Composition and Resolution of Forces, &c. 91 
any three of its points which are not in the same straight line, the 
position of each point will depend on its three coordinates, .*. nine’ 
coordinates will be necessary, to determine the situation of the three 
points, but three of these will be given in terms of the other six by 
means of the distances of the three points from each other, .*. the 
distances of the three points from each other are equivalent to three 
coordinates,.". they are equivalent to three of the equations (a), (b): 
(c). Again, the coordinates of any body of the system are given in 
terms of the coordinates of the given points by means of its distan-. 
ces from the points, and of their distances from each other,.’. the dis- 
tances of each body from the three points are equivalent to its three 
coordinates, or to three of the equations (a),(b),(c); hence there re- 
main but six undetermined coordinates, or but six of the equations 
(a),(b),(c), or neglecting these and using (f) and () instead of them 
for the reasons before given, we shall have (f) and (g) for the equa- 
tions of equilibrium of any rigid system, and they are sufficient with- 
out using any of the equations (a),(b),(c) ; which is in conformity 
with what has been said of the equations of condition when consid- 
ering systems in general. 
We shall begin by supposing that the system does not contain any 
fixed point, also we shall suppose the forces to be positive, and that. 
their directions are determined by the (well known,) algebraic rules: 
for the signs of the cosines of the angles which we shall suppose 
their directions to make with the positive directions of the axes of x, 
y, z. Let then F denote the resultant of all the foreign forces which 
affect a unit of m, a, b, c the angles which its direction makes with 
those of x, y, z severally, F’ a’, b’, c’ the corresponding quantities for 
a unit of m’, and so on; decomposing the forces in the directions of 2, 
y, z we have F cos. a, F cos. b, F cos. c, F’ cos. a’, &c. to be substi- 
tuted for P, Q, R, P’, &c. in (f) and (g), hence they become SmF 
cos. a=0, SmF cos. b=0, SmF cos. c=0, (h); SmF(y cos. a—a 
cos.b)=0, SmF(zcos. a—axcos.c)=0, SmF'(y cos. c—z cos. b)=0 
(i); which are the equations of equilibrium when the system is free. 
If the system is to be in equilibrium about a fixed point to which it 
is firmly attached, then fixing the origin of the coordinates at the 
point, we shall have (i) for the equations of equilibrium of the ap- 
plied forces. Put SmF cos. a=X, SmF cos. b=Y, SmF cos. c= Z, 
M=the mass of the system, MR,=R=the reaction of the point 
supposed positive, and A BC for the angles which its direction 
makes with those of x, y, z; then decomposing R in the directions of 
