If we write x = x + ξ, y = y + η, z = z + ζ, so that ξ, η, ζ denote co-ordinates relative to the mass-centre G, we have from (6)
Σ(mξ) = 0, Σ(mη) = 0, Σ(mζ) = 0.
(7)
One or two special cases may be noticed. If three masses α, β, γ be situate at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′ in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then divide AA′ so that α·AG = (β + γ) GA′. It is easily proved that
α : β : γ = ΔBGA : ΔGCA : ΔGAB;
also, by giving suitable values (positive or negative) to the ratios α : β : γ we can make G assume any assigned position in the plane ABC. We have here the origin of the “barycentric co-ordinates” of Möbius, now usually known as “areal” co-ordinates. If α + β + γ = 0, G is at infinity; if α = β = γ, G is at the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find
α : β : γ : δ = tetn GBCD : tetn GCDA : tetn GDAB : tetn GABC,
and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If α + β + γ + δ = O, G is at infinity; if α = β = γ = δ, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB : ΔABC, G is at the centre of the inscribed sphere.
If we have a continuous distribution of matter, instead of a system of discrete particles, the summations in (6) are to be replaced by integrations. Examples will be found in textbooks of the calculus and of analytical statics. As particular cases: the mass-centre of a uniform thin triangular plate coincides with that of three equal particles at the corners; and that of a uniform solid tetrahedron coincides with that of four equal particles at the vertices. Again, the mass-centre of a uniform solid right circular cone divides the axis in the ratio 3 : 1; that of a uniform solid hemisphere divides the axial radius in the ratio 3 : 5.
It is easily seen from (6) that if the configuration of a system of particles be altered by “homogeneous strain” (see [Elasticity]) the new position of the mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.