for 
ate of equa, therefore, of a folid body, will 
be eeotie ag uations 
cn XW A be =.0 ‘ 
yk yn + Y¥" 4+ &e. = ot (M) 
Tha Za Zl 4 &e = . 
KX! yf — Vil XM yl — Vx 4&0). ; 
Xat — Zit XM elt — Zl sl + < =o pr (N) 
Z! y! — Y'al + Zi yh — Ye sl a 
If the fyftem contains a fixed point or ae axi as it may. 
e in equilibrium without the forces pe ae ing each other ; 
it would be fuffic =~ = the — force to be diredted 
to this axis or this 
th eir 
lar planes paflitg throu eb this 
_ If th m is retanied by a tele sone (the origin ‘of 
e co-ordinates) the above epee mutt - fubf Ane are 
notfufficient; the forees parallel to’ muft no ‘Tou 
negle&ted, and eon refulting forces mut pafs chrouch ae 
origin or fixed poi 
Of the equations "(M), ( (N); the former are called e equa 
ecaufe, when th hau the body 
otion of tranflat 
ct 
ae 
does n ie ta bit there is one refulting’ force only 
as R; then if R be refolyed into X Y Z, the'fix a will 
fubhit, aneanes forces — > tee — Zar are.a ‘to. the. 
reft. The. th ft € ob lems, 
a en Le | 
