32] MOTION OF CENTER OF MASS. 91 



Returning to the equations 39), whether there is a force -function 

 or not, A, [i, v, being the same for each term of the summation, 

 may be taken out from under the summation sign and being arbitrary, 

 the equation 39) is equivalent to the three 



or as before 



46) d ^Z r m r = Z r X r , ^Z r m r = Z r Y r , ^Z r m r 



that is: The center of mass of any system of the kind specified 

 moves as if all the forces applied to its various parts were applied 

 at the center of mass to a single particle whose mass is equal to 

 the mass of the whole system. 



This principle of the motion of the center of mass reduces the 

 problem of the motion of the system to that of finding the motion 

 of a single particle together with that of the motion of the parts of 

 the system with respect to the center of mass. 



A rigid body is a system of particles coming under the case 

 here treated, since the only constraints are such as render all the 

 mutual distances of individual points constant. Therefore the only 

 new principles required in order to treat the motion of a rigid body 

 are such as determine its motion relatively to its center of mass. 



If the center of mass is to remain at rest or move uniformly, 

 we must have 



47) 2; r x r = o, .z r r r = o, z r z r = o. 



This will always be the case as shown above for mutually attracting 

 particles, since to. every action there is an equal and opposite 

 reaction. The three equations 47) furnish three necessary con- 

 ditions for the equilibrium of a rigid body. 



If we introduce the relative coordinates of the particles with 

 respect to the center of mass into the expression for kinetic energy 

 it assumes a remarkable form. Let us put 



x r = x + | r , y r = y + ij rf s r = ~8 4- 

 then 



dx r dx di- r 

 ~dt"' = ~di^ ~dt' 



dt ~ dt dt 



z r dz d 



dt = = ~dt ~^~ ~dt 



