its representative vector is the same whatever point O be chosen. Secondly, we have an angular momentum whose components are
Σ {m (yż − zẏ) }, Σ {m (zẋ − xż) }, Σ {m (xẏ − yẋ) },
(2)
these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + δt. Thus
| Σ ( m· | PP> | ) = Σ(m)· | GG′> | . |
| δt | δt |
(3)
Analytically we have
| Σ(mẋ) = | d | Σ(mx) = Σ(m)· | dx | , |
| dt | dt |
(4)