276 PROFESSOR TAIT ON THE ROTATION OF A 



§§ 15-60. Kinetics of a Rigid Body icitli one Point Fixed. 



15. Having premised these kinematical theorems, we pass to the consi- 

 deration of the motion of a rigid mass. It was of course at once obvious to 

 Hamilton (Proc. R. I. A. 1847), that if «r be (as in § 7) the vector of the 

 portion m of the mass referred to the fixed point, /3 the vector-force acting at w, 

 Lagrange's general equation of motion takes in quaternions the form 



2 . Yir(mw — j8) = 0, 

 or, if we put 



4 = 2- Wis 



so that \|/ denotes the vector-couple acting on the body, 



2 . mV^ = 4 (11). 



This is our sole dynamical equation. 



16. Integrating once with respect to t, we have, putting 



y=f*dt (12), 



2 . mVffi- = y (13), 



where, if we please, we may omit the V, as zr^ is necessarily a vector. 



Now, by the kinematical relation in § 1, if g be the vector-instantaneous axis, 

 we may write (13) as 



2 . mzrXi-sr = y ..... (14). 



17. From these equations Hamilton has deduced, in an extremely simple 

 way, many known results of great interest. For instance, if \^ vanish, i.e., if 

 there be no applied forces, 7 is a constant vector, and (operating on (14) or (13) 



by S . gj 



Ssy = 2 . jn(V««-) 2 = 2mi- 2 = - h? .... (15), 



a constant, by the principle of conservation of energy. 

 Of these equations 



2m(V«r) 2 = -A 2 



denotes obviously an ellipsoid fixed in the body, and such that g is a radius- vector 

 of it. The tangent plane to it at the extremity of g is easily seen to be the fixed 

 plane 



Hence we have at once Poinsot's beautiful construction of the motion, by the 

 rolling of the central ellipsoid on the invariable plane. But this, although 

 extremely elegant, is not well adapted to assist us in the determination of the 

 position of the body in space after a given time. 



18. In most of the investigations which follow, we shall use the form (14) 



