(17)
The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes. The formula (17) is due to J. L. Lagrange; it may be written
| Σ(m · OP2) | = | Σ(m · GP2) | + OG2, |
| Σ(m) | Σ(m) |
(18)
and expresses that the mean square of the distances of the particles from O exceeds the mean square of the distances from G by OG2. The mass-centre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make O coincide with P1, P2, ... Pn in succession, we obtain
| 0 | + m2·P1P22 | + ... | + mn·P1Pn2 | = Σ(m · GP2) + Σ(m) · GP12, |
| m1·P2P12 | + 0 | + ... | + mn·P2Pn2 | = Σ(m · GP2) + Σ(m) · GP22, |
| ......... | ||||
| m1·PnP12 | + m2·PnP22 | + ... | + 0 | = Σ(m · GP2) + Σ(m) · GPn2. |
(19)
If we multiply these equations by m1, m2 ... mn, respectively, and add, we find
ΣΣ (mrms · PrPs2) = Σ (m) · Σ (m · GP2),
(20)