(11)
where ρ = OJ. This relation will be of use to us presently (§ 19).
The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose co-ordinates relative to O are x, y, z are then
u + qz − ry, v + rx − pz, w + py − qx
(12)
by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the kinetic energy is given by
2T = M0 (u2 + v2 + w2) + Ap2 + Bq2 + Cr2 − 2Fqr − 2Grp − 2Hpg,
(13)
where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf. § 15 (9).
The components ξ, η, ζ of linear momentum are