223.] GENERAL PRINCIPLES. I2i 



forces acting on the rigid body and the reversed effective forces 

 form a system in equilibrium ; and this is d'Alembert's Principle 

 for the rigid body. 



It must, however, not be forgotten that the displacements $>x, 

 Sj>, &z should be so selected as to be compatible with the nature 

 of the rigid body; i.e. with the conditions that the distances 

 between trie particles should not be disturbed. 



222. The number of conditions expressing the invariability 

 of the distances between n particles is 3/2 6. For if there 

 were but 3 particles, the number of independent conditions 

 would evidently be 3 ; for every additional particle, 3 additional 

 conditions are required. Hence, the total number of condi- 

 tions is 3 + 3(-3) = 3^-6. 



It follows that if a rigid body be subject to no other con- 

 straining conditions, the number of its equations of motion 

 must be $n ($n 6) =6. Hence, a free rigid body has six 

 independent equations of motion. (Comp. Part I., Art. 37.) 



223. The six equations of motion of the rigid body can be 

 obtained as follows. 



Imagine the equations (2), viz. 



mx=X, my= Y, mz = Z, 



written down for every particle, and add the corresponding 



equations. This gives the first 3 of the 6 equations of motion: 



^mx=^X 1 ^my = ^Y y 2mz = 2Z. (4) 



As the internal forces cancel in the summation, the right-hand 

 members of these equations represent the components R x , R y , R g 

 of the resultant R of all the external forces acting on the body. 

 The left-hand members can be put into the form d$mx)/dt, 

 d@.my)/dt, d(*Lmz)/dt ; these are the time derivatives or fluxes 

 of the sums of the linear momenta of all the particles parallel 

 to the axes. The equations (4) can therefore be written in the 

 form 



R r (5) 



a ,, 



dt dt dt 



