Finite Rotations of a Rigid Body

Extract

The following problem was recently proposed by Professor E. H. Thompson: A rigid body is freely pivoted at a point O through which pass three fixed axes, OA, OB, OC; if the body is given one finite rotation about each of these axes, in the order stated, what conditions must be satisfied by the angles between the axes if these three rotations are to be capable of moving the body into every position consistent with O remaining fixed?

It is clearly sufficient that OA and OC coincide with a perpendicular to OB, for this is the familiar configuration. for the three Euler angles which specify rotations about two fixed intersecting perpendicular axes.