TWISTS AND WRENCHES. 7 



6. Restriction of the Forces. It is first necessary 

 to point out the restrictions which we shall impose 

 upon the forces. The rigid body M y whose motion we 

 are considering, is presumed to be acted upon by the same 

 forces whenever it occupies the same position. This will 

 necessitate that the surrounding bodies are fixed whence 

 the forces acting on M emanate. Forces such as those 

 due to a resisting medium are excluded, because such 

 forces do not depend on the position of the body, but on 

 the manner in which the body is moving through that 

 position. The same consideration excludes friction which 

 depends on the direction in which the body is moving 

 through the position under consideration. 



But the condition that the forces shall be defined, when 

 the position is given, is still not sufficiently precise. We 

 might include, in this restricted group, forces which could 

 have no existence in nature. We shall, therefore, add 

 the condition that the system is to be one in which the con- 

 tinual creation of energy is impossible. 



7. The Energy of Position. An important con- 

 sequence of this restriction is stated as follows : The 

 quantity of energy necessary to compel the body M to 

 move from the position A to the position B, is indepen- 

 dent of the route by which the change has been effected. 



Let L and M be ;two such routes, and suppose that 

 less energy was required to make the change from A to 

 B via L than via M. Make the change via Z, with the 

 expenditure of a certain quantity of energy, and then 

 allow the body to return via M. Now, since at every 

 stage of the route M the forces acting on the body are the 

 same whichever way the body be moving, it follows, that 

 in returning from B to A via M, the forces will give out 

 exactly as much energy as would have been required to 

 compel the body to move from A to B via M ; but by 

 hypothesis this exceeds the energy necessary to make 



