Composition and Resolution of Forces, &c. 87 
the reaction LING ca IN | we ve have ee M is in ae 
zum, X+ ne Ne ma YN Nag aay ZN’ =+ 
Ain 
N’” —-=0, (22); and by eliminating N’, N” from (22), we shall 
dz y S 
du du’ du dw 'S du’ du a) ee du’ 
dy dz dz a dz dx dx dz dx dy 
have x 
du du! mle 
dy da =0, (23), for the equation of equilibrium required ; which 
with (21) will enable us to find the coordinates x, y, z of the point 
on the line where M must be placed, to bein equilibrium. _It is ev- 
ident that we shall have VW X2+Y2+2Z2=the force with which 
the surface must react, in order to destroy the resultant of the appli- 
ed forces; Mec. Cel. pp. 9, 10, &c. 
Equilibrium of a System of Bodies. 
We will now consider the conditions of equilibrium of a system of 
bodies; whose quantities of matter are denoted by m, m’, m’’, &c., 
supposing the unit of masses to be a portion of matter so small that 
it may be considered as a particle: we shall also suppose the bodies" 
m, m’, &c. to be so small that every unit of each, may. be considered 
as acted on by the forces, (which are supposed to affect them,) with 
the same intensity. 
Let the system be referred to the fixed rectangular axes 2, y, 2, 
drawn (at pleasure,) through any given point for their origin; sup- 
posing x, y, z to be the coordinates of m, x’, y’, 2’ those of m’, and so 
on. Weshall suppose the reactions of the surfaces or lines, on which 
any of the bodies may be supposed to be in equilibrium; and the 
reactions of any fixed points which may be supposed to be in the 
system; together with the forces with which the bodies are suppo- 
sed, or made to act (whatever may be the cause,) on each other, are 
included among the forces. 
Let P, Q, R, be the sums formed by resolving each foree (as at p. 
308, Vol. xxvz,) which affects a unit of m, in the directions ofa, y, 2 
severally ; and P’, Q’, R’ the corresponding quantities for a unit of 
m’ ; and soon. 
Then for the equilibrium of m, we must have (as at p. 308,) 
~P=0, Q=0, R=0; and for that of m’, P/=0, Q’=0, R’=0; and 
so on for all the bodies of the system ; ihewes when the system is in 
