3770 FREE SYSTEM. 211 



by forces and that these constraining forces are included among 

 the forces X, Y, Z, acting on the particles. Thus in par- 

 ticular, the general equations of motion of a rigid body, viz. 

 (4) or (5), Art. 223, and (6) or (7), Art. 224, hold for a variable 

 system. For they express the necessary, though not in general 

 sufficient, conditions of equilibrium of the forces acting on the 

 particles with the reversed effective forces of these particles ; 

 and this equilibrium is not changed by making the distances 

 between the particles invariable ; i.e. by what is sometimes 

 called solidifying the system. But it should be observed that 

 the reductions of the systems of momenta and effective forces, 

 given in the chapter on the rigid body, do not in general hold 

 for a variable system. 



376. Let F be the resultant of all the external and internal 

 forces acting on one of the n particles ; X, Y, Z its components 

 along a system of fixed rectangular axes ; x, y, z the co-ordinates 

 of the particle, and m its mass. Just as in Arts. 219, 220, we 

 have the equations of motion of the particle 



mx=X, my=- Y, mz = Z. (i) 



There are 3 such equations for each particle, and hence 3 for 

 the whole system. These $n equations express the equilibrium 

 of the system of forces composed of the external, internal, and 

 reversed effective forces. 



377. Applying the principle of virtual work to this system 

 of forces, we find d'Alemberfs equation 



^(-mx + X}x+-$(-my+ Y)ty + 2(-mz + Z)Sz = o, (2) 

 in which Bx, By, 82 are the components of an arbitrary displace- 

 ment Bs of the particle m. As there are 3/2 such arbitrary 

 component displacements, the equation (2) is equivalent to the 

 3 72 equations (i). 



If written in the form 



, (3) 



