

56, 57] WORK AND ENERGY. 107 



(15) 



...... . 



oz n oz n cz n 



Eliminating from these the k quantities X, we have 3ft - A; 

 equations expressing the conditions of equilibrium, being as many 

 as the system has degrees of freedom. The equation (12), or as 

 we may write it 



( 1 6) 2 (XSx + YSy + ZSz) = 0, 



expresses the fact that the work in a virtual displacement vanishes, 

 and is the condition for equilibrium. This is the Principle of 

 Virtual Work. 



57. D'Alembert's Principle. The equations of motion of a 

 point are (51) 



(17) ' r 



m r 

 or 

 (18) 



Multiplying the equations (18) respectively by the arbitrary 

 quantities 8# r , fy r > ZT adding, and taking the sum for all values of 

 the suffix r, 



2) / ' ^ ^ r V \ 5- 

 r 1 \ m r ~T^r ^ r) Or ' 



