385.] CONSTRAINED SYSTEM. 21$ 



and hence the motion of the particle, will not be changed. But 

 we can now combine Fand mj'to a resultant F". Since F, F', 

 7/z/are in equilibrium, the forces .F'and F" are in equilibrium ; 

 />. F" is equal and opposite to F', as appears from the figure. 



The figure also shows that F can be resolved into the com- 

 ponents mj and F" ; the former component, mj, produces the 

 actual motion of the particle, while the latter, F", is consumed 

 in overcoming the internal reactions and constraints repre- 

 sented by F'. This component F" of F is therefore called by 

 d'Alembert the lost force. As F' + F" = o at every particle of the 

 system, d'Alembert's principle can be expressed by saying that, 

 at every moment during the motion, the lost forces are in equilib- 

 rium with the constraints of the system. 



If the constraints, instead of being expressed by means of 

 forces, are given by equations of condition, we may express 

 the same idea by saying that, owing to the given conditions, the 

 .lost forces form a system in equilibrium. 



385. We shall now assume that the constraints or conditions 

 to which the system is subject are expressed by means of 

 equations (the case of conditions expressed by inequalities is 

 excluded) between the co-ordinates x t y. z of the particles and 

 the time t. In the most general case these equations might 

 also contain the velocities of the particles ; this case, however, 

 will not be considered here. 



Let there be k conditions 



<l>(t, *i, JV*i, *2> " )=o, -^(t, *i, y\> *i x v ' )> ( r ) 



for a system of n points. Then the number of the indepen- 

 dent equations of motion will be ^n k. For these equations 

 must express the equilibrium of the given forces, together with 

 the reversed effective forces, under the given conditions ; and 

 for this equilibrium it is sufficient that the virtual work should 

 -vanish for any displacement compatible with the conditions, the 

 Avork of the reactions and constraining forces being zero for 



