The  pitch  pa  of  the  screw  a  is  given  by  the  equation  (§  38),


the  six  screws  of  reference  being  co-reciprocals,  the  function  pa  must  retain
the  same  form  after  the  transformation  of  the  axes.  The  discriminant  of
the  function


equated  to  zero  will  give  six  values  of  X  ;  these  values  of  X  will  determine

the  coefficients  of  U  in  the  required  form.  I  do  not,  however,  enter  further

into  the  discussion  of  this  question,  which  belongs  to  the  general  theory  of
linear  transformations.
270  THE   THEORY    OF    SCREWS.  [254-

The  transformation   having  been   effected,  an    important   result   is   im
mediately  deduced.     Let  the  transformed  function  be  denoted  by


then  we  have

ft=-
PI


PI

whence  it  appears  that  the  six  screws  of  reference  are  the  common  screws  of
the  two  systems.     We  thus  find  that  in  this  special  case  of  homography

The  six  common  screws  of  the  two  systems  are  co-reciprocal.
The  correspondence  between  impulsive  screws  and  instantaneous  screws
is  a  particular  case  of  the  type  here  referred  to.  The  six  common  screws  of
the  two  systems  are  therefore  what  we  have  called  the  principal  screws  of
inertia,  and  they  are  co-reciprocal.
255.     Correspondence  of  a  Screw  and  a  system.
We  shall  sometimes  have  cases  in  which  each  screw  of  a  system  cor
responds  not  to  a  single  screw  but  to  a  system  of  screws.  For  the  sake  of
illustration,  suppose  the  case  of  a  quiescent  rigid  body  with  two  degrees  of
freedom  and  let  this  receive  an  impulsive  wrench  on  some  screw  situated
anywhere  in  space.  The  movement  which  the  body  can  accept  is  limited.
