12 THE THEORY OF SCREWS. [8 



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 continual creation 

 of energy is impossible. 



An important consequence 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 independent 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 L, 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 the change via L, and hence, on the return of the body to 

 A, there is a clear gain of a quantity of energy, while the position of the body 

 and the forces are the same as at first. By successive repetitions of the 

 process an indefinite quantity of energy could be created from nothing. This 

 being contrary to experience, compels us to admit that the quantity of energy 

 necessary to force the body from A to B is independent of the route 

 followed. 



It follows that the amount of work done in a number of twists against 

 a wrench is equal to the work that would be done in the resultant twist. 



For, the work done in producing a given change of position is independent 

 of the route. 



We may calculate the work done in a twist against a wrench by deter 

 mining the amount of work done against three forces which are equivalent 

 to the wrench, in consequence of the movements of their points of application 

 which are caused by the twist. 



We shall assume the two lemmas 1st. The work done in the displace 

 ment of a rigid body against a force is the same at whatever point in its line 

 of application the force acts. 2nd. The work done in the displacement of a 

 point against a number of forces acting at that point, is equal to the work 

 done in the same displacement against the resultant force. 



The theorem to be proved is as follows : The amount of work done in a 

 given twist against a number of wrenches, is equal to the work done in the 

 same twist against the resultant wrench. 



