426 
ME.  A.  CAYLEY’S  EOTJETH  MEMOIR  ITOX  QUA^TICS. 
order  m — 1,  viz.  treating  the  coefficients  of  the  adjoint  quantic  as  the  facients  of  the 
cobezoutiant,  the  cobezoutiant  is  an  m-ary  quadric,  the  coefficients  of  which  are  lineo- 
linear  functions  of  the  coefficients  of  the  two  quantics,  and  the  number  of  the  cobezou- 
tiants  is  \m  or  according  asm  is  even  or  odd. 
87.  If  the  two  quantics  are  the  differential  coefficients,  or  first  derived  fimctions  (with 
respect  to  the  facients)  of  a single  quantic 
(«,  h, . .Jx,  yY, 
then  we  have  what  are  termed  the  Cohezoutoids  of  the  single  quantic,  viz.  the  cobezou- 
toid  is  an  invariant  of  the  single  quantic  of  the  order  m,  and  of  an  adjoint  quantic  of 
the  order  (m— 2);  and  treating  the  coefficients  of  the  adjoint  quantic  as  facients,  the 
cobezoutoid  is  an  (yn — l)ary  quadric,  the  coefficients  of  which  are  quadric  functions  of 
the  coefficients  of  the  given  quantic.  The  number  of  the  cohezoutoids  is  \{in — 1)  or 
^{m — 2),  according  as  m is  odd  or  even. 
88.  Consider  any  two  quantics  of  the  same  order, 
{a,..Jx,yY,  {a!,,.Xx,yY, 
and  introducing  the  new  facients  (X,  Y),  form  the  quotient  of  determhiants. 
{a,  ..Jx  ,y  Y,  («',  ..Jx  ,y  Y 
X ,y 
(a,  ..XX,  Y)-,  {a!,  ..JX,  Y)- 
X,  Y 
which  is  obviously  an  integral  function  of  the  order  {in — 1)  in  each  set  of  facients  sepa- 
rately, and  lineo-linear  in  the  coefficients  of  the  two  quantics ; for  instance,  if  the  two 
quantics  are 
{a,  b,  c,  djx,  y)\ 
(a',  b',  c\  d'Jx,  y)\ 
the  quotient  in  question  may  be  written 
("  ^ab'—a'b),  S{ad—a'c)  , ad'—o'd  ^x,yy{'K,Yy 
^ad  —a'c),  ad'—a'd-\-9{bc'—b'c),  ?>{bd'—b’d) 
ad'—a'd,  2>{bd!—b'd)  , ?>{cd'—c'd) 
The  function  so  obtained  may  be  termed  the  Bezoutic  Einanant  of  the  two  quantics. 
89.  The  notion  of  such  function  was  in  fact  suggested  to  me  by  Bezout’s  ahbrewated 
process  of  elimination,  viz.  the  two  quantics  of  the  order  m being  put  equal  to  zero,  the  , 
process  leads  to  {in — 1)  equations  each  of  the  order  1):  these  equations  are  nothing  i 
else  than  the  equations  obtained  by  equating  to  zero  the  coefficients  of  the  ditferent 
terms  of  the  series  (X,  in  the  Bezoutic  emanant,  and  the  result  of  the  elimination  , 
is  consequently  obtained  by  equating  to  zero  the  determinant  formed  mth  the  matrix  ! 
which  enters  into  the  expression  of  the  Bezoutic  emanant.  In  other  words,  this  deter-  | 
minant  is  the  Besultant  of  the  tAvo  quantics.  Thus  the  resultant  of  the  last-mentioned 
two  cubics  is  the  determinant 
