166 WORK 



For, let us give the system a small displacement, which consists 

 in turning the body in question through a further angle dO about 

 the selected line, so that 6 becomes changed into -f- d6. The 



dW 



increase in potential energy is r- d6, while the work performed 



GV 



is, by the theorem of 121, equal to G d6, where G is the moment 

 about the axis of all the forces acting on the rigid body. 



M 



so that e = -^' 



cu 



the result required. 



135. THEOREM. In a position of equilibrium of a system of 

 bodies, the potential energy W is either a maximum or a minimum 



The potential energy is a function of all the coordinates of al 

 the particles of which the system of bodies is composed, say 



x if z ' x ii z ' etc 



If the position is one of equilibrium, each particle is in equilib- 

 rium, so that the components of the forces acting on each particle 

 vanish separately by 33. By 133 the condition for this is 



dW A 



= 0, etc. 



dx z 



But these are exactly the conditions that W shall be a maximum 

 or a minimum. 



136. The converse of this theorem is also true. 



THEOREM. If the potential energy of a system of bodies is either 

 a maximum or a minimum in any configuration, then the con- 

 figuration is one of equilibrium. 



For, with the notation of the previous section, if W is a maxi- 

 mum or a minimum, it follows that 



n SW ~Q ?*-_ 

 ", \J) v. 



dy l dz l 



