The relative position of the two sets of axes is given by means of Euler’s unsymmetrical angles θ, φ, ψ, such that the successive turning of the axes Ox, Oy, Oz through the angles (i.) ψ about Oz, (ii.) θ about OE, (iii.) φ about OZ, brings them into coincidence with OX, OY, OZ, as shown in fig. 11, representing the concave side of a spherical surface.
The component angular velocities about OD, OE, OZ are
ψ sin θ, θ, φ + ψ cos θ;
(1)
so that, denoting the components about OX, OY, OZ by P, Q, R,
| P = | θ cos φ | + ψ sin θ sin φ, |
| Q = | −θ sin φ | + ψ sin θ cos φ, |
| R = | φ | + ψ cos θ. |
(2)
Consider, for instance, the motion of a fly-wheel of preponderance Mh, and equatoreal moment of inertia A, of which the axis OC is held in a light ring ZCX at a constant angle γ with OZ, while OZ is held by another ring zZ, which constrains it to move round the vertical Oz at a constant inclination θ with constant angular velocity μ, so that
θ = 0, ψ = μ;
(3)