251.] THE PRINCIPLE OF VIRTUAL WORK. ^5 



251. Whether the system be in equilibrium or not, the quan- 

 tity 2/^=0 2(-&r+ YSy + ZSz) represents the element of work 

 8 W done by the forces in a virtual displacement. For an actual 

 displacement of the system B should be replaced by d, and we 

 have 



Ydy+Zdz) 



for the work done in the infinitesimal displacement. 



Most of the forces occurring in nature are such as to make 

 this quantity an exact differential. In this case the forces are 

 said to form a conservative system, and the equation can be 

 integrated from any initial position or configuration of the sys- 

 tem to any final position or configuration without reference to 

 |the intermediate positions. 



Taking the initial position as the standard of reference, the 

 Iresult can be written in the form 



(where U is a function of the co-ordinates (or other quantities) 

 letermining the position and configuration of the system, while 

 \U Q is the initial value of U. 



Leaving the standard of reference indeterminate, the equa- 

 :ion can be written in the form 



w=u+c> 



\C being merely a symbol for the constant of integration. The 

 [unction Wis called the work function or force function. 



If the final position of the system be taken as standard of 

 iference,' and /i be the value of U in the final position, the 



luation takes the form 



'here V is called the potential energy of the forces with refer- 

 ence to the final position. 



