]VIE.  A.  CAYLEY’S  FOEETH  MEMOIE  UPON  QUANTICS. 
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S(ab'  —a'b),  ^ac'—a'c)  , ad'—a'd 
3(ac'—a'c),  ad'—a'd-\-9(bc'—b'c),  ^bd'  — b'd) 
ad'-a'd , S{bd'-b'd)  , S(cd'-dd) 
90.  If  the  two  quantics  are  the  differential  coefficients  or  first  derived  functions  (with 
respect  to  the  facients)  of  a single  quantic  of  the  order  m,  then  we  have  in  like  manner 
the  Bezoutoidal  Emanant  of  the  single  quantic;  this  is  a function  of  the  order  m — 2 in 
each  set  of  facients,  and  the  coefficients  whereof  are  quadric  functions  of  the  coefficients 
of  the  single  quantic.  Thus  the  Bezoutoidal  emanant  of  the  quartic 
{a,  b,  c,  d,  ejx,  yj 
( 3(«c  — 5^),  ?,{ad—bc)  , ae—bd 
2>{ad~bc),  9c^  ?>{be—cd) 
ae—bd,  ?>{be—cd)  , '^{ce  — d^) 
and  of  course  the  determinant  formed  with  the  matrix  which  enters  into  the  expression 
of  the  Bezoutoidal  Emanant,  is  the  discriminant  of  the  single  quantic. 
91.  Professor  Sylvester  forms  with  the  matrix  of  the  Bezoutic  emanant  and  a set  of 
m facients  (w,  v,  . . .)  an  m-ary  quadric  function,  which  he  terms  the  Bezoutiant.  Thus 
the  Bezoutiant  of  the  hefore-mentioned  two  cubics  is 
( Z{aV—cib),  Z{ad—dc)  , ad’—a'd  \u,v,ioy-, 
?>{ac' — dc),  ad! — dd-\-9{bc' — b’c),  S(bd'—b'd) 
ad' —dd  , ?>bd'—b'd  , ^cd'  — c'd) 
and  in  like  manner  with  the  Bezoutoidal  emanant  of  the  single  quantic  of  the  order  m 
and  a set  of  (m— 1)  new  facients  (u,  v,  . . .),  an  (m — l)ary  quadric  function,  which  he 
terms  the  Bezoutoid.  Thus  the  Bezoutoid  of  the  before-mentioned  quartic  is 
( 3(ac— ?>{ad—bc)  , ae—bd  yu,v,wf-, 
?>{ad—bc),  ae+8JfZ— 9c^  ?>{be—cd) 
ae—bd  , ?>{be—cd)  , S(ce—d’^) 
And  to  him  is  due  the  important  theorem,  that  the  Bezoutiant  is  an  invariant  of  the 
two  quantics  of  the  order  m and  of  the  adjoint  quantic  (u,  v,  . being  in 
fact  a linear  function  with  mere  numerical  coefficients,  of  the  invariants  called  Cobe- 
zoutiants,  and  in  like  manner  that  the  Bezoutoid  is  an  invariant  of  the  single  quantic. 
of  the  order  m and  of  the  adjoint  quantic  (u,.v,  . .y^y,  — being  a linear  function 
with  mere  numerical  coefficients  of  the  invariants  called  Cobezoutoids. 
The  modes  of  generation  of  a covariant  are  infinite  in  number,  and  it  is  to  be  antici- 
pated that,  as  new  theories  arise,  there  will  be  frequent  occasion  to  consider  new  pro- 
cesses of  derivation,  and  to  single  out  and  to  define  and  give  names  to  new  covariants. 
But  I have  now,  I think,  established  the  greater  part  by  far  of  the  definitions  which  are 
for  the  present  necessary. 
