TWISTS AND WRENCHES. 9 



done against the 3/2 original forces, equals the sum of the 

 works done against the gn components. It, therefore, 

 appears from the ist lemma, that Ww'ill still be the 

 sum of the works done against the gn components, ot 

 which 3/2 act at P, $n at Q y $n at R. Finally, by the 

 2nd lemma, Wwill also be the sum of the works done 

 by the original twist against the three resultants formed 

 by compounding each group at P y Q, R. But these re- 

 sultants constitute the resultant wrench, whence the 

 theorem has been proved. 



10. Theorem. From a comparison of the two last 

 articles, we easily deduce the following theorem, which 

 we shall find of great service throughout the essay. 



If a series of twists A ly &c., A m , would compound into 

 one twist^4, and a series of wrenches B^ &c., B n , would 

 compound into one wrench JB y then the energy that would 

 be expended or gained when the rigid body performs the 

 twist A, under the influence of the wrench B y is equal to 

 the algebraic sum of the mn quantities of energy that 

 would be expended or gained when the body performs 

 severally each twist A iy &c., under the influence of each 

 wrench J3 } , &c. 



ii. Concluding Remarks. We have now explained 

 the conceptions, and the language in which the solu- 

 tion of any problem in the Dynamics of a rigid body 

 may be presented. A complete solution of such a pro- 

 blem must provide us, at each epoch, with a screw, by a 

 twist about which of an amplitude also to be specified, the 

 body can be brought from a standard position to the po- 

 sition occupied at the epoch in question. It will also be 

 of much interest to know the instantaneous screw about 

 which the body is twisting at each epoch, as well as its 

 twist velocity. Nor can we regard the solution as quite 

 complete, unless we also have a clear conception of the 

 screw on which all the forces acting on the body consti- 



