104 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. 



in order that the constraint may hold we must have for each c/> 



Zr + bz, ....... ) = 0, 



and if <j> be a continuous function, developing by Taylor's Theorem, 



and accordingly, taking account only of the terms of the first order 

 in the small quantities &z? r > 8y r , &z r , and using equations (6), we 

 have 



If a number of particles are displaced, we must take the sum 

 of expressions like the above for all the particles, or 



as the conditions which must be satisfied by all the displacements 

 8x rt Sy r , Sz r . There must be one such equation for each function 

 <. Such displacements, which are purely arbitrary, except that 

 they satisfy the equations of condition, are called virtual, being 

 possible, as opposed to the displacements that actually take place 

 in a motion of the system. 



The Principle of Virtual Work is an analytical statement of the 

 conditions for equilibrium of a system. A system is in equilibrium 

 when the forces acting on its various particles, together with the 

 constraints, balance each other in such a way that there is no 

 tendency toward motion of any part of the system. If the system 

 consists of a single free point, in order for it to be in equilibrium, 

 the resultant of all the forces applied to it, whose components 

 are X, F, Z, must vanish, 



(9) X=Y = Z=0. 



If we multiply these equations respectively by the arbitrary 

 small quantities Sx, By, Bz and add, we get 



(10) XSx+YSy + ZSz = 0, 



which expresses that the work done in an infinitesimal displace- 

 ment of a point from its position of equilibrium vanishes. The 



