MOMENTS AND CENTROIDS 45 
therefore P’, Q’, R’, and S’ must have the same sense. But if the sense 
of Q were opposite to that of P, then the sense of Q’ would be opposite to 
that of P’. 
In applying the above method to the determination of the centre of a 
system of parallel forces, it is usually most convenient to take the two 
directions of the forces at right angles to one another. 
In determining the eentre of a system of parallel forces by caleula- 
tion, it is most convenient to apply the principle of moments. weg let 
Pes; P,, Ps, etc., be parallel forces in a plane acting at fixed points 1, 2, 3, 
etc., in a rigid body ; ; choose a point X in the plane of the forces, and let 
the ‘perpendicular distances of X from P,, P,, Ps, etc., be 2, 22, %,, etc., 
respectively. Let R be the resultant of the forces, and a the per- 
pendicular distance of its line of action from X, then 
Re= Pa, + Pw, + Pyx, + ete., and ages Egat gee : 
Care must be taken to give the proper signs to the products P,z,, P,7., 
P.a,, etc. If one force tending to turn the body in one direction about 
X be considered as having a positive moment, then another force tending 
to turn the body in the opposite direction about the point X is to be 
considered as having a negative moment. A line parallel to the direc- 
tions of the forces and at a distance 2 from X will be the line of action 
of R. Turning all the lines of action of all forces through the same 
angle in the original plane, and repeating the calculation with reference 
to the same point X, or any other point in the plane of the forces, a new 
line of action of R is determined which intersects the first at the centre 
of the given system of forces. 
If the fixed points, and therefore the lines of action of the forces, are 
not in the same plane, the procedure may be as follows. Select three axes, 
X, Y, and Z, perpendicular to one another. Take the lines of action of 
the forces in turn parallel to the axes Y, Z, and X, and in turn take 
moments about the axes X, Y, and Z to determine z, 2, and y the 
distances from X, Y, and Z respectively of three planes parallel to 
XY, YZ, and ZX respectively. The point of intersection of these three 
planes is the centre required. 
64. Centres of Gravity or Centroids.—The particles of which any 
body is made up are attracted to the earth by forces which are propor- 
tional to the masses of these particles. For all practical purposes these 
forces may be considered to be parallel, and their resultant will act 
through the centre of these parallel forces. In this case the centre of the 
parallel forces is called the centre of gravity or centroid of the body, and 
the determination of a centroid resolves into finding the centre of a 
system of parallel forces. 
The centre of gravity of a body may also be defined as that point 
from which if the body is suspended it will balance in any position. 
When the term centre of gravity is applied to a line, the line is 
supposed to be made of indefinitely thin wire ; and when the centre of 
gravity of a surface is spoken of, the surface is supposed to be made of 
indefinitely thin substance. 
The following results, which are not difficult to prove, should be 
noted :— 
