540 
THE  EEV.  G.  SALMON  ON  CTJEVES  OF  THE  THIRD  OEDEE. 
A^(AC-Q^)-QX,  Ay(AC-Q^)-Qy,  Az{AC-Q^)-Qz* 
The  condition,  then,  that  these  coordinates  should  be  the  original  ar,  y,  2,  is  Q=0. 
But  Q denotes  the  nine  lines  joining  the  points  of  contact  of  tangents  drawn  from  the 
point  of  inflexion.  This,  then,  coincides  with  M.  Pluckek’s  determination,  already 
referred  to,  of  the  points  of  contact  of  osculating  conics. 
Osculating  Cubics. 
If  through  eight  consecutive  points  of  a cubic  several  cubics  be  drawn,  these  all  meet 
the  cm’ve  again  in  a fixed  ninth  point,  which  can  easily  be  determined.  In  fact,  if  we 
consider  the  nine  points  of  intersection  of  two  cubics,  the  opposite  (see  page  537)  of  any 
four  lies  on  the  same  conic  with  the  remaining  five.  For  let  the  equation  of  one  be 
AU— BV  and  of  the  other  CU=DV,  where  A,  B,  C,  D represent  right  lines  and  U,  T 
conics ; then  the  intersections  of  UV  are  points  common  to  both  cubics,  and  AB,  CD 
the  opposite  points  in  each  cubic.  But  by  combining  the  equations,  we  get  for  the 
equation  of  the  conic  through  the  remaining  five  points  of  intersection  AD=BC,  which 
passes  through  the  two  opposite  points.  Q.  E.  D. 
The  theorem  may  be  otherwise  stated  in  what  is  easily  seen  to  be  an  equivalent  form : 
the  opposites  of  any  two  sets  of  four  out  of  the  nine  points  of  intersection  lie  in  a right 
line  with  the  ninth. 
Now  we  have  already  proved  that,  in  the  case  of  four  consecutive  points,  the  opposite 
is  the  second  tangential  to  the  original  point.  Hence,  in  the  case  of  eight  consecutive 
points,  the  point  through  which  all  cubics  through  these  meet  the  curve  again,  is 
simply  the  third  tangential  of  the  original  point.  In  other  words,  at  the  given  point 
A draw  a tangent  meeting  the  curve  again  in  B ; at  B draw  a tangent  meeting  the  ciu've 
in  C ; at  C draw  a tangent  meeting  it  again  in  D : then  D is  the  point  through  which 
all  osculating  cubics  must  pass.  If  AC  meet  the  curve  again  in  E,  it  has  been  already 
shown  that  E is  the  point  through  which  all  osculating  conics  must  pass ; and  it  is  to 
be  observed  that  the  intersection  of  AD  and  BE  lies  on  the  cubic. 
It  may  be  proposed  to  determine  the  points  at  which  it  shall  be  possible  to  draw 
cubics  osculating  the  given  curve  in  nine  points ; or  in  other  words,  such  that  the  third 
tangential  D may  coincide  with  the  original  point  A.  It  is  evident  that  in  this  case  the 
points  E and  C will  coincide ; that  therefore  the  coordinates  of  E,  given  at  the  top  of 
this  page,  must  reduce  to  X,  Y,  Z ; and  it  has  been  proved  that  the  condition  that  this 
should  happen  is 
A(AC-Q^)=0. 
Since  A is  the  sum  of  three  squares,  it  is  evident  that  A=0  can  denote  no  real  locus. 
* These  coordinates  are  of  the  25th  degree  in  the  original  coordinates.  Mr.  Catlet  has  informed  me 
that  Mr.  Stlvesteu  has  established  that  the  degree  of  the  coordinates  of  every  derivative  point  is  neces- 
sarily a square  number.  I am  led  by  induction  to  believe  that  in  the  case  of  three  derivative  points  in  a 
right  line,  the  sum  of  the  square  roots  of  their  degrees  taken  with  proper  signs  is  always  cypher. 
