THE  EEV.  G.  SALMON  ON  CTJEVES  OF  THE  THIED  OEDEE. 
537 
is  necessary,  then,  in  the  first  place  to  determine  so  that  pS+2 — jp  inay  be  divisible 
by  P,  which  we  do  simply  by  equating  to  0 the  discriminant  of  this  equation.  It  will 
be  observed  that  2— denotes  a pair  of  lines  drawn  through  x'y'z'  to  touch  2 : hence 
in  the  discriminant  that  part  vanishes  which  is  not  multiphed  by  yj ; and  because  these 
two  lines  intersect  on  S,  the  part  multiplied  by  also  vanishes.  Likewise  the  coefficient 
of  in  the  discriminant  of  |a/S+2  vanishes,  and  the  coefiicient  of  jOi^inthe  discriminant 
of  ///S+n^  is  the  function  which  Mr.  Cayley  has  called  0*.  Hence  the  discriminant 
1.0^ 
of  ^S+2— jpn^  is  simply  and  therefore,  if  this  conic  will  break 
up  into  two  right  lines,  and  we  may  write 
^S+2-g,n»=/3P. 
By  the  help  of  this  equation,  the  equation  of  the  cubic  can  be  transformed  from  the  form 
ns-P2=o 
to  the  form 
(n+;t.P)(s-xP-Ypn)=F|(3-|^,n-x(n+^p)|. 
The  form  of  the  equation  shows  that  H+jW/P  is  the  tangent  at  the  point  of  the  cubic 
which  is  tangential  to  the  given  one,  and  that  j3— ^,n  passes  through  the  point  where 
that  tangent  meets  the  cubic  agahl,  or,  as  we  shall  call  it,  through  the  second  tangential 
of  the  given  point. 
The  theorem  contained  in  the  last  equation,  viz.  “ that  if  a conic  pass  through  four 
consecutive  points  of  a cubic  at  a^y’z',  the  chord  joining  the  remaining  points  passes 
through  the  second  tangential  of  ci^y'z',"  may  easily  be  deduced  independently.  In  fact, 
if  abc=def  he  the  equation  of  any  cubic,  any  conic  ab=(jtide  meets  the  cubic  again  in 
two  points  whose  chord  (J^c=f  passes  through  the  fixed  point  cf.  And  hence,  as  is  well 
known,  all  conics  through  four  points  on  a cubic  meet  the  curve  again  in  a chord  pass- 
ing through  a fixed  point  which  I call  the  opposite  of  the  four  given  points,  and  which 
is  constructed  as  follows : Let  the  line  joining  the  points  1 and  2 meet  the  curve  in  a 
point  5,  let  the  line  joining  the  points  3 and  4 meet  the  curve  in  a point  6,  then  the 
line  joining  5,  6 meets  the  curve  in  the  point  7 required.  Now  when  I and  2,  3 and  4 
coincide,  the  points  5 and  6 both  coincide  with  the  tangential  of  that  point,  and  con- 
sequently 7 is  the  second  tangential. 
It  follows  immediately  that  the  conic  through  five  consecutive  points  meets  the  curve 
again  in  a point  which  is  found  by  joining  the  original  point  to  its  second  tangential, 
and  taking  the  point  where  the  joining  line  meets  the  curve  again.  We  deduce  hence 
at  once  M.  Pluckee’s  determination  of  the  points  at  which  a conic  can  osculate  a 
cubic  in  six  points.  In  fact,  if  the  point  just  determined  were  to  coincide  with  the 
* Third  Memoir  on  Quantics,  p.  642. 
4 B 2 
