ME.  A.  CAYLEY’S  EOUETH  MEMOIE  UPON  QUANTICS. 
421 
76.  When  the  two  quantics  are  the  first  derived  functions  of  the  same  quantic  of  any 
odd  order,  the  lineo-linear  invariant  does  not  vanish,  but  it  is  not  an  invariant  of  the 
single  quantic.  Thus  the  lineo-linear  invariant  of 
(«,  h,  cX^,  yf 
and 
{b,  c,  djx,  yf 
IS 
{ad—2bc-\-cb=)ad—hc. 
which  is  not  an  invariant  of  the  cubic 
(«,  5,  c,  djx,  yf. 
But  for  two  quantics  which  are  the  first  derived  functions  of  the  same  quantic  of  any  even 
order,  the  hneo-hnear  invariant  is  the  quadrinvariant  of  the  single  quantic.  Thus  the 
lineo-linear  invariant  of 
and 
is 
{a,  b,  c,  dXx,  yf 
(J,  c,  d,  ejx,  yf 
{ae—?>bd-\-2>c^—db=-)ae—^bd-\-2>(f^ 
which  is  the  quadrinvariant  of  the  quartic 
(a,  b,  c,  d,  yf- 
77.  I do  not  stop  to  consider  the  theory  of  the  lineo-linear  covariants  of  two  quantics, 
but  I derive  the  quadiicovariants  of  a single  quantic  directly  from  the  quadi’invariant. 
Imagine  a quantic  of  any  order  even  or  odd.  Its  successive  even  emanants  will  be  in 
regard  to  the  facients  of  emanation  quantics  of  an  even  order,  and  they  will  each  of 
them  have  a quadrinvariant,  which  will  be  a quadricovariant  of  the  given  quantic.  The 
emanants  in  question,  beginning  with  the  second  emanant,  are  (in  regard  to  the  facients 
of  the  given  quantic  assumed  to  be  of  the  order  m)  of  the  orders  m — 2,  m — 4,  ..  down 
to  1 or  0,  according  as  m is  odd  or  even,  or  writing  successively  2^-t-l  and  Ijy  in  the 
place  of  m,  and  taking  the  emanants  in  a reverse  order,  the  emanants  for  a quantic  of 
any  odd  order  are  of  the  orders  1,  3,  5 ...  2^  — 1,  and  for  a quantic  of  any  even 
order  2^,  they  are  of  the  orders  0,  2,  4 ..  2p  — 2.  The  quadricovariants  of  a quantic  of 
an  odd  order  2j9  + l,  are  consequently  of  the  orders  2,  6,  10  ...  4^  — 2,  and  the  quadri- 
covariants of  a quantic  of  an  even  order  2^,  are  of  the  orders  0,  4,  8...4p  — 4.  We 
might  in  each  case  cany  the  series  one  step  further,  and  consider  a quadricovariant  of 
the  order  4p+2,  or  (as  the  case  may  be)  4j9,  which  arises  from  the  0th  emanant  of  the 
given  quantic ; such  quadricovariant  is,  however,  only  the  square  of  the  given  quantic. 
78.  In  the  case  of  a quantic  of  an  evenly  even  order  (but  in  no  other  case)  we  have  a 
quadricovariant  of  the  same  order  with  the  quantic  itself.  We  may  in  this  case  form 
the  lineo-hnear  invariant  of  the  quantic  and  the  quadricovariant  of  the  same  order ; such 
hneo-linear  invariant  is  an  invariant  of  the  given  quantic,  and  it  is  of  the  degree  3 in  the 
