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XXIV.  On  the  Tangential  of  a Cubic.  By  Aethue  Cayley,  Esg.,  F.R.S. 
Eeceived  Eebraary  11, — Bead  March  18,  1858. 
!>'  my  “ Memoir  on  Curves  of  the  Third  Order  I had  occasion  to  consider  a derivative 
which  may  be  termed  the  “ tangential  ” of  a cubic,  viz.  the  tangent  at  the  point  [x,  y,  z) 
of  the  cubic  curve  {^fx,  y,  2)^=0  meets  the  curve  in  a point  (|,  -e,  Q,  which  is  the  tan- 
gential of  the  first-mentioned  point ; and  I showed  that  when  the  cubic  is  represented 
in  the  canonical  form  xf‘-\-'f-\-z^-\-%lxyz=0,  the  coordinates  of  the  tangential  may  be 
taken  to  be  x(y^—z^)  :y{z^ — F)  : z{F — y^).  The  method  given  for  obtaining  the  tangen- 
tial may  be  apphed  to  the  general  form  {a,  b,  c,f  g,  h,  i,j,  k,  Ifx,  y,  zf:  it  seems  desirable, 
in  reference  to  the  theory  of  cubic  forms,  to  give  the  expression  of  the  tangential  for  the 
general  formf;  and  this  is  what  I propose  to  do,  merely  indicating  the  steps  of  the 
calculation,  which  was  performed  for  me  by  Mr.  Ceeedy. 
The  cubic  form  is 
{a^  b,  c,f  g,  A,  i^y^  A,  Tfx^  ?/,  2) , 
which  means 
aaf‘-\-by^-\-cz^-\-?>fy'^z-\-?>gz^x-\-^hx'^y-\-  2»iyz^ -\-?>jzx^ -\-‘^hxy'^ ^Ixyz ; 
and  the  expression  for  | is  obtained  from  the  equation 
h cJij,f  c,  i,  g,  Tfx,  y,  z)\  — {h,  b,  i,f,  I,  kjx,  y,zff 
—{a,  b,  c,f  g,  h,  i,j,  k,  1%^,  y,  zf{€x+M), 
where  the  second  line  is  in  fact  equal  to  zero,  on  account  of  the  first  factor,  which 
vanishes.  And  C,  denote  respectively  quachic  and  cubic  functions  of  (y,  2),  which 
are  to  be  determined  so  as  to  make  the  right-hand  side  divisible  by  x^;  the  resulting 
value  of  I may  be  modified  by  the  adjunction  of  the  evanescent  term 
{2x-\-hy+gz){a,  b,  c,f,  g,  h,  i,j,  k,  IJx,  y,  z)\ 
where  a,  g,  h are  arbitraiy  coetficients ; but  as  it  is  not  obvious  how  these  coefficients 
should  be  determined  in  order  to  present  the  result  in  the  most  simple  form,  I have  given 
the  result  in  the  form  in  which  it  was  obtained  without  the  adjunction  of  any  such  term. 
Write  for  shortness 
V = (]c,l  Xy,z), 
Q=(5,/,  i 
* Philosopbical  Transactions,  vol.  cxlvii.  1857. 
t At  the  time  when  the  present  paper  was  written,  I was  not  aware  of  Mr.  Salmon’s  theorem  (Higher 
Plane  Curves,  p.  156),  that  the  tangential  of  a point  of  the  cubic  is  the  intersection  of  the  tangent  of  the 
cubic  with  the  first  or  line  polar  of  the  point  with  respect  to  the  Hessian ; a theorem,  which  at  the  same 
time  that  it  affords  the  easiest  mode  of  calculation,  renders  the  actual  calculation  of  the  coordinates  of  the 
tangential  less  important.  Added  7th  October,  1858.' — A.  C. 
