Composition and Resolution of Forces, &c. 93 
rest, and directly opposite to it, which are evidently the conditions 
of equilibrium of a free system, when acted on by parallel forces. 
If SmF'=0, but the numerators of (p) are not =O, then evidently 
any one of the forces equals the resultant of all the rest, but is not 
directly opposite to it; .’. there is properly nocenter of parallel for- 
ces in this case, although it is usual to say that the resultant =0, and 
is situated at an infinite distance. If the forces F, F’, &c. have the 
same sign, and are equal to each other, (p) become X= GT, Y= 
Sa oe A Sh eT Sn 
Sm? 2= Sm? (4)> which are the well known formule for finding 
the center of gravity of a system of bodies; the resultant of the 
forces=FSm (=the weight of the system if F=gravity.) Put S 
m=M-=the mass of the system, then by (q) M? X?=(Sma)? =m? 
v2 + malt m/ta/2+ &e. +2mm’ax'!-+-Qmm ve”! + 2Qm/m" a! x” 
+&e.; but Qra’ =x? 4 2!* — (a — 2)2, Qaa’ =a? +4? — (a —2)*, 
Qr/x! =a? + 2/2 — (a/’— <a’)? and so on; by substituting these val- 
ues we have M* X* =MSmx* — Smm'(2’-—«x)?; hence and by (q) 
Keg veg ze Smerty te) Sm (wmayetiyy)eH(eo2)") 
(r) ; which gives the distance of the center of gravity from the ori- 
gin of the coordinates by means of the distances of the bodies from 
the same point, and of their distances from each other ; by finding 
in this way the distances of the center from three given points, 
which are not in the same straight line, its position in space becomes 
known; Mec. Cel. vol. 1. p. 45. 
We will now suppose that the system is to be in equilibrium about 
two fixed points which are invariably attached toit. Let the points 
be denoted by the letters m and n, then suppose the origin of the co- 
ordinates to be at m, R its reaction, ‘A, B, C the angles which the 
direction of R makes with the axes of x, y, z (as before,) R’ the re- 
action of n, A’, B’, C’ the angles which its direction makes with those 
of x, y, z; X, Y, Z the coordinates of n: then by including R and 
R’ among the forces in (h) and (i) they become SmF cos. a+-R cos. 
A+R’ cos. A’=0, SmF cos. 6+ R cos. B+ R’ cos. B’=0, 
SmF cos. ¢-++R cos. C+ R’ cos. C’/=0, (8); SmF (y cos. a— 2 cos. 
6) + RY cos. A’—X cos. B’)=0, SmF (z cos. a— x cos. c)+ 
R(Z cos: A’ — X cos. C’)=0, SmF (y cos. ¢ — z cos. b) +R’ cos. 
C’~Z cos. B/)=0, (t). which are the equations of equilibrium, be- 
tween the applied forces and the reactions of the fixed points. Put 
