418 
ME.  A.  CAYLEY’S  EOUETH  ]MEMOIE  UPOX  QTJAXTICS. 
A cubinvariant  only  exists  for  quantics  of  an  evenly  even  order,  and  for  each  such 
quantic  there  is  one,  and  only  one,  cubinvariant. 
67.  But  a quartic  has  tvro  invariants,  vrhich  are  of  the  degrees  2 and  3 respectively^ 
and  the  number  of  quartinvariants  of  a quantic  of  the  degree  m is  equal  to  the  number 
of  ways  in  which  m can  be  made  up  -with  the  parts  2 and  3.  AMien  m is  even,  there  is 
of  course  a quartinvariant  which  is  the  square  of  the  quadiinvaiiant,  and  which,  if  we 
attend  only  to  the  irreducible  quartinvariants,  must  be  excluded  from  consideration. 
The  preceding  number  must  therefore,  when  m is  even,  be  diminished  by  unitv'.  The 
result  is  easily  found  to  be 
Quartinvariants  exist  for  a quantic  of  any  order,  even  or  odd,  whatever,  the  quadric 
and  the  quartic  alone  excepted ; and  accordmg  as  the  order  of  the  quantic  is 
6^,  6^+2,  6^+3,  6^+4,  6^+5, 
the  number  of  quartinvariants  is 
9^  9 ^ 9 ^ 9+^^  9 > 9+'^- 
In  particular,  for  the  orders 
2,  3,  4,  5;  6,  7,  8,  9,  10,  11;  12,  &c., 
the  numbers  are 
0,  1,  0,  1;  1,  1,  1,  2,  1,  2;  2,  &c. 
So  that  the  nonic  is  the  lowest  quantic  which  has  more  than  one  quartinvariant. 
68.  But  the  whole  theory  of  the  invariants  or  covariants  of  the  degrees  2,  3,  4 is  most 
easily  treated  by  the  method  above  alluded  to,  contained  in  the  second  part  of  my 
original  memoir ; and  indeed  the  method  appears  to  be  the  appropriate  one  for  the 
treatment  of  the  theory  of  the  invariants  or  covariants  of  any  given  degi’ee  whatever, 
although  the  application  of  it  becomes  difficult  when  the  degree  exceeds  4.  I remark, 
in  regard  to  this  method,  that  it  leads  naturally,  and  in  the  first  instance,  to  a special 
class  of  the  co variants  of  a system  of  quantics,  viz.  these  co variants  are  linear  fimctions 
of  the  derived  functions  of  any  quantic  of  the  system.  (It  is  hardly  necessary  to  remark 
that  the  derived  functions  referred  to  are  the  derived  functions  of  any  order  of  the 
quantic  with  regard  to  the  facients.)  Such  covariants  may  be  termed  tantijjartite 
covariants ; but  when  there  are  only  two  quantics,  I use  in  general  the  term  liMO-linear. 
The  tantipartite  covariants,  while  the  system  remains  g-eneral,  are  a special  class  of 
covariants,  but  by  particularizing  the  system  we  obtam  all  the  covariants  of  the  par- 
ticularized system.  The  ordinary  case  is  when  all  the  quantics  of  the  system  reduce 
themselves  to  one  and  the  same  quantic,  and  the  method  then  gives  all  the  covariants 
of  such  single  quantic.  And  while  the  order  of  the  quantic  remains  indefinite,  the 
method  gives  covariants  (not  invariants) ; but  by  particularizing  the  order  of  the  quantic 
in  such  manner  that  the  derived  functions  become  simply  the  coefficients  of  the  quantic. 
the  covariants  become  invariants : the  like  applies  of  course  to  a system  of  two  or  more 
quantics. 
