25] CONDITION OF EQUILIBRIUM. 61 



end, the other being fixed. If now the forces are in equilibrium, 

 an arbitrary small displacement of all the blocks will neither raise 

 nor lower the weight at the end of the string. Thus by the applica- 

 tion of the property of the pulley the principle was proved. 1 ) 



We shall not here undertake to give a more formal proof of 

 the principle, which may be given by an analysis of the various 

 kinds of constraint, such a proof is found in Appell, Traite de 

 Mecanique Bationelle. Tom. I, Chap. 7. 



If the forces X 19 T 19 Z act upon the particle 1, X 2 , Y 2 , Z 2 upon 

 the particle 2, etc., the condition of equilibrium is 



12) X, dx, + YJyi + Z^gt 

 or as we may write it, 



is) 2j ( Xdx + Yd y + Zd ^ = - 



This is the analytical expression of the Principle of Virtual Work. 

 If the particles satisfy the equations of constraint 5) the dis- 

 placements must satisfy the equations 





Multiplying the equations 14) respectively by J^ 9 Ag, . . . A*, and adding 

 to 12) we have 



2n = 0. 



Of the 3n quantities dx lf . . . d# n , only 3n Jc are arbitrary, we 

 may however determine the Jc multipliers A so that the coefficients 



1) Ibid. 18. 



