When  a  rigid  body  has  freedom  of  the  fifth  order,  the  screws  about  which
the  body  can  be  twisted  are  all  reciprocal  to  a  single  screw  p.  In  general,  p
does  not  lie  on  the  system  prescribed  by  the  equation  which  the  co-ordinates
of  all  possible  77  screws  have  to  satisfy.  It  is  therefore,  in  general,  not
possible  that  the  reaction  of  the  constraints  can  provide  an  77.  There  are,
however,  three  screws  in  any  five-system  which  possess  the  property  of
permanent  screws  without  however  making  any  demand  on  the  reaction  of
the  constraints.  The  existence  of  these  screws  is  thus  demonstrated : —
Through  the  centre  of  inertia  of  the  body  draw  the  three  principal  axes,
then,  on  each  of  these  axes  one  screw  can  always  be  found  which  is  reciprocal
to  p.  Each  of  these  will  belong  to  the  five-system,  and  it  is  obvious  from
the  property  of  the  principal  axes,  that  if  the  body  be  set  twisting  about  one
of  these  screws  it  will  have  no  tendency  to  depart  therefrom.
A  body  which  has  freedom  of  the  sixth  order  is  perfectly  free.  Any  screw
on  one  of  the  principal  axes  through  the  centre  of  inertia  is  a  permanent
screw,  and,  consequently,  there  is  in  this  case  a  triply  infinite  number  of
permanent  screws.
395.  Summary.
The  results  obtained  show  that  for  a  rigid  body  with  the  several  degrees
of  freedom  the  permanent  screws  are  as  follows  : —

No.  of  Permanent  Screws


Freedom    I

1

II.
1

Ill.
3

IV..
3

V.
3

VI Triply  infinite


CHAPTER  XXVI.
AN   INTRODUCTION   TO   THE   THEORY   OF   SCREWS   IN   NON-EUCLIDIAN    SPACE.
396.  Introduction.
