269.  We  next  suppose  the  quantics  to  have  common  two
surfaces  having  i  points  of  intersection.  The  method  would
be  the  same  if  there  were  several  surfaces.  Let  the  last
quantic  be  a  complex  one,  consisting  of  Z'  which  passes
through  the  first  surface  and  Z"  which  passes  through  the
second.  Then  the  system  Z7,  F,  U7",  F,  Z\  have  common  the
surface  Xfiv  and  the  curve  Xpv's',  which  have  i  points  common,
and  they  will  therefore  intersect  in  points  which  lie  on  neither

Imnrs  -  Xfiv  {(I  +  m  +  n  +  r)  s  +  2m  +  &c.}  +  /3  (I  +  m  +  n  +  r  +  s')

- 7 - X'fju'vs  (I  +  m  +  n  +  r  +  *')  +  Xfi'v's'  (X  +  p  +  v  +  s)  +  Si

In  like  manner  the  points  common  with  Z"  are

Imnrs" -  X/W  (l+m  +  n  +  r  +  s")  +  X/W  (X  +  H>  +  v  +  s")

-XpV{{l+m  +  n  +  r)8"  +  lm  +  &c.}+P'(l+m  +  n  +  r  +  8n)+3u.-

HH


234  OKDER  OF  RESTRICTED  SYSTEMS  OF   EQUATIONS.
Adding  which  we  have

Imnr  (s'  +  s")  —  {Xjjlv  -f  V//V)  {(I  +  m  +  n  -+■  r)  [s  +  s")  +  hn  +  &c.}
+  (^  +  /3')(Z  +  m  +  w-+r  +  s'  +  s")-7  +  6/-7.
In  other  words,  the  combined  effect  of  the  two  surfaces  is
equal  to  the  sum  of  the  effects  of  the  surfaces  separately  con-
sidered, diminished  by  six  times  the  number  of  their  common
points.  When  there  are  only  four  variables,  two  surfaces
always  must  have  common  points  of  intersection.
