As  an  example,  we  may  take  the  simple  case  of  the  cylindroid,  in  which
I,  m,  n,  r  being  each  unity  the  locus  is  of  the  third  degree.  The  screws  of

16—2


244  THE  THEORY  OF  SCREWS.  [228

a  three-system  which  satisfy  an  equation  of  the  nth  degree  must  have  as
their  locus  a  surface  of  degree  not  exceeding  3n.  The  most  important
application  of  this  is  when  n  =  2,  in  which  case  the  screws  form  a  quadratic
two-system.  The  degree  of  this  surface  cannot  exceed  six,  on  the  other
hand,  if  the  quadratic  condition  which  we  may  write


A0?  +  B02*  +  CB?  +  2F0A  +  200,6,  +  2H0A  =  0,

should  break  up  into  two  linear  factors  each  of  these  linear  factors  will
correspond  to  a  cylindroid,  i.e.  a  surface  of  the  third  degree.  Hence  the
degree  of  the  surface  must  in  general  be  neither  less  than  six  nor  greater
than  six,  and  hence  we  learn  that  the  surface  which  is  the  locus  of  the
screws  of  a  quadratic  two-system  is  of  the  sixth  degree.
A  particular  case  of  special  importance  arises  when  the  pitches  of  all  the
screws  on  the  surface  are  to  be  the  same.  The  statement  of  this  condition
is  of  course  one  equation  of  the  second  degree  in  the  co-ordinates  of  the
screw.  In  the  case  of  canonical  co-reciprocals,  this  equation  would  be


But  the  condition  that  Q  and  a  shall  intersect  will  now  submit  to
modification.  We  sacrifice  no  generality  by  making  a  of  zero  pitch,  so
that  if  6  has  a  given  pitch  pe,  the  condition  that  a  and  0  shall  intersect
is  no  longer  of  the  third  degree.  It  is  the  linear  equation

2t3-ae  =  pe  cos  (a0).
