240.] THE PRINCIPLE OF VIRTUAL WORK. 147 



and sufficient conditions of equilibrium of a single particle are 

 that the sum of the virtual zvorks of all forces must be zero for 

 any three virtual displacements not all in tJie same plane. 

 This is the principle of virtual work for a single particle. 



If the particle be referred to a rectangular system of co-ordi- 

 nates, its displacement 8s can be resolved into three com- 

 ponent displacements &r, by, Sz, parallel to the axes. The 

 forces acting on the particle being replaced by their compo- 

 nents X, Y, Z, the sum of their virtual works, for the displace- 

 ment Bs is 2-Y-&M-2K-5y + 2Z.&sr. Hence the analytical 

 expression for the principle of virtual work : 



$X- &r+2 Y- Sy + 2Z-Sz = 0. (5) 



As the displacements &r, fy, 8z are independent of each other, 

 and perfectly arbitrary, this single equation is equivalent to the 



three equations 



o, 2F=o, 



which are the ordinary conditions of equilibrium of a single 

 particle. 



240. The principle of virtual work is particularly useful in 

 eliminating the unknown reactions arising from constraints. 



Suppose the particle be constrained to a smooth surface or 

 curve. After introducing the normal reaction of the surface 

 or curve the particle can be regarded as free ; and the equation 

 of virtual work can be used to express the conditions of equi- 

 librium. This equation will, in general, contain the unknown 

 reaction. But as this reaction has the direction of the normal, 

 it will be eliminated if the virtual displacement be selected 

 along a tangent. Hence, immhe case of constrainment to a sur- 

 face, the two conditions of equilibrium independent of the reaction 

 are found by forming the^^uiation of virtual work for virtual 

 displacements along any tw^^mLgents to the surface ; and in the 

 case of constrainment to a 40^ the one suck condition is found 

 from a virtual displacement along the tangent. 



If it be required to find the normal pressure on the surface 



