Also there cannot be a distinct point G′ such that Σ(m·G′P) = 0, for we should have, by subtraction,
Σ {m(GP> + PG>′)} = 0, or Σ(m)·GG′ = 0;
(4)
i.e. G′ must coincide with G. The point G determined by (1) is called the mass-centre or centre of inertia of the given system. It is easily seen that, in the process of determining the mass-centre, any group of particles may be replaced by a single particle whose mass is equal to that of the group, situate at the mass-centre of the group.
If through P1, P2, ... Pn we draw any system of parallel planes meeting a straight line OX in the points M1, M2 ... Mn, the collinear vectors OM>1, OM>2 ... OM>n may be called the “projections” of OP>1, OP>2, ... OP>n on OX. Let these projections be denoted algebraically by x1, x2, ... xn, the sign being positive or negative according as the direction is that of OX or the reverse. Since the projection of a vector-sum is the sum of the projections of the several vectors, the equation (2) gives
| x = | Σ(mx) | , |
| Σ(m) |
(5)
if x be the projection of OG>. Hence if the Cartesian co-ordinates of P1, P2, ... Pn relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (xn, yn, zn), the mass-centre (x, y, z) is determined by the formulae
| x = | Σ(mx) | , y = | Σ(my) | , z = | Σ(mz) | . |
| Σ(m) | Σ(m) | Σ(m) |
(6)