243-] THE PRINCIPLE OF VIRTUAL WORK. 149 



It may here be noted that, in general, the principle of virtual 

 work must be understood to mean that the sum of the works 

 of the forces differs from the work of their resultant by an 

 infinitesimal of an order higher than that of the virtual dis- 

 placement. It does not mean that the difference is absolutely 

 zero. 



243. Owing to the proposition proved in the preceding article, 

 the sum of the works of all the forces acting on a rigid body is 

 equal to the sum of the works of the resultant R and the result- 

 ing couple H for any infinitesimal displacement of the body, 

 and the work of the forces is not changed by such a displace- 

 ment. 



It follows that the necessary and sufficient conditions of 

 equilibrium of a rigid body (Art. 201), viz. 



^ = 0, H=o, 



can be expressed by saying that the sum of the virtual works 

 of all the forces must be zero for any infinitesimal displacement 

 of the body. 



For when the forces are in equilibrium, this condition is 

 evidently fulfilled. To prove that there must be equilibrium 

 whenever this condition is fulfilled, it is only necessary to show 

 that both R and H must vanish if the sum of their works is 

 zero for any infinitesimal displacement. A 



To see this, consider first a displacement of translation, Ss, 

 parallel to R. The work of R will be RSs while the works of 

 the two forces constituting H are equal and opposite, so that 

 the work of H is zero. As the sum of the works of R and H 

 must vanish by hypothesis, it follows that R = o. 



Next consider a displacement of rotation S0 about an axis 

 parallel to the vector H. Taking this axis so as to intersect 

 R and bisect the arm/ of ttffcouple //", the work of R will be 

 zero while that of each of forces F of the couple H will 

 hence the whole work of H is FW = HW. As 





