[ 535  ] 
XXVII.  On  Curves  of  the  Third  Order.  By  the  Rev.  Geoege  Salmon,  Trinity  College, 
Dublin.  Communicated  by  Akthue  Cayley,  Esg.,  F.B.S. 
Keceived  May  20, — Eead  June  17,  1858. 
The  following  Notes  are  intended  as  supplementary  to  Mr.  Cayley’s  Memoir  on  Curves 
of  the  Third  Order*. 
If  in  a cubic  U we  substitute  x-\-'kx’,  y-\-\y’,  z-\-'kz'  for  x\  y',  z\  let  the  result  be  written 
U+3aS+3aT+A®U', 
where  S and  P are  evidently  the  polar  conic  and  polar  line  of  with  respect  to 
the  cubic,  and 
QQ  ,dV  ,dJ]  (d\]\  , (d\]\’  , (dJ]\ 
In  like  manner  let  the  result  of  a similar  substitution  in  H be  written 
H+3A2+3A='n+A^H', 
where  2 and  11  are  the  polar  conic  and  polar  line  of  x'y’z',  with  respect  to  the  Hessian. 
Then  the  identical  equation  which  Mr.  Cayley  has  given,  p.  442,  may  with  advantage 
be  replaced  by  the  following, 
3(Sn-2P)=H'U-U'H, 
an  equation  which  can  be  easily  verified  by  the  help  of  the  canonical  form 
U = +3/^ + 5 Ixyz. 
When  ody'z'  is  on  the  curve  U,  the  identical  equation  just  written  enables  us  to  write 
the  equation  U=0  in  the  form 
sn-P2=o, 
a transformation  from  which  many  important  consequences  may  be  deduced. 
In  the  first  place,  the  equation  shows  that  the  lines  P,  TI  intersect  on  the  cubic ; 
hence  the  tangential  of  the  point  x'y'z,  that  is  to  say,  the  point  where  the  tangent  P 
meets  the  cubic  again,  is  the  intersection  of  P with  H,  the  polar  of  x'y'z'  with  regard  to 
the  Hessian. 
Again,  the  points  of  contact  of  tangents  from  cdy’z'  are  known  to  be  the  intersections 
of  S with  the  cubic ; and  the  equation  shows  that  the  points  in  question  are  the  inter- 
sections of  S,  2f. 
Further,  the  equation  shows  that  S=^2  and  Y=gjll  (where  gj  is  arbitrary)  intersect 
on  the  cubic.  But  the  second  of  these  equations  is  the  polar  of  x'y'z'  with  respect  to  the 
former : hence  the  cubic  may  be  generated  as  the  locus  of  the  points  of  contact  of 
tangents  from  a point  x'y'z'  to  a system  of  conics  passing  through  four  fixed  points. 
* Philosophical  Transactions,  1857,  p.  415  t See  Mr.  Caylet’s  Memoir,  p.  443. 
MDCCCLVIII.  4 B 
