8 TWISTS AND WRENCHES. 



the change via Z, 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 pro- 

 cess an indefinite quantity of energy could be created 

 from nothing. This being contrary to nature, compels 

 us to admit that the quantity of energy necessary to force 

 the body from A to B is independent of the route fol- 

 lowed. 



8. Theorem. The sum of the works done in a 

 number of twists against a wrench is equal to the work 

 that would be done in the resultant twist. 



For, by the last article, the work done in producing a 

 given change of position is independent of the route. 



9. Theorem. We first define that by the work done in 

 a twist against a wrench is to be understood the sum of the 

 works done against the three forces which constitute the 

 wrench in the movements of their points of application 

 which are caused by the twist. 



We shall assume the two lemmas ist. The w r ork 

 done in the displacement 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 offerees acting at that point, 

 equals the work done in the same displacement against 

 the resultant force. 



The theorem to be proved is as follows : The sum of 

 the works done in a given twist against a number of 

 wrenches, equals the work done in the same twist against 

 the resultant wrench. 



Let n wrenches, which consist of $n forces acting at 

 A } , &c., A 3n , compound into one wrench, of which the 

 three forces act at P, Q, jR. The force at each point A k 

 may be decomposed into three forces along PAk, QA^ 

 RA k . By the 2nd lemma the sum of the works (W\ 



