§ 11. Theory of Mass-Systems.—This is a purely geometrical subject. We consider a system of points P1, P2 ..., Pn, with which are associated certain coefficients m1, m2, ... mn, respectively. In the application to mechanics these coefficients are the masses of particles situate at the respective points, and are therefore all positive. We shall make this supposition in what follows, but it should be remarked that hardly any difference is made in the theory if some of the coefficients have a different sign from the rest, except in the special case where Σ(m) = 0. This has a certain interest in magnetism.
In a given mass-system there exists one and only one point G such that
Σ(m·GP>) = 0.
(1)
For, take any point O, and construct the vector
| OG> = | Σ(m·OP>) | . |
| Σ(m) |
(2)
Then
Σ(m·GP>) = Σ {m(GO> + OP>)} = Σ(m)·GO> + Σ(m)·OP> = 0.
(3)