The  correspondence  of  this  result  with  that  of  Art.  249,
may  be  made  more  manifest  by  writing  the  symmetric  func-
tion a2  +  aft  +  /32  ■+  &c.  ^2 ;  a3  +  a*  ft  +  aft2  +  ft'6  +  &c.  ps,  &c.  then
the  result  of  Art.  249  is  Q+pP+p2,  and  of  this  article  is
It  +p  Q  -\-pJP  +iV  And  we  are  led  by  induction  to  the  con-
clusion, that  the  order  of  a  system  where  the  columns  exceed
the  rows  by  three  will  be  S+pB+jp2Q  + p^P+p^  and  so  on
generally.  I  have  not  actually  carried  the  calculation  further
than  the  case  considered  in  the  present  article,  but  the  following
articles  will  show  how  the  process  here  used  can  be  extended
to  the  higher  cases.
In  all  cases  the  weight  of  the  system  may  be  deduced  from
its  order  by  the  rule  already  given.
263.  The  formula  of  the  last  article  may  be  applied  to
calculate  the  order  of  the  system  of  conditions,  that  the  equa-
tions atm  +  &c,  at  +  &c.  may  have  three  common  roots.  The
conditions  are  formed  by  a  system  of  determinants,  the  matrix
for  which  is  formed  as  in  Art.  252 ;  save  that  the  line  a,  #,  c
is  repeated  n  —  2  times,  and  the  line  a',  &',  c',  m  —  2  times.
The  matrix  consists  of  m  +  n  —  2  columns  and  m  +  n  —  4  rows.
