538 
THE  EEV.  G.  SALMON  OH  CHEYES  OE  THE  THIED  OEDEE. 
original  point,  the  first  and  second  tangential  should  coincide;  or  in  other  words,  the 
tangential  should  be  a point  of  infiexion.  There  are  then  twenty-seven  conics  which  can 
meet  a cubic  in  six  consecutive  points,  and  the  points  of  contact  are  the  points  of  con- 
tact of  the  tangents  drawn  from  the  nine  points  of  infiexion. 
We  return  to  complete  the  algebraic  solution  of  the  problem  to  determine  the  equa- 
tion of  the  conic  meeting  a cubic  in  five  consecutive  points  at  and  it  is  obvious 
that  what  remains  to  be  done  is  to  determine  X so  that  the  line 
(3-|in'->.(n+^p) 
may  pass  through  x'y'z'.  The  only  difficulty  is  to  determine  the  result  of  substituting 
x'y'z'  in  (3.  Now  if  we  differentiate,  with  regard  to  x,  y,  or  z,  the  equation 
;aS+2-in’=/3P, 
and  substitute  x',  y\  z'  for  x,y,  z in  the  result,  we  get  (3'=2[Jj.  Hence,  since  n'=H', 
P'=0,  we  have 
which  determines  X in  terms  of  (m,  which  has  been  found  akeady. 
The  equation,  then,  of  the  conic  having  five-point  contact  with  U is 
(n+;..p)|s-ip(n+;xP)|=p{/3-|-,n-^,(n+p,P)J. 
It  appears  then  that  the  tangent  at  the  first  tangential,  is  the  chord  of  mter- 
section  of  the  five-point  conic  with  the  polar  conic  S.  We  are  thus  also  able  to  con- 
struct the  five-point  conic  geometrically,  since  five  points  of  it  are  given,  namely,  the  two 
points  of  intersection  of  the  tangent  at  the  first  tangential  with  S ; the  point  where  the 
line  joining  the  original  point  to  the  second  tangential  meets  the  curve  again;  and  of 
course  the  original  point  and  its  consecutive  one. 
Working  with  the  equation 
a;®  +3/® + -f-  6 Ixyz  = 0 , 
Mr.  Cayley  has  calculated  the  equation  of  the  five-point  conic,  and  thence  the  coordi- 
nates of  the  point  where  it  meets  the  curve  again.  I shall  now  form  the  coordinates  of 
the  same  point,  deriving  them  from  the  geometrical  construction  for  that  point  which  I 
have  just  given.  Let  xyz  be  the  coordinates  of  the  point  of  contact,  and  XYZ  those  of 
its  second  tangential ; then  the  coordinates  of  the  point  requu-ed  must  be 
where 
X=:.r(X®+2^YZ)+3/(Y®+2/ZX)+;s(Z®+2S:Y) 
yj=.'X{x^-\-2lyz)  -\-Y{y'^-\-2lzx)  -\-Z{z''~-{-2lxy). 
I write  for  abbreviation  a,  j3,  y instead  of  x^,  3/®,  2® ; then  the  coordinates  of  the  first 
tangential  are 
^{(^—7%  y{7—<^\  z(ci—(3); 
