236.] GENERAL PRINCIPLES. 



The left-hand member of this equation evidently represents 

 the differential of the kinetic energy 



r=%^ 2 =%/^ 2 +> 2 +^ 2 ) (14) 



of the body, while the right-hand member is the elementary 

 work of the external forces. Hence equation (13) expresses 

 the principle of kinetic energy for a free rigid body, viz. the 

 proposition that, for any infinitesimal displacement of the body, 

 the increase of the kinetic energy is equal to the sum of the works 

 done by all the external forces. 



235. By introducing the co-ordinates of the centroid, i.e. by 

 putting x=x+% y y=y^-r), =# + , as in Art. 228, the expres- 

 sion for the kinetic energy assumes the form (since ^m^=o, 



(is) 



where v is the velocity of the centroid and u the relative veloc- 

 ity of any particle m with respect to the centroid. 



Thus, it appears that the kinetic energy of a free rigid body 

 consists of two parts, one of which is the kinetic energy of the cen- 

 troid (the whole mass being regarded as concentrated at this 

 point), while the other may be called the relative kinetic energy 

 with respect to the centroid. 



236. By the same substitution the right-hand member of 

 equation (13), i.e. the elementary work ^(Xdx+ Ydy + Zdz), 

 resolves itself into the two parts 



(dx $X+ dy 2 F-f dz 2Z) + 2 (Xd% 4- Ydri + Zd). 



The first parenthesis contains the work that would be done by 

 all the external forces if they were applied at the centroid ; it is 

 therefore equal to the kinetic energy of the centroid, that is to 

 mfyMv*). The equation of kinetic energy (13) reduces, there- 

 fore, to the following : 



(16) 



PART m-9 ^- 



