81]  THE   PRINCIPAL    SCREWS    OF   INERTIA.  7i

80.     Impulsive  Screws  and  Instantaneous  Screws.
If  a  free  quiescent  rigid  body  receive  an  impulsive  wrench  on  a  screw  77,
the  body  will  immediately  commence  to  twist  about  an  instantaneous  screw
a.  The  co-ordinates  of  a  being  given  for  the  six  screws  of  reference  just
denned,  we  now  seek  the  coordinates  of  77.
The  impulsive  wrench  on  77  of  intensity  77'"  is  to  be  decomposed  into  com
ponents  of  intensities  tj'"r}1,  ...  i}'"rj6  on  (»l,  ...  ws.  The  component  on  con
will  generate  a  twist  velocity  about  o>H  amounting  to

jL  W

M      Pn     '

but  if  a  be  the  twist  velocity  about  a  which  is  finally  produced,  the  expression
just  written  must  be  equal  to  aan,  and  hence  we  have  the  following  useful
result  :  —

If  the  co-ordinates  of  the  instantaneous  screw  be  proportional  to  al}  ...  afi>
then  the  co-ordinates  of  the  corresponding  impulsive  screw  are  proportional  to


81.     Conjugate  Screws  of  Inertia.
Let  a  be  the  instantaneous  screw  about  which  a  quiescent  body  either
free  or  constrained  in  any  way  will  commence  to  twist  in  consequence  of
receiving  an  impulsive  wrench  on  any  screw  whatever  77.  Let  ft  be  the
instantaneous  screw  in  like  manner  related  to  another  impulsive  screw  £

We  have  to  prove  that  if  £  be  reciprocal  to  a  then  shall  77  be  reciprocal
to  j3.
