536 
THE  EEV.  G.  SALMON  ON  CHRYES  OE  THE  THIRD  OEDEE. 
Let  the  conic  S=/:a2  break  up  into  two  right  lines;  then  obriously  passes 
through  the  intersection  of  these  lines:  this  intersection  is  therefore  a point  on  the 
cubic  and  P— the  tangent  at  it.  Hence  the  four  points  of  contact  of  tangents  to 
the  cubic  from  ci^y'z'  form  a quadrangle,  the  three  centres  of  which  are  on  the  cubic, 
and  are  the  three  points  co-tangential  with  a^y’z\  that  is  to  say,  having  the  same 
tangential 
Formation  of  the  Equation  of  the  Conic  through  five  consecutive  Points  of  a Cubic. 
Mr.  Cayley  has  communicated  to  me  an  investigation  of  the  equation  of  the  above 
conic  in  the  case  where  the  cubic  is  given  by  the  equation 
P-{-y^-^z^-\-^lxyz={). 
The  investigation  of  the  general  case  presents  no  greater  difSculty,  by  the  help  of  the 
identity  with  which  we  commenced. 
Since  S touches  the  cubic  and  P is  the  common  tangent,  the  general  equation  of  a 
conic  touching  U must  be  of  the  form  S— aP  (where  ot,=Ax-\-'Qy-\-Cz  denotes  any  right 
line  whatever).  But  by  the  identity  referred  to,  the  equation  of  the  cubic  may  be  written 
n(S-aP)=P(2-«n). 
Hence  the  four  points  where  S— a P meets  the  cubic  again  are  its  intersections  with 
2 — csH,  and  if  the  latter  conic  pass  through  x'y'z\  the  former  will  pass  through  three 
consecutive  points  of  the  cubic.  But  on  substituting  x'y'z'  for  xyz,  we  have  2'=n'  = H'. 
and  the  condition  that  2— aH  shall  pass  through  Fy'z'  is  simply  a'=l. 
In  order  that  S — aP  may  pass  through  four  consecutive  points,  2 — aH  must  have 
P for  a tangent  at  the  point  aiy'z'.  Now  the  tangent  to  2— aH  (being  the  polar  of  x'y'z 
with  respect  to  this  conic)  is 
2n-a'n-an', 
or,  since 
a'=l,  n'=H',  is  H-aH'. 
But  this  quantity  must  be  proportional  to  P.  Hence  we  have 
c& = ?».P  -|“  H. 
The  general  equation  therefore  of  a conic  through  four  consecutive  points  is 
S-XP-^PH, 
and 
2-xpn-j^n^ 
passes  through  the  two  points  where  the  former  conic  meets  the  cubic  again.  Since 
these  two  conics  have  P for  a common  tangent,  it  will  be  possible,  by  adding  the 
equations  (multiplied  by  suitable  constants),  to  obtain  a result  divisible  by  P,  and  the 
quotient  will  be  the  line  joining  the  points  where  the  conic  meets  the  cubic  again.  It 
* See  Mr.  Cayley’s  Memoir,  p.  444,  and  my  ‘ Higher  Plane  Curves,’  p.  134. 
