88 Composition and Resolution of Forces, &c. 
equilibrium, we must have P=0, P’=0, P’=0, &c. (a); Q=0, 
‘=0, Q”’=0, &c. (b); R=0, R’=0, R’=0, &c. (c); where 
there are three times as many equations as there are bodies, there 
being three for each body ; or as many as there are coordinates, 
vy z, vy! 2’, &c., as evidently ought to be the case, for it is indif- 
ferent, whether the position of m is determined by the forces P,Q, R 
or by the coordinates xyz; and the same remarks are applicable 
to each body. It may not be improper to remark, that in forming 
P, Q, R, &c., we may neglect the consideration of the algebraic signs 
of the cosines of the angles, which the directions of the component 
forces make withthe directions of the axes of x, y, z, provided, ifwe 
regard those forces which tend to increase the coordinates as posi- - 
tive, we consider those which tend to decrease them as negative ; 
and reciprocally. 
We will now consider the forces which any two bodies of the sys- 
tem, as m and m’, exert on each other. Imagine m and m’ to be joim- 
ed by the straight line f, then the equilibrium will evidently not be 
disturbed by, supposing f to be rigid; now if the conditions of the 
system cause m and m’ to act on each other with any forces, they 
must act along f; for otherwise they will give f an angular or par- 
allel motion, and the equilibrium will be disturbed. Let p denote 
the whole force which a unit of m exerts on a unit of m’, then a unit 
of m’ must act on a unit of m with the force —p, which is directly 
opposite to p; for otherwise f will be moved in the direction of its 
length, and the equilibrium will be disturbed. Hence mp = the 
whole force with which m acts ona unit of m’, and —m/p= the 
whole force of the consequent reaction of m’ on a unit of m; by re- 
solving these forces in the directions of the axes of x, y, z, we shall 
oa! lat , rot 
have F mp, a a ba F mp, severally, for the com- 
ponents of P’, Q’, R’, which depend on the force mp; also 
ig 5 aa y-y z—2! 7 
_ rs mp, — a ie mp, ra ao mp, are the components 
of P, Q, R, which depend on the force ~ m’p: hence (for simplicity,) 
Ree 5d 
considering these forces only, we shall have P=— ("| mp, 
a=-('F* mp, p=(— = = mp, Qe (ee Lo ) mp, . me + 
mn P* =0, (d); also Py—-Qz =(-(") (>) i’'p = 
