156 STATICS. [252. 



252. If the work function 



W=U+C 



be given as a known function of the co-ordinates determining 

 the configuration of the system, the positions of equilibrium of 

 the system can be found from the condition 



which expresses that the work function W is a maximum, or a 

 minimum (or stationary). It can be shown without difficulty 

 that the equilibrium is stable when the work function is a maxi- 

 mum, and unstable when the work function is a minimum. 



As the potential energy by the formulae of Art. 248 is equal 

 to the work function, but of opposite sign, it follows that 

 the equilibrium is stable or unstable according as the potential 

 energy is a minimum or a maximum. 



253. The special case when the only forces acting are the weights of 

 the particles constituting the system is worth mentioning. 



Let m be the mass of one of the particles, mg its weight, and z its 

 height above a horizontal plane of reference. Then the virtual work of 

 the weights is 



If z be the height of the centroid of the system above the plane of refer- 

 ence, we have ^mz = ^m z ; hence ^m Bz = ^m 8z. The work func- 

 tion is, therefore, 



W=g$m-z+ C, 



and this becomes a maximum or minimum according as z is a minimui 

 or maximum ; i.e. the equilibrium is stable or unstable according as tht 

 centroid of the system is at its least or greatest height. 



254. It is the object of every machine to do work in a certain pre- 

 scribed way, i.e. to exert force, or overcome a resistance, through a cer- 

 tain distance. The various forces of nature, such as the muscular foi 

 of man and other animals, the force of gravity, the pressure of the windj 

 electricity, the expansive force of steam or gas, etc., are called upon fo^ 

 this purpose. In most cases it would not do to apply these forces* 



