ME.  A.  CAYLEY’S  FOUETH  MEMOIE  UPON  QUANTICS. 
425 
which  is  equivalent  to 
—fx, 
yxr^, 
—x^ 
a , 
b , 
c , 
d 
h. 
c , 
d , 
e 
c , 
d , 
e , 
f 
and  a like  transformation  exists  with  respect  to  the  canonisant  of  a quantic  of  any  odd 
order  whatever.  The  canonisant  and  the  lambdaic  (which  includes  of  course  the  cata- 
lecticant)  form  the  basis  of  Professor  Stlvestee’s  theory  of  the  Canonical  Forms  of 
quantics  of  an  odd  and  an  even  order  respectively. 
86.  There  is  another  family  of  covariants  which  remains  to  be  noticed.  Consider  any 
two  quantics  of  the  same  order, 
{a,h,.  .Jx,  yY, 
(«',  h\  . yY, 
and  join  to  these  a quantic  of  the  next  inferior  order, 
{u,v,..Jjy,—xY~\ 
where  the  coefficients  (i«,  v, are  considered  as  indeterminate,  and  which  may  be  spoken 
of  as  the  adjoint  quantic. 
Take  the  odd  lineo-linear  covariants  (viz.  those  which  arise  from  the  odd  emanants) 
of  the  tw’o  quantics;  the  term  arising  from  the  (2^+l)th  emanants  is  of  the  form 
(A,  B, 
where  (A,  B,  . .)  are  lineo-linear  functions  of  the  coefficients  of  the  two  quantics. 
Take  also  the  quadricoyariants  of  the  adjoint  quantic;  the  term  arising  from  the 
(2?— m)th  emanant  is  of  the  form 
(u,  V,  ..x^, 
where  (U,  V,  ..)  are  quadric  functions  of  the  indeterminate  coefficients  (u.v,..).  We 
may  then  form  the  quadrin variant  of  the  two  quantics  of  the  order  2(m — 1 — 2i):  this 
^vill  be  an  invariant  of  the  two  quantics  and  the  adjoint  quantic,  lineo-linear  in  the  coeffi- 
cients of  the  two  quantics  and  of  the  degree  2 in  regard  to  the  coefficients  (w,  ^7, . .)  of  the 
adjoint  quantic;  or  treating  the  last-mentioned  coefficients  as  facients,  the  result  is  a 
lineo-linear  m-ary  quadric  of  the  form 
..Xu,v, 
viz.  m this  expression  the  coefficients  91,  33,  . • are  lineo-linear  functions  of  the  coefficients 
of  the  two  quantics.  And  giving  to  i the  different  admissible  values,  viz.  from  z=0  to 
i—\m — 1 or  \{m — 1)  — 1,  according  as  m is  even  or  odd,  the  number  of  the  functions 
obtained  by  the  preceding  process  is  \ni  or  ^{ni — 1),  according  as  m is  even  or  odd.  The 
functions  in  question,  the  theory  of  which  is  altogether  due  to  Professor  Sylvester,  are 
termed  by  him  Cohezoutiants ; we  may  therefore  say  that  a cobezoutiant  is  an  invariant 
of  two  quantics  of  the  same  order  m,  and  of  an  adjoint  quantic  of  the  next  preceding 
