If  a  series  of  twists  Al,...Am,  would  compound  into  one  twist  A,  and
a  series  of  wrenches  B1,...Bn,  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  Alt...Am  under  the
influence  of  each  wrench  Bl,...Bn.
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.
