92 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



48) r= 



9 ^T-L 9^ ^!4.0^f ^ 

 dt d< + d< d* + dt dt 



r \* /Jfl r \ /<lft.\M 



j + Urj.+ (lit) }' 



and taking outside of the sign of summation the factors common to 



,, ., , . dx dy dz 



all the terms involving -^> -~i -^> 



- i[((f )'+() 



+ 



Now in the last three terms we may write 



if J is the a;- coordinate of the center of mass in the |, 17, g system. 

 But since the center of mass is the origin of the relative coordinates 

 %,ij,, this is equal to zero. Similarly for the terms in rj r and r . 

 Thus we have remaining if we write M for the mass of the whole 

 system, 



The first term is the kinetic energy of a particle whose mass is equal 

 to the total mass of the system placed at the center of mass, while 

 the second is the relative kinetic energy of the system with respect 

 to the center of mass. Thus the absolute kinetic energy is always 

 greater than its relative kinetic energy with respect to the center of 

 mass (unless the center of mass be at rest). The center of mass is 

 the only point for which such a decomposition of the kinetic energy 

 is generally possible. 



If the principle of the conservation of motion of the center of 

 mass holds we have 



dx dy , dz _ 



~dt~ a > ^di > 'di = * c > 



