Composition and Resolution of Forces, &c. 87 
the reaction of u’, N’=L/,N ; we shall have a M is in pens 
' du du’ du 
rium, X+N +N" =0, YING LNG <0, ZN =f 
du’ 
N’7-=90, (22); and by eliminating N’, N” from (22), we shall 
du du’ du al (. du’ du nA ( du’ 
have X (Fs oe ie ay ie ay 
du du 
dy o) =0, (23), for the equation of equilibrium required ; which 
dz dx dx dz 
with (21) will enable us to find the coordinates x, y, z of the point 
on the line where M must be placed, to be in equilibrium. It is ev- 
ident that we shall have “/ X2?+Y2+2Z?=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, z, 
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 force (as at p. 
308, Vol. xxv1,) which affects a unit of m, in the directions of 2, y, z 
pte A, : we P’, Q’, R’ the corresponding quantities for a unit of 
; and so 
Then for ‘tis 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 Nodies of the system; hence when the system 1s in 
