55, 56] WORK AND ENERGY. 103 



The components of the force are supposed to be given as 



functions of s and the derivatives -y- , -~ , 7 are known as func- 



ds as as 



tions of s from the equations of the path. 

 Understanding this, we may write 



(4) W AB = [ Xdx + Ydy + Zdz. 



J A 



56. Virtual Work. Suppose that we have a system of n 

 material points. If they are entirely free to move, they require 3n 

 coordinates for their specification. They may be mechanically 

 constrained, however, in such a manner that there must be certain 

 relations satisfied by their coordinates. Let these equations of 

 condition or constraint be 



<i Oi, 2/1, *i, #2, 2/2, * 2 , #n, y n , -O = o, 



02 (#1, 2/i> ^' ^2, 2/2.- * 2 , #n> 2/n, *n) = 0, 



(5) 



</>&Ol, 2/1, 2l, #2, 2/2, *2, #n> 2/, %) = 0. 



Such constraints may be imposed by causing the particles to 

 lie on certain surfaces. For instance, if two particles 1 and 2 are 

 connected by a rigid rod of length I, either particle must move on 

 a sphere of radius I of which the other is the center, and we have 

 the equation of condition 



*0 = <X -tf 2 ) 2 + (y, - ytf + (z, - ztf - / 2 = 0. 



(We might have constraints defined by inequalities, e.g., if a 

 particle were obliged to stay on or within a spherical surface of 

 radius I the constraint would be only from without, and we should 

 have 



0-a) 2 + (y-&) 2 + 0-c) 2 - Z 2 ^0. 



We shall assume that the constraint is toward both sides, and 

 is defined by an equation.) 



If any particle at a? r , y r , z r is displaced by a small amount so 

 that it has the coordinates 



x r + Sx r , y r + Sy r , z r + Sz r , 



* The sign = is to be read is identically i.e., is for all possible values of the 

 variables, or is defined as. 



