120 KINETICS OF A RIGID BODY. [220. 



along the same axes are X, Y, Z, the equation (i) can be 

 replaced by the following three : 



=o, my + Y=o, mz + Z=o. (2) 



Such a set of three equations can be written down for each 

 particle ; hence, if the body consist of n particles, there would 

 be in all 3^ equations. 



220. For the solution of particular problems these 3/2 equa- 

 tions are of little use, not only because their number would 

 in general be very great and may even be infinite, but mainly 

 because the forces X, Y, Z include the unknown reactions 

 between the particles. It is, however, possible to deduce cer- 

 tain general propositions from these equations. 



The 3 equations express the equilibrium of the system 

 formed by all the forces, both internal and external, acting on 

 the particles, and the reversed effective forces. To apply the 

 principle of virtual work to this system, let us multiply the 

 three equations (2) by the components Sx, Sj/, z of some virtual 

 displacement of the particle m ; let the same thing be done 

 for every other particle of the body ; and let all the resulting 

 equations be added : 



2(-0*jr+-Y)&r+2(-*y+ F)fy + 2(-*2+Z)&sr=o. (3) 



221. It is important to notice that the internal reactions 

 between the particles which make the body rigid occur in 

 pairs of equal and opposite forces, and form, therefore, a 

 system which is in equilibrium by itself. Hence, while these 

 internal forces enter into the equations (2), they do not appear 

 in equation (3), since the equal and opposite forces cancel in 

 the summation. Thus, equation (3) expresses that the external 



Derivatives were called fluxes by Newton; thus the component of the acceleration 

 of a point in any direction is the time-flux of its velocity in that direction; the com- 

 ponent of its effective force in any direction is the time-flux of its momentum; 

 and so on. 



