8] TWISTS AND WRENCHES. 13 



Let n wrenches, which consist of 3w forces acting at A lt ... A 3n , compound 

 into one wrench, of which the three forces act at P, Q, R. The force at A k 

 may generally be decomposed into three forces along PA^, QA/c, RA^. By 

 the 2nd lemma the amount of work ( W) done against the 3w original forces, 

 equals the amount of work done against the 9w components. It, therefore, 

 appears from the 1st lemma, that W will still be the amount of work done 

 against the 9n components, of which 3/i act at P, 3w at Q, 3n at R. Finally, 

 by the 2nd lemma, W will also be the amount of work done by the original 

 twist against the three resultants formed by compounding each group at 

 P, Q, R. But these resultants constitute the resultant wrench, whence 

 the theorem has been proved. 



We thus obtain the following theorem, which we shall find of great 

 service throughout this book. 



If a series of twists A l ,...A m , would compound into one twist A, and 

 a series of wrenches B 1 ,...B n , would compound into one wrench B, then 

 the energy that would be expended or gained when the rigid body per 

 forms the twist A, under the influence of the wrench B, 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 lt ...A m under the 

 influence of each wrench B l ,...B n . 



We have now explained the conceptions, and the language in which 

 the solution of any problem in the Dynamics of a rigid body may be pre 

 sented. A complete solution of such a problem 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 position 

 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 constitute a wrench of which we should also know the 

 intensity. 



There is one special feature which characterises that portion of Dynamics 

 which is discussed in the present treatise. We shall impose no restrictions 

 on the form of the rigid body, and but little 011 the character of the con 

 straints by which its movements are limited, or on the forces to which the 

 rigid body is submitted. The restriction which we do make is that the body, 

 while the object of examination, remains in, or indefinitely adjacent to, its 

 original position. 



As a consequence of this restriction, we here make the remark that the 

 amplitude of a twist is henceforth to be regarded as a small quantity. 



