ME.  A.  CAYLEY’S  EOUETH  MEMOIE  UPON  QUANTICS. 
419 
69.  To  take  the  simplest  example,  in  seeking  for  the  covariants  of  a single  quantic  U, 
we  in  fact  have  to  consider  two  quantics  U,  V.  An  expression  such  as  12  UV  is  a lineo- 
linear  covariant  of  the  two  quantics ; its  developed  expression  is 
which  is  the  Jacobian.  In  the  particular  case  of  two  linear  functions  («,  b'^x,  y)  and 
(a',  S'X-^5  y)i  the  lineo-linear  covariant  becomes  the  lineo-linear  invariant  db'—dh^  which 
is  the  Jacobian  of  the  two  linear  functions. 
In  the  example  we  cannot  descend  from  the  two  quantics  U,  V to  the  single  quantic 
U (for  putting  V= U the  covariant  vanishes) ; but  this  is  merely  accidental,  as  appears 
by  considering  a different  hneo-linear  covariant  l^UV,  the  developed  expression  of 
which  is 
. B^V-2B  AU . . B^V. 
In  the  particular  case  of  two  quadrics  (a,  Z*,  yY,  {a!,  b',  c'y^x,  yY^  the  lineo-linear 
covariant  becomes  the  lineo-linear  invariant 
ad  — 2bb'  -\-ca! . 
If  we  have  V= U,  then  the  lineo-linear  covariant  gives  the  quadricovariant 
B^U.B^U-(B.B,U)^ 
of  the  single  quantic  U (such  quadricovariant  is  in  fact  the  Hessian) ; and  if  in  the  last- 
mentioned  fonnula  we  put  for  U the  quadric  {a,  b,  c^x,  yY,  or  what  is  the  same  thing, 
if  in  the  expression  of  the  lineo-linear  invariant  ad  — IbV  -\-cd ^ we  put  the  two  quadrics 
equal  to  each  other,  we  have  the  quadrinvariant 
ac—b^ 
of  the  single  quadric. 
70.  The  lineo-linear  invariant  ab' — db  of  two  linear  functions  may  be  considered  as 
ghing  the  lineo-linear  covariant  B^U.Bj,V-— Bj,U.B^V  of  the  two  quantics  U and  V, 
and  in  like  manner  the  lineo-linear  invariant  ad  — 2bV-\-cd  may  be  considered  as  giving 
the  lineo-linear  covariant  B^U.B^V  — 2B^B^U.B^B^V+B^U.B^V  of  the  quantics  U,  V. 
And  generally,  any  invariant  whatever  of  a quantic  or  quantics  of  a given  order  or 
orders  leads  to  a covariant  of  a quantic  or  quantics  of  any  higher  order  or  orders : viz. 
the  coefficients  of  the  original  quantic  or  quantics  are  to  be  replaced  by  the  derived 
functions  of  the  quantic  or  quantics  of  a higher  order  or  orders. 
71.  The  same  thing  maybe  seen  by  means  of  the  theory  of  Emanants.  In  fact,  con- 
sider any  emanants  whatever  of  a quantic  or  quantics ; then,  attending  only  to  the 
facients  of  emanation,  the  emanants  will  constitute  a system  of  quantics  the  coefficients 
of  which  are  derived  functions  of  the  given  quantic  or  quantics ; the  invariants  of  the 
system  of  emanants  will  be  functions  of  the  derived  functions  of  the  given  quantic  or 
quantics,  and  they  will  be  covariants  of  such  quantic  or  quantics ; and  we  thus  pass 
from  the  invariants  of  a quantic  or  quantics  to  the  covariants  of  a quantic  or  quantics 
of  a higher  order  or  orders. 
