ORDER  OF   RESTRICTED   SYSTEMS   OF  EQUATIONS.  213

system  of  common  values  of  the  variables,  and  the  system  of
equations  denotes  an  infinity  of  "  points."  Such  a  system  of
equations  we  shall  speak  of  as  denoting  a  curve.  If  with  the
given  system  of  k  —  1  equations,  we  combine  any  arbitrary
equation  of  the  first  degree,  we  have  then  data  enough  to
determine  points,  which  will  be  equal  in  number  to  the  product
of  the  degrees  of  the  equations.  We  shall  define  the  order  of  a
curve  as  the  number  of  points  which  are  obtained  when  with  the
equations  which  denote  the  curve,  we  combine  an  arbitrary
equation  of  the  first  degree.
When  we  are  given  a  system  of  k  —  2  equations,  these
denote  a  doubly  infinite  series  of  points,  since  we  cannot  deter-
mine any  points  unless  we  are  given  two  other  equations.
Such  a  system  we  shall  speak  of  as  denoting  a  surface.  If
with  the  system  of  k  —  2  equations  we  combine  an  arbitrary
equation  of  the  first  degree,  we  shall  have  a  "curve"  whose
order  is  the  product  of  the  degrees  of  the  k—  2  equations.  In
general  by  the  order  of  a  surface,  we  mean  either  the  order
of  the  Gfxrve  obtained  by  combining  with  the  given  equations  an
equation  of  the  first  degree,  or,  what  comes  to  the  same  thing,
the  number  of  points  obtained  by  combining  with  the  given
equations  two  equations  of  the  first  degree.
