CHAPTER XI. 



THE DYNAMICS OF A RIGID BODY, WHICH HAS FREEDOM 

 OF THE THIRD ORDER. 



107. Introduction. The dynamics of a rigid body 

 which has freedom of the third order, possesses a special 

 claim to attention, for, included as a particular case, we 

 have the celebrated problem of the rotation of a rigid 

 body about a fixed point. In the theory of screws the 

 screw complex of the third order is characterised by the 

 feature that the reciprocal screw complex is also of the 

 third order, and this is a fertile source of interesting 

 theorems. 



We shall first study the screw complex of the third 

 order, and its reciprocal. We shall then show how the 

 instantaneous screw, corresponding to a given impulsive 

 screw, can be determined for a rigid body whose move- 

 ments are prescribed by any screw complex of the third 

 order. We shall also point out the three principal screws 

 of inertia, of which the three principal axes are only 

 special cases, and we shall determine the kinetic 

 energy acquired by a given impulse. Finally, we shall 

 determine the three harmonic screws, and we shall 

 apply these principles to the discussion of the small 

 oscillations of a rigid body about a fixed point under the 

 influence of gravity. 



A screw complex of the first order consists of course 

 of one screw. A screw complex of the second order con- 

 sists of all the screws on a certain ruled surface (the 

 cylindroid). Ascending one step higher, we find that in 

 a screw complex of the third order the screws are so 



