| dξ | = X, | dη | = Y, | dζ | = Z, |
| dt | dt | dt |
| dλ | = L, | dμ | = M, | dν | = N, |
| dt | dt | dt |
(17)
| Fig. 79. |
where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the co-ordinate axes.
An exception occurs, however, in the case of a solid which is kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform sphere. The equations then take the forms
| M0u̇ = X, | M0v̇ = Y, | M0ẇ = Z, |
| Cṗ = L, | Cq̇ = M, | Cṙ = N, |
(18)
where C is the constant moment of inertia about any axis through the mass-centre. Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the equation of the latter is z = −a. We will suppose that the extraneous forces consist of a known force (X, Y, Z) at the centre, and of the reactions (F1, F2, R) at the point of contact. Hence
M0u̇ = X + F1, M0v̇ = Y + F2, 0 = Z + R,
Cṗ = F2a, Cq̇ = −F1a, Cṙ = 0.