373-1 CONTINUOUS MOTION OF FREE BODY. 2OQ 



If q, q denote the radii of inertia of the body for the instan- 

 taneous axis / and the parallel centroidal axis 7, we have 



where x is the distance between / and L Hence denoting by 

 v the velocity of the centroid, v = V.z/ 2 + aPx*, we find 



T= \M(u* + o> 2 ^ 2 ) + \ Mrfq* = \M& + \ M^co*. (2) 



It thus appears that the kinetic energy consists of two parts, 

 = 7\+ 7*2, one of which, 



may be called the kinetic energy of the centroid (the whole mass 

 M being regarded as concentrated at the centroid), while the 

 other, 



the so-called kinetic energy of rotation, is the kinetic energy 

 which the body would possess if it were rotating with the 

 angular velocity o>, not about the instantaneous axis /, but 

 about the parallel centroidal axis / The developments of 

 Arts. 322 and 323 apply without change to T z . 



373. Numerous exercises and applications will be found in the works 

 of Price, Besant, Williamson and Tarleton, Walton, quoted in Art. 159 ; 

 but above all in E. J. ROUTH, Dynamics of a system of rigid bodies, Ele- 

 mentary part, fifth edition, 1891 ; Advanced part, fourth edition, 1884; 

 London and New York, Macmillan. Illustrations and examples, as well 

 as further developments of the theory, will also be found in the works of 

 Schell and Budde (Art. 159) ; in the French collections of problems by 

 M. Jullien and by A. de Saint-Germain ; in J. PETERSEN, Lehrbuch der 

 Dynamik fester Korper, deutsch von R. von Fischer-Benzon, Kopeh- 

 hagen, Host, 1887; E. BOUR, Cours de mecanique et machines, IIP 

 fascicule, Paris, Gauthier-Villars, 1874; and the original memoirs of 

 L. POINSOT, in particular his Theorie nouvelle de la rotation des corps, 

 Paris, Bachelier, 1852 (also in Liouville's Journal de mathematiques , 

 Vol. XVI.), and his Precession des equinoxes, Paris, Mallet-Bachelier, 

 1857. Among the numerous French treatises on theoretical mechanics 

 those of Poisson, Sturm, Resal, Collignon deserve especially to be 

 mentioned here. 



PART III 14 



