THE  EEV.  a.  SALMON  ON  CUEVES  OF  THE  THIED  OEDEE. 
541 
Also,  since  4AC— B^=3Q^  the  condition  AC— Q*=0  may  be  written  Q*=0,  and 
breaks  up  into  the  factors  B+Q=0,  or 
a^y + z\o(^ —'ity  i (2® — ^) = 0, 
or 
0,  and  .r*^®+3/V+zV—  3^®3/V= 0. 
Hence  there  are  in  general  seventy-two  real  or  imaginary  points  of  contact  of  osculating 
cubics. 
When  a point  on  a cubic  is  determined  such  that  it  is  its  own  third  tangential,  it, 
together  with  its  first  and  second  tangential,  determines  a system  of  right  lines,  in 
terms  of  which  the  cubic  can  be  transformed  to  the  new  canonical  form 
x^y-\-iyz-\-z‘^x-{-  2mxyz=.^, 
in  which  form  the  points  of  inflexion  are  determined  by  the  right  lines 
^ + 3/* + 2:®  — 3^yz  = 0 . 
It  may  be  observed  that,  according  to  circumstances,  one  of  the  two  canonical  forms  is 
simpler  than  the  other.  Thus  for  the  case  S=0,  when  the  Hessian  is  one  real  and  two 
imaginary  lines,  the  canonical  forms  are 
afy-\-y^z-{-z^x=^,  and  x^-\-y^-\-z^-\-^xyz=0, 
the  former  being  the  simpler.  On  the  contrary,  when  the  Hessian  is  three  real  lines, 
the  form  oif-\-y^+z^  is  simpler  than  x^y-\-y‘^z-\-z^x-\-'2mxyz,  where  m^=  — 3. 
