24, 25] GEOMETRICAL CONSTRAINTS. 57 



Understanding this, we may write 



B 



4) W AB =fxdx + Ydy -f Zdz. 



A 



25. Statics. 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. If however they are subjected 

 to geometrical constraint, as explained in 21 for a single particle, 

 there must he certain relations satisfied by their coordinates. Let 

 these equations of condition or constraint be 



9>1 fa> Vl> g l> X *> 2/2, % -- Xn> yn, *) = 0, 



92 0*i, ft, *!, x*, 2/2, *a> - x n, y, *) = 0, 



9k i, ft, *i, x* 9 2/2, *a> x*> y*> = - 



Such constraints may be caused in a great variety of ways. 

 Particles may be caused to lie on certain fixed or moving surfaces, 

 may be connected by inextensible strings which may pass over 

 pulleys, or by rigid links variously jointed. 



For instance, if two particles 1 and 2 are connected by a rigid 

 rod of length ?, either particle must move on a sphere of radius I of 

 which the other is the center, and we have the equation of condition 



9 ~ (, - xtf + (y, - ytf + ( Zi - stf -l* = 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 



(x - a)* + (y- &) 2 + (g - c) 2 - I 2 ^ 0. 



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

 is defined by an equation.) 



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

 that it has the coordinates 



x r + $x r , y r + dy r , z r + $z r , 



in order that the constraints may hold we must have for each 9, 

 y r , s r , . . .) = 0, 



(f (X r + dx r , y r + Sy r , 2r + Z ry . . .) = 0, 



and if g? be a continuous function, developing by Taylor's Theorem, 



r\ 



V (X r + dX r , y r + 8y r , i + 8*r, ' ) = 9 (r, yr, Z r , . . ) + 8 XT * + 



